Continuous groups of transversal gates for quantum error
correcting codes from finite clock reference frames
Mischa P. Woods
1
and
´
Alvaro M. Alhambra
2
1
Institute for Theoretical Physics, ETH Zurich, Switzerland
2
Perimeter Institute for Theoretical Physics, Waterloo, Canada
Following the introduction of the task of ref-
erence frame error correction [1], we show
how, by using reference frame alignment with
clocks, one can add a continuous Abelian
group of transversal logical gates to any error-
correcting code. With this we further explore
a way of circumventing the no-go theorem of
Eastin and Knill, which states that if local er-
rors are correctable, the group of transversal
gates must be of finite order. We are able to
do this by introducing a small error on the de-
coding procedure that decreases with the di-
mension of the frames used. Furthermore, we
show that there is a direct relationship be-
tween how small this error can be and how
accurate quantum clocks can be: the more ac-
curate the clock, the smaller the error; and the
no-go theorem would be violated if time could
be measured perfectly in quantum mechanics.
The asymptotic scaling of the error is stud-
ied under a number of scenarios of reference
frames and error models. The scheme is also
extended to errors at unknown locations, and
we show how to achieve this by simple major-
ity voting related error correction schemes on
the reference frames. In the Outlook, we dis-
cuss our results in relation to the AdS/CFT
correspondence and the Page-Wooters mecha-
nism.
1 Introduction and Overview of Re-
sults
In order to build a functional universal quantum com-
puter, full fault-tolerance must be achieved. The idea
behind fault-tolerance is that the errors that occur
at particular points during the computation do not
propagate or amplify along the whole computation to
the point of being uncorrectable. Due to fundamental
physical constraints such as no-cloning, achieving this
is a notoriously challenging task, with a number of dif-
ferent requirements on how to prepare, manipulate,
and protect the quantum states with error-correcting
codes. One of the most desirable features of the codes
used in fault-tolerant computation is the ability to ap-
ply logical gates transversally, which one can imple-
ment while still being able to correct for local errors.
The framework for error correction is based on con-
sidering three spaces — a logical H
L
, physical H
P
and a code H
Co
⊆ H
P
space.
1
Logical states ρ
L
containing quantum information are encoded via an
encoding map E : B(H
L
) → B(H
Co
) onto the code
space, which is a subspace of some larger physical
space where errors — represented via error maps
{E
j
}
j
: B(H
Co
) → B(H
P
) — can occur. Decoding
maps {D
j
}
j
: B(H
P
) → B(H
L
) can then retrieve the
information while correcting for errors; outputting the
logical state ρ
L
. That is:
ρ
L
E
E
j
D
j
ρ
L
,
(1)
for all j and for all states ρ
L
∈ S (H
L
). Depending
on the error model, the index j indicating which error
occurred may or may not be known. If it is unknown,
the decoding map D
j
cannot depend on j. We say
that a logical gate V
L
can be applied transversally if
for any state ρ, the encoder E is such that
E(V
L
ρV
†
L
) = V
⊗K
Co
E(ρ)V
†⊗K
Co
, (2)
where the tensor product structure “⊗K” represents
the division of the code into different subsystems or
“blocks” in which errors can be independently cor-
rected. This condition means that the action of the
encoding map commutes with the action of the logical
gate V
L
, which is represented by V
⊗K
Co
in the physi-
cal space. An interesting case to consider is that of
codes E and groups G for which all group elements
(indexed by g) can be applied transversally using a
unitary group representation U
L
(g),
E(U
L
(g)ρU
L
(g)
†
) = U
Co
(g)
⊗K
E(ρ)U
Co
(g)
†⊗K
∀g ∈ G.
(3)
We will refer to codes whose encoding has this prop-
erty as covariant codes. There are a number of results
that restrict the existence of such codes, in particu-
lar for stabilizer codes [2–6]. Most notably, the no-go
theorem of Eastin and Knill [7] states that in any
finite-dimensional code (not necessarily stabilizer) in
which local errors can be corrected, the groups of log-
ical gates that can be applied transversally must be
1
Some authors use the convention of not considering the
code space explicitly, in which case one sets H
Co
= H
P
.
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arXiv:1902.07725v4 [quant-ph] 19 Mar 2020
finite. This thus excludes dense sets of logical gates,
as well as any continuous Lie subgroup of U(d).
Since this no-go result imposes a fundamental limi-
tation on the possible transversal gates, it is interest-
ing to find ways of circumventing it, via schemes that
do not satisfy all of the assumptions. A number of
alternatives have been thoroughly explored, includ-
ing the protocols of magic state distillation [8], and
other more specific schemes (see for instance [9–14]),
many of which propose a relaxation of the transver-
sality condition in some fault-tolerant way.
Recently, a new kind of circumvention was put for-
ward in [1], where they show examples of codes with
physical spaces of infinite dimension that allow for
the transversal implementation of Lie groups. The
need for infinite dimensions (and seemingly infinite
energy too, as we will soon discuss) limits their prac-
tical relevance, but the idea motivates the following
question: do there exist large (but finite) covariant
codes in which approximate error correction can be
performed? In other words, can we circumvent the
no-go results by allowing for small errors that decrease
with the size of the code?
Here we explore this question using the notion of
reference frames and clocks. We construct imperfect
codes in which Abelian U(1) groups can be imple-
mented transversally. To that aim, we use a simple fi-
nite dimensional version of the encoding map from [1],
which shows that perfect covariant codes are possible
provided one has access to a perfect reference frame.
A perfect reference frame [15] is defined as a quan-
tum system that encodes, without error, information
about a particular group element. That is, given a
group representation U(g), such that U (0) = , the
state |ψi is a perfect reference frame iff ∀g
hψ|U(g) |ψi = δ
0,g
, (4)
where δ
0,g
is a Kronecker delta for finite groups and
Dirac delta in the case of Lie groups. Hence, each
point of the orbit |ψ(g)i = U(g) |ψi is orthogonal to
all the others (and thus perfectly distinguishable).
This connects with the work by Pauli on quan-
tum clocks in the case of an Abelian U(1) groups.
If the group {U(g)}
g
is a one-parameter compact Lie
group generated by solving the Schr¨odinger equation,
namely U(g) = e
−ig
ˆ
H
for some Hamiltonian
ˆ
H, where
g = t is the time transcribed, then the constraint Eq.
(4) is analogous to requiring the existence of a perfect
time operator
ˆ
t. Indeed, defining
ˆ
t =
Z
G
dg g |ψ(g)ihψ(g)|, (5)
where the integral is over the Haar measure, one finds
ˆ
t |ψ(g)i = g |ψ(g)i for all g ∈ G. Pauli [16, 17] already
concluded that such time operators require quantum
systems that — while mathematically well defined —
cannot exist as they require infinite energy.
2
These
are known as Idealised clocks. However, this does not
rule out the existence of approximate time operators
ˆ
t
and initial clock states |ψi which serve as a reference
frame for the observable t, [19–21] either in the form
of a stopwatch [22] or ticking clock [23–25]
Here we explore how certain finite-sized reference
frames (which we call “clocks”) can be used to build
imperfect covariant codes, and we give upper bounds
to the errors induced by their finite size. We use the
construction of Quasi-Ideal clocks [21] and Salecker-
Wigner-Peres (SWP) clocks [19, 20], to provide sim-
ple encoding and decoding protocols based on the task
of reference frame alignment. We show using Quasi-
Ideal clock states entangled over L subsystems, that
the worst-case entanglement fidelity between the in-
put ρ
L
and decoded output D
j
(E
j
(E
cov
(ρ
L
))) denoted
f
worst
, and defined in Section 2.2, in our protocol sat-
isfies (up to log factors) the lower bound 1 −f
worst
≤
O(1/(Ld
C
)
2
) where L is the number of entangled
Quasi-Ideal clocks and d
C
is their dimension. In [26]
the generic upper bound 1 −f
worst
≥ O(1/(Ld
C
)
2
) is
derived for all covariant encoding maps generated by
isometries. A direct consequence of the combination
of our results with those of [26] adapted to our setting,
is that all error correcting codes {E, {E
j
, D
j
}
j
}, can
be made covariant w.r.t. the U(1) groups considered
here with an optimal fidelity f
worst
(largest possible
value) satisfying
O
1
(Ld
C
)
2
≤ 1 −f
worst
≤ O
ln
6
(Ld
C
)
(Ld
C
)
2
, (6)
for a large Ld
C
.
In order to study the role of different resources in
our protocols, we define t-incoherent clock states ρ
C
as those for which there exits g ∈ G such that their
group evolved initial state, U
C
(g)ρ
C
U
†
C
(g), commutes
with the projective measurements used in our protocol
to measure them.
3
As we later explain, these states
require minimal coherent resources to be created in
comparison with other clock states. We find that all
codes which use t-incoherent clock states satisfy the
upper bound 1 − f
worst
≥ O(1/(Ld
C
)). We then con-
sider the more commonly used SWP clocks (see for
instance [19, 20, 27–30] and references therein), which
belong to this class, and show that they yield an er-
ror of order 1 −f
worst
= O(1/(Ld
C
)) hence saturating
the bound for t-incoherent clocks. However, we also
show that while coherent clock states are necessary to
achieve 1 − f
worst
≤ O(1/(Ld
C
)), they are not suffi-
cient, since finite dimensional analogues of coherent
2
The infinite energy manifests itself either with unbounded
from below Hamiltonians (when t belongs to an unbounded in-
terval) or infinitely strong potentials producing strict confine-
ment of the wave function (for when t belongs to an bounded
interval) [18].
3
See text around Eq. (17) for full definition.
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states can only achieve the same scaling as the SWP
clocks.
The results discussed so far consider codes in which
the error model on the clock is just erasure at known
locations. However, what if one cannot discern the
location at which the error occurred? Our final result
is to prove that one can also correct local unknown
phase errors which occur at an unknown location in
the clocks. For this case, we are able to achieve 1 −
f
worst
≤ O(1/(Ld
C
)) up to log factors.
2 Covariant codes based on reference
frames
We now outline the generic error correction scheme we
consider, which is based on a generalisation of the ref-
erence frame-based scheme in [1]. Let {E, {E
j
, D
j
}
j
}
be any perfect error correcting code satisfying Eq. (1)
(not necessarily covariant).
Now, let ρ
(M)
F
be a reference frame for a group
isomorphic to {U
L
(g)}
g
and {U
Co
(g)}
g
, with unitary
group representation {U
⊗M
F
(g)}
g
. We define the fol-
lowing for all g ∈ G
E
g
(·) := U
Co
(g)
⊗K
E(U
†
L
(g)(·)U
L
(g))U
Co
(g)
†⊗K
. (7)
The covariant code is then:
E
cov
(·) :=
Z
G
dg E
g
(·) ⊗U
F
(g)
⊗M
ρ
(M)
F
U
F
(g)
†⊗M
, (8)
where dg is the uniform or Haar measure over the
group.
4
Note that E
cov
is from H
L
to H
P
⊗H
F
while
usually the encoding map takes logical states to states
in the physical space. As such, it is convenient to
think of the reference frames as an extension of the
code space to H
e
Co
:= H
Co
⊗H
F
. Moreover, the chan-
nel E
cov
is now covariant in the sense of Eq. (3) if
the symmetry group on the r.h.s. is defined in the ex-
tended physical space, namely U
Co
(g)
⊗K
⊗U
F
(g)
⊗M
,
so that U
L
(g) can be implemented transversally. This
way, the notion of covariance defined in Eq. (3) is now
over H
e
Co
rather than H
Co
alone. We note, however,
that unlike the code subspace, none of the logical in-
formation is encoded directly into frames F, and they
merely serve as a reference for g, for which both the
group {U
⊗M
F
}
g
and initial frame state ρ
(M)
F
can be
chosen at our convenience to optimise our protocol.
The purpose of the error maps E
j
is that they take in-
formation from the code space and “mix” it with the
rest of physical space resulting in errors. On the other
hand, the reference frames play the following role in
the encoding map E
cov
: the logical information is still
encoded into the code space, but now the information
4
It is noteworthy that all our results hold for all K — the
number of gates being applied transversally. In fact, one can
choose K = 1 when considering a physical space in which the
gates are not applied locally.
about which transversal gate is applied to it is en-
coded in the reference frames, hence without knowl-
edge from the reference frames during the decoding,
it would not be possible to discern which gates had
been applied. If the reference frames are damaged via
an error (which is a reasonable assumption since they
are now part of the extended code space), this will
inhibit the ability to decode correctly resulting in a
worse decoding of the logical information.
The protection of the information to be encoded re-
lies on the channel E, which is arbitrary. On the other
hand, the protection for the reference frame in the en-
conding of Eq. (8) will come merely from having a few
copies of them which may or may not be entangled. In
the case of no entanglement, ρ
(M)
F
= ρ
⊗M
F
, the states
are only classically correlated due to the twirling over
G in Eq. (4), and thus bears strong analogies with a
classical repetition code.
The encoding-error-decoding procedure consists of
the following steps:
1) An arbitrary state ρ
L
is encoded as E
cov
(ρ
L
).
2) Errors occur: The error maps considered here are
now of the form E
j,q
= E
j
⊗ξ
q
, with ξ
q
acting on
the reference frames and the index q — as with
j — may or may not be known depending on the
error model. This may include the loss of up to
M − 1 frames, or in the case of Theorem 4, an
unknown phase error at an unknown location.
3) Frames are measured: In the case of erasure er-
rors at known locations of the reference frames,
only the N non erased clocks are measured. The
remaining N frames are measured projectively
in some basis, and after tracing out the reference
frames, we write
tr
F
[E
j,q
(E
cov
(ρ
L
))] = E
j
(E
g
0
(ρ
L
)) +
ˆ
E
00
g
0
, (9)
where
ˆ
E
00
g
0
represents the error term and g
0
is cho-
sen based on the measurement outcomes, the ini-
tial clock states and the error maps ξ
q
. If the
reference frames are good, the error
ˆ
E
00
g
0
will be
small since the measurements will be able to dis-
tinguish approximately the group elements. In-
deed, in the case of perfect reference frames Eq.
(4) or equivalently the Idealised clock Eq. (5),
the optimal choice of g
0
leads to no error,
ˆ
E
00
g
0
= 0.
4) Finally, we apply the decoding map
¯
D
j,g
0
defined
as
¯
D
j,g
(·) = (10)
U
L
(g)E
−1
U
†⊗K
Co
(g)E (D
j
(·)) U
⊗K
Co
(g)
U
†
L
(g),
where D
j
(E
j
(·)) = E
−1
(·), with E
−1
being the
inverse channel for the encoder E (that is, the
channel that undoes the encoding when no errors
occur).
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This achieves
¯
D
j,g
0
tr
F
[E
j,q
(E
cov
(ρ
L
))]
= ρ
L
+
ˆ
E
g
0
, (11)
where
ˆ
E
g
0
is the final error term.
The circuit diagram is as follows.
encoding
error
Decoding
ρ
L
E
E
cov
E
j
U
Co
(g
0
)
⊗K
¯
D
j,g
|0i
⊗K−1
ρ
L
+
ˆ
E
g
0
| i
| i
| i
This scheme is closely related to the task of
reference frame alignment, in which the group of
“transversal gates” corresponds to the unknown ro-
tation that happens between two parties, here the
encoder and decoder. This is such that the chan-
nel connecting them is effectively a decoherence chan-
nel. The unknown rotation is inferred by measuring
the frame (see [15, 31] for further details). Refer-
ence frame alignment can in fact be thought of as the
particular case where the encoding, error and decod-
ing are identity channels. We discuss this connection
further in Supplementary material I, where we show
that the error probability using the Quasi-Ideal clock
is (ln d
C
)
3
/d
2
C
— a quadratic improvement over the
SWP clock, which only achieves ∝ 1/d
C
as shown in
[31].
Another question of relevance is what happens
if the original encoding E already had a group
of transversal gates {V
L
, V
Co
} satisfying Eq. (2).
If [V
L
, U
L
(g)] = 0 and [V
Co
, U
Co
(g)] = 0 for
all g ∈ G then these gates are also transver-
sal for E
cov
. Moreover, while these gates are
not transversal for E
cov
in general, the gates
{U
L
(g
0
)V
L
U
†
L
(g
0
), U
Co
(g
0
)V
Co
U
†
Co
(g
0
)}, which can be
viewed as merely a change of basis on the original
gates, are — up to a small error E
00
g
0
— pairs of
transversal gates on E
cov
after the reference frames
have been measured and a value g
0
has been discerned
in step 3); which is performed before decoding. See
Supplementary material A for details.
2.1 Finite-sized clocks
We now detail the unitary group representations to
which our results apply to. For the logical and phys-
ical spaces, we consider all compact representations
of the Abelian U(1) group. These can be written in
the form U
L
(t) = e
−it
ˆ
H
L
, U
⊗K
Co
(t) = e
−it
ˆ
H
Co
respec-
tively, where using the shorthand S ∈ {L, Co}, the
generators are
ˆ
H
S
=
P
d
S
−1
n=0
ωh
S,n
|nihn|
S
for a d
S
di-
mensional space, for some frequency ω > 0. They
have fixed gaps but arbitrary degeneracy, so that
h
S,n
∈ {0, 1, 2, . . . , ∆h
S
} with 3 ∆h
S
≤ d
S
. As
we will see, the performance improves as the range
∆h
Co
decreases, so in fact the best choice is a trivial
generator
ˆ
H
Co
∝ , for which we can set ∆h
Co
= 0.
As motivated by our discussion of Pauli’s findings,
in this context, one can think of the group elements
U
F
(g) as providing the dynamics of quantum clocks
ρ
F
where g = t is time. We will therefore refer to
the reference frame as a “clock” and use the labels F
and C interchangeably in this context. The clock is
chosen as the compact non-degenerate representation,
namely U
C
(t) = e
−it
ˆ
H
C
with
ˆ
H
C
=
P
d
C
−1
r=0
ωr|rihr|
C
where {|ri
C
} is an arbitrary orthonormal basis. Phys-
ically, energy gaps which are all integer multiples of ω
are required to create degeneracies between systems
L, Co, C. Technically, it means that the group repre-
sentations on L, Co, C are compact, and that the ir-
reps of U
C
(g) contain those of U
L
(g) and U
Co
(g). The
compactness allows replacement of the integration
range in Eq. (8) with [0, T
0
] where T
0
= 2π/ω is the
recurrence time and the measure becomes dg = dt/T
0
.
The main clocks we use in Eq. (8) are based upon
the Quasi-Ideal clocks [21] and the SWP clocks [20].
They both share the same Hamiltonian
ˆ
H
C
. The def-
inition of SWP clock states are simple, they are any
one of the pure states of the Fourier-transformed basis
of eigenbasis of
ˆ
H
C
, namely ρ
SWP
= |θ
k
ihθ
k
| with
|θ
k
i =
1
√
d
C
d
C
−1
X
j=0
e
−i2πnj/d
C
|ji
C
, (12)
where |θ
k
i = |θ
k mod d.
i, k ∈ ; and is referred to as
the time basis. On the other hand, the Quasi-Ideal
clock states ρ
QI
= |Ψ(k
0
1
)ihΨ(k
0
1
)| are defined as a
coherent superposition of SWP clocks,
|Ψ(k
0
1
)i =
X
k∈S
d
C
(k
0
1
)
Ae
−
π
σ
2
(k−k
0
1
)
2
e
i2πn
0
(k−k
0
1
)/d
C
|θ
k
i,
where A is a normalization constant, S
d
C
(k
0
1
) is the
set of d
C
consecutive integers centred about k
0
1
∈
and ωn
0
∈ (0, ωd
C
) is approximately the mean energy
of the clock. k
0
1
can be thought of as an approximate
initial time marked by the clock. The parameters n
0
,
k
0
1
and σ ∈ (0, d
C
) can be tuned to our convenience.
Both the Quasi-Ideal and SWP clocks are usually
associated with the same projective operators in the
“time” basis, defined as
ˆ
t
C
=
d
C
−1
X
k=0
k
T
0
d
C
|θ
k
ihθ
k
|. (13)
When measuring individual clocks, we will do so in
this time basis. When the clocks are entangled, we
will use a time operator of this form but with the
projectors |θ
k
ihθ
k
| replaced with non-local projectors
over the multiple entangled clocks, as explained later.
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For a detailed description of the measurement proto-
cols, see Supplementary material B.
The Quasi-Ideal clock states have been shown to
yield a good performance in the context of quantum
control [21] and measurement of time [25]. On the
other hand, in [21, 25] the SWP clocks appeared to
be suboptimal. For the task at hand, we will prove
that this difference of performance still occurs.
2.2 Entanglement fidelity
The figure of merit we will use for quantifying the
performance of codes which cannot decode perfectly
is the worst-case entanglement fidelity. Given a se-
quence of encoding, error and decoding which we la-
bel as the channel K : H
L
→ H
L
, the entanglement
fidelity is defined as
f
worst
(K) = min
φ
hφ|K ⊗ I(|φihφ|) |φi, (14)
where I is an identity channel acting on a copy of H
L
,
and the optimization is over all bipartite states on
which the map K⊗I acts.
5
Having a perfect code as
those satisfying Eq. (1), is equivalent to f
worst
(K) =
1, which is achieved with perfect reference frames or
Idealised clocks. For a justification of why this is a
good measure of approximate codes, see for instance
[32, 33].
The channel K that we consider here is for the error
correction codes {E
cov
, {E
j,q
,
˜
D
j,q
}
j,q
} as described in
section 2, in which we use clocks that are not idealised.
3 Results
For simplicity, we will only state our bounds to lead-
ing order in d
C
, d
P
, d
L
, L. Here d
P
≥ d
Co
is the di-
mension of the physical space and L is introduced
later. Except for Theorem 4, explicit bounds —
not just the leading order terms — can be found in
the supplementary material. Furthermore, in all the-
orems and corollaries in which the Quasi-Ideal clock is
involved, the width σ scales logarithmically with d
C
.
The exact value can be found in the proofs.
Our first result covers the case in which at least one
of the clocks is left untouched by erasure errors.
Theorem 1. Consider the covariant code E
cov
in Eq.
(8), with M Quasi-Ideal clocks, namely
E
cov
(·) =
1
T
0
Z
T
0
0
dt E
t
(·) ⊗ U
C
(t)
⊗M
ρ
⊗M
QI
U
C
(t)
†⊗M
,
(15)
with error channels {E
j,q
= E
j
⊗ ξ
q
}
j,q
, where {ξ
q
}
q
are erasure at known locations of at most M −1 clocks
and {E
j
}
j
are the error channels for the code E. Then
for all M ∈
+
, and for all error correcting codes
5
Note that it is sufficient to take an ancillary space of size
d
L
.
{E, {E
j
, D
j
}
j
}, there exists decoding channels
˜
D
j,q
for
E
cov
such that the worst-case entanglement fidelity of
the covariant encoding satisfies
f
worst
≥ 1 −
3π
√
d
L
d
P
4
ln
3
(d
C
)
d
C
2
(∆h
L
+ ∆h
Co
)
2
+ O
p
d
L
d
P
ln
3/2
(d
C
)
d
C
3
!
. (16)
For the proof, we need to write the encoding-
error-decoding scheme in the form described in
supplementary material B when we have a single clock
unaffected by errors. After working out the decod-
ing protocol explicitly, we find a bound that, to-
gether with the results on the entanglement fidelity
of supplementary material C allows us to derive the
Theorem in supplementary material D, after choos-
ing σ = ln
3
(d
C
) for the single clock we measure.
In order to study the role that the different re-
sources play in the current scheme, consider the fol-
lowing definitions. We call a clock state ρ
inc
∈ S(H
L
),
with time operator
ˆ
t
C
and unitary group representa-
tion {U
C
(t)}
t∈
, t-incoherent if there exists a state
in its periodic orbit which commutes with the time
operator, namely if there exists t
0
∈ such that
[U
C
(t
0
)ρ
inc
U
†
C
(t
0
),
ˆ
t
C
] = 0, (17)
where
ˆ
t
C
is the operator used in step 3) to measure
the clock.
6
Conversely, clock states which are not t-
incoherent are called t-coherent.
Observe that if one can construct the code E and
decoders D
j
, then for a given initial clock state ρ
(M)
C
,
in order to construct the covariant code E
cov
in Eq.
(8), one needs to be able to apply the transversal gates
to the clock and code E, and create classical correla-
tions (a separable state) between them. For the de-
coders
˜
D
j
, the only additional resource required to
apply them is the ability to measure the clocks pro-
jectively in the time basis. The ability to perform such
operations is always required for the construction of
any error correction code {E
cov
, {E
j,q
,
˜
D
j,q
}
j,q
}. In
the case of t-incoherent clocks, the initial clock state
ρ
(M)
C
can then easily be constructed by simply measur-
ing any state on the clock Hilbert space and applying
the appropriate traversal gate. However, in the case
that the initial clock state ρ
(M)
C
is t-coherent (such
as in the case of Quasi-Ideal clock states in Theorem
1), one cannot constructed it with any of the above
mentioned operations since they do not allow for the
creation of the necessary coherence (that is, with re-
spect to the basis of
ˆ
t
C
). Therefore, from a resource-
theoretic standpoint, it is interesting to characterise
how good codes can be if they only use t-incoherent
clocks:
6
Note that in the case of time operators which are not pro-
jective POVMs, t-incoherent clocks can still be defined by re-
placing
ˆ
t
C
by the 1st moment operator of the POVM. We do
not need to consider such generalisations here however.
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Theorem 2. Consider the covariant code E
cov
in Eq.
(8), with one d
C
dimensional t-incoherent clock ρ
inc
,
with time operator
ˆ
t
C
defined in Eq. (13),
E
cov
(·) =
1
T
0
Z
T
0
0
dt E
t
(·) ⊗ U
C
(t) ρ
inc
U
†
C
(t). (18)
Allow for no errors on the clock, {E
j,q
= E
j
⊗
ξ
q
= E
j
⊗ I}
j,q
. Then for all error correcting codes
{E, {E
j
, D
j
}
j
}, the entanglement fidelity that can be
achieved is upper bounded by
f
worst
≤ 1 −
C
d
C
, (19)
where C > 0 is independent of d
C
. Moreover,
there exists a scheme using the Salecker-Wigner-Peres
clock, ρ
inc
= ρ
SWP
that achieves
f
worst
= 1 −
C
∗
d
C
, (20)
where C
∗
≥ C is also independent of d
C
.
The full proof is shown in supplementary material
E and it goes as follows: we write explicitly the state
of the code after the clock has been measured, and
we apply an arbitrary decoder. Then we write the
entanglement fidelity explicitly and show that the er-
ror term decays at best linearly with the dimension
of the clock. The performance achieved by the SWP
clock is calculated by looking at the scaling of the
errors in the decoding scheme of Section 2.
The setup in the above Theorem is the same as in
Theorem 1 in the special case that M = 1 and one ex-
changes the Quasi-Ideal clock for a t-incoherent one.
So by comparing Eqs. (16) and (19) we see that there
is essentially a quadratic improvement in the scaling
w.r.t. the clock dimension d
C
. This demonstrates the
necessity of t-coherent clock states to achieve the im-
proved scaling in our protocols. Furthermore, in all
our protocols, the final state of the clock after apply-
ing a decoder
˜
D
j
is t-incoherent and thus in protocols
in which the initial clock state was t-coherent, the
coherence was “used up” in the process.
Interestingly, we have observed the same effec-
tive quadratic advantage by using the Quasi-Ideal
clock rather than the SWP clock for the related
task of reference frame alignment (See Section 2 and
Supplementary material I) and ticking clocks [25].
While t-coherent clock states are necessary to
achieve the improved scaling, they are clearly not suf-
ficient. A case in point are clock states which are
incoherent in the energy basis. Indeed, by inverting
Eq. (12) one observes that they are t-coherent yet
since they do not evolve in time are useless for this
task.
Changing the width σ of the Quasi-Ideal clock state
allows one to understand these differences better. The
uncertainty in both the energy and time basis, de-
noted ∆E and ∆t respectively, satisfy ∆E∆t = 1/2
to leading order in d
C
for all Quasi-Ideal clock states.
They are thus approximately minimum uncertainty
states. By changing the value of σ/
√
d
C
from small
to large one can achieve the limiting cases of a time
eigenstate ∆E ∆t, and an energy eigenstate ∆E
∆t. Eq. (16) in Theorem 1 corresponds to the opti-
mal value of σ = ln
3
(d
C
), which corresponds to states
which are time squeezed (∆E ≥ ∆t), but not by too
much — the states do not become (t-incoherent) time
eigenstates. On the other hand, Quasi-Ideal clock
states which are energy squeezed (∆E ≤ ∆t), yield
an entanglement fidelity f
worst
which scales with d
C
even worse than t-incoherent states. In-between, one
finds the non squeezed (∆E = ∆t) Quasi-Ideal states
which have the same d
−1
C
scaling as the t-incoherent
SWP clock.
The latter Quasi-Ideal clock states behave anal-
ogously to “classical” coherent states — their ex-
pectation values in the time and energy bases oscil-
late like simple harmonic oscillators, and they min-
imize the Heisenberg uncertainty with equal uncer-
tainty in each basis. However, the analogy ends here.
The time squeezed Quasi-Ideal clock states remain
time squeezed under the application of U
C
(t) for all
t ∈ , while squeezed coherent states in one quadra-
ture (e.g. position) become squeezed in the comple-
mentary quadrature (e.g. momentum) under evolu-
tion of its Hamiltonian — broadening in the initial
quadrature basis. Intuitively, this is expected to be
an important difference between squeezed coherent
states and squeezed Quasi-Ideal clock states, at least
regarding the present task. This is because good de-
coding maps would require the states to be squeezed
during the entire periodic orbit in the basis in which
they are measured in step 3) of the decoding protocol.
We also find a significant improvement to the en-
tanglement fidelity when one has access to a larger
number of clocks. To achieve it, we embed a large
entangled clock within the Hilbert space of L ∈
+
smaller ones, effectively creating a clock of dimension
d(L) := L(d
C
− 1) + 1.
Theorem 3. Given a covariant error correction code
{E
cov
, {E
j,q
,
˜
D
j,q
}
j,q
} as described in section 2, with
entanglement fidelity f(d
C
) ≤ f
worst
≤ f
0
(d
C
) in
which a single d
C
∈ S
N
⊆
+
dimensional clock
is used, there exists another covariant error correc-
tion code with the same error channels {E
j,q
}
j,q
in
which L clocks of dimension d
C
are entangled, such
that f(L(d
C
− 1) + 1) ≤ f
worst
≤ f
0
(L(d
C
− 1) + 1),
for all L, d
C
∈
+
, s.t. L(d
C
− 1) + 1 ∈ S
N
.
The embedding needed for this theorem leads natu-
rally to an entangled discrete Fourier transform basis
on the Hilbert space of L clocks
|θ
k
(L)i =
1
p
d(L)
d(L)
X
n=0
e
−i2πnk/d(L)
|E
n,1
i, (21)
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where |E
n,1
i are a non-degenerate set of L
eigenvectors of the generator
L
L
i=1
ˆ
H
C
7
(see
supplementary material F for details).
When this embedding is applied to the SWP clock
and Quasi-Ideal clock, it gives rise to a L-site SWP
Entangled clock state, ρ
SWPE,L
= |θ
k
(L)ihθ
k
(L)|
and an L-site Quasi-Ideal Entangled clock, ρ
QIE,L
=
|Ψ
L
(k
0
1
)ihΨ
L
(k
0
1
)|,
|Ψ
L
(k
0
1
)i =
X
k∈S
d(L)
(k
0
1
)
Ae
−
π
σ
2
(k−k
0
1
)
2
e
i2πn
0
(k−k
0
1
)
d(L)
|θ
k
(L)i,
with the new time operator,
ˆ
t
C
(L) =
d
C
−1
X
k=0
k
T
0
d
C
|θ
k
(L)ihθ
k
(L)|. (22)
The combination of Theorem 3 with Theorems 1,
2, yields an important corollary:
Corollary 1. Consider the setup in Theorem 1, but
with M L-site Quasi-Ideal Entangled clocks ρ
⊗M
QIE,L
,
rather than M Quasi-Ideal clocks and the erasure now
being on at most M −1 of the L-site Quasi-Ideal En-
tangled clocks. The worst-case entanglement fidelity
of the covariant encoding now satisfies
f
worst
≥ 1 −
3π
√
d
L
d
P
4
ln
3
(L d
C
)
L d
C
2
(∆h
L
+ ∆h
Co
)
2
+ O
p
d
L
d
P
ln
3/2
(L d
C
)
(L d
C
)
3
!
. (23)
Similarly, consider the setup in Theorem 2, but with a
d
C
dimensional t-incoherent clock state ρ
inc,L
, with a
time operator
ˆ
t
C
(L), rather than a SWP clock with
time operator
ˆ
t
C
. The worst-case entanglement fi-
delity of the covariant encoding now satisfies
f
worst
≤ 1 −
C
L d
C
, (24)
where C is independent of d
C
, L; and equality is ob-
tained for the L-site SWP Entangled clock ρ
SWPE,L
.
If one compares the scaling with the number of
copies L in Eqs. (23),(24) one finds effectively a
quadratic advantage in the case of the t-coherent
states. The difference in scaling between the two cases
is the same as that found in metrology when compar-
ing the classical shot noise scaling with the quantum
Heisenberg scaling [34].
The bound Eq. (23) in Corollary 1 effectively satu-
rates the upper bound derived in [26], which is proven
to hold for all covariant error correction codes gener-
ated by isometries. Applying it to our constructions,
it takes the form
f
worst
≤ 1 −
∆h
2
L
16(∆h
Co
+ Ld
C
)
2
. (25)
7
The symbol
L
denotes the Kronecker sum.
For the details about how this inequality follows from
the results of [26] see Supplementary material G.
8
If we keep the code parameters ∆h
Co
, ∆h
L
, d
P
, d
L
fixed and scale up the clock size and the number of
clocks, the combination of lower and upper bounds
Eqs. 23, 25 prove that our construction achieves an
optimal scaling with both the dimension of the clock
d
C
and the number of them L, up to the logarithmic
factors. Furthermore, this proves that the bound is
tight for all choices of {E, {E
j
, D
j
}
j
}.
So far we have only considered erasure errors at
known locations on the clock. However, what if one
cannot detect where the error occurred without dam-
aging the code? This is the case in the most elemen-
tary error correcting codes. If one has many clocks
and the error occurs with an approximately uniform
error probability distribution over the clock locations,
one simple approach would be to choose one of the
clocks, erase the other clocks, and perform error cor-
rection with this clock. If the probability of the error
occurring on this clock is low, then this would work
well on average. However, this approach is wasteful
since it does not use the encoded information in the
other clocks and requires a low probability of error on
a particular clock. Now we will investigate how well
we can recover in the case of unknown phase errors at
unknown locations which also works well for a small
number of clocks. Consider the case in which one
clock (or an entangled block of L clocks in the sense
of Theorem 3) whose location is unknown has a ran-
dom phase applied to it (i.e. an un-known group ele-
ment U
⊗L
C
(t
ph
), t
ph
∈ ). We call this a 2-unknown
phase error and denote it ξ
ph,q
(t
ph
), where q ∈ and
t
ph
∈ are the unknown site location and phase re-
spectively. The following result shows that such errors
are correctable up to an arbitrarily small error.
Theorem 4. Consider the covariant code E
cov
in
Eq. (8), with 3 blocks of L-site Quasi-Ideal Entan-
gled clocks, namely
E
cov
(·) =
1
T
0
Z
T
0
0
dt E
t
(·)⊗U
C
(t)
⊗3L
ρ
⊗3
QIE,L
U
C
(t)
†⊗3L
,
(26)
with error channels {E
j,q
= E
j
⊗ ξ
ph,q
(t
ph
)}
j,q
, q ∈
1, 2, 3 where {E
j
}
j
are the error channels for the code
E, and ξ
ph,q
is a 2-unknown phase error acting on
one of the three L-site Quasi-Ideal Entangled clocks,
ρ
QIE,L
. Then for all L ∈
+
, q ∈ {1, 2, 3}, t
ph
∈
and error correcting codes {E, {E
j
, D
j
}
j
}, there exists
a decoding channel
˜
D
j
for E
cov
, which is independent
of the unknown block q and phase t
ph
, such that
f
worst
≥ 1−
p
d
L
d
P
10π(∆h
L
+ ∆h
Co
)
√
3
ln
7
(L d
C
)
L d
C
+ O
p
d
L
d
P
ln
11
(L d
C
)
(L d
C
)
2
. (27)
8
Note that the extra additive factor ∆h
P
in Eq. (25) is to
be expected since the code E could itself contain a clock.
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Figure 1: Illustration of 3 clocks superimposed with a
2-unknown phase error. All clocks are set to have the
same initial time, k
0
1
= k
0
2
= k
0
3
(red glowing hand).
The three measurement outcomes of the three clocks (black
hands) attain similar values with high probability. The yel-
low shaded region represents possible values of k
α
for this
example, where k
α
= g
0
(d
C
− 1)/(2πT
0
), and g
0
is described
in step 3) of the decoding protocol in Section 2. An un-
known phase error occurs on an unknown clock (for the
purpose of illustration, this is the 2
nd
clock). It has the
effect of shifting the initial time of the 2
nd
clock by an un-
known amount k
ph
. The apparent elapsed times are ∆t
1
=
2π(k
1
−k
0
1
)T
0
/(d
C
−1), ∆t
2
= 2π(k
2
−k
0
2
−k
ph
)T
0
/(d
C
−1),
∆t
3
= 2π(k
3
− k
0
3
)T
0
/(d
C
− 1), however, due to the 2-
unknown phase error, ∆t
2
will give the incorrect prediction.
Nevertheless, in this example, the phase error is small and no
correction is needed.
This result extends trivially to the more general
case in which one has M blocks (rather than 3) of
L-site Quasi-Ideal Entangled clocks in the covari-
ant code and has erasure errors at up to M − 3
known locations, and the 2-unknown phase error on
one of the 3 blocks of the remaining copies. See
supplementary material H for proof.
To gain an intuitive picture of how the protocol
works, it is best to consider the case L = 1 (the gen-
eral L case is then a direct consequence of Theorem
3). For L = 1 the protocol is similar to our previous
ones: we measure the 3 clocks locally in the time basis,
and based upon the information from the 3 measure-
ment outcomes, we calculate a time g
0
= t
0
[step 3)]
and apply a corresponding decoding map
¯
D
j,t
0
[step
4)] on the physical space. Due to the classical corre-
lations between the code and clocks, the outcomes of
the 3 clocks are correlated. This is such that, if there
were no 2-unknown phase errors, the clocks would all
indicate approximately the same “elapsed time” and
one could correct the code up to an error of order
Eq. (27) with the knowledge of only one of the 3
measurement outcomes. However, when a phase er-
ror occurs in the q
th
clock, its initial time k
0
q
shifts by
an unknown phase, making it an unknown variable
and the reported elapsed time by the clock measure-
ment of the q
th
clock, is thus incorrect. Since the
value of q is also unknown, one cannot simply ignore
the corresponding measurement outcome, so our pro-
tocol is to order the three measured elapsed times in
ascending order and apply a unitary corresponding to
the elapsed time which is neither the smallest nor the
largest out of the three. There are two possibilities:
1) there was no phase error on this clock, in which
case the marked elapsed time is approximately cor-
rect. In this case, the phase error could have been
large. 2) the phase error occurred on this clock. In
this case, the phase error must have been small, since
the (incorrect) measured elapsed time is upper and
lower bounded by that of the other two clocks (which
must both be correct, since there is at most one 2-
unknown phase error). This corresponds to the case
of Fig. 1.
This processing of the measurement outcomes is
very closely related to majority voting used in a clas-
sical repetition code. Here the main difference is that
the outcomes of the clocks are not binary and are un-
likely to agree exactly.
Note how our protocol does not rely on any assump-
tions about the probability distributions over blocks
q = 1, 2, 3 and phases t
ph
∈ over which the 2-
unknown phase error occurs. However, if one does
assume that the probability of the 2-unknown phase
error ξ
ph,q
(t
ph
) is small and allows for an arbitrary
number of 2-unknown phase errors to occur in an in-
dependent and identically distributed manner, then
the above protocol will also work with high fidelity
since the probability of two or more phase errors oc-
curring on two or three clocks will be very small.
Finally, what about infinite dimensional covariant
codes of finite energy? We leave the motivation to the
Outlook (Section 4) and state our conclusions here.
One can embed our finite dimensional clock into an
infinite dimensional space, and express d
C
in terms
of the mean energy of the clock, denoted h
ˆ
H
C
i. For
all the clocks considered here, this corresponds to the
mapping
d
C
7→ 2h
ˆ
H
C
i/ω + 1, (28)
(up to additive higher order corrections in h
ˆ
H
C
i,
for the case of Quasi-Ideal clocks). Our theorems
and corollary; when framed in this context, provide
bounds as a function of energy for how well the errors
on covariant codes can be corrected. In this context,
due to the infinite dimensions of the physical space,
the Eastin-Knill no-go theorem does not apply, so in
principle one could have perfect covariant error cor-
recting codes. However, as we have seen in Section 2,
due to the insights of Pauli, covariant error correcting
codes based on reference frames can only be decodable
without errors (i.e. f
worst
= 1) iff they use Idealised
clocks which necessitate infinite energy. We conjec-
ture that this is true in general. Specifically, that
all error correcting codes which are covariant w.r.t.
any faithful representation of a non-trivial Lie group,
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necessitate infinite energy. This would be an exten-
sion of the Eastin-Knill no-go theorem since all finite
dimensional systems have finite energy but not vice-
versa.
4 Outlook
Given any error correcting code, we have seen how
to construct simple classes of approximate codes in
which compact U(1) groups can be implemented
transversally. We study a specific scheme using Quasi-
Ideal clocks [21] that saturates optimal bounds, ex-
plore the performance of alternative schemes, and also
extend to an error model beyond errors at known lo-
cations. The present codes are based on the task
of reference frame alignment, which has been stud-
ied quite extensively [15] and also formalized within
the resource theory of asymmetry [35, 36], which we
hope will be of further use in the context of error cor-
rection [1, 26]. Furthermore, since the clocks need to
be measured projectively in our protocols, a preferred
basis naturally arises and necessary conditions w.r.t.
this basis for a quadratic advantage are identified.
An obvious extension of our results is to groups be-
yond U(1). This requires the construction of more
involved reference frames, such as finite-sized quan-
tum “gyroscopes” (for the case of SU(2)). We hope
that our techniques may serve as a starting point for
generalising the results found here in this direction
— perhaps together with the construction of approx-
imate frames for general groups from [31].
An important question is to understand and char-
acterize the sort of error models and implementations
for which the present scheme is adequate. We have
mostly explored the case of erasure errors at known
locations, and a type of dephasing errors on the clocks
at unknown locations. While these are quite natural
error models, it may well be that with more involved
schemes of codes involving clocks, other types of er-
rors can be dealt with.
Also, the results of [26] show tight bounds that can-
not be overcome only for erasure errors at known lo-
cations, and it would be interesting to know if their
bounds are also tight for other error models. We
suspect that the error scaling in Theorem 4 is op-
timal, even though it is only as good as that of the t-
incoherent clock states when the error model was that
of erasure at know locations. Intuitively, the limiting
factor in this case is that the 3 blocks of clocks are
only classically correlated (separable states). While
one can easily construct entangled counterparts, it is
not clear how this can help when the errors occur over
the unknown blocks.
One key point to be determined is whether the er-
ror bounds shown here, even if they are essentially
optimal, are not too large to be useful in practice for
some type of architecture. In the case of Theorem
1 and Corollary 1, explicit bounds on the entangle-
ment fidelity f
worst
for finite d
C
have been derived in
the supplementary material which can be evaluated
numerically for experimentally feasible parameters.
Were this the case, we would hope that the reference
frames used here might be useful in some near-term
applications of quantum technologies in which error
correction has started playing a role, such as quan-
tum metrology [37–39]. In particular, given Eq. (25),
the Quasi-Ideal clock states appear to be almost op-
timally distiguishable along a whole U (1) orbit. This
suggests that a setting like that of Theorem 4 could
be of use in the construction of a metrological scheme
aiming to determine unknown parameters (such as the
time t of the operator U
C
(t)).
Covariant infinite dimensional error correcting
codes are also of interest. Perhaps most prominently
is the example of the hypothesized AdS/CFT dual-
ity between quantum gravity in asymptotically AdS
space (known as the bulk) and a conformal field the-
ory on the boundary, where the theory on the bulk
and boundary are related via quantum error correct-
ing codes [40]. Specifically, the bulk constitutes the
logical space while the boundary is the physical/code
space. Global symmetries in the bulk should cor-
respond to symmetries on the boundary which con-
serve the local structure of the theory — in a simi-
lar spirit to how transversal gates act globally in the
logical space while locally in the physical space [c.f.
Eq. (3)]. It is argued that variants of the Eastin-
Knill no-go theorem have important interpretations
in AdS/CFT: all global symmetries in the bulk (both
continuous and discrete) are ruled out, since their
existence would be in contradiction with the struc-
ture of the correspondence
9
[41, 42]. A related is-
sue is time dynamics which is given by a U(1) sym-
metry with representations e
−it
ˆ
H
blk
and e
−it
ˆ
H
bdy
for
Hamiltonians
ˆ
H
blk
,
ˆ
H
bdy
on the bulk and boundary
respectively. Often
ˆ
H
blk
,
ˆ
H
bdy
are quasi-local and
thus the corresponding U(1) group action has to also
preserve this local structure. An approximate U(1)
covariant code with these properties for finite dimen-
sional Hamiltonians has been developed in [43] us-
ing techniques from [44]. By choosing the number
of code blocks to be one i.e. K = 1, our protocols
allow for such bulk-boundary time dynamics covari-
ance for arbitrary (e.g. quasi-local) finite dimensional
Hamiltonians on H
L
and H
Co
. An open question is
whether further work may allow for clocks with inter-
acting quasi-local Hamiltonians too.
Since both theories of the duality are infinite-
dimensional but of finite energy locally, further quan-
titative variants of the Eastin-Knill no-go theorem for
infinite dimensional physical spaces, seem more ap-
9
Local CFT operators acting on a spatial region R in the
boundary, can only access information in a limited region in
the bulk via entanglement wedge reconstruction. This is in-
compatible with the existence of charged localised objects in
the bulk.
Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 9
propriate (for instance, by taking into account the
“average energy density” of the code space). Our re-
sults suggest that given sufficient energy, the preser-
vation of local symmetries in the boundary is at least
approximately possible.
Finally, it is worth noting that the extension of
a physical Hilbert space by including clocks has
been proposed in a different context before. The
aim of the Wheeler-DeWit equation is to describe
a time-static theory of quantum gravity in which
locally one finds dynamics [45]. This motivated
the Page-Wooters mechanism for describing how a
quantum clock can allow for time evolution to follow
from a static universe using Idealised clocks [46]. In
supplementary material J we show that when the
logical space has the trivial group representation of
the U(1) symmetry, our formalism is an example
of an approximate Page-Wooters mechanism where
the approximation comes from using non Idealised
clocks.
Note on Related Work: Around the time this
work was being developed, it was realized by the au-
thors that a complementary approach to characteriz-
ing how well the Eastin-Knill no-go theorem can be
circumvented by allowing for a small error, was being
developed independently by the authors of [26]. We
thank them for their community spirit.
Acknowledgements
We thank useful conversations with Benjamin Brown,
Philippe Faist, Daniel Harlow, Tamara Kohler and
Rob Spekkens. This work was initiated at QIP 2018
hosted by QuTech at Delft University of Technol-
ogy. M.P.W. acknowledges funding from the Swiss
National Science Foundation (AMBIZIONE Fellow-
ship PZ00P2 179914) in addition to the NCCR QSIT.
This research was supported in part by Perimeter In-
stitute for Theoretical Physics. Research at Perimeter
Institute is supported by the Government of Canada
through the Department of Innovation, Science and
Economic Development and by the Province of On-
tario through the Ministry of Research, Innovation
and Science.
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A Compatibility with the transversal gates of E
In our schemes we start with a code E(·) which is in principle arbitrary. We can ask what happens if this code
already has a group of transversal or covariant gates (possibly of finite order as granted by [7]). Does that
covariance remain after the reference frame is added? Let {V
L
, V
Co
, V
F
} be an element of a representation of
the group of transversal gates of E on the logical, physical and reference frame spaces. Thus:
E
V
L
(·)V
†
L
= V
⊗K
Co
E(·)V
†⊗K
Co
, (29)
where ⊗K represents the tensor product structure of the physical space of E. Does the same symmetry hold for
E
cov
? Let us write
E
cov
(V
L
(·)V
†
L
) =
Z
G
dg E
g
(V
L
(·)V
†
L
) ⊗ U
F
(g)
⊗M
ρ
(M)
C
U
F
(g)
†⊗M
, (30)
whereas we have, on the other hand
V
⊗K
Co
⊗ V
F
E
cov
(·)
V
†⊗K
Co
⊗ V
†
F
=
Z
G
dg V
⊗K
Co
E
g
(·)V
†⊗K
Co
⊗ V
F
(U
F
(g)
⊗M
ρ
(M)
C
U
F
(g)
†⊗M
)V
†
F
, (31)
It is clear that Eqs. (30) and (31) coincide when i) both commutations [V
Co
, U
Co
(g)
⊗K
] = 0 and [V
L
, U
L
(g)] = 0
hold for all g ∈ G and ii) V
F
acts trivially on the clocks. Otherwise, the covariance/transversality is lost. While
the latter can be obtained by choosing V
F
to act trivially on the clocks, the former seems much more restrictive.
However, there is a sense in which we might still have transversal gates in the code, if we are able to set the
representation U
Co
(g)
⊗K
on the physical space to the trivial case. After measuring the clocks and obtaining
some outcome g
0
, one can apply the gates V
⊗K
P
⊗ V
F
. Then, applying the decoder, the resulting state is
V
L
U
†
L
(g
0
)ρ
L
U
L
(g
0
))V
†
L
. (32)
If [V
L
, U
L
(g)] = 0 we may just apply U
L
(g) and end up with the desired state to which the gate has been applied
transversally. Otherwise, what has happened is that we have applied the gate V
L
U
†
L
(g
0
). We have knowledge
Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 12
of g
0
, and so to end up with V
L
ρ
L
V
†
L
we may just apply the unitary V
L
U
L
(g
0
)V
†
L
. This assumes that one can
apply the logical gate both at the logical and the physical level.
This discussion gives a further example, together with the errors of the main results, of why it is advantageous
to choose the generator
ˆ
H
Co
to be trivial
ˆ
H
Co
∝
P
, if possible.
B General encoding-decoding error with finite clocks
Here we explain the form for the encoding-error-decoding protocol for all the schemes of the present work.
We refer to this discussion repeatedly in the different proofs. We will here assume that the M clocks are of
product form. We will see later that this construction is general enough for our purposes even when the clock
are entangled, by considering further divisions of the local sites considered here.
We define the dimensions of the input and output of the error correcting code E to be d
L
, d
P
. The dimension
of the ith clock is d
i
. If all clocks have the same dimension we use the notation d
C
:= d
1
= d
2
= . . .. If we do
not write the range over a summation, it will be over the full range.
For our case, the unitary representation is U
C
(t) = e
−it
ˆ
H
C
where
ˆ
H
C
is defined in Section 2.1. The integral
measure over the group is dg = dt/T
0
on the interval [0, T
0
]. The encoded state in which M clocks in states
ρ
C,i
with i ∈ {1, ..M} are used has the form of Eq. (8)
E
cov
(ρ
L
) =
1
T
0
Z
T
0
0
dt E
t
(ρ
L
)
M
O
i=1
U
C
(t)ρ
C,i
U
†
C
(t). (33)
Let us first calculate the integral over t by writing the unitary operators U
C
(t), U
L
(t), U
Co
(t) in their
eigenbasis:
E
U
†
L
(t)ρ
L
U
L
(t)
=
d
L
−1
X
n,n
0
=0
e
−iωt(h
L,n
−h
L,n
0
)
E (
L
hn
0
|ρ
L
|ni
L
|n
0
ihn|) , (34)
and thus,
E
t
(ρ
L
) =
d
P
−1
X
q,q
0
=0
d
L
−1
X
n,n
0
=0
e
−iωt(h
Co,q
−h
Co,q
0
+h
L,n
−h
L,n
0
)
E
q,q
0
,n,n
0
(ρ
L
) |qihq
0
|
P
, (35)
with
E
q,q
0
,n,n
0
(ρ
L
) :=
P
hq|E (
L
hn
0
|ρ
L
|ni
L
|n
0
ihn|) |q
0
i
P
. (36)
For simplicity, let us define Q := h
Co,q
− h
Co,q
0
+ h
L,n
− h
L,n
0
. Therefore
E
cov
(ρ
L
) =
1
T
0
Z
T
0
0
dt E
t
(·) ⊗ ρ
C,1
(t) ⊗ ρ
C,2
(t) ⊗ . . . ⊗ ρ
C,M
(t), (37)
=
d
P
−1
X
q,q
0
=0
d
L
−1
X
n,n
0
=0
X
~r
0
M
,~r
0
M
1
T
0
Z
T
0
0
dt e
−iωt(Q+r
1
−r
0
1
+...+r
M
−r
0
M
)
E
q,q
0
,n,n
0
(ρ
L
) |qihq
0
| (38)
⊗ hr
1
|ρ
C,1
|r
1
i ⊗ . . . ⊗ hr
M
|ρ
C,M
|r
M
i (39)
=
d
P
−1
X
q,q
0
=0
d
L
−1
X
n,n
0
=0
X
~r
M
,~r
0
M
δ
Q+r
1
−r
0
1
+...+r
M
−r
0
M
,0
E
q,q
0
,n,n
0
(ρ
L
) |qihq
0
| (40)
⊗ hr
1
|ρ
C,1
|r
0
1
i|r
1
ihr
0
1
| ⊗ . . . ⊗ hr
M
|ρ
C,M
|r
0
M
i|r
M
ihr
0
M
|, (41)
where {|r
i
i} is the eigenbasis of the generator
ˆ
H
C
of the i-th clock, δ
(·,·)
is the Kronecker-Delta function and
X
~r
M
,~r
0
M
=
d
1
−1
X
r
1
,r
0
1
,=0
. . .
d
M
−1
X
r
M
,r
0
M
,=0
. (42)
Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 13
Since by assumption the last N + 1, . . . , M clocks may be lost due to erasure errors, we can trace them out:
tr
C
N+1
...C
M
[E
cov
(ρ
L
)] =
d
P
−1
X
q,q
0
=0
d
L
−1
X
n,n
0
=0
X
~r
M
,~r
0
M
δ
Q+r
1
−r
0
1
+...+r
N
−r
0
N
,0
E
q,q
0
,n,n
0
(ρ
L
) |qihq
0
| (43)
⊗hr
1
|ρ
C,1
|r
0
1
i|r
1
ihr
0
1
| ⊗ . . . ⊗ hr
N
|ρ
C,N
|r
0
N
i|r
N
ihr
0
N
|(hr
N+1
|ρ
C,N+1
|r
N+1
i. . . hr
M
|ρ
C,M
|r
M
i) (44)
=
d
P
−1
X
q,q
0
=0
d
L
−1
X
n,n
0
=0
X
~r
N
,~r
0
N
δ
Q+r
1
−r
0
1
+...+r
N
−r
0
N
,0
E
q,q
0
,n,n
0
(ρ
L
) |qihq
0
| (45)
⊗hr
1
|ρ
C,1
|r
0
1
i|r
1
ihr
0
1
| ⊗ . . . ⊗ hr
N
|ρ
C,N
|r
0
N
i|r
N
ihr
0
N
|. (46)
So by comparing Eqs. (46), (41), we observe that after tracing out the additional clocks, the resultant channel
is of the same form as the original channel, when evaluated for the renaming number of clocks.
We now assume that an error due to the environment occurs. This means that we apply a CPTP error map
E
j
: H
P
→ H
P
corresponding to an error j. We thus denote
E
j
q,q
0
,n,n
0
(ρ
L
) :=
P
hq|E
j
(E (
L
hn
0
|ρ
L
|ni
L
|n
0
ihn|)) |q
0
i
P
. (47)
We now measure the remaining clocks, performing projective measurements in the “time” basis
(
|θ
k
i
m
=
1
√
d
m
d
m
−1
X
n=0
e
−i2πnk/d
m
|ni
m
)
d
m
k=0
(48)
on the mth clock. Let us drop the m subindex for simplicity of notation. The state after the outcome
~
k :=
k
1
, k
2
, . . . , k
N
is:
ρ
~
k
P
⊗ |θ
k
1
ihθ
k
1
| ⊗ . . . ⊗ |θ
k
N
ihθ
k
N
| := (49)
|θ
k
1
ihθ
k
1
| ⊗ . . . ⊗ |θ
k
N
ihθ
k
N
|tr
C
N+1
...C
M
[E
j
E
cov
(ρ
L
)] |θ
k
1
ihθ
k
1
| ⊗ . . . ⊗ |θ
k
N
ihθ
k
N
|
p
~
k
(50)
=
1
p
~
k
d
P
−1
X
q,q
0
=0
d
L
−1
X
n,n
0
=0
X
~r
N
,~r
0
N
δ
Q+r
1
−r
0
1
+...+r
N
−r
0
N
,0
E
j
q,q
0
,n,n
0
(ρ
L
) |qihq
0
| (51)
hr
1
|ρ
C,1
|r
0
1
ihθ
k
1
|r
1
ihr
0
1
|θ
k
1
i. . . hr
N
|ρ
C,N
|r
0
N
ihθ
k
N
|r
N
ihr
0
N
|θ
k
N
i
!
⊗ |θ
k
1
ihθ
k
1
| ⊗ . . . ⊗ |θ
k
N
ihθ
k
N
|. (52)
Thus, after postselecting on a particular outcome represented with the vector
~
k, we obtain the following state
on the Hilbert space of the output of E
j
E(·)
ρ
~
k
P
=
1
p
~
k
d
P
−1
X
q,q
0
=0
d
L
−1
X
n,n
0
=0
X
~r
N
,~r
0
N
δ
Q+r
1
−r
0
1
+...+r
N
−r
0
N
,0
E
j
q,q
0
,n,n
0
(ρ
L
) |qihq
0
| (53)
hr
1
|ρ
C,1
|r
0
1
ihθ
k
1
|r
1
ihr
0
1
|θ
k
1
i. . . hr
N
|ρ
C,N
|r
0
N
ihθ
k
N
|r
N
ihr
0
N
|θ
k
N
i. (54)
An outcome labeled with
~
k happens with a probability given by
p
~
k
= tr
"
|θ
k
1
ihθ
k
1
| ⊗ . . . ⊗ |θ
k
N
ihθ
k
N
|tr
C
N+1
...C
M
[E
j
E
cov
(ρ
in
)] |θ
k
1
ihθ
k
1
| ⊗ . . . ⊗ |θ
k
N
ihθ
k
N
|
#
(55)
=
1
T
0
Z
T
0
+t
0
t
0
dt tr [E
j
E
t
(ρ
L
)] hθ
k
1
|ρ
C,1
(t)|θ
k
1
i. . . hθ
k
N
|ρ
C,N
(t)|θ
k
N
i (56)
=
1
T
0
Z
T
0
+t
0
t
0
dt hθ
k
1
|ρ
C,1
(t)|θ
k
1
i. . . hθ
k
N
|ρ
C,N
(t)|θ
k
N
i ∀ t
0
∈ . (57)
Now, let us define the following function.
F
Q
(
~
k) :=
1
T
0
Z
T
0
+t
0
t
0
dt e
−iωQt
hθ
k
1
|ρ
C,1
(t)|θ
k
1
i. . . hθ
k
N
|ρ
C,N
(t)|θ
k
N
i, ∀ t
0
∈ , Q ∈ , (58)
where for simplicity we will assume that all the clock dimensions are equal d
1
= d
2
= . . . = d
N
=: d
C
. This
definition has the following properties:
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1) Since the integrand has period T
0
, F
Q
(
~
k) is independent of t
0
and we can set it to any preferred real number
to help perform the calculations.
2) We can perform the change of variable t = t
0
+ a, a = −l
2π
ωd
C
, l ∈ , to show
F
Q
(
~
k) =
e
i2πlQ/d
T
0
Z
T
0
+t
0
−l
2π
ωd
C
t
0
−l
2π
ωd
C
dt
0
e
−iωQt
hθ
k
1
+l
|ρ
C,1
(t)|θ
k
1
+l
i. . . hθ
k
N
+l
|ρ
C,N
(t)|θ
k
N
+l
i (59)
= e
i2πlQ/d
F
Q
(
~
k + l), (60)
where the jth vector component of
~
k+l is defined by [
~
k+l]
j
:= [
~
k]
j
+l for all vector component j. Note how
this expression is invariant under l 7→ l + jd
C
, j ∈ . So w.l.o.g. we can exchange
~
k + l with [
~
k + l]
(mod. d)
,
where the map acts element-wise on vectors and (mod. d) denotes modular d arithmetic.
3) By inspection, we see that F
Q
is invariant under any pairwise interchanges
k
q
↔ k
r
and ρ
C,q
(t) ↔ ρ
C,r
(t), q, r ∈ 1, . . . , N, (61)
where k
q
and k
r
are the qth and pth vector elements of
~
k.
4) By Eq. (57), it follows that F
Q
encodes the measurement outcome probabilities,
p
~
k
= F
0
(
~
k). (62)
This has important consequences when property 2) is taken into account. For example, in the case of one
clock, it implies that all measurement outcomes are equally likely, so
p
~
k
=
1
d
C
∀k
1
= 0, . . . , d
C
− 1. (63)
5) By making the substitution U
m
(t) =
P
d
m
−1
r
m
=0
e
−iωr
m
t
|r
m
ihr
m
|, we find
F
Q
(
~
k) =
X
~r
N
,~r
0
N
δ
Q+r
1
−r
0
1
+...+r
N
−r
0
N
,0
(64)
hr
1
|ρ
C,1
|r
0
1
ihθ
k
1
|r
1
ihr
0
1
|θ
k
1
i. . . hr
N
|ρ
C,N
|r
0
N
ihθ
k
N
|r
N
ihr
0
N
|θ
k
N
i. (65)
Thus using property 5), we can write ρ
~
k
P
in Eq. 52 as
ρ
~
k
P
=
1
p
~
k
d
P
−1
X
q,q
0
=0
d
L
−1
X
n,n
0
=0
X
~r
N
,~r
0
N
δ
Q+r
1
−r
0
1
+...+r
N
−r
0
N
,0
E
j
q,q
0
,n,n
0
(ρ
L
) |qihq
0
| (66)
hr
1
|ρ
C,1
|r
0
1
ihθ
k
1
|r
1
ihr
0
1
|θ
k
1
i. . . hr
N
|ρ
C,N
|r
0
N
ihθ
k
N
|r
N
ihr
0
N
|θ
k
N
i. (67)
=
d
P
−1
X
q,q
0
=0
d
L
−1
X
n,n
0
=0
F
Q
(
~
k)
F
0
(
~
k)
E
j
q,q
0
,n,n
0
(ρ
L
) |qihq
0
|. (68)
It is convenient to introduce an arbitrary phase k
α
∈ which for our purposes has to be defined modulo d
C
.
It will depend on
~
k, the measurement outcomes. We will choose it later depending on the particular covariant
error correcting code set-up and protocol. We can now write
ρ
~
k
P
=
d
P
−1
X
q,q
0
=0
d
L
−1
X
n,n
0
=0
F
Q
(
~
k)
F
0
(
~
k)
E
j
q,q
0
,n,n
0
(ρ
L
) |qihq
0
|. (69)
=
d
P
−1
X
q,q
0
=0
d
L
−1
X
n,n
0
=0
"
F
Q
(
~
k)
F
0
(
~
k)
e
2πik
α
Q/d
C
− 1
#
+ 1
!
(70)
U
⊗K†
P
(t
α
)E
j
q,q
0
,n,n
0
U
L
(t
α
)ρ
L
U
†
L
(t
α
)
|qihq
0
|U
⊗K
P
(t
α
) (71)
=U
⊗K†
P
(t
α
)
ˆ
E
0
~
k
(˜ρ
L
) + E
j
(˜ρ
L
)
U
⊗K
P
(t
α
), (72)
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where
˜ρ
L
:= U
L
(t
α
)ρ
L
U
†
L
(t
α
), t
α
:= k
α
T
0
d
C
, (73)
and
ˆ
E
0
~
k
(·) :=
d
P
−1
X
q,q
0
=0
d
L
−1
X
n,n
0
=0
"
F
Q
(
~
k)
F
0
(
~
k)
e
2πik
α
Q/d
C
− 1
#
E
j
q,q
0
,n,n
0
(·)|qihq
0
| (74)
In order to proceed with the decoding, let us define the function p(Q,
~
k) as:
p(Q,
~
k) = 1 −
F
Q
(
~
k)
F
0
(
~
k)
e
2πik
α
Q/d
C
, (75)
where p(Q,
~
k) is a complex number for which we expect |p(Q,
~
k)| 1 (how small will determine the size of the
error of the particular sheme). The phase k
α
determines the group elements we should apply in the decoding
procedure, as per Eq. (73).
Since by assumption the group action only acts non-trivially in the physical space, U
⊗K
P
(t
α
) = U
Co
(t
α
)⊕
P/Co
,
the decoder may take the form
¯
D
j,t
α
(·) = U
L
(t
α
)E
−1
(U
Co
(t
α
)E (D
j
(·)) U
Co
(t
α
)) U
†
L
(t
α
), (76)
where E
−1
(·) = D
j
(E
j
(·)) is the inverse channel for the encoder E and D
j
the decoding map of E.
With it we see that if we apply the decoder to the state ρ
~
k
P
, we obtain
ρ
~
k
P
= ρ
L
−
ˆ
E
~
k
(ρ
L
), (77)
where we define
ˆ
E
~
k
(ρ
L
) =
d
P
−1
X
q,q
0
=0
d
L
−1
X
n,n
0
=0
p(Q,
~
k)
¯
D
j,t
α
E
j
q,q
0
,n,n
0
(ρ
L
) U
†⊗K
P
(t
α
)|qihq
0
|U
⊗K
P
(t
α
)
. (78)
Since each of the outcomes
~
k occurred with probability F
0
(
~
k), the final decoded state, averaged over all mea-
surement outcomes, is of the form
K(ρ
L
) = ρ
L
−
X
~
k
F
0
(
~
k)
ˆ
E
~
k
(ρ
L
). (79)
In Appendix C we show how a bound on the entanglement fidelity of the code follows from this expression. The
last fact that then needs to be shown is the form of the RHS of Eq. (75) for particular cases of clocks.
C Calculation of the entanglement fidelity
We here give the bounds on the entanglement fidelity which will be used to prove the main results. Again, it is
defined as
f
worst
(K) = min
φ
hφ|K ⊗ I(|φihφ|) |φi, (80)
where the minimization is over all the pure bipartite states |φi.
Motivated by the discussion in Appendix B leading to Eq. (79), let us start with the assumption that after
the encoding, error and decoding steps, the map has the following form when applied to a state ρ.
K(ρ
L
) = ρ
L
−
ˆ
E(ρ
L
), (81)
A result from [32] allows us to lower bound the entanglement fidelity in terms of
ˆ
E(ρ
L
). If for all pure states
on a single system on H
d
in
the following holds
hφ|K(|φihφ|) |φi ≥ 1 −, (82)
then we have that f
worst
(K) ≥ 1 −
3
2
. Given that hφ|
ˆ
E(|φihφ|) |φi ≤ ||
ˆ
E(|φihφ|)||
1
, we can choose =
max
|φihφ|
||
ˆ
E(|φihφ|)||
1
, where the optimization is over all pure states, thus obtaining
f
worst
(K) ≥ 1 −
3
2
max
|φihφ|
||
ˆ
E(|φihφ|)||
1
. (83)
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The next step is to give an upper bound to the 1-norm of
ˆ
E that holds for any pure state. First, by the triangle
inequality
||
ˆ
E||
1
= ||
X
~
k
F
0
(
~
k)
ˆ
E
~
k
||
1
≤
X
~
k
F
0
(
~
k)||
ˆ
E
~
k
||
1
. (84)
As seen in Appendix B, in all of the cases considered here, the operator
ˆ
E
~
k
is defined as
ˆ
E
~
k
=
¯
D
j,t
α
ˆ
E
0
~
k
where
¯
D
j,t
α
is the decoder, and
ˆ
E
0
~
k
=
d
out
−1
X
q,q
0
=0
d
in
−1
X
n,n
0
=0
U
†⊗K
Co
(t
α
)|qihq
0
|U
⊗K
Co
(t
α
) hq|E
j
E
0
(|nihn|hn|ρ
L
|n
0
i) |q
0
ip(Q,
~
k), (85)
where p(Q,
~
k) is in principle constrained by complete positivity and trace preservation (as defined in Eq. (78)).
By contractivity of CPTP maps, ||
ˆ
E
~
k
||
1
≤ ||
ˆ
E
0
~
k
||
1
. Now, we can write
||
ˆ
E
0
~
k
||
1
= max
q
0
X
q
|hq|
d
in
−1
X
n,n
0
=0
E
j
(|nihn
0
|hn|ρ
L
|n
0
i) |q
0
ip(Q,
~
k)| (86)
= max
q
0
X
q
tr
|q
0
ihq|
d
in
−1
X
n,n
0
=0
E
j
(|nihn
0
|hn|ρ
L
|n
0
i)p(Q,
~
k)
(87)
≤ max
q
0
X
q
||E
j
(
d
in
−1
X
n,n
0
=0
|nihn
0
|hn|ρ
L
|n
0
ip(Q,
~
k))||
1
(88)
≤ ||E
j
||
1−1
max
q
0
X
q
||
d
in
−1
X
n,n
0
=0
|nihn
0
|hn|ρ
L
|n
0
ip(Q,
~
k)||
1
(89)
where the third line follows from the inequality [47],
|tr[BA
†
]| ≤ ||A||
p
||B||
r
, (90)
with 1 = 1/p + 1/r and choosing p = 1, r = ∞, with B = |q
0
ihq| and A the rest. The fourth line follows from
the definition of the 1 −1 norm for CPTP maps, which is ||E
j
||
1−1
= sup
X
||E
j
(X)||
1
||X||
1
, where the optimization is
over operators on the Hilbert space of the input of E
j
. By contractivity, we have that ||E
j
||
1−1
≤ 1. The last
step is to bound the left-over 1-norm. By using ||A||
1
≤
√
d
in
||A||
2
we can bound the 2 norm instead, as
||
d
in
−1
X
n,n
0
=0
|nihn|hn|ρ
L
|n
0
ip(Q,
~
k)||
2
2
= tr
d
in
−1
X
n,n
0
=0
|nihn|hn|ρ
L
|n
0
ip(Q,
~
k)
2
(91)
=
d
in
−1
X
n,n
00
=0
hn|ρ
L
|n
00
ihn
00
|ρ
L
|nip(h
Co,q
− h
Co,q
0
+ h
L,n
− h
L,n
00
,
~
k)p(h
Co,q
− h
Co,q
0
+ h
L,n
− h
L,n
00
,
~
k) (92)
≤ max
Q
|p(Q,
~
k)|
2
d
in
−1
X
n,n
00
=0
hn|ρ
L
|n
00
ihn
00
|ρ
L
|ni = max
Q
|p(Q,
~
k)|
2
tr[ρ
2
L
] = max
Q
|p(Q,
~
k)|
2
,
since ρ
L
= |φihφ|. Generically in our examples the optimization is over the range Q ∈ {−(∆h
L
+
∆h
Co
), . . . , ∆h
L
+ ∆h
Co
} as defined in Sec. 2.1, but if the representation U
†
Co
(t) is trivial (that is, the generator
is the identity) then Q ∈ {−∆h
L
, ∆h
L
}, which yields the best performance. Finally we have
||
ˆ
E
~
k
||
1
≤
p
d
in
max
q
0
X
q
max
Q
|p(Q,
~
k)| =
p
d
in
d
out
max
Q
|p(Q,
~
k)|, (93)
which holds for any |φihφ|. Since outcome
~
k occurs with probability p
~
k
= F
0
(
~
k), we have
f
worst
(K) ≥ 1 −
3
2
p
d
in
d
out
X
~
k
F
0
(
~
k) max
Q
|p(Q,
~
k)|. (94)
For particular sets of clocks and schemes, we will give bounds for F
0
(
~
k) max
Q
|p(Q,
~
k)|, and then use Eq. (94) to
estimate the entanglement fidelity.
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D Proof of Theorem 1, a bound for a single remaining clock
Following the discussion of appendices B and C, here we provide the proof of the bound on the quantity that we
use to bound the entanglement fidelity for the case in which, after erasure errors, only a single clock is ensured
not to be erased. For simplicity of notation, let us use the shorthand d = d
C
.
The first step is to notice that the discussion of Appendices B and C shows we only have to compute a bound
on max
Q
|p(Q,
~
k)|, where
−p(Q,
~
k) =
F
Q
(
~
k)
F
0
(
~
k)
e
2πik
α
Q/d
− 1, (95)
as defined in Eq. (75). Then, any upper bound on |p(Q,
~
k)| ∀Q,
~
k will yield a lower bound on the worst-case
entanglement fidelity as per Eq. (94). We now compute this ratio explicitly for a single Quasi-Ideal clock, which
we do following the notation of Appendix B. Let us recall that
F
Q
(
~
k) =
1
T
0
Z
T
0
+t
0
t
0
dt e
−iωQt
hθ
k
1
|ρ
C,1
(t)|θ
k
1
i, ∀ t
0
∈ , Q ∈ . (96)
In this case of this proof, we let k
α
= k
1
− k
0
1
. After measuring the single clock, and right before applying the
decoding, the state is that of Eq. (72), that is
ρ
~
k
P
=
d
P
−1
X
q,q
0
=0
d
L
−1
X
n,n
0
=0
"
F
Q
(
~
k)
F
0
(
~
k)
e
2πik
α
Q/d
− 1
#
+ 1
!
(97)
U
⊗K†
Co
(t
α
)E
j
q,q
0
,n,n
0
U
L
(t
α
)ρ
L
U
†
L
(t
α
)
|qihq
0
|U
⊗K
Co
(t
α
). (98)
Using property 2) of F
Q
(
~
k) (see Eq. (60)), we observe that the first line of Eq. (98) is invariant under the
change of variable k
1
7→ k
1
+ l (Since F
Q
(
~
k) maps to e
−i2πlQ/d
F
Q
(
~
k) while k
α
to k
α
+ l). Thus the situation
is the same for all measurement outcomes k
1
. As such we will only need to concern ourselves with one of the
measurement outcome k
1
, which we choose for convenience to be k
1
= k
Ma
:= max{S
d
1
(k
0
1
)}, where the set
S
d
(k
0
1
) is described in the main text is defined to be
S
d
(k
0
1
) =
n
k : k ∈ , −
d
2
≤ k
0
1
− k <
d
2
o
, (99)
for k
0
1
∈ . Thus k
Ma
= k
0
1
+ d/2 if k
0
1
+ d/2 is integer (we will assume this is the case here, but one can find
analogous results for when it is half integer)
10
We are now interested in bounding the overlap
hθ
k
Ma
|e
−it
ˆ
H
c
|ψ
nor
(k
0
1
)i. (100)
This can be bounded with the results in Theorem 8.1 on page 151 of [21]. The theorem tells us that for all
t ∈
e
−it
ˆ
H
c
|Ψ
nor
(k
0
1
)i = |Ψ
nor
(k
0
1
+ t
d
T
0
)i + |εi, (101)
where
|Ψ
nor
(k
0
1
)i =
X
k∈S
d
(k
0
1
)
ψ
nor
(k
0
1
; k) |θ
k
i, (102)
with
ψ
nor
(k
0
1
; k) = Ae
−
(
π
σ
)
2
(
k−k
0
1
)
2
e
i2πn
1
k−k
0
1
d
. (103)
The error term ε
c
:= |||εi||
2
satisfies
ε
c
(t) < |t|
d
T
0
ε
total
+
|t|
d
T
0
+ 1
ε
step
+ ε
nor
, (104)
10
Recall that k
0
1
is the value at which the Gaussian amplitude of the initial Quasi-Ideal clock state is centred.
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where
ε
total
= 2πAd
2σ
α
0
2
+
1
2πσ
2
+
1
1 − e
−πσ
2
α
0
e
−
πσ
2
4
α
2
0
+
1
1 − e
−
πd
σ
2
+
1
1 − e
−
πd
2
σ
2
+
d
2σ
2
+
1
2πd
!
e
−
πd
2
4σ
2
!
(105)
and A is defined in Eq. (124), and α
0
∈ (0, 1] is defined in [21] to be:
Definition 1. (Distance of the mean energy from the edge of the spectrum) We define the parameter α
0
∈ (0, 1]
as a measure of how close n
0
∈ (0, d − 1) is to the edge of the energy spectrum, namely
α
0
=
2
d − 1
min{n
0
, (d − 1) − n
0
} (106)
= 1 −
1 − n
0
2
d − 1
∈ (0, 1]. (107)
The maximum value α
0
= 1 is obtained for n
0
= (d − 1)/2 when the mean energy is at the mid point of the
energy spectrum, while α
0
→ 0 as n
0
approaches the edge values 0 or d − 1.
Furthermore,
ε
step
< 2Ae
−
πd
2
4σ
2
(108)
ε
nor
≤
40
√
2
3σ
e
−
πd
2
2σ
2
1 − e
−
2πd
σ
2
+
σ
√
2
e
−
πσ
2
2
1 − e
−πσ
2
(109)
where on the r.h.s. of the inequality in Eq. (109) we have assumed σ ≥ 1 and d = 2, 3, 4, . . . (tighter bounds
can be found in Section E.A.2 of [21]).
We can now get back to estimating the overlap Eq. (100). We find that for all t ∈ [0, T
0
],
hθ
k
Ma
|e
−it
ˆ
H
c
|ψ
nor
(k
0
1
)i = hθ
k
Ma
|e
−it
ˆ
H
c
|ψ
nor
(k
0
1
)i + hθ
k
Ma
|ε(t)i (110)
hθ
k
Ma
|
X
k∈S
d
(k
0
1
)
ψ
nor
(k
0
1
+ td/T
0
; k)|θ
k
i + hθ
k
Ma
|ε(t)i (111)
=ψ
nor
(k
0
1
+ td/T
0
; k
Ma
) + hθ
k
Ma
|ε(t)i. (112)
Thus when ρ
C,1
= |ψ
nor
(k
0
1
)ihψ
nor
(k
0
1
)|, and t ∈ [0, T
0
],
hθ
k
Ma
|ρ
C,1
(t)|θ
k
Ma
i = hθ
k
Ma
|e
−it
ˆ
H
c
|ψ
nor
(k
0
1
)ihψ
nor
(k
0
1
)|e
−it
ˆ
H
c
|θ
k
Ma
i (113)
=|ψ
nor
(k
0
1
+ td/T
0
; k
Ma
)|
2
+ |hθ
k
Ma
|ε(t)i|
2
(114)
+ 2 Re
ψ
nor
(k
0
1
+ td/T
0
; k
Ma
) hε(t)|θ
k
Ma
i
(115)
=|ψ
nor
(k
0
1
+ td/T
0
; k
Ma
)|
2
+ ε
1
(t/T
0
) (116)
=|Ae
−π
(
d
σ
)
2
t
T
0
−
1
2
2
e
−i2πn
1
t
T
0
−
1
2
|
2
+ ε
1
(t/T
0
) (117)
=A
2
e
−2π
(
d
σ
)
2
t
T
0
−
1
2
2
+ ε
1
(t/T
0
). (118)
Thus using ω = 2π/T
0
and the change of variable x = t/T
0
− 1/2, and setting t
0
= 0 such that integral is over
t ∈ [0, T
0
], we find
F
Q
(
~
k) =
1
T
0
Z
T
0
0
dt e
−iωQt
hθ
k
Ma
|ρ
C,1
(t)|θ
k
Ma
i (119)
=
1
T
0
Z
T
0
0
dt e
−iωQt
A
2
e
−2π
(
d
σ
)
2
t
T
0
−
1
2
2
+ ε
1
(t/T
0
) (120)
= e
−πiQ
Z
1
2
−
1
2
dx e
−i2πQx
A
2
e
−2π
(
d
σ
)
2
x
2
+ ε
1
(x + 1/2) (121)
= e
−πiQ
Z
∞
−∞
dx e
−i2πQx
A
2
e
−2π
(
d
σ
)
2
x
2
+ ε
2
(122)
= e
−πiQ
A
2
σ
√
2d
e
−
π
2
(
σ
d
)
2
Q
2
+ ε
2
, (123)
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Recalling that F
0
(
~
k) = 1/d and that, following Section E.1.1 of [21] we have
A
2
=
√
2/σ + ε
A
, (124)
ε
A
is specified later in Eq. (131), we can write
F
Q
(
~
k)
F
0
(
~
k)
e
2πik
α
Q/d
=
A
2
σ
√
2
e
−
π
2
(
σ
d
)
2
Q
2
+ dε
2
e
πi(2k
α
−d)Q/d
(125)
=
(1 +
σ
√
2
ε
A
)e
−
π
2
(
σ
d
)
2
Q
2
+ dε
2
e
πi(2k
α
−d)Q/d
(126)
= (1 +
σ
√
2
ε
A
)e
−
π
2
(
σ
d
)
2
Q
2
+ dε
2
, (127)
where in the last line we have used k
α
= k
1
−k
0
1
= d/2 in order to get rid of the phase. Finally, we need to find
bounds for both ε
2
and ε
A
. Let us start with ε
2
, defined as
ε
2
=
Z
1
2
−
1
2
ε
1
(x + 1/2)dx + A
2
e
−
π
2
d
2
σ
2
, (128)
where ε
1
is defined in Eq. (118). By inspection, we have
ε
1
(t) ≤ 2ε
c
(t) + ε
c
(t)
2
(129)
where ε
c
is reproduced in Eq. (104). Hence going back to the definition of
2
in Eq. (128), we can then conclude
that
|ε
2
| ≤ 2 (dε
total
+ (d + 1)ε
step
+ ε
nor
) + (dε
total
+ (d + 1)ε
step
+ ε
nor
)
2
+ A
2
e
−
π
2
d
2
σ
2
(130)
To bound the error term ε
A
, we use the results from Eq. (477-483) from Section E.1.1 on page 84 of [21]. The
bound is
|ε
A
| ≤
¯ε
1
+ ¯ε
2
σ
√
2
σ
√
2
− ¯ε
1
− ¯ε
2
, (131)
where
¯ε
1
:=
2e
−
πd
2
2σ
2
1 − e
−
2πd
σ
2
, ¯ε
2
:=
σ
√
2
2e
−
πσ
2
2
1 − e
−πσ
2
. (132)
Thus, the leading contribution to ε
A
comes from ¯ε
2
, and the leading contribution to ε
2
comes from the first
term of ε
total
and the last of ε
nor
. It can be seen in Eq. (105) that the first term of the sum only decays as
√
σde
−
πσ
2
4
α
2
0
. Thus, we can write
˜
F
Q
(
~
k)
F
0
(
~
k)
e
2πik
α
Q/d
= e
−
π
2
(
σ
d
)
2
Q
2
1 + O
e
−
πσ
2
2
+ O(e
−d
2
)
+
O(d
3
e
−
πsσ
2
4
) + O(e
−
πσ
2
2
) + O(e
−d
2
)
. (133)
Going back to the definition of Eq. (75), this gives the following bound on max
Q
|p(Q,
~
k)|,
max
Q
|p(Q,
~
k)| ≤ 1 − min
Q
e
−
π
2
(
σ
d
)
2
Q
2
1 + O
e
−
πσ
2
2
+ O(e
−d
2
)
+
O(d
3
e
−
πsσ
2
4
) + O(e
−
πσ
2
2
) + O(e
−d
2
)
,
(134)
≤ 1 −e
−
π
2
(
σ
d
)
2
(∆h
L
+∆h
Co
)
2
1 + O
e
−
πσ
2
2
+ O(e
−d
2
)
+
O(d
3
e
−
πsσ
2
4
) + O(e
−
πσ
2
2
) + O(e
−d
2
)
,
(135)
from which, given the discussion of Appendix C, the bound on the entanglement fidelity follows.
To finalise the proof, we need to pick an optimal value of σ. Choosing σ = ln
3
(d), we obtain the best scaling
of the leading error term in d, which gives
max
Q
|p(Q,
~
k)| ≤ 1 − min
Q
e
−
π
2
ln
3
(d)
d
2
Q
2
+ O
ln
3/2
(d)
d
3
!
(136)
=
π
2
ln
3
(d)
d
2
(∆h
L
+ ∆h
Co
)
2
+ O
ln
3/2
(d)
d
3
!
, (137)
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from which, given the discussion of Appendix C, the bound on the entanglement fidelity follows.
The choice σ = ln
3
(d) corresponds to time squeezed Quasi-Ideal lock state (i.e. ∆t < ∆E, where ∆t and ∆E
are the standard deviation in the time and energy bases as described in the main text).
E Proof of bound for clocks diagonal in the time eigenbasis
We here give a proof of Theorem 2, which is a bound on the entanglement fidelity of clocks which are, at some
point in their periodic orbit, diagonal in the basis in which the are measured — the time eigenbasis {|θ
k
i},
conjugate to the energy eigenbasis. That is, we assume that there exits t
0
∈ such that the initial clock state
ρ
inc
satisfies
U
C
(t
0
)ρ
inc
U
†
C
(t
0
) =
d
C
−1
X
k=0
|A
k
|
2
|θ
k
ihθ
k
| := ˜ρ
C,1
, (138)
for some probability amplitudes {|A
k
|}
d
C
−1
k=0
. Therefore,
hr
1
|˜ρ
C,1
|r
0
1
i =
d
C
−1
X
k=0
|A
k
|
2
hr
1
|θ
k
ihθ
k
|r
1
i =
d
C
−1
X
k=0
|A
k
|
2
d
C
e
−i2πk(r
1
−r
0
1
)/d
C
. (139)
Unlike in the previous case of an optimal clock, the above equation only depends on the difference r
1
−r
0
1
rather
than r
1
, r
0
1
individually (c.f. the Quasi-Ideal clock case). Before preceding, we need to write the covariant
encoding channel in a way which reflects the form of Eq. (138). Note that Eq. (18) in the theorem has the
property
E
cov
(·) =
1
T
0
Z
T
0
0
dt E
t
(·) ⊗ U
C
(t) ρ
inc
U
†
C
(t) =
1
T
0
Z
T
0
+t
0
t
0
dt E
t
(·) ⊗ U
C
(t) ρ
inc
U
†
C
(t), (140)
for all t
0
∈ . This is easily provable via a change of variable and is a consequence of the fact that the integrand
is periodic and we are integrating over one period. Hence via the change x = t −t
0
and setting t
0
= t
0
, we find
E
cov
(ρ
L
) =
1
T
0
Z
T
0
0
dt E
t
(·) ⊗ U
C
(t) ρ
inc
U
†
C
(t) =
1
T
0
Z
T
0
0
dt E
t+t
0
(˜ρ
L
) ⊗ U
C
(t) ˜ρ
C,1
U
†
C
(t), (141)
where ˜ρ
L
= U
†
L
(t
0
)ρ
L
U
L
(t
0
) ∈ H
L
. The above equation shows how to write the covariant encoding channel in
terms of the clock state ˜ρ
C,1
which is diagonal in the time basis. Since our results are stated in terms of the
entanglement fidelity which is independent of any particular input, the change in inputs ρ
L
to ˜ρ
L
in the above
equation is irrelevant and we will thus hence ignore this difference. The only relevant difference is thus that
the encoding map E
t
is shifted to E
t+t
0
. This small difference can easily be accounted for at the decoding stage
without extra complication.
From Eq. (35), we see that changing E
t
for E
t+t
0
is equivalent to changing E
q,q
0
,n,n
0
to
e
−iωt
0
(h
Co,q
−h
Co,q
0
+h
L,n
−h
L,n
0
)
E
q,q
0
,n,n
0
or equivalently, changing E
j
q,q
0
,n,n
0
to
˜
E
j
q,q
0
,n,n
0
:= e
−iωt
0
(h
Co,q
−h
Co,q
0
+h
L,n
−h
L,n
0
)
E
j
q,q
0
,n,n
0
(142)
Hence we can use Eq. 52 by making the replacements E
j
q,q
0
,n,n
0
(defined in Eq. (47)) with
˜
E
j
q,q
0
,n,n
0
(·) and ρ
C,1
with ˜ρ
C,1
, followed by plugging in Eq. (139). Hence recalling the short hand notation Q := h
Co,q
− h
Co,q
0
+
h
L,n
− h
L,n
0
, we find
ρ
~
k
P
=
1
p
~
k
d
P
−1
X
q,q
0
=0
d
L
−1
X
n,n
0
=0
d
C
−1
X
r
1
,r
0
1
=0
δ
Q+r
1
−r
0
1
,0
˜
E
j
q,q
0
,n,n
0
(ρ
L
) |qihq
0
| (143)
hr
1
| ˜ρ
C,1
|r
0
1
ihθ
k
1
|r
1
ihr
0
1
|θ
k
1
i (144)
=
1
p
~
k
d
P
−1
X
q,q
0
=0
d
L
−1
X
n,n
0
=0
d
C
−1
X
r
1
,r
0
1
=0
δ
Q+r
1
−r
0
1
,0
˜
E
j
q,q
0
,n,n
0
(ρ
L
) |qihq
0
| (145)
d
C
−1
X
k=0
|A
k
|
2
d
2
C
e
−i2π(k
1
−k)(r
1
−r
0
1
)/d
C
. (146)
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Now applying
d−1
X
r,r
0
=0
f(r − r
0
) =
d−1
X
x=−(d−1)
(d − |x|)f(x) ∀f : → (147)
to Eq. (146), we have
ρ
~
k
P
=
1
p
~
k
d
P
−1
X
q,q
0
=0
d
L
−1
X
n,n
0
=0
d
C
−1
X
x
1
=−(d
C
−1)
d
C
−1−|x
1
|
X
p
1
=0
δ
Q+x
1
,0
˜
E
j
q,q
0
,n,n
0
(ρ
L
) |qihq
0
| (148)
d
C
−1
X
k=0
|A
k
|
2
d
2
C
e
−i2π(k
1
−k)x
1
/d
C
(149)
=
1
p
~
k
d
P
−1
X
q,q
0
=0
d
L
−1
X
n,n
0
=0
d
C
−1
X
x
1
=−(d
C
−1)
(d
C
− |x
1
|) δ
Q+x
1
+...+x
N
,0
˜
E
j
q,q
0
,n,n
0
(ρ
L
) |qihq
0
| (150)
d
C
−1
X
k=0
|A
k
|
2
d
2
C
e
−i2π(k
1
−k)x
1
/d
C
. (151)
=
1
p
~
k
d
P
−1
X
q,q
0
=0
d
L
−1
X
n,n
0
=0
(d
C
− |Q|)
˜
E
j
q,q
0
,n,n
0
(ρ
L
) |qihq
0
| (152)
d
C
−1
X
k=0
|A
k
|
2
d
2
C
e
−i2π(k
1
−k)Q/d
C
. (153)
As all the outcomes are equally likely, we have that p
~
k
= 1/d
C
. Hence
ρ
~
k
P
=
d
P
−1
X
q,q
0
=0
d
L
−1
X
n,n
0
=0
1 −
|Q|
d
C
d
C
−1
X
k=0
|A
k
|
2
ˆ
O
q,q
0
,n,n
0
(t
k−k
1
), (154)
where we have defined
ˆ
O
q,q
0
,n,n
0
(t
k−k
1
) := U
⊗K†
Co
(t
k−k
1
)
˜
E
j
q,q
0
,n,n
0
U
L
(t
k−k
1
)ρ
L
U
†
L
(t
k−k
1
)
U
⊗K
Co
(t
k−k
1
) (155)
with t
k−k
1
=
2π
ω
(k − k
1
). It is convenient to write this as
ρ
~
k
P
=
d
C
−1
X
k=0
|A
k
|
2
ρ
P
(t
k−k
1
) +
1
d
C
ˆ
δ(t
k−k
1
)
, (156)
where
ρ
P
(t) :=
d
P
−1
X
q,q
0
=0
d
L
−1
X
n,n
0
=0
ˆ
O
q,q
0
,n,n
0
(t) = U
⊗K†
Co
(t + t
0
)
E
j
E
U
L
(t)ρ
L
U
†
L
(t)
U
⊗K
Co
(t + t
0
), (157)
ˆ
δ(t) := −
d
P
−1
X
q,q
0
=0
d
L
−1
X
n,n
0
=0
|Q|
ˆ
O
q,q
0
,n,n
0
(t). (158)
Observe that if we can apply the map
U
†
L
(t
k−k
1
)D
j
U
⊗K
Co
(t
k−k
1
+ t
0
)(·)U
⊗K†
Co
(t
k−k
1
+ t
0
)
U
L
(t
k−k
1
) (159)
to ρ
P
(t
k−k
1
), we have perfect error correction, i.e.
U
†
L
(t
k−k
1
)D
j
U
⊗K
Co
(t
k−k
1
+ t
0
)ρ
P
(t
k−k
1
)U
⊗K†
Co
(t
k−k
1
+ t
0
)
U
L
(t
k−k
1
) = ρ
L
∀t
0
, t
k−k
1
∈ . (160)
as such we define the error term
ˆ
E(t) := U
†
L
(t)D
j
U
⊗K
Co
(t + t
0
)δ(t)U
⊗K†
Co
(t + t
0
)
U
L
(t). (161)
Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 22
Let us assume we apply an arbitrary decoder D to ρ
~
k
P
. The fidelity with an arbitrary initial pure state |ψihψ| is
hψ|
d
C
−1
X
k=0
|A
k
|
2
D
ρ
P
(t
k−k
1
) +
1
d
C
ˆ
δ(t
k−k
1
)
|ψi ≤ max
k
hψ|D
ρ
P
(t
k−k
1
) +
1
d
C
ˆ
δ(t
k−k
1
)
|ψi (162)
≤ 1 −
1
d
C
min
k
hψ|D
ˆ
δ(t
k−k
1
)
|ψi (163)
which shows that a the Salecker-Wigner-Peres clock with |A
k
|
2
= δ
k,k
0
is the optimal choice.
Crucially, the term in the optimization hψ|D
ˆ
δ(t
k−k
1
)
|ψi now only depends on parameters of the encoding
E and decoding D independently of the clock and its dimension d
C
. Now, let us assume there exists a sequence
of encoding-decoding schemes such that for some fixed d
P
,
lim
l→∞
min
k
hψ|D
l
ˆ
δ
l
(t
k−k
1
)
|ψi = 0. (164)
If this were true, one could achieve an arbitrarily large entanglement fidelity for all pure states without increasing
the size of the clock. This contradicts the existing no-go results [7, 26], so there must exist a constant C, such
that for all D and
ˆ
δ(t
k−k
1
) with fixed d
P
min
k
hψ|D
ˆ
δ(t
k−k
1
)
|ψi ≥ C > 0, (165)
which can be chosen such that the inequality is saturated for some D and
ˆ
δ(t
k−k
1
). Hence, we have that
f
worst
(K) ≤ 1 −
C
d
C
. (166)
On the other hand, since the Salecker-Wigner-Peres clock saturates the inequalities (162) and (163) we can
apply the decoder U
L
(t)D
j
U
†
L
(t) in (159) to obtain
f
worst
(K) = 1 −
C
∗
d
C
, (167)
for some C
∗
≥ C, as the only dependence with the dimension of the clock is in the term
1
d
C
ˆ
δ(t
k−k
1
).
F Proof of Theorem 3
We just need to show that with L d
C
-dimensional clocks we can construct a clock state with the same properties
as a single clock of dimension L(d
C
−1)+1. Let us take the Hamiltonian of L non-interacting clocks of dimension
d
C
,
H
tot
=
L
M
l=1
H
C
=
L(d
C
−1)
X
r=0
ωr
X
α
|E
r,α
ihE
r,α
| (168)
which has energy levels {0, L(d
C
−1)}, and α is the degeneracy index, as this new Hamiltonian has a very large
degeneracy. Let us choose an arbitrary non-degenerate subspace of maximal dimension d(L) := L(d
C
− 1) + 1,
such that H
tot
= H
clock
ˆ
⊕H
⊥
.
11
This can be done for instance by choosing all the eigenvectors with a fixed
α = 1, in which case
H
clock
=
d
L
−1
X
r=0
ωr|E
r,1
ihE
r,1
|. (169)
This defines a subspace with dimension d(L) and with a Hamiltonian with equally-spaced energy levels with gap
ω. Now given any single clock state ρ
C
=
P
d
C
−1
r
1
,r
2
=0
A
r
1
,r
2
|r
1
ihr
2
|
C
∈ S(H
C
) in a d
C
dimensional space, we can
construct a d(L) dimensional space via the mappings d
C
7→ d(L) and ρ
C
7→ ρ
0
:=
P
d(L)−1
r
1
,r
2
=0
A
r
1
,r
2
|E
r
1
,1
ihE
r
2
,1
| ∈
S(H
⊗L
C
). This large dimensional clock state will now be a superposition of product states (the energy eigenstates)
and will thus be entangled. Thus, in any scheme that uses a clock with dimension d
C
, such that it achieves a
fidelity bounded by f(d
C
) (either from above or below), one can now use this d(L) dimensional clock, to achieve
a fidelity bounded by f(d(L)) (again, either from above or below). This way, we can use a large amount of
clocks to vastly improve the performance of the codes.
11
Here
ˆ
⊕ denotes the Direct sum.
Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 23
G The converse bound from [26]
Here we show how to adapt the bound from [26] to our setting
12
. To understand why the bound cannot be
straightforwardly applied to our setting, first recall our definition of the covariant code E
cov
(·). For all t ∈ R,
and a given encoding E(·) and group representations U
L
(t), U
Co
(t)
⊗K
, we define
E
t
(·) := U
Co
(t)
⊗K
E(U
†
L
(t)(·)U
L
(t))U
Co
(t)
†⊗K
. (170)
The covariant encoding that we use throughout our work is then the CPTP map:
E
cov
(·) :=
1
T
0
Z
T
0
0
dt E
t
(·) ⊗ U
C
(t)
⊗M
ρ
(M)
C
U
C
(t)
†⊗M
. (171)
This construction is by definition not an isometry, and in general the output E
cov
(·) is a mixed state even for
pure inputs. However, the theorem of [26] assumes that the encoding map is an isometry, and as such it does
not directly apply to our results. However, since in our setup we assume that the encoding of Eq. (171) is
covariant, it is possible to construct a Stinespring dilation with an isometry to which the bound of [26] can be
directly applied.
We can construct the Stinespring dilation with an isometry directly for Eq. (171). However, since in our
set-up M − N clocks are lost due to erasure errors, and the erased clocks are inaccessible, we can trace out
M −N clocks and work with the resultant effective encoding map instead. Since we also assumed no correlations
between the erased and non-erased clocks, this leads to the effective channel
tr
C
N+1
...C
M
[E
cov
(·)] =
1
T
0
Z
T
0
0
dt E
t
(·) ⊗ U
C
(t)
⊗N
ρ
(N)
C
U
C
(t)
†⊗N
. (172)
Finally, since in some instances the generator of the unitary group on the clock can be exchanged for an effective
generator w.l.o.g. (noticeably Theorem 3), we will assume in Eq. (172) that the unitary representation of the
compact U(1) group on the clock U
C
(t)
⊗N
, takes on the generic form U
C
(t)
⊗N
= e
−it
¯
H
C
and specialise it later
in this section to specific cases.
We can now proceed with the isometry. Notice that E(·) can be dilated to an isometry V
L→CoA
defined such
that the representation of the symmetry group acts trivially on system A. Also, the state ρ
(N)
C
may not be a
pure state, but it can also be dilated to |Ψ
C
¯
C
ihΨ
C
¯
C
|
(N)
, again such that the symmetry group again acts trivially
on system
¯
C. We thus define
¯
E
cov
(·) :=
1
T
0
Z
T
0
0
dtV
t
(·) ⊗
U
C
(t)
⊗N
⊗
⊗N
¯
C
|Ψ
C
¯
C
ihΨ
C
¯
C
|
U
C
(t)
†⊗N
⊗
⊗N
¯
C
, (173)
where now we have
V
t
(·) :=
U
Co
(t)
⊗K
⊗
A
V
L→CoA
(U
†
L
(t)(·)U
L
(t))V
†
L→CoA
U
Co
(t)
†⊗K
⊗
A
. (174)
The only reason why
¯
E
cov
(·) is not yet an isometry is the twirling over the group, for which we are also able
to define a dilation with the help of the Choi-Jamiolkowski isomorphism. First, let {|ki
L
} be the eigenbasis
of L with
ˆ
H
L
=
P
k
ωh
L,k
|kihk|
L
, and let |Φ
L
¯
L
i =
P
k
|ki
L
⊗ |ki
¯
L
be an unnormalized maximally entangled
state between the logical space L and a copy
¯
L. Moreover, let us define
ˆ
H
¯
L
=
P
k
−ωh
L,k
|kihk|
¯
L
such that
ˆ
H
L
⊗
¯
L
+
L
⊗
ˆ
H
¯
L
|Φ
L
¯
L
i = 0. The Choi-Jamiolkowski representation of channel
¯
E
cov
is (we now omit writing
the trivial representations such as
A
,
⊗N
¯
C
for simplicity of notation)
¯
E
cov
(|Φ
L
¯
L
ihΦ
L
¯
L
|) =
1
T
0
Z
T
0
0
dt V
t
(|Φ
L
¯
L
ihΦ
L
¯
L
|) ⊗
U
C
(t)
⊗N
|Ψ
C
¯
C
ihΨ
C
¯
C
|U
C
(t)
†⊗N
(175)
=
1
T
0
Z
T
0
0
dt U
CoC
¯
L
(t)
V
L→CoA
|Φ
L
¯
L
ihΦ
L
¯
L
|V
†
L→CoA
⊗ |Ψ
C
¯
C
ihΨ
C
¯
C
|
U
†
CoC
¯
L
(t), (176)
where in the second line we have used the properties of the maximally entangled state, and we define U
CoC
¯
L
(t) :=
U
Co
(t)
⊗K
⊗ U
C
(t)
⊗N
⊗ U
¯
L
(t) = e
−it(
ˆ
H
Co
⊕
¯
H
C
⊕
ˆ
H
¯
L
)
. Thus, we can write
¯
E
cov
in terms of the projectors onto the
degenerate eigenspaces of
ˆ
H
Co
⊕
¯
H
C
⊕
ˆ
H
¯
L
as
¯
E
cov
(|Φ
L
¯
L
ihΦ
L
¯
L
|) =
X
x
Π
x
CoC
¯
L
h
V
L→CoA
|Φ
L
¯
L
ihΦ
L
¯
L
|V
†
L→CoA
⊗ |Ψ
C
¯
C
ihΨ
C
¯
C
|
i
Π
x
CoC
¯
L
. (177)
12
We thank Philippe Faist for sharing this argument with us and allowing us to reproduce it here.
Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 24
Since the operators
ˆ
H
Co
,
¯
H
C
,
ˆ
H
¯
L
act on different subsystems, we can decompose Π
x
CoC
¯
L
as
Π
x
CoC
¯
L
=
X
k,l:
−h
L,k
+h
CoC,l
=x
Π
l
CoC
⊗ Π
k
¯
L
, (178)
where h
CoC,l
are the eigenvalues of
ˆ
H
Co
⊕
¯
H
C
and h
L,k
are the eigenvalues of
ˆ
H
¯
L
. Furthermore we can write
Π
k
¯
L
= |kihk|
¯
L
. If we write the corresponding projector in L as Π
k
L
= |kihk|
L
, using the definition of |Φ
L
¯
L
ihΦ
L
¯
L
|
we have that
¯
E
cov
(|Φ
L
¯
L
ihΦ
L
¯
L
|) =
X
k,l,k
0
,l
0
−h
L,k
+h
CoC,l
=−h
L,k
0
+h
CoC,l
0
Π
l
CoC
h
V
L→CoA
Π
k
0
L
|Φ
L
¯
L
ihΦ
L
¯
L
|Π
k
L
V
†
L→CoA
⊗ |Ψ
C
¯
C
ihΨ
C
¯
C
|
i
Π
l
0
CoC
.
(179)
Now we can define an extra system B on a Hilbert space spanned by orthonormal basis vectors {|xi
B
}, and
such that
ˆ
H
B
=
P
x∈{h
CoC,l
−h
L,k
}
(x)|xihx|
B
, so that every single energy difference h
CoC,l
−h
L,k
appears in the
sum only once (note that by assumption this set is finite). With this, we can define the following isometry from
the logical space labeled by L to BCoAC
¯
C.
W
L→BCoAC
¯
C
=
X
x
|xi
B
⊗
X
k,l
h
CoC,l
−h
L,k
=x
Π
l
CoC
(V
L→CoA
Π
k
L
) ⊗ |Ψ
C
¯
C
i
. (180)
This is indeed an isometry since W
†
L→BCoAC
¯
C
W
L→BCoAC
¯
C
=
L
. It is covariant, in the sense that by construction
W
L→BCoAC
¯
C
e
−it
ˆ
H
L
=
e
−it(
ˆ
H
Co
⊕
¯
H
C
⊕
ˆ
H
B
)
⊗
A
¯
C
W
L→BCoAC
¯
C
. (181)
Moreover, it can be easily computed that
tr
B
h
W
L→BCoAC
¯
C
|Φ
L
¯
L
ihΦ
L
¯
L
|W
†
L→BCoAC
¯
C
i
=
¯
E
cov
(|Φ
L
¯
L
ihΦ
L
¯
L
|) (182)
and subsequently
tr
BA
¯
C
h
W
L→BCoAC
¯
C
|Φ
L
¯
L
ihΦ
L
¯
L
|W
†
L→BCoAC
¯
C
i
= tr
C
N+1
...C
M
[E
cov
(|Φ
L
¯
L
ihΦ
L
¯
L
|)] , (183)
so W
L→BCoAC
¯
C
is a covariant Stinespring dilation of channel tr
C
N+1
...C
M
[E
cov
(·)], which we can see as an
encoding isometry for which the error is i) first the loss to the environment of systems BA
¯
C, and then ii)
an erasure error channel for E
cov
. The result of [26] states that for these isometries, the entanglement fidelity
achieved by any decoding scheme is bounded by
f
worst
≤ 1 −
∆h
2
L
4N
2
∆h
2
loss
, (184)
where ∆h
L
is the range of values of the set {h
L,k
} of
ˆ
H
L
, N is the number of subsystems in the encoding
that can be erased independently by the error model, and where ∆h
loss
is the largest energy difference in the
Hamiltonians of all the subsystems that are lost to the environment. We have that BA
¯
C are lost and since
A,
¯
C have trivial generators we just have to look at
ˆ
H
B
, so that ∆h
loss
= ∆h
B
. The eigenvalues of
ˆ
H
B
are of
the form h
CoC,l
−h
L,k
, so that the range of
ˆ
H
B
is ∆h
B
= ∆h
Co
+ ∆h
C
(the terms with h
L,k
always contribute
negatively and hence ∆h
L
does not enter here).
We will now specialise the bound to the first case considered in Corollary 1. Here we have erasure of up to
M −1 blocks of L entangled clocks so N = 1. Furthermore, the effective clock generator on the remaining block
of L entangled clocks is
¯
H
C
= H
clock
(recall Eq. (169) for an expression for H
clock
). This effective clock system
has ∆h
C
= Ld
C
. Therefore ∆h
loss
= ∆h
Co
+ Ld
C
. To compute N, first notice that BA
¯
C counts as a single
system as it is always lost to the environment. Moreover, as mentioned above, the M −1 clocks that are lost to
the environment do not appear in the decoding procedure at all, and as detailed above are such that effectively
they were not there in the first place. Finally, the error channel on P is later corrected by its own decoding map
D
j
(·) and is independent of whether we make the code covariant in the first place, and the block of L entangled
clocks that is left (which we have taken to be the whole C system) does not get erased, so we can also count
both these two as a single subsystem (which gets erased with probability 0). Hence we have N = 2. Putting
everything together, we finally obtain
f
worst
≤ 1 −
∆h
2
L
16(∆h
Co
+ Ld
C
)
2
. (185)
Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 25
The appearance of ∆h
Co
as an additive contribution to the effective clock dimension Ld
C
in the upper bound
on the fidelity is to be expected, since in our setup one could have chosen the encoding map E to be a clock
whose decoding map D
j
measures this clock in the same way that the decoding map
˜
D
j,q
measures the clocks on
C. This scenario would help to make the encoding channel E
cov
increase its decoding fidelity, since its effective
clock dimension would be ∆h
Co
+Ld
C
. Contrarily, if one considers encoding and decoding channels E, D
j
which
do not help to make the encoding channel E
cov
reduce its decoding errors at all, then the value of ∆h
Co
should
not be expected to play a role in the decoding fidelity f
worst
.
Finally, comparing Eq. (185) with the lower bound of Eq. (23), we see that up to logarithmic factors, the
L entangled Quasi-Ideal clocks achieve the optimal error scaling with both their dimension d
C
and number of
clocks L.
H Proof of Theorem 4
In this section, we will prove Theorem 4. We will prove it only for the special case that the three blocks consist
of one clock each, i.e. L = 1. Once Theorem 4 is proven for this specialised case, the result for larger L is a
direct consequence of Theorem 3. Again for simplicity of notation, we will label d = d
C
.
For simplicity, we will start assuming, as for a single clock, that d is even (d odd follows analogously), and
that σ satisfies
(0, d) 3 σ → ∞ and
d
σ
→ ∞ as d → ∞. (186)
The first goal will be to work out an explicit expression for F
Q
(
~
k) evaluated for the case of three Quasi-Ideal
clocks in which a 2-unknown phase error applied to one of them. To indicate this difference (i.e. that un
unknown phase has now been applied), we add a tilde to F . Specifically, we have
˜
F
Q
(k
1
, k
2
, k
3
) := (187)
1
T
0
Z
T
0
0
dt e
−iωQt
hθ
k
1
|ρ
C,1
(t +
˜
t
ph,1
)|θ
k
1
ihθ
k
2
|ρ
C,2
(t +
˜
t
ph,2
)|θ
k
2
ihθ
k
3
|ρ
C,3
(t +
˜
t
ph,3
)|θ
k
3
i, (188)
where
˜
t
ph,q
=
(
t
ph
if q = r,
0 otherwise,
(189)
and r = 1, 2, 3 denotes the clock to which the phase is applied. The variables r and t
ph
are assumed to be
unknown.
Applying Theorem 8.1 (Eq. 71) in [21] we find
hθ
k
q
|ρ
C,q
(t +
˜
t
ph,q
) |θ
k
q
i (190)
= hθ
k
q
|e
−i(t+
˜
t
ph,q
)
ˆ
H
C
|ψ
nor
(k
0
q
)ihψ
nor
(k
0
q
)|
q
e
i(t+
˜
t
ph,q
)
ˆ
H
C
|θ
k
q
i (191)
= hθ
k
q
|e
it
ˆ
H
C
|ψ
nor
(
˜
k
0
q
)i
q
+ |ε
c
(
˜
t
ph
)i
q
hε
c
(
˜
t
ph
)|
q
+ hψ
nor
(
˜
k
0
q
)|
q
e
−it
ˆ
H
C
|θ
k
q
i (192)
= hθ
k
q
|ρ
C,q
(t)|θ
k
q
i
k
0
q
7→
˜
k
0
q
+ ε
I0
, (193)
where we have used the Quasi-Ideal clock states ρ
C,q
= |ψ
nor
(k
0
q
)ihψ
nor
(k
0
q
)|
q
and defined
˜
k
0
q
=
(
k
0
q
+ t
ph
d/T
0
if q = r,
k
0
q
otherwise.
(194)
In the last line of Eq. (193) we have defined
|ε
I0
| =
hθ
k
q
|e
it
ˆ
H
C
|ψ
nor
(
˜
k
0
q
)i
q
hε
c
(
˜
t
ph
)|
q
e
−it
ˆ
H
C
|θ
k
q
i (195)
+ hθ
k
q
|e
it
ˆ
H
C
|ε
c
(
˜
t
ph
)i
q
hψ
nor
(
˜
k
0
q
)|
q
e
−it
ˆ
H
C
|θ
k
q
i + hε
c
(
˜
t
ph
)|
q
e
−it
ˆ
H
C
|θ
k
q
ihθ
k
q
|e
it
ˆ
H
C
|ε
c
(
˜
t
ph
)i
q
(196)
≤ 2ε
c
(
˜
t
ph
, d) + ε
2
c
(
˜
t
ph
, d), (197)
where ε
c
is defined in Theorem 8.1 (Eq. 71) in [21]. Thus from Eqs. (193) and (58), it follows that if the clocks
ρ
C,q
in question are Quasi-Ideal clocks, then
˜
F
Q
(k
1
, k
2
, k
3
) = F
Q
(k
1
, k
2
, k
3
)
{k
0
q
7→
˜
k
0
q
}
3
q=1
+ 7 ε
I0
, (198)
Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 26
so we see that a 2-unknown phase error, up to a small error ε
c
, simply re-maps the initial time of one of the
clocks to an unknown value. Given the relation Eq. (198), our 1st task will be to find an explicit expression for
the integral (58). We do this in the following section.
H.1 An Expression for the Integral in F
Q
We start by deriving a general expression for hθ
k
|ρ
C,q
(t)|θ
k
i for k ∈ S
d
(k
0
q
), q ∈ {1, 2, 3} for Quasi-Ideal clock
states. For this we need to consider the overlaps
hθ
k
q
|ψ
nor
(k
0
q
+ td/T
0
)i = hθ
k
q
|
X
k∈S
d
(k
0
q
+td/T
0
)
ψ
nor
(k
0
q
+ td/T
0
; k) |θ
k
i (199)
hθ
k
q
|
X
k∈S
d
(k
0
q
+td/T
0
)
ψ
nor
(k
0
q
; k − td/T
0
) |θ
k
i. (200)
Let t
q
be defined via the relation
k
q
− t
q
d/T
0
= min{S
d
(k
0
q
)}. (201)
Using Eq. (99) we find min{S
d
(k
0
q
)} = bk
0
q
c −
d
2
+ 1, giving
t
q
=
k
q
− bk
0
q
c +
d
2
− 1
T
0
d
. (202)
Therefore,
hθ
k
q
|ψ
nor
(k
0
q
+ td/T
0
)i = (203)
(
hθ
k
q
|
P
k∈S
d
(k
0
q
)
ψ
nor
(k
0
q
; k − td/T
0
) |θ
k
i = ψ
nor
(k
0
q
; k
q
− td/T
0
) if t ∈ [0, t
q
]
hθ
k
q
|
P
k∈S
d
(k
0
q
+d)
ψ
nor
(k
0
q
; k − td/T
0
) |θ
k
i = ψ
nor
(k
0
q
; k
q
+ d − td/T
0
) if t ∈ (t
q
, T
0
]
(204)
Now consider Eq. (58) with measurement outcomes k
1
≤ k
2
≤ k
3
, with identical clocks other then their
starting times denoted k
0
1
, k
0
2
, k
0
3
respectively. We consider this special case w.l.o.g. since the other cases can be
reconstructed later using property 3) in Section B. Thus using Theorem 9.1 in [21] we have
F
Q
(k
1
, k
2
, k
3
) =
1
T
0
Z
T
0
0
dt e
−iωQt
hθ
k
1
|ρ
C,1
(t)|θ
k
1
ihθ
k
2
|ρ
C,2
(t)|θ
k
2
ihθ
k
3
|ρ
C,3
(t)|θ
k
3
i (205)
=
1
T
0
Z
T
0
0
dt e
−iωQt
Aψ
nor
(k
0
1
;
ˆ
k
1
(t, k
1
) − td/T
0
) + hθ
k
1
|ε
c
(t)i
2
(206)
Aψ
nor
(k
0
2
;
ˆ
k
2
(t, k
2
) − td/T
0
) + hθ
k
2
|ε
c
(t)i
2
(207)
Aψ
nor
(k
0
3
;
ˆ
k
3
(t, k
3
) − td/T
0
) + hθ
k
3
|ε
c
(t)i
2
(208)
=
Z
1
0
dx e
−i2πxQ
Aψ
nor
(k
0
1
;
ˆ
k
1
(xT
0
, k
1
) − xd) + hθ
k
1
|ε
c
(xT
0
)i
2
(209)
Aψ
nor
(k
0
2
;
ˆ
k
2
(xT
0
, k
2
) − xd) + hθ
k
2
|ε
c
(xT
0
)i
2
(210)
Aψ
nor
(k
0
3
;
ˆ
k
3
(xT
0
, k
3
) − xd) + hθ
k
3
|ε
c
(xT
0
)i
2
, (211)
where in the last line we performed the change of variable x = t/T
0
and used ω = 2π/T
0
and defined t
m
=
k
m
− bk
0
m
c +
d
2
− 1
T
0
d
and for m = 1, . . . , N
ˆ
k
m
(t, k
m
) =
(
k
m
if t ∈ [0, t
m
],
k
m
+ d if t ∈ (t
m
, T
0
].
(212)
From Eq. (211) we have
F
Q
(k
1
, k
2
, k
3
) = A
6
Z
1
0
dx e
−i2πxQ
ψ
2
nor
k
0
1
;
ˆ
k
1
(xT
0
, k
1
) − xd
(213)
ψ
2
nor
k
0
2
;
ˆ
k
2
(xT
0
, k
2
) − xd
ψ
2
nor
k
0
3
;
ˆ
k
3
(xT
0
, k
3
) − xd
(214)
+ ε
I1
. (215)
Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 27
Using the triangle inequality and the identity |R + C|
2
= R
2
+ ε where |ε| ≤ |C|(2R + |C|) for R ≥ 0, C ∈ ,
we can bound the ε
I1
term,
|ε
I1
| ≤ A
6
Z
1
0
dx 5 max
y∈
ψ
2
nor
(0; y) ε
c
(T
0
, d) ≤ 5A
6
ε
c
(T
0
, d), (216)
where ε
c
(T
0
, d) is an upper bound to k|ε
c
(t)ik
2
evaluated at t = T
0
. This quantity, (which already appeared in
Appendix D) is given in Theorem 8.1 of [21] and satisfies
ε
c
(T
0
, d) =
T
0
T
0
O
d
2
σ
1/2
e
−
π
4
σ
2
α
2
0
+ O
1 +
d
3
σ
3/4
e
−
π
4
(
d
σ
)
2
+ O
e
−
π
4
(
d
σ
)
2
+ O
e
−
π
4
σ
2
(217)
= O
d
2
σ
1/2
e
−
π
4
σ
2
α
2
0
+ O
d
3
σ
3/4
e
−
π
4
(
d
σ
)
2
(218)
where α
0
∈ (0, 1] is defined in Def. 2 in [21] and explained in Eq. (106).
Using A
2
=
√
2/σ + ε
A
= O(1/σ) from Eq. 483 in [21], and Eq. (216) we conclude
|ε
I1
| =O
d
2
σ
5/2
e
−
π
4
σ
2
α
2
0
+ O
d
3
σ
15/4
e
−
π
4
(
d
σ
)
2
. (219)
Dividing the integral in Eq. (215) into subintervals and substituting for ψ
nor
we find
F
Q
(k
1
, k
2
, k
3
) = A
6
Z
t
1
/T
0
0
dx e
−i2πxQ
e
−
2π
σ
2
[
(k
0
1
−k
1
+xd)
2
+(k
0
2
−k
2
+xd)
2
+(k
0
3
−k
3
+xd)
2
]
(220)
+ A
6
Z
t
2
/T
0
t
1
/T
0
dx e
−i2πxQ
e
−
2π
σ
2
[
(k
0
1
−k
1
−d+xd)
2
+(k
0
2
−k
2
+xd)
2
+(k
0
3
−k
3
+xd)
2
]
(221)
+ A
6
Z
t
3
/T
0
t
2
/T
0
dx e
−i2πxQ
e
−
2π
σ
2
[
(k
0
1
−k
1
−d+xd)
2
+(k
0
2
−k
2
−d+xd)
2
+(k
0
3
−k
3
+xd)
2
]
(222)
+ A
6
Z
1
t
3
/T
0
dx e
−i2πxQ
e
−
2π
σ
2
[
(k
0
1
−k
1
−d+xd)
2
+(k
0
2
−k
2
−d+xd)
2
+(k
0
3
−k
3
+xd)
2
]
(223)
+ ε
I1
. (224)
For the rest of the proof, it is convenient to work with
˜
F
Q
rather than F
Q
. Substituting the above into Eq.
(198) and performing the mapping k
0
q
7→
˜
k
0
q
, q = 1, 2, 3, we find
˜
F
Q
(k
1
, k
2
, k
3
) = (225)
A
6
Z
¯
k
1
/d+1/2−1/d
0
dx e
−i2πxQ
e
−2π
(
d
σ
)
2
h
x−
¯
k
1
d
+
∆k
0
1
d
2
+
x−
¯
k
2
d
+
∆k
0
2
d
2
+
x−
¯
k
3
d
+
∆k
0
3
d
2
i
(226)
+ A
6
Z
¯
k
2
/d+1/2−1/d
¯
k
1
/d+1/2−1/d
dx e
−i2πxQ
e
−2π
(
d
σ
)
2
h
x−1−
¯
k
1
d
+
∆k
0
1
d
2
+
x−
¯
k
2
d
+
∆k
0
2
d
2
+
x−
¯
k
3
d
+
∆k
0
3
d
2
i
(227)
+ A
6
Z
¯
k
3
/d+1/2−1/d
¯
k
2
/d+1/2−1/d
dx e
−i2πxQ
e
−2π
(
d
σ
)
2
h
x−1−
¯
k
1
d
+
∆k
0
1
d
2
+
x−1−
¯
k
2
d
+
∆k
0
2
d
2
+
x−
¯
k
3
d
+
∆k
0
3
d
2
i
(228)
+ A
6
Z
1
¯
k
3
/d+1/2−1/d
dx e
−i2πxQ
e
−2π
(
d
σ
)
2
h
x−1−
¯
k
1
d
+
∆k
0
1
d
2
+
x−1−
¯
k
2
d
+
∆k
0
2
d
2
+
x−1−
¯
k
3
d
+
∆k
0
3
d
2
i
(229)
+ ε
I1
+ 7 ε
I0
. (230)
where we have defined initial clock time independent measurement outcomes,
¯
k
p
:= k
p
− b
˜
k
0
p
c ∈ S
d
(
˜
k
0
p
) − b
˜
k
0
p
c =
−
d
2
+ 1, −
d
2
+ 2, . . . ,
d
2
, p = 1, 2, 3, (231)
and 1 > ∆k
0
p
:=
˜
k
0
p
−b
˜
k
0
p
c ≥ 0, p = 1, 2, 3. Before proceeding further, we write explicitly bounds for the ε terms.
Using Eq. (218) we find
|ε
I0
| ≤ 2ε
c
(
˜
t
ph
, d) + ε
2
c
(
˜
t
ph
, d) = O
d
2
σ
1/2
e
−
π
4
σ
2
α
2
0
+ O
d
3
σ
3/4
e
−
π
4
(
d
σ
)
2
. (232)
Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 28
Thus taking into account (219), we conclude
|ε
I1
| + 7|ε
I0
| = O
d
2
σ
1/2
e
−
π
4
σ
2
α
2
0
+ O
d
3
σ
3/4
e
−
π
4
(
d
σ
)
2
. (233)
In order to simplify the equations further, we will set k
1
to the smallest measurement outcome possible, i.e.
k
1
= b
˜
k
0
1
c − d/2 + 1. We can easily generate the other cases by employing Eq. (60), which we postpone to
Section H.4. Substituting into Eq. (230) gives us
˜
F
Q
(k
1
, k
2
, k
3
) = (234)
A
6
Z
¯
k
2
/d+1/2−1/d
0
dx e
−i2πxQ
e
−2π
(
d
σ
)
2
h
x−
1
2
−
1−∆k
0
1
d
2
+
x−
¯
k
2
d
+
∆k
0
2
d
2
+
x−
¯
k
3
d
+
∆k
0
3
d
2
i
(235)
+ A
6
Z
¯
k
3
/d+1/2−1/d
¯
k
2
/d+1/2−1/d
dx e
−i2πxQ
e
−2π
(
d
σ
)
2
h
x−
1
2
−
1−∆k
0
1
d
2
+
x−1−
¯
k
2
d
+
∆k
0
2
d
2
+
x−
¯
k
3
d
+
∆k
0
3
d
2
i
(236)
+ A
6
Z
1
¯
k
3
/d+1/2−1/d
dx e
−i2πxQ
e
−2π
(
d
σ
)
2
h
x−
1
2
−
1−∆k
0
1
d
2
+
x−1−
¯
k
2
d
+
∆k
0
2
d
2
+
x−1−
¯
k
3
d
+
∆k
0
3
d
2
i
(237)
+ ε
I1
+ 7 ε
I0
. (238)
The intuition is that each of the three above integrals will be small whenever the the corresponding term in
square brackets in the exponent does not pass through zero in the interval over which it is being integrated. We
now solve for
¯
k
2
,
¯
k
3
for when this happens. We start with the 1st integral. From line (235), we observe that
the term
"
x −
1
2
−
1 − ∆k
0
1
d
2
+
x −
¯
k
2
d
+
∆k
0
2
d
2
+
x −
¯
k
3
d
+
∆k
0
3
d
2
#
(239)
can only be zero iff x = 1/2 + 1/d − ∆k
0
1
/d,
¯
k
2
/d = 1/2 + 1/d +
∆k
0
2
− ∆k
0
1
/d, and
¯
k
3
/d = 1/2 + 1/d +
∆k
0
3
− ∆k
0
1
/d. In this case, x ∈ [0,
¯
k
2
/d + 1/2 − 1/d] = [0, 1 + +
∆k
0
2
− ∆k
0
1
/d], so x passes through
x = 1/2 + 1/d − ∆k
0
1
/d. Observe that this values of
¯
k
2
/d,
¯
k
3
/d are outside the domain of
¯
k
2
/d,
¯
k
3
/d by an
additive factor of 1/d +
∆k
0
2
− ∆k
0
1
/d and 1/d +
∆k
0
3
− ∆k
0
1
/d respectively. However, since this quantity
tends to zero as d becomes large, the integral on line (235) will provide a significant contribution. For this case,
we will make the parametrization
¯
k
2
/d = 1 + 1/d − k/d,
¯
k
3
/d = 1 + 1/d − l/d, k, l = 1, . . . , d.
Similarly for the integral on line (236), we find that
"
x −
1
2
−
1 − ∆k
0
1
d
2
+
x − 1 −
¯
k
2
d
+
∆k
0
2
d
2
+
x −
¯
k
3
d
+
∆k
0
3
d
2
#
(240)
can only be zero iff x = 1/2 + 1/d − ∆k
0
1
/d,
¯
k
2
/d = −1/2 + 1/d +
∆k
0
2
− ∆k
0
1
/d, and
¯
k
3
/d = 1/2 + 1/d +
∆k
0
3
− ∆k
0
1
/d. In this case, x ∈ [0,
¯
k
2
/d + 1/2 − 1/d] = [0, 1 +
∆k
0
2
− ∆k
0
1
/d] For this case, we will make
the parametrization
¯
k
2
/d = −1/2 + 1/d + m/d,
¯
k
3
/d = 1 + 1/d − l/d, m = 0, . . . , d − 1, l = 1, . . . , d.
Similarly for the integral on line (237), we find that
"
x −
1
2
−
1 − ∆k
0
1
d
2
+
x − 1 −
¯
k
2
d
+
∆k
0
2
d
2
+
x − 1 −
¯
k
3
d
+
∆k
0
3
d
2
#
(241)
can only be zero iff x = 1/2 + 1/d − ∆k
0
1
/d,
¯
k
2
/d = −1/2 + 1/d +
∆k
0
2
− ∆k
0
1
/d, and
¯
k
3
/d = −1/2 + 1/d +
∆k
0
3
− ∆k
0
1
/d with x ∈ [
¯
k
3
/d + 1/2 −1/d, 1] = [
∆k
0
3
− ∆k
0
1
/d, 1].
Intuitively, the three cases represent all possibilities (satisfying our assumptions k
1
≤ k
2
≤ k
3
) in which
approximately either all three measured elapsed times (these are all proportional to k
1
− k
0
1
, k
2
− k
0
2
, k
3
− k
0
3
)
coincide with the measured elapsed time of the 1st clock (proportional to k
1
−k
0
1
), or the three measured elapsed
times correspond to some combination of either the measured elapsed time of the first clock or approximately
the time corresponding to one period T
0
later. In all cases, these correspond approximately to the initial time,
due to the periodic motion of the clock, i.e. the clock resets back to the initial time after one period of its
motion. So in all cases, we should expect that to a good approximation, the outcomes which are most likely are
those when all the clocks mark (at least approximately) the same elapsed time.
We will now find a good approximation for the last integral (line (237)), the others can be approximated in
the same way.
Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 29
H.2 Approximating the integral in line (237).
We will start by substituting the parametrizations
¯
k
2
/d = −1/2 + 1/d + m/d,
¯
k
3
/d = −1/2 + 1/d + p/d,
m, p = 0, . . . , d − 1 into the integral on line (237), obtaining
Int
3
(m, p) :=
Z
1
¯
k
3
/d+1/2−1/d
dx e
−i2πxQ
e
−2π
(
d
σ
)
2
h
x−
1
2
−
1
d
+
∆k
0
1
d
2
+
x−1−
¯
k
2
d
+
∆k
0
2
d
2
+
x−1−
¯
k
3
d
+
∆k
0
3
d
2
i
(242)
=
Z
1
p/d
dx e
−i2πxQ
e
−2π
(
d
σ
)
2
h
x−
1
2
−
1
d
+
∆k
0
1
d
2
+
x−
1
2
−
1
d
+
∆k
0
2
d
−
m
d
2
+
x−
1
2
−
1
d
+
∆k
0
3
d
−
p
d
2
i
. (243)
We now consider the case in which the integral may contain a significant contribution to Eq. (238), and write
Int
3
(m, p) =
Z
∞
−∞
dx e
−i2πxQ
e
−2π
(
d
σ
)
2
h
x−
1
2
−
1
d
+
∆k
0
1
d
2
+
x−
1
2
−
1
d
+
∆k
0
2
d
−
m
d
2
+
x−
1
2
−
1
d
+
∆k
0
3
d
−
p
d
2
i
(244)
+ ε
L
3
(m, p) + ε
R
3
(m, p), (245)
where
ε
L
3
(m, p)
=
−
Z
p/d
−∞
dx e
−i2πxQ
e
−2π
(
d
σ
)
2
h
x−
1
2
−
1
d
+
∆k
0
1
d
2
+
x−
1
2
−
1
d
+
∆k
0
2
d
−
m
d
2
+
x−
1
2
−
1
d
+
∆k
0
3
d
−
p
d
2
i
(246)
≤
Z
p/d
−∞
dx e
−2π
(
d
σ
)
2
h
x−
1
2
−
1
d
+
∆k
0
1
d
2
+
x−
1
2
−
1
d
+
∆k
0
2
d
−
m
d
2
+
x−
1
2
−
1
d
+
∆k
0
3
d
−
p
d
2
i
. (247)
We now employ the relation
(x + A)
2
+ (x + A −y)
2
+ (x + A −z)
2
= −
1
3
(y + z)
2
+ y
2
+ z
2
+ 3
x + A −
1
3
(y + z)
2
(248)
for x, y, z, A ∈ followed by introducing the condition (later we will workout a bound for when this condition
is not satisfied)
p
d
−
1
2
−
1
d
−
p + m −∆k
0
1
− ∆k
0
2
− ∆k
0
3
3d
≤ −ε
L
3,1
for some ε
L
3,1
> 0, (249)
we find
ε
L
3
(m, p)
≤e
−2π
(
d
σ
)
2
h
−
1
3
p+m+2∆k
0
1
−∆k
0
2
−∆k
0
3
d
2
+
p+∆k
0
1
−∆k
0
3
d
2
+
m+∆k
0
1
−∆k
0
2
d
2
i
(250)
Z
p/d
−∞
dx e
−6π
(
d
σ
)
2
x−
1
2
−
1
d
+
∆k
0
1
d
−
p+m+2∆k
0
1
−∆k
0
2
−∆k
0
3
3d
2
(251)
≤e
−2π
(
d
σ
)
2
h
−
1
3
p+m+2∆k
0
1
−∆k
0
2
−∆k
0
3
d
2
+
p+∆k
0
1
−∆k
0
3
d
2
+
m+∆k
0
1
−∆k
0
2
d
2
i
(252)
Z
p/d
−∞
dx
x −
1
2
−
1
d
−
p+m−∆k
0
1
−∆k
0
2
−∆k
0
3
3d
p
d
−
1
2
−
1
d
−
p+m−∆k
0
1
−∆k
0
2
−∆k
0
3
3d
e
−6π
(
d
σ
)
2
x−
1
2
−
1
d
−
p+m−∆k
0
1
−∆k
0
2
−∆k
0
3
3d
2
(253)
=e
−2π
(
d
σ
)
2
h
−
1
3
p+m+2∆k
0
1
−∆k
0
2
−∆k
0
3
d
2
+
p+∆k
0
1
−∆k
0
3
d
2
+
m+∆k
0
1
−∆k
0
2
d
2
i
(254)
−1
2π6
σ
d
2
e
−6π
(
d
σ
)
2
p
d
−
1
2
−
1
d
−
p+m−∆k
0
1
−∆k
0
2
−∆k
0
3
3d
2
p
d
−
1
2
−
1
d
−
p+m−∆k
0
1
−∆k
0
2
−∆k
0
3
3d
(255)
≤
1
2π6
σ
d
2
e
−2π
(
d
σ
)
2
h
p
d
−
1
2
−
1
d
+
∆k
0
1
d
2
+
p
d
−
1
2
−
1
d
+
∆k
0
2
d
−
m
d
2
+
p
d
−
1
2
−
1
d
+
∆k
0
3
d
−
p
d
2
i
ε
L
3,1
(256)
≤
1
2π6
σ
d
2
e
−2π
(
d
σ
)
2
1
2
+
1−∆k
0
3
d
2
ε
L
3,1
(257)
≤
1
2π6
σ
d
2
e
−
π
2
(
d
σ
)
2
ε
L
3,1
, ∀m, p = 0, . . . , d − 1. (258)
Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 30
where in the penultimate line we have used Eq. (248) and (249). Analogously, we can bound ε
R
3
(m, p). We find
ε
R
3
(m, p)
=
−
Z
∞
1
dx e
−i2πxQ
e
−2π
(
d
σ
)
2
h
x−
1
2
−
1
d
+
∆k
0
1
d
2
+
x−
1
2
−
1
d
+
∆k
0
2
d
−
m
d
2
+
x−
1
2
−
1
d
+
∆k
0
3
d
−
p
d
2
i
(259)
≤
1
2π6
σ
d
2
e
−
π
2
(
d
σ
)
2
(
1−
2
d
)
2
ε
R
3,1
, m, p = 0, . . . , d − 1. (260)
where we have introduced the constraint
1 −
1
2
−
1
d
−
p + m −∆k
0
1
− ∆k
0
2
− ∆k
0
3
3d
≥ ε
R
3,1
for some ε
R
3,1
> 0. (261)
Thus computing the integral in Eq. (245) and combining the epsilons into a single term, we have
Int
3
(m, p) =
1
√
6
σ
d
e
−iπQ
e
iπ
2
3
∆k
0
T
Q/d
e
−iπ
2
3
(3+m+p)Q/d
(262)
e
−π
4
3
(
d
σ
)
2
[∆k
0
1
−∆k
0
2
+m]
2
+[∆k
0
1
−∆k
0
3
+p]
2
−[∆k
0
1
−∆k
0
2
+m][∆k
0
1
−∆k
0
3
+p]
d
2
e
−
π
6
(
σ
d
)
2
Q
2
+ ε
3
(m, p), (263)
=
1
√
6
σ
d
e
−i2π
˜
kQ/d
e
−iπ
2
3
(m+p)Q/d
(264)
e
−π
4
3
(
d
σ
)
2
[∆k
0
1
−∆k
0
2
+m]
2
+[∆k
0
1
−∆k
0
3
+p]
2
−[∆k
0
1
−∆k
0
2
+m][∆k
0
1
−∆k
0
3
+p]
d
2
e
−
π
6
(
σ
d
)
2
Q
2
+ ε
3
(m, p), (265)
where in the last line we have multiplied by 1 = e
2πiQ
and defined the quantities
˜
k := −
1
2
+
1
d
−
∆k
0
T
3d
, (266)
∆k
0
T
:= ∆k
0
1
+ ∆k
0
2
+ ∆k
0
3
. (267)
Note that if inequalities (249) and (261) are both satisfied, then
|ε
3
(m, p)| ≤
1
2π6
σ
d
2
1
ε
L
3,1
+
1
ε
R
3,1
!
e
−
π
2
(
d
σ
)
2
(
1−
2
d
)
2
, ∀m, p = 0, . . . , d − 1. (268)
Whenever inequality (249) and/or inequality (261) are/is not satisfied, we can bound the integral by the largest
value of the integrand, which is exponentially small in (d/σ)
2
. In this case since the point at which the Gaussian
takes on its maximum value lays outside of the integration region. However, it turns out that (m
2
+p
2
−mp)/d
2
is uniformly bounded away from zero in this case and thus the 1st term in (265) is exponentially small in (d/σ)
2
.
We will prove this result, since it will provide a more useful expression.
We start by defining three functions for x ∈
C
3L,p
(x) := −p +
1
2
+
1
d
−
∆k
0
3
d
+
1
3
(p + x) −ε
L
3,1
, (269)
C
3R,p
(x) := 1 −
1
2
−
1
d
+
∆k
0
1
d
−
1
3
(p + x) −ε
R
3,1
(270)
F
3,p
(x) := x
2
+ p
2
− x p. (271)
With the identification m = (∆k
0
1
− ∆k
0
2
+ m)/d and p = (∆k
0
1
− ∆k
0
3
+ p)/d, we see that the constraints
in Eqs. (249), (261) are the conditions C
3L,p
(m) ≤ 0 and C
3R,p
(m) ≥ 0 respectively, while F
3,p
(m) is the
exponent in Eq. (265) which we aim to prove is uniformly bounded away from zero whenever C
3L,p
(m) ≤ 0 and
C
3R,p
(m) ≥ 0 is not satisfied, i.e. when
C
3L,p
(m) > 0 and/or C
3R,p
(m) < 0. (272)
Given the range of p = 0, . . . , d − 1, it is easy to verify that both Eqs, (272) cannot simultaneously hold
since C
3L,p
(m) > 0 implies C
3R,p
(m) ≥ 0 and C
3R,p
(m) < 0 implies C
3L,p
(m) ≤ 0. Therefore, it suffices to
consider Eq. (272) with the “or” case only. It is useful to think of these functions as functions of m which are
Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 31
parametrized by p, as the chosen notation suggests. Furthermore,
d
2
dx
2
F
3,p
(x) = 2, and thus F
3,p
(x) is a convex
function for all p. As such, we can lower bound F
3,p
by any tangent straight line.
First consider the case that C
3L,p
(m) > 0 holds. We will proceed to lower bound F
3,p
(x) by the tangent line
A
p
+ B
p
x which intersects F
3,p
(x) at x = m
?
, where m
?
is defined by C
3L,p
(m
?
) = 0.
It follows that
m
?
= 3
ε
L
3,1
+ p −
1
2
−
1
d
+
∆k
0
3
d
− p, (273)
from which we find
B
p
=
d
dx
F
3p
(x)
x=m
?
= 2m
?
− p = 6
ε
L
3,1
−
1
2
−
1
d
+
∆k
0
3
d
+ 3p (274)
≤ 6
ε
L
3,1
−
1
2
−
1
d
+
∆k
0
3
d
+ 3
1 −
1
d
+
∆k
0
1
d
−
∆k
0
3
d
. (275)
It is convenient to demand B
p
≤ 0. We thus set
ε
L
3,1
:=
3
2
1
d
−
1
2
∆k
0
1
+ ∆k
0
3
d
>
1
2d
, (276)
so that B
p
≤ 0 for all p = 0, . . . , d − 1. To find A
p
, we need to solve the equation
A
p
+ B
p
m
?
= F
3p
(m
?
), (277)
giving
A
p
= F
3p
(m
?
) − B
p
m
?
= −(m
?
)
2
+ p
2
. (278)
Therefore, putting it all together we have for all m, p = 0, 1/d, . . . , 1 − 1/d,
F
3p
(m) ≥ A
p
+ B
p
m = −(m
?
)
2
+ p
2
+ (2m
?
− p) m. (279)
Taking into account B
p
≤ 0, we have F
3p
(m) ≥ A
p
+ B
p
m ≥ A
p
+ B
p
m
?
for all m < m
?
. Furthermore, by
construction, C
3L,p
(m) > 0 holds iff m < m
?
. Thus simplifying, we find
F
3p
(m) ≥ (m
?
)
2
+ p
2
− p m
?
=
9
4d
2
d − 1 + ∆k
0
1
− ∆k
0
3
2
−
9
2d
d − 1 + ∆k
0
1
− ∆k
0
3
p + 3p
2
, (280)
if C
3L,p
(m) > 0 holds. The R.H.S. is a convex function in p, thus calculating its stationary point, it can be
lower bounded by
F
3p
(m) ≥
9
16
1 −
1
d
+
∆k
0
3
− ∆k
0
1
d
2
≥
9
16
1 −
2
d
2
, (281)
if C
3L,p
(m) > 0 holds. Performing the analogous procedure for the case that C
3R,p
(m) < 0 holds, and defining
ε
R
3,1
:=
1
2d
1 + ∆k
0
1
+ ∆k
0
3
≥
1
2d
, (282)
we find
F
3p
(m) ≥
9
16
1 −
3
d
2
. (283)
Thus summarising the two cases, we see that
F
3p
(m) ≥
9
16
1 −
3
d
2
. (284)
if the statement “Inequalities (249), (261) are both satisfied” is false. Thus defining ε
F
by the equation
ε
F
:= Int
3
(m, p) −
1
√
6
σ
d
e
−i2π
˜
kQ/d
e
−iπ
2
3
(m+p)Q/d
(285)
e
−π
4
3
(
d
σ
)
2
[∆k
0
1
−∆k
0
2
+m]
2
+[∆k
0
1
−∆k
0
3
+p]
2
−[∆k
0
1
−∆k
0
2
+m][∆k
0
1
−∆k
0
3
+p]
d
2
e
−
π
6
(
σ
d
)
2
Q
2
, (286)
Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 32
we have that
|ε
F
| ≤
Int
3
(m, p)
(287)
+
1
√
6
σ
d
e
−π
4
3
(
d
σ
)
2
[∆k
0
1
−∆k
0
2
+m]
2
+[∆k
0
1
−∆k
0
3
+p]
2
−[∆k
0
1
−∆k
0
2
+m][∆k
0
1
−∆k
0
3
+p]
d
2
e
−
π
6
(
σ
d
)
2
Q
2
(288)
≤
Int
3
(m, p)
+
1
√
6
σ
d
e
−π
3
4
(
d
σ
)
2
(
1−
3
d
)
2
, (289)
if “Inequalities (249), (261) are both satisfied” is false. We now bound
Int
3
(m, p)
. From Eq. (243) it follows
Int
3
(m, p)
≤
Z
1
p/d
dx e
−2π
(
d
σ
)
2
h
x−
1
2
−
1
d
+
∆k
2
1
d
2
+
x−
1
2
−
1
d
−
m
d
+
∆k
0
2
d
2
+
x−
1
2
−
1
d
−
p
d
+
∆k
0
3
d
2
i
(290)
≤ e
−2π
(
d
σ
)
2
h
−
1
3
p+m+2∆k
0
1
−∆k
0
2
−∆k
0
3
d
2
+
p+∆k
0
1
−∆k
0
3
d
2
+
m+∆k
0
1
−∆k
0
2
d
2
i
(291)
Z
1
p/d
dx e
−6π
(
d
σ
)
2
x−
1
2
−
1
d
−
p+m−∆k
0
T
3d
2
(292)
≤ e
−2π
(
d
σ
)
2
h
−
1
3
p+m+2∆k
0
1
−∆k
0
2
−∆k
0
3
d
2
+
p+∆k
0
1
−∆k
0
3
d
2
+
m+∆k
0
1
−∆k
0
2
d
2
i
(293)
1 −
p
d
e
−6π
(
d
σ
)
2
min
x∈[p/d,1]
x−
1
2
−
1
d
−
p+m−∆k
0
T
3d
2
. (294)
where we have used Eq. (248). Since the above minimisation is over a parabola, where the smallest value at
x =
1
2
+
1
d
+
p+m−∆k
0
T
3d
lays outside the interval [p/d, 1] whenever “Inequalities (249), (261) are both satisfied” is
false, the solution to the minimization is achieved at one of the boundary points, x = p/d or x = 1. Specifically
we find
min
x∈[p/d,1]
x −
1
2
−
1
d
−
p + m −∆k
0
T
3d
2
=
∆
3,p
−
1
2
−
1
d
−
p + m −∆k
0
T
3d
2
, (295)
where
∆
3,p
=
(
p
d
if C
3L,p
(m) ≤ 0 is false
1 if C
3R,p
(m) ≥ 0 is false.
(296)
Thus from Eq. (294), it follows
Int
3
(m, p)
≤ e
−2π
(
d
σ
)
2
h
−
1
3
p+m+2∆k
0
1
−∆k
0
2
−∆k
0
3
d
2
+
p+∆k
0
1
−∆k
0
3
d
2
+
m+∆k
0
1
−∆k
0
2
d
2
i
(297)
e
−6π
(
d
σ
)
2
∆
3,p
−
1
2
−
1
d
−
p+m−∆k
0
T
3d
2
(298)
≤
1
d
e
−2π
(
d
σ
)
2
h
∆
3,p
−
1
2
−
1
d
+
∆k
0
1
d
2
+
∆
3,p
−
1
2
−
1
d
+
∆k
0
3
d
−
p
d
2
+
∆
3,p
−
1
2
−
1
d
+
∆k
0
2
d
−
m
d
2
i
(299)
≤
1
d
e
−2π
(
d
σ
)
2
(
1
2
−
2
d
)
2
=
1
d
e
−
π
2
(
d
σ
)
2
(
1−
4
d
)
2
, d = 4, 6, 8, . . . , (300)
if “Inequalities (249), (261) are both satisfied” is false. Hence, finally, taking into account Eq. (265) and Eqs.
(286), (300), we find that for all m, p = 0, . . . , d − 1, such that m ≤ p,
13
Int
3
(m, p) =
1
√
6
σ
d
e
−i2π
˜
kQ/d
e
−iπ
2
3
(m+p)Q/d
(301)
e
−π
4
3
(
d
σ
)
2
[
m+∆k
0
1
−∆k
0
2
]
2
+
[
p+∆k
0
1
−∆k
0
3
]
2
−
[
m+∆k
0
1
−∆k
0
2
][
p+∆k
0
1
−∆k
0
3
]
d
2
e
−
π
6
(
σ
d
)
2
Q
2
+ ε
3T
, (302)
13
This last condition is simply to ensure that
¯
k
2
≤
¯
k
3
, as has been assumed from the outset.
Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 33
where for d = 4, 6, 8 . . .
|ε
3T
| ≤ max
n
|ε
F
|, |ε
3
(m, p)|
o
(303)
≤ max
1
√
6
σ
d
e
−π
3
4
(
d
σ
)
2
(
1−
3
d
)
2
+
1
d
e
−
π
2
(
d
σ
)
2
(
1−
4
d
)
2
,
1
3π
σ
2
d
e
−
π
2
(
d
σ
)
2
(
1−
2
d
)
2
(304)
= O
σ
2
d
e
−
π
2
(
d
σ
)
2
(
1−
4
d
)
2
as d → ∞, (305)
where we have used Eqs. (289), (268) and the values of ε
L
3,1
, ε
R
3,1
from Eqs. (282), (276).
H.3 Approximating the integrals in lines (235), (236).
The steps taken in the previous section for approximating the integral in line (237), can be repeated for the
integrals in lines (235), (236). We summarise here the final results, starting with the integrals in line (235): For
the parametrization
¯
k
2
/d = 1/2 + 1/d − k/d,
¯
k
3
/d = 1/2 + 1/d − l/d, k, l = 1, . . . , d.
Int
1
(k, l) :=
Z
¯
k
2
/d+1/2−1/d
0
dx e
−i2πxQ
e
−2π
(
d
σ
)
2
h
x−
1
2
−
1−∆k
0
1
d
2
+
x−
¯
k
2
d
+
∆k
0
2
d
2
+
x−
¯
k
3
d
+
∆k
0
3
d
2
i
(306)
=
1
√
6
σ
d
e
−2iπ
˜
kQ
e
iπ
2
3
(k+l)Q/d
(307)
e
−π
4
3
(
d
σ
)
2
[
k+∆k
0
2
−∆k
0
1
]
2
+
[
l+∆k
0
3
−∆k
0
1
]
2
−
[
k+∆k
0
2
−∆k
0
1
][
l+∆k
0
3
−∆k
0
1
]
d
2
e
−
π
6
(
σ
d
)
2
Q
2
+ ε
1T
, (308)
where
|ε
1T
| = O
σ
2
d
e
−
π
2
(
d
σ
)
2
(
1−
4
d
)
2
as d → ∞. (309)
In the case of the integral in line Eq. (236), for the parametrization
¯
k
2
/d = −1/2 + 1/d + m/d,
¯
k
3
/d =
1/2 + 1/d −n/d, m = 0, . . . , d − 1, n = 1, . . . , d; we find
Int
2
(m, n) :=
Z
¯
k
3
/d+1/2−1/d
¯
k
2
/d+1/2−1/d
dx e
−i2πxQ
e
−2π
(
d
σ
)
2
h
x−
1
2
−
1−∆k
0
1
d
2
+
x−1−
¯
k
2
d
+
∆k
0
2
d
2
+
x−
¯
k
3
d
+
∆k
0
3
d
2
i
(310)
=
1
√
6
σ
d
e
−i2π
˜
kQ/d
e
−iπ
2
3
(m−n)Q/d
(311)
e
−π
4
3
(
d
σ
)
2
[
m−∆k
1
2
+∆k
0
1
]
2
+
[
n+∆k
1
3
−∆k
0
1
]
2
+
[
n+∆k
1
3
−∆k
0
1
][
m−∆k
1
2
+∆k
0
1
]
d
2
e
−
π
6
(
σ
d
)
2
Q
2
+ ε
2T
, (312)
where
|ε
2T
| = O
σ
2
d
e
−
π
2
(
d
σ
)
2
(
1−
4
d
)
2
as d → ∞. (313)
H.4 Final expression for
˜
F
Q
(k
1
, k
2
, k
3
)
With the results from the previous section, we can finally formulate a useful approximation for
˜
F
Q
(k
1
, k
2
, k
3
).
Plugging Eqs. (308), (312), (302), into Eq. (238) and after some re-parametrization, we find for
¯
k
1
/d =
−1/2 + 1/d,
¯
k
2
/d = −1/2 + 1/d + m/d,
¯
k
3
/d = −1/2 + 1/d + p/d, m, p = 0, . . . , d − 1, with
¯
k
1
≤
¯
k
2
≤
¯
k
3
,
˜
F
Q
(k
1
, k
2
, k
3
) =
A
6
√
6
σ
d
e
−i2π
˜
kQ/d
e
−
π
6
(
σ
d
)
2
Q
2
G
Q
(d; m, p) + ε
T
, (314)
where
G
Q
(d; m, p) :=e
−iπ
2
3
(m+p)Q/d
e
−π
4
3
(
d
σ
)
2
H
1
(d,m,p)
+ e
−iπ
2
3
(m+p−d)Q/d
e
−π
4
3
(
d
σ
)
2
H
2
(d,m,p)
(315)
+ e
−iπ
2
3
(m+p−2d)Q/d
e
−π
4
3
(
d
σ
)
2
H
3
(d,m,p)
, (316)
Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 34
and
H
1
(d, m, p) :=
[
m+∆k
0
1
−∆k
0
2
]
2
+
[
p+∆k
0
1
−∆k
0
3
]
2
−
[
m+∆k
0
1
−∆k
0
2
][
p+∆k
0
1
−∆k
0
3
]
d
2
, (317)
H
2
(d, m, p) :=
[
m−∆k
0
2
+∆k
0
1
]
2
+
[
−p+d+∆k
0
3
−∆k
0
1
]
2
+
[
−p+d+∆k
0
3
−∆k
0
1
][
m−∆k
0
2
+∆k
0
1
]
d
2
, (318)
H
3
(d, m, p) :=
[
−m+d+∆k
0
2
−∆k
0
1
]
2
+
[
−p+d+∆k
0
3
−∆k
0
1
]
2
−
[
−m+d+∆k
0
2
−∆k
0
1
][
−p+d+∆k
0
3
−∆k
0
1
]
d
2
. (319)
The small error term ε
T
in Eq. (314) is defined via
ε
T
= A
6
(ε
1T
+ ε
2T
+ ε
3T
) + ε
I1
+ 7ε
I0
, (320)
and bounded by
|ε
T
| =
A
6
(ε
1T
+ ε
2T
+ ε
3T
) + ε
I1
+ 7ε
I0
(321)
≤
√
2
σ
+ |ε
A
|
3
|ε
1T
| + |ε
2T
| + |ε
3T
|
+ ε
I1
+ 7ε
I0
(322)
= O
1
d σ
e
−
π
2
(
d
σ
)
2
(
1−
4
d
)
2
+ ε
I1
+ 7ε
I0
as d → ∞, (323)
where we have used A
2
=
√
2/σ + ε
A
= O(1/σ) (from page 483 in [21]) in line (322), followed by Eqs. (309),
(313). Finally, using Eq. (233) we conclude
|ε
T
| = O
1
d σ
e
−
π
2
(
d
σ
)
2
(
1−
4
d
)
2
+ O
d
2
σ
5/2
e
−
π
4
σ
2
α
2
0
as d → ∞. (324)
We can now easily generalise this result to any measurement outcomes. Define k
0
1
, k
0
2
, k
0
3
∈ I
d
= {0, 1, . . . , d −
1}, by
k
0
q
:= [k
q
+ l]
(mod. d)
=
−
d
2
+ 1 + b
˜
k
0
1
c + l
(mod. d)
if q = 1,
−
d
2
+ 1 + b
˜
k
0
2
c + l + m
(mod. d)
if q = 2,
−
d
2
+ 1 + b
˜
k
0
3
c + l + p
(mod. d)
if q = 3,
(325)
for l, m, p = 0, . . . , d − 1. Rather than the previous case where the measurement outcomes k
q
where chosen
k
q
∈ S
d
(
˜
k
0
q
), q = 1, 2, 3; in the present case of the measurement outcomes k
0
1
, k
0
2
, k
0
3
, for simplicity we will use
the convention k
0
1
, k
0
2
, k
0
3
∈ I
d
= {0, 1, . . . , d − 1}
14
, with outcome k
0
q
associated with projective measurement
|θ
k
0
q
ihθ
k
0
q
|
q
, q = 1, 2, 3.
Now employing Eqs. (60), (314) we achieve
˜
F
Q
(k
0
1
, k
0
2
, k
0
3
) = e
−i2πlQ/d
˜
F
Q
(k
1
, k
2
, k
3
) (326)
=
A
6
√
6
σ
d
e
−i2π
˜
k
l
Q/d
e
−
π
6
(
σ
d
)
2
Q
2
G
Q
(d; m, p) + ε
T
e
−i2πlQ/d
, if m ≤ p (327)
Here we have defined the common phase,
˜
k
l
:= −
d
2
+ 1 −
∆k
0
T
3
+ l, (328)
where recall ∆k
0
T
is defined in Eq. (267). Finally, we note that not all values of k
0
1
, k
0
2
, k
0
3
∈ I
d
are achievable
due to the constraint m ≤ p. However, we can remedy this by using property 3) [see Eq. (61)]. Since in this case
the clocks only differ by the values
˜
k
0
1
,
˜
k
0
2
,
˜
k
0
3
, interchanging ρ
C,2
(t) with ρ
C,3
(t) is equivalent to interchanging
˜
k
0
2
with
˜
k
0
3
. Furthermore, recalling k
2
=
¯
k
2
+ b
˜
k
0
2
c = −d/2 + 1 + m + b
˜
k
0
2
c, k
3
=
¯
k
3
+ b
˜
k
0
3
c = −d/2 + 1 + p + b
˜
k
0
3
c,
we see that interchanging
˜
k
0
2
with
˜
k
0
3
and k
2
with k
3
is equivalent to interchanging
˜
k
0
2
with
˜
k
0
3
and m with p.
Finally, by construction
¯
k
2
>
¯
k
3
iff m > p. Thus using Eq. (327) we find that if m > p,
˜
F
Q
(k
0
1
, k
0
2
, k
0
3
) =
"
A
6
√
6
σ
d
e
−i2π
˜
k
l
Q/d
e
−
π
6
(
σ
d
)
2
Q
2
G
Q
(d; m, p) + ε
T
e
−i2πlQ/d
#
˜
k
0
2
↔
˜
k
0
3
m ↔p
(329)
=
A
6
√
6
σ
d
e
−i2π
˜
k
l
Q/d
e
−
π
6
(
σ
d
)
2
Q
2
h
G
Q
(d; p, m)
i
˜
k
0
2
↔
˜
k
0
3
+ ˜ε
T
, (330)
14
Note that this simply corresponds to a shifting the values of all the measurement outcomes by a constant value.
Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 35
where |˜ε
T
| satisfies the same upper bound in Eq. (323) that |ε
T
| satisfies. Thus, in general, from Eqs. (327)
and (330) we conclude
˜
F
Q
(k
0
1
, k
0
2
, k
0
3
) =
A
6
√
6
σ
d
e
−i2π
˜
k
l
Q/d
e
−
π
6
(
σ
d
)
2
Q
2
˜
G
Q
(d; m, p) + ¯ε
T
, (331)
where
˜
G
Q
(d; m, p) :=
G
Q
(d; m, p) if m ≤ p
h
G
Q
(d; p, m)
i
˜
k
0
2
↔
˜
k
0
3
if p < m
(332)
and |¯ε
T
| is upper bounded by the r.h.s. of Eq. (323).
H.5 Bounding k
ˆ
Ek
1
Now that we have from the previous section an expression for
˜
F
Q
for all measurement outcomes of the 3 clocks,
the next task is to bound k
ˆ
Ek
1
defined in 72 to be
||
ˆ
E||
1
≤d
L
p
d
P
X
k
0
1
,k
0
2
,k
0
3
˜
F
0
(k
0
1
, k
0
2
, k
0
3
) max
Q
|p(Q,
~
k)|
2
(333)
=d
L
p
d
P
X
k
0
1
,k
0
2
,k
0
3
˜
F
0
(k
0
1
, k
0
2
, k
0
3
) max
Q
˜
F
Q
(k
0
1
, k
0
2
, k
0
3
)
˜
F
0
(k
0
1
, k
0
2
, k
0
3
)
e
2πik
α
Q/d
− 1
(334)
=d
L
p
d
P
max
Q
X
k
0
1
,k
0
2
,k
0
3
˜
F
Q
(k
0
1
, k
0
2
, k
0
3
) e
2πik
α
Q/d
−
˜
F
0
(k
0
1
, k
0
2
, k
0
3
)
(335)
=d
L
p
d
P
A
6
√
6
σ
d
max
Q
(336)
d−1
X
l,m,p=0
e
−
π
6
(
σ
d
)
2
Q
2
˜
G
Q
(d; m, p)e
2πi(k
α
−
˜
k
l
)Q/d
−
˜
G
0
(d; m, p) + ˜ε
T
+ ¯ε
T
. (337)
In all cases, the optimization is over Q ∈ {−(∆h
Co
+ ∆h
L
), . . . , (∆h
Co
+ ∆h
L
)}.
In order to proceed further, we need to specify k
α
for this protocol. If there were no phase errors, it turns
out that defining it equal to either of the three quantities
¯
k
0
1
:= −d/2 + 1 + l,
¯
k
0
2
:= −d/2 + 1 + l + m,
¯
k
0
3
:= −d/2 + 1 + l + p, would suffice. Note how doing so would only require information from one clock
measurement — the other two clock results would be redundant information. Intuitively, this is because with
high probability, all three clocks will have measurement outcomes corresponding to approximately the same
elapsed time (if there were no phase errors). For the case of no phase error, it is convenient to prove that the
protocol works up to the specified error when k
α
is chosen to be “any angle
15
in-between the three angles
¯
k
0
1
,
¯
k
0
2
,
¯
k
0
3
” when
¯
k
0
1
,
¯
k
0
2
,
¯
k
0
3
belong to a to-be specified domain, and “any angle otherwise”. Later in section H.6 we
will show how such a result is sufficient in the case of the unknown phase error. Specifically, we define
k
α
= −
d
2
+ 1 + γ
l,m,p
(α) (338)
where γ
l,m,p
are functions indexed by l, m, p = 0, . . . , d − 1 of the form
γ
l,m,p
(α) = α. (339)
Due to the modular arithmetic, depending on which sector l, m, p belong to, the domain of γ
l,m,p
changes.
Specifically, for R ∈ [1, d/4] we define the sets
S
11
(R) :=
(m, p) ∈
2
| 0 ≤ m, p ≤ R − 1
, (340)
S
31
(R) := {(m, p) ∈
2
| 0 ≤ m ≤ R − 1 & d − R ≤ p ≤ d − 1}, (341)
S
13
(R) := {(m, p) ∈
2
| d −R ≤ m ≤ d − 1 & 0 ≤ p ≤ R − 1}, (342)
S
33
(R) := {(m, p) ∈
2
| d −R ≤ m, p ≤ d − 1}. (343)
15
Throughout this proof, we call “angle” to real numbers which only need to be specified up to modulo d. These angles can be
converted to radian by multiplying them by 2π/d.
Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 36
Figure 2: Dom(γ
l,m,p
) for (m, p) in different intervals and l = 0, 1, . . . , d − 1.
Case 1): Yellow highlighted interval represents Dom(γ
l,m,p
) for (m, p) ∈ S
11
(R).
Case 2): Yellow highlighted interval represents Dom(γ
l,m,p
) for (m, p) ∈ S
31
(R).
Case 3): Yellow highlighted interval represents Dom(γ
l,m,p
) for (m, p) ∈ S
33
(R).
The case Dom(γ
l,m,p
) for (m, p) ∈ S
13
(R) is the same as case 2) above under the interchange m ↔ p. In all four cases,
CoDom(2πγ
l,m,p
/d) is “any angle between angles l, m, p.”
Now we can define Dom(γ
l,m,p
):
Dom(γ
l,m,p
) :=
l, l + 1, . . . , l + max{m, p} if (m, p) ∈ S
11
(R)
l + p, l + p + 1, . . . , l + m + d if (m, p) ∈ S
31
(R)
l + m, l + m + 1, . . . , l + p + d if (m, p) ∈ S
13
(R)
l + min{m, p}, l + min{m, p} + 1, . . . , l + d if (m, p) ∈ S
33
(R)
l, l + 1, . . . , l + d − 1 otherwise
(344)
Observer that l, (l + m)
(mod. d)
, (l + p)
(mod. d)
∈ Dom(γ
l,m,p
)
(mod. d)
always holds. In particular, the function
γ
m,p
(α) covers up to (but not more than), all angles 0, 1, . . . , d−1 which are between the three angles l, l+m, l+p
whenever (m, p) ∈ S
11
(R) ∪ S
31
(R) ∪ S
13
(R) ∪ S
33
(R) and l ∈ I
d
. See Fig. 2.
Writing Eq. (337) in terms of this new notation, we have
||
ˆ
E||
1
≤d
L
p
d
P
A
6
√
6
σ
d
max
Q∈{−(∆h
Co
+∆h
L
),...,(∆h
Co
+∆h
L
)}
(345)
d−1
X
l,m,p=0
e
−
π
6
(
σ
d
)
2
Q
2
˜
G
Q
(d; m, p)e
2πi
γ
m,p
(α)+
∆k
T
3
Q/d
−
˜
G
0
(d; m, p)
+ 2d |ˆε
T
|
(346)
=d
L
p
d
P
A
6
√
6
σ
d
max
Q∈{−(∆h
Co
+∆h
L
),...,(∆h
Co
+∆h
L
)}
d−1
X
l,m,p=0
∆F (d; l, m, p) + 2d |ˆε
T
|
, (347)
Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 37
where |ˆε
T
| = max
k
0
1
|˜ε
T
+ ¯ε
T
|/2 is upper bounded by the r.h.s. of Eq. (323). It is now convenient to bound the
summand ∆F (d; l, m, p). Using Eqs. (316) and (331), we find
∆F (d; l, m, p) :=
e
−
π
6
(
σ
d
)
2
Q
2
˜
G
Q
(d; m, p)e
2πi
γ
m,p
(α)−l+
∆k
T
3
Q/d
−
˜
G
0
(d; m, p)
(348)
≤
e
−
π
6
(
σ
d
)
2
Q
2
e
−iπ
2
3
(m+p)Q/d
e
2πi
γ
m,p
(α)−l+
∆k
T
3
Q/d
− 1
e
−π
4
3
(
d
σ
)
2
˜
H
1
(d,m,p)
(349)
+
e
−
π
6
(
σ
d
)
2
Q
2
e
−iπ
2
3
(m+p−d)Q/d
e
2πi
γ
m,p
(α)−l+
∆k
T
3
Q/d
− 1
e
−π
4
3
(
d
σ
)
2
˜
H
2
(d,m,p)
(350)
+
e
−
π
6
(
σ
d
)
2
Q
2
e
−iπ
2
3
(m+p−2d)Q/d
e
2πi
γ
m,p
(α)−l+
∆k
T
3
Q/d
− 1
e
−π
4
3
(
d
σ
)
2
˜
H
3
(d,m,p)
, (351)
where for q = 1, 2, 3, we have defined
˜
H
q
(d, m, p) :=
H
q
(d, m, p) if m ≤ p
h
H
q
(d, p, m)
i
k
0
2
↔k
0
3
if p < m
(352)
The evaluation of the summations over m, p for ∆F (d; l, m, p) is more complicated than the summation over
l and it is convenient to perform the summation separately over subsets of the domain of m, p. With this in
mind, we define for R ∈ [1, d/4] the following sets.
S
tot
= S
11
(R) ∪
¯
S
12
(R) ∪ S
13
(R) ∪
¯
S
21
(R) ∪
¯
S
22
(R) ∪
¯
S
23
(R) ∪ S
31
(R) ∪
¯
S
32
(R) ∪ S
33
(R). (353)
where S
tot
= {(m, p) ∈
2
| 0 ≤ m, p ≤ d − 1} is the set m and p run over in the summation in Eq. (347) and
S
11
(R), S
31
(R), S
11
(R), S
33
(R), are the sets which when R is chosen to scale with d appropriately, will include
all measurement outcomes k
0
2
, k
0
3
which, as a whole, occur with high probability. On the other hand, the sets
¯
S
12
(R) := {(m, p) ∈
2
| R ≤ m ≤ d − 1 − R & 0 ≤ p ≤ R − 1} (354)
¯
S
21
(R) := {(m, p) ∈
2
| 0 ≤ m ≤ R − 1 & R ≤ p ≤ d −1 −R} (355)
¯
S
22
(R) := {(m, p) ∈
2
| R ≤ m, p ≤ d − 1 − R} (356)
¯
S
23
(R) := {(m, p) ∈
2
| d −R ≤ m ≤ d − 1 & R ≤ p ≤ d − 1 − R} (357)
¯
S
32
(R) := {(m, p) ∈
2
| R ≤ m ≤ d − 1 − R & d − R ≤ p ≤ d − 1}, (358)
correspond to all measurement outcomes k
0
2
, k
0
3
which, as a whole, occur with very low probability. See graphical
depiction Fig. 3.
In addition to the limits Eqs. (186) assumed thought, we now demand that R satisfies the following limits.
The exact choice of R will be determined later.
lim
d→+∞
R
σ
= +∞, lim
d→+∞
R
d
Q
max
= 0, (359)
16
where Q
max
:= max
Q∈{−(∆h
P
+∆h
L
),...,(∆h
P
+∆h
L
)}
|Q| = ∆h
L
+∆h
P
. We now want to bound the ∆F (d, l, m, p)
in Eq. (348) for measurement outcomes in sets Eqs. (340) to (343). We start with the set S
12
. For all
(m, p) ∈ S
12
(R), we have that lim
d→∞
˜
H
2
(d, m, p) = lim
d→∞
˜
H
3
(d, m, p) = 1, so the terms in lines Eqs.
(350),(351) are exponentially small in (d/σ)
2
. On the other hand, lim
d→∞
˜
H
2
(d, m, p) converges to zero and
the common factor in line (349) is significant. For the term in line Eq. (349), it is the terms within the absolute
value which are small. Specifically, for all α
l,m,p
∈ Dom(γ
l,m,p
) and for all (m, p) ∈ S
12
(R), l ∈ I
d
,
e
−
π
6
(
σ
d
)
2
Q
2
e
−iπ
2
3
(m+p)Q/d
e
2πi
γ
l,m,p
(α
l,m,p
)−l+
∆k
T
3
Q/d
− 1
(360)
=
−
π
6
σ
d
2
Q
2
− iπ
2
3
(m + p)Q/d + 2πi
γ
l,m,p
(α
l,m,p
) − l +
∆k
T
3
Q/d
(361)
+ O
−
π
6
σ
d
2
Q
2
− iπ
2
3
(m + p)Q/d + 2πi
γ
l,m,p
(α
l,m,p
) − l +
∆k
T
3
Q/d
2
(362)
≤
π
4
3
R
d
Q
max
+ 2π
R
d
Q
max
+ O
π
4
3
R
d
Q
max
+ 2π
R
d
Q
max
2
(363)
=π
10
3
R
d
Q
max
+ O
π
10
3
R
d
Q
max
2
, (364)
16
Note that Q
max
≥ 1 uniformly in d and thus it follows from Eq. (359) that lim
d→∞
R/d = 0.
Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 38
Figure 3: Diagrammatic view of the nine distinct sets used in the proof. Note that all pairs have the empty set as their
intersection and the union of all sets is the set S
tot
given by Eq. (353).
where we have used that for (m, p) ∈ S
11
(R), Dom(γ
l,m,p
) = l, . . . , l + max{m, p}. Thus we conclude that for
all α
l,m,p
∈ Dom(γ
l,m,p
) and for all (m, p) ∈ S
12
(R), l ∈ I
d
∆F (d; l, m, p) ≤ π
10
3
R
d
Q
max
+ O
π
10
3
R
d
Q
max
2
+ O
2e
−π
4
3
(
d
σ
)
2
. (365)
Now consider the set S
33
. For all (m, p) ∈ S
33
(R), we have that lim
d→∞
˜
H
1
(d, m, p) = lim
d→∞
˜
H
2
(d, m, p) = 1,
so the terms in lines Eqs. (349),(350) are exponentially small in (d/σ)
2
. For the other term, i.e. line Eq. (351),
we proceed by bounding the part within absolute values. Specifically, for all α
l,m,p
∈ Dom(γ
l,m,p
) and for all
(m, p) ∈ S
33
(R), l ∈ I
d
,
e
−
π
6
(
σ
d
)
2
Q
2
e
−iπ
2
3
(m+p−2d)Q/d
e
2πi
γ
l,m,p
(α
l,m,p
)−l+
∆k
T
3
Q/d
− 1
(366)
=
e
−
π
6
(
σ
d
)
2
Q
2
e
−iπ
2
3
(m+p−2d)Q/d
e
2πi
γ
l,m,p
(α
l,m,p
)−l−d+
∆k
T
3
Q/d
− 1
(367)
≤
π
4
3
R
d
Q
max
+ 2π
R
d
Q
max
+ O
π
4
3
R
d
Q
max
+ 2π
R
d
Q
max
2
(368)
=π
10
3
R
d
Q
max
+ O
π
10
3
R
d
Q
max
2
, (369)
where we have used that for (m, p) ∈ S
33
(R), CoDom(γ
l,m,p
(α
l,m,p
) − l − d) = min{m, p} − d, . . . , 0 and
−R ≤ min{m, p}−d, giving us |γ
l,m,p
(α
l,m,p
) −l −d| ≤ R. Thus we conclude that for all α
l,m,p
∈ Dom(γ
l,m,p
)
and for all (m, p) ∈ S
33
(R), l ∈ I
d
∆F (d; l, m, p) ≤ π
10
3
R
d
Q
max
+ O
π
10
3
R
d
Q
max
2
+ O
2e
−π
4
3
(
d
σ
)
2
. (370)
In general, for every set S
xy
(R), x, y ∈ {1, 3}, two out of the three terms
˜
H
1
(d; m, p),
˜
H
2
(d; m, p),
˜
H
3
(d; m, p)
converge to unity as d tends to infinity, resulting in two of the lines Eqs. (349), (350), (351) being exponentially
small in (d/σ)
2
. The terms in the remaining line can be expended in a power expansion similarly to Eq. (369),
resulting in an order Q
max
R/d contribution.
We therefore conclude that the following is true for all x, y ∈ {1, 3}: For all α
l,m,p
∈ Dom(γ
l,m,p
) and for all
(m, p) ∈ S
xy
(R), l ∈ I
d
,
∆F (d; l, m, p) ≤ π
10
3
R
d
Q
max
+ O
π
10
3
R
d
Q
max
2
+ O
2e
−π
4
3
(
d
σ
)
2
. (371)
Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 39
From the definition of the sets x, y ∈ {1, 3} with m, p ∈ S
xy
(R) we see that they all have cardinality R
2
. Thus
from Eq. (371), it follows that for all α
l,m,p
∈ Dom(γ
l,m,p
) and for all (m, p) ∈ S
11
(R)∪S
13
(R)∪S
31
(R)∪S
33
(R),
l ∈ I
d
,
X
l∈I
d
X
(m,p)∈S
11
(R)∪S
13
(R)∪S
31
(R)∪S
33
(R)
∆F (d; l, m, p) (372)
≤ dR
2
π
10
3
R
d
Q
max
+ O
π
10
3
R
d
Q
max
2
+ O
2e
−π
4
3
(
d
σ
)
2
!
. (373)
We will now bound ∆F (d; l, m, p) for (m, p) belonging to the sets Eqs. (354) to (354). We start with the set
¯
S
12
. For all (m, p) ∈
¯
S
12
(R), we have for sufficiently large d
d
σ
2
˜
H
1
(d, m, p) ≥
R
σ
2
, (374)
d
σ
2
˜
H
2
(d, m, p) ≥
R
σ
2
, (375)
d
σ
2
˜
H
3
(d, m, p) ≥
d
σ
2
. (376)
Thus from Eq. (348) we conclude that for all α
l,m,p
∈ Dom(γ
l,m,p
) and for all (m, p) ∈
¯
S
12
(R), l ∈ I
d
,
∆F (d; l, m, p) ≤ O
2 e
−π
4
3
(
d
σ
)
2
+ O
4 e
−π
4
3
(
R
σ
)
2
. (377)
Similarly, one finds that Eq. (377) holds for all α
l,m,p
∈ Dom(γ
l,m,p
) and for all (m, p) ∈
¯
S
xy
(R), l ∈ I
d
, for
xy = 21, 23, 32. In the case of the set
¯
S
22
(R) we find that for all α
l,m,p
∈ Dom(γ
l,m,p
) and for all (m, p) ∈
¯
S
22
(R),
l ∈ I
d
,
∆F (d; l, m, p) ≤ O
6 e
−π
4
3
(
R
σ
)
2
. (378)
Finally, since the cardinality of the sets
¯
S
12
(R),
¯
S
21
(R),
¯
S
22
(R),
¯
S
23
(R),
¯
S
32
(R) are all upper bounded by d
2
,
we have that for all α
l,m,p
∈ Dom(γ
l,m,p
), and for all (m, p) ∈
¯
S
12
(R) ∪
¯
S
21
(R) ∪
¯
S
22
(R) ∪
¯
S
23
(R) ∪
¯
S
32
(R),
l ∈ I
d
,
X
l∈I
d
X
(m,p)∈
¯
S
12
(R)∪
¯
S
21
(R)∪
¯
S
22
(R)∪
¯
S
23
(R)∪
¯
S
32
(R)
∆F (d; l, m, p) (379)
≤ O
d
3
e
−π
4
3
(
d
σ
)
2
+ O
d
3
e
−π
4
3
(
R
σ
)
2
. (380)
Hence, combining Eqs. (373) and (380), and plugging into Eq. (347) we find for all α
l,m,p
∈ Dom(γ
l,m,p
) and
for all (m, p) ∈ S
tot
, l ∈ I
d
||
ˆ
E||
1
≤
p
d
L
d
P
A
6
σ
√
6
"
R
2
π
10
3
R
d
Q
max
+ O
R
d
Q
max
2
+ O
e
−π
4
3
(
d
σ
)
2
!
(381)
+ O
d
2
e
−π
4
3
(
d
σ
)
2
+ O
d
2
e
−π
4
3
(
R
σ
)
2
+ 2d
2
|ˆε
T
|
#
(382)
=
p
d
L
d
P
A
6
σ
√
6
"
R
2
π
10
3
R
d
Q
max
+ O
R
d
Q
max
2
!
(383)
+ O
d
2
e
−π
4
3
(
R
σ
)
2
+ 2d
2
|ˆε
T
|
#
, (384)
where, to find the higher order terms, we have used in the last equality
lim
d→∞
R/σ
d/σ
= 0, (385)
Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 40
which follows from Eq. (359). Finally, using A
2
=
√
2/σ + ε
A
= O(1/σ) from page 483 in [21] and the upper
bound on in Eq. (324), we find for all α
l,m,p
∈ Dom(γ
l,m,p
) and for all (m, p) ∈ S
tot
, l ∈ I
d
,
||
ˆ
E||
1
≤
p
d
L
d
P
2
√
3
"
R
σ
2
π
10
3
R
d
Q
max
+ O
R
d
Q
max
2
!
(386)
+ O
d
σ
2
e
−π
4
3
(
R
σ
)
2
!
+ O
d
4
σ
9/2
e
−
π
4
σ
2
α
2
0
#
, as d → ∞. (387)
We have the error term ||
ˆ
E||
1
in terms of R, and σ. These are free parameters that we can choose. The
optimal scaling is given by
σ = ln
5/2
d R = ln
4
d, (388)
as the errors become
||
ˆ
E||
1
≤
p
d
L
d
P
2
√
3
"
10π ln
7
(d)Q
max
3d
+ O
ln
11
(d)
d
2
Q
2
max
(389)
+ O
1
d ln
5/2
(d)
!
+ O
1
d ln
45/4
(d)
!#
, as d → ∞, (390)
for all α
l,m,p
∈ Dom(γ
l,m,p
) and for all (m, p) ∈ S
tot
, l ∈ I
d
. Asymptotically, the first two error terms will be
the dominant ones, and thus we conclude that for all α
l,m,p
∈ Dom(γ
l,m,p
) and for all (m, p) ∈ S
tot
, l ∈ I
d
,
||
ˆ
E||
1
≤
p
d
L
d
P
2
√
3
"
10π ln
7
(d)Q
max
3d
+ O
ln
11
(d)
d
2
Q
2
max
#
, as d → ∞, (391)
making the error effectively scale as O
d
L
√
d
P
ln
7
(d)Q
max
d
. Finally using Eq. (83), we achieve for all α
l,m,p
∈
Dom(γ
l,m,p
) and for all (m, p) ∈ S
tot
, l ∈ I
d
.
f
2
worst
≥ 1 −
p
d
L
d
P
"
10π ln
7
(d)Q
max
√
3d
+ O
ln
11
(d)
d
2
Q
2
max
#
, as d → ∞, (392)
where Q
max
= ∆h
L
+ ∆h
Co
.
H.6 The Protocol
Here we provide the protocol (based on the results from the previous subsection) which will achieve the statement
of Theorem 4 . Recall that the general protocol for all set-ups in this paper is described in Appendix B, thus
here we only need to specificity how to choose the parameter k
α
given the clock measurement outcomes.
After obtaining the measurement outcomes k
0
1
, k
0
2
, k
0
3
∈ {0, 1, . . . , d − 1}, the second task will be to work out
the quantities
˜
l,
˜
l + ˜m,
˜
l + ˜p which are defined via the relations
k
0
q
=
−
d
2
+ 1 + bk
0
q
c +
˜
Γ
q
(mod. d)
,
˜
Γ
q
=
˜
l if q = 1,
˜
l + ˜m if q = 2,
˜
l + ˜p if q = 3,
(393)
where
˜
l, ˜m, ˜p ∈ {0, 1, . . . , d −1} and I
d
= {0, 1, . . . , d −1}. Importantly
˜
Γ
1
,
˜
Γ
2
,
˜
Γ
3
are always calculable since d,
k
0
1
, k
0
2
, k
0
3
are known.
Next, out of the three angles
˜
Γ
1
, [
˜
Γ
2
]
(mod. d)
, [
˜
Γ
3
]
(mod. d)
∈ [0, d − 1], find which one of these is the middle
angle; denoted α
middle
. Specifically, this angle can be calculated as follows:
1. First, for q
1
< q
2
with q
1
, q
2
∈ {1, 2, 3}, calculate the quantities
∆
q
1
,q
2
:=
[
˜
Γ
q
1
]
(mod. d)
− [
˜
Γ
q
2
]
(mod. d)
if
[
˜
Γ
q
1
]
(mod. d)
− [
˜
Γ
q
2
]
(mod. d)
≤
d
2
− 1
d −
[
˜
Γ
q
1
]
(mod. d)
− [
˜
Γ
q
2
]
(mod. d)
if
[
˜
Γ
q
1
]
(mod. d)
− [
˜
Γ
q
2
]
(mod. d)
≥
d
2
(394)
Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 41
2. Second, find the set containing the smallest two out of the values ∆
1,2
, ∆
1,3
, ∆
2,3
:
Case 2.1) {∆
1,3
, ∆
2,3
} if ∆
1,2
≥ ∆
1,3
, ∆
1,2
≥ ∆
2,3
(395)
Case 2.2) {∆
1,3
, ∆
1,2
} if ∆
2,3
≥ ∆
1,2
, ∆
2,3
≥ ∆
1,3
(396)
Case 2.3) {∆
1,2
, ∆
2,3
} if ∆
1,3
≥ ∆
1,2
, ∆
1,3
≥ ∆
2,3
. (397)
3. Third, find the common index w of the smallest two:
w :=
3 in case 2.1)
1 in case 2.2)
2 in case 2.3)
(398)
4. Fourth, the middle angle is
α
middle
=
˜
Γ
w
. (399)
We call w ∈ 1, 2, 3 in Eq. (399) corresponding to the middle angle, the location of the middle angle.
Set α in Eq. (338) to the middle angle, α = α
middle
,
k
α
= −
d
2
+ 1 + γ
l,m,p
(α
middle
) = −
d
2
+ 1 + α
middle
. (400)
Since Eq. (392) holds for all α
l,m,p
∈ Dom(γ
l,m,p
) and for all (m, p) ∈ S
tot
, l ∈ I
d
, the protocol would
be complete, since calculation of the r.h.s. of Eq. (400) only requires the knowledge of known parameters
d, k
0
1
, k
0
2
, k
0
3
, k
0
1
, k
0
2
, k
0
3
. However, we are not quite done, since what is left to prove is that α
middle
∈ Dom(γ
l,m,p
)
indeed holds true for all (m, p) ∈ S
tot
, l ∈ I
d
, since this has been assumed in Eq. (400) without justification.
To do so, we have to analyse two possible scenarios. First, however, recall Eq. (325); reproduced here for
convenience:
k
0
q
:= [k
q
+ l]
(mod. d)
=
−
d
2
+ 1 + b
˜
k
0
1
c + Γ
q
(mod. d)
, Γ
q
=
l if q = 1,
l + m if q = 2,
l + p if q = 3,
(401)
where I
d
= {0, 1, . . . , d − 1} and
˜
k
0
q
=
(
k
0
q
+ t
ph
d/T
0
if q = r,
k
0
q
otherwise,
(402)
with r ∈ 1, 2, 3 the unknown location of the unknown phase t
ph
∈ . The two scenarios are:
Case 1) The unknown location r ∈ 1, 2, 3 does not coincide with the location of the middle angle (w 6= r):
In this case, by comparing Eqs. (393), (401), we see that the angle α
middle
corresponds to one of the angles Γ
1
,
Γ
2
, Γ
3
. By definition, all three of these angles belong to the domain of γ
m,p
for all (m, p) ∈ S
tot
and for all
l ∈ I
d
.
Case 2) The unknown location r ∈ 1, 2, 3 coincides with the location of the middle angle (w = r): In this
case, it follows that the two angles out of the three angles Γ
1
, Γ
2
, Γ
3
which do not correspond to the middle
angle α
middle
, (namely angles Γ
h
, Γ
k
, where h 6= k 6= r) have not had an unknown phase applied to them (since
by definition, there is only one unknown phase). Hence from Eqs. (393), (401) we have Γ
h
=
˜
Γ
h
, Γ
k
=
˜
Γ
k
. Hence
since the middle angle is always between the other two angles Γ
h
, Γ
k
, and Dom(γ
l,m,p
) includes all angles which
are between all three angles Γ
1
, Γ
2
, Γ
3
(See Fig. 2), it follows that the middle angle α
middle
=
˜
Γ
r
∈ Dom(γ
l,m,p
)
for all (m, p) ∈ S
tot
, l ∈ I
d
.
Hence cases 1) and 2) together prove the assumption made in Eq. (400), namely that α
middle
∈ Dom(γ
l,m,p
)
indeed holds true for all (m, p) ∈ S
tot
, l ∈ I
d
. This completes the proof.
I Quantum communication without a shared reference frame
Throughout the paper we have shown that the Quasi-Ideal clock states |ψ
nor
(k
0
1
)i achieve the best bounds for
the task of maximizing the entanglement fidelity of a covariant code. Here, following [31], we also show how
Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 42
they can improve on a pre-existing scheme for another related task: the transmission of quantum states between
parties which do not share a common reference frame.
The setting is as follows: Alice and Bob are connected by a noiseless quantum channel, but separated by an
indeterminate amount of “time”. This means that the channel connecting then takes the form
E(·) = G(·) := lim
T →∞
Z
T
0
dt
T
U(t)(·)U
†
(t), (403)
where U(t) is the time translation of the systems that are sent. This is a decoherence map in the basis of the
generator of U(t). In order to damp the loss of information from the channel, what Alice can do is to send
the message ρ
M
together with a reference frame F, as G(ρ
M
⊗ |ψ
F
ihψ
F
|) which Bob then uses to “decode”
the message. To decode, Bob measures the clock to learn the time that separates them, which can be used to
retrieve the message up to some error (we label this decoding procedure with D). In [31] it was shown that for
a qubit, the resultant channel is given by
D ◦ E(ρ) = lim
T →∞
Z
T
0
dt
T
|hψ|U
F
(t) |ψi|
2
U
M
(t)ρU
†
M
(t) = (1 − p)I + pG(ρ), (404)
where the probability of fully decohering p is given by
p =
A
1
− A
2
A
1
, (405)
where
A
1
= lim
T →∞
Z
T
0
dt
T
|hψ|U
F
(t) |ψi|
2
, (406)
A
2
= lim
T →∞
Z
T
0
dt
T
|hψ|U
F
(t) |ψi|
2
e
−i
2πt
T
. (407)
In [31], it was found that for the Salecker-Wigner-Peres clock, p
P SW
=
1
d
C
+1
with d the dimension of the clock.
We now show how this changes when one uses the Quasi-Ideal clock |ψ(0)i as defined in the main text. In that
case, it can be shown that
|hψ|U
F
(t) |ψi|
2
∝
e
−
π
2σ
2
2πd
C
t
ω
2
+ e
−
π
2σ
2
d
C
−
2πd
C
t
ω
2
2
+
1
, (408)
where ω is the frequency at which the qubit oscillates, and
|
1
| ≤ O(e
−c
1
σ
2
) + O(e
−c
2
d
2
C
σ
2
), (409)
where c
1
, c
2
are positive constants independent of d
C
and σ. Let us now sketch how to derive Eq. (408). First
one writes out the characteristic function |hψ|U
F
(t) |ψi|, as a finite sum of terms corresponding to the elements
of the eigenbasis {|φ
k
i}. This is
|hψ|U
F
(t) |ψi| = A
2
X
k∈{−
d
C
2
+1,−
d
C
2
+b
2πd
C
t
ω
c}
e
−
π
σ
2
[k
2
+(k−
2πd
C
t
ω
)
2
]
+ A
2
X
k∈{−
d
C
2
+b
2πd
C
t
ω
c+1,
d
C
2
}
e
−
π
σ
2
[k
2
+(k+d
C
−
2πd
C
t
ω
)
2
]
(410)
= A
2
X
k∈Z
e
−
π
σ
2
[k
2
+(k−
2πd
C
t
ω
)
2
]
+ e
−
π
σ
2
[k
2
+(k+d
C
−
2πd
C
t
ω
)
2
]
+ O(e
−c
2
d
2
C
σ
2
) (411)
= A
2
X
m∈Z
σ
√
2d
C
e
−
π
2
h
2πd
C
t
ω
2
+m
2
σ
2
−2id
2
C
m
2πd
C
t
ω
)σ
2
i
(412)
+
σ
√
2d
C
e
−
π
2
h
d
C
−
2πd
C
t
ω
2
+m
2
σ
2
−2id
2
C
m(d
C
−
2πd
C
t
ω
)σ
2
i
+ O(e
−c
2
d
2
C
σ
2
) (413)
= A
2
σ
√
2d
C
e
−
π
2
2πd
C
t
ω
2
+ e
−
π
2
d
C
−
2πd
C
t
ω
2
+ O(e
−c
1
σ
2
)
+ O(e
−c
2
d
2
C
σ
2
), (414)
Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 43
where in going from the second to the third line we have used the Poisson summation formula (see for instance
Corollary C.0.2 from [21]), and in the third to the fourth we have kept the m = 0 term and bounded the size
of the others.
With that, we obtain
A
1
/C =
σ
d
C
+ O
σ
d
C
4
!
+ O(e
−c
1
σ
2
) + O(e
−c
2
d
2
C
σ
2
) (415)
A
2
/C =
σ
d
C
− π
σ
3
d
3
C
+ O
σ
d
C
4
!
+ O(e
−c
1
σ
2
) + O(e
−c
2
d
2
C
σ
2
), (416)
with C > 0 independent of d
C
and σ. Thus, to leading order, it gives a probability of decoherence of
p
id
= π
σ
2
d
2
C
+ O
σ
d
C
3
!
+ O(e
−c
1
σ
2
) + O(e
−c
2
d
2
C
σ
2
). (417)
The best choice for the width of the clock is σ = (ln d
C
)
3
2
, which gives a scaling of
p
id
= π
(ln d
C
)
3
2
d
C
!
2
+ O
(ln d
C
)
3
2
d
C
!
3
, (418)
which (up to the logarithmic factor) is quadratically smaller than p
P SW
. This is essentially the same advantage
as the one obtained in covariant error correction as shown in the main text.
J Evolution without evolution: connection with the Page-Wootters mechanism
In this Section we briefly re-cap the Page-Wooters mechanism (Section III. “Evolution in a Stationary Universe”
in [46]) and show how our formalism and the results in this paper fit into that picture. These authors consider
a bipartite setup consisting of a clock and a system S of interest. The system and clock are considered as a
closed system evolving under a Hamiltonian of the form
ˆ
H =
ˆ
H
S
⊗
clock
+
S
⊗
ˆ
H
clock
. (419)
Their idea is to show that there is a global state ρ
Sclock
which does not evolve in time, i.e. [ρ
Sclock
,
ˆ
H] = 0 yet
the dynamics of the system, after conditioning on the clock being in a state
|ψ(τ)i
clock
= e
−τi
ˆ
H
clock
|ψ(0)i
clock
, (420)
at time τ, is given by the free evolution of the system S according to its own Hamiltonian
ˆ
H
S
. Namely,
tr
clock
[
ˆ
P
clock
(τ)ρ
Sclock
ˆ
P
clock
(τ)] = e
−iτ
ˆ
H
S
tr
clock
[ρ
Sclock
]e
iτ
ˆ
H
S
, (421)
where
ˆ
P
clock
(τ) :=
S
⊗|ψ(τ )ihψ(τ )|
clock
is the projector onto the clock state at time τ . So in the above sense,
even though the global state ρ
Sclock
is stationary, the state of the system conditioned on the clock being at a
particular time, is evolving according to the Schr¨odinger equation for its own Hamiltonian
ˆ
H
S
.
Before seeing how this is connected to our work, a few remarks are pertinent. Firstly, time is assumed to
be continuous, i.e. if one wishes Eq. (421) to hold for all τ in some finite subinterval of the real line, then
one needs to use an Idealised clock (these are discussed in Section 1 in the main text). These, however, are
unphysical in the sense of requiring infinite energy. To the best knowledge of the authors, while many aspects
of the Page-Wooters mechanism have been explored (see [48–51] and references therein), it remains an open
question to how well finite dimensional clocks (or infinite dimensional clocks with finite energy) can realize this.
It is also noteworthy that in our setup, the clock and system are only classically correlated, and thus our results
demonstrate that quantum entanglement between the clock and system S is not necessary to fulfil the Page-
Wooters mechanism. We now find the Hamiltonian and time invariant state which realise the Page-Wooters
mechanism up to a quantifiable error incurred due to using physical clocks.
Firstly, we can identify the physical space with the system in the Page-Wooters model and the clocks in
our setup with those in Page-Wooters model. The total Hamiltonian on the system and the clock will be the
Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 44
Kronecker sum of the generators of the physical space Lie group and those of the clock’s Lie group, namely (c.f.
Eq. (419))
ˆ
H =
ˆ
H
P
⊗
⊗M
C
+
P
⊗
M
M
i=1
ˆ
H
C
. (422)
Secondly, we use our covariant encoding channel E
cov
(·) in Eq. (8) to realise the stationary state ρ
Sclock
of the
Page-Wooters mechanism
ρ
Sclock
= E
cov
(ρ
L
), (423)
for any ρ
L
∈ S (H
L
) and for the choice of trivial representation on the Logical space, namely U
L
(t) =
L
for all
t ∈ . To see how this works, note that 1) the unitary group representation on the physical and clock space
has the Hamiltonian in Eq. (422) as its generator, and 2) that the encoding map is covariant. Specifically
e
−it
ˆ
H
E
cov
(ρ
L
)e
it
ˆ
H
= U
Co
(t) ⊗ U
⊗M
C
(t) E
cov
(ρ
L
) U
†
Co
(t) ⊗ U
⊗M†
C
(t) = E
cov
(U
L
(t)ρ
L
U
†
L
(t)) = E
cov
(ρ
L
) ∀t ∈ ,
(424)
where in the last line, we have used U
L
(t) =
L
for all t ∈ . Hence differentiating w.r.t. t on both sides on
Eq. (424), we achieve the Page-Wooters condition,
[
ˆ
H, E
cov
(ρ
L
)] = 0 ∀t ∈ . (425)
Thirdly, our clock state must, at least approximately, satisfy Eq. (421). We have already calculated the state
of E
cov
(ρ
L
) when the clocks were projected onto the time basis {|θ
k
ihθ
k
|}
d
C
−1
k=0
in Section B. Now, we want to
perform a similar calculation but for when the clock is projected onto the state of the clock at time τ. In
the case of one Quasi-Ideal clock, this corresponds to covariant positive-operator valued measure, which up
to a vanishing error in the large d
C
/σ and σ limit, are {
ˆ
P
QI
(τ) := U
C
(τ)|ψ
nor
(k
0
1
)ihψ
nor
(k
0
1
)|U
C
(τ)
†
}
τ∈[0,T
0
]
.
Proceeding analogously to Section B, but this time not applying the error map E
j
, we find
ρ
P
(τ) := tr
C
h
ˆ
P
QI
(τ)E
cov
(ρ
L
)
ˆ
P
QI
(τ)
i
=U
⊗K†
P
(τ)
ˆ
E
0
τ
(ρ
L
) + E(ρ
L
)
U
⊗K
P
(τ), (426)
where
ˆ
E
0
τ
(·) :=
d
P
−1
X
q,q
0
=0
d
L
−1
X
n,n
0
=0
F
Q
(τ)
F
0
(τ)
e
2πiτ Q/T
0
− 1
E
q,q
0
,n,n
0
(·)|qihq
0
|, (427)
with E
q,q
0
,n,n
0
give by Eq. (36) and F
Q
(τ) given by
F
Q
(τ) :=
1
T
0
Z
T
0
0
dte
−iωQt
hψ
nor
(k
0
)|U
†
C
(τ)ρ
C,1
(t)U
C
(τ) |ψ
nor
(k
0
)i (428)
=
1
T
0
Z
T
0
0
dte
−i2πQt/T
0
|hψ
nor
(k
0
)|U
C
(t − τ) |ψ
nor
(k
0
)i|
2
, (429)
where |ψ
nor
(k
0
)i is the Quasi-Ideal clock described previously in the manuscript. Note that Eq. (426) is
analogous to Eq. (72) but with a trivial error map E
j
= I (no clock errors), ˜ρ
L
= U
L
(t
α
)ρ
L
U
†
L
(t
α
) = ρ
L
(trivial
logical group representation), and a time τ rather than the discrete times t
α
:= k
α
T
0
d
C
(since we are not projecting
onto the time basis any-more). One can bound Eqs. (429) and (427), analogously to the calculations in Sections
D C. We will not perform such calculations here for brevity, but one finds that F
Q
(τ)/F
0
(τ) ≈ e
−2πiτQ/T
0
where
the approximation becomes exact in the large d
C
limit. So one can see that the quantity in square brackets in
Eq. (427) vanishes in said limit. Hence for large d
C
, from Eq. (426)
tr
C
h
ˆ
P
QI
(τ)E
cov
(ρ
L
)
ˆ
P
QI
(τ)
i
≈ U
⊗K†
Co
E(ρ
L
)U
⊗K
Co
(τ) = e
−iτ
ˆ
H
Co
E(ρ
L
)e
iτ
ˆ
H
Co
∀τ ∈ , (430)
where the approximate equality becomes exact in the large d
C
limit. This is an approximate Page-Wooters
condition (Eq. (420)).
Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 45