Hamiltonians for one-way quantum repeaters
Filippo M. Miatto
1,2
, Michael Epping
1
, and Norbert L
¨
utkenhaus
1
1
Institute for Quantum Computing and Department of Physics and Astronomy, University of Waterloo, 200 University Ave. W, N2L 3G1
Waterloo, Ontario, Canada
2
T
´
el
´
ecom ParisTech, LTCI, Universit
´
e Paris Saclay, 46 Rue Barrault, 75013 Paris, France
June 29, 2018
Quantum information degrades over distance
due to the unavoidable imperfections of the
transmission channels, with loss as the leading
factor. This simple fact hinders quantum com-
munication, as it relies on propagating quantum
systems. A solution to this issue is to introduce
quantum repeaters at regular intervals along a
lossy channel, to revive the quantum signal. In
this work we study unitary one-way quantum
repeaters, which do not need to perform mea-
surements and do not require quantum memo-
ries, and are therefore considerably simpler than
other schemes. We introduce and analyze two
methods to construct Hamiltonians that gener-
ate a repeater interaction that can beat the fun-
damental repeaterless key rate bound even in
the presence of an additional coupling loss, with
signals that contain only a handful of photons.
The natural evolution of this work will be to ap-
proximate a repeater interaction by combining
simple optical elements.
1 Introduction
Quantum physics plays an increasingly important role
in the development of future technologies. The trans-
mission of information is an example of this trend,
where the benefit of exploiting quantum effects has mo-
tivated several ideas, such as quantum money, quan-
tum key distribution (QKD), quantum fingerprinting,
quantum digital signatures, and several others [3, 2, 7,
12, 11, 19]. Eventually, we will build a quantum in-
ternet: a communication infrastructure that will enable
distributed quantum information tasks on a global scale
[13, 17].
It would be very convenient if we could rely as much
as possible on the current infrastructure, such as the op-
tical fiber network, to propagate quantum states of light
as the main carrier of quantum information. Light is
cheap, fast, and it does not require vacuum or low tem-
peratures. However, at the present moment, we cannot
send photons reliably to distant parties: optical fibers
are “leaky” and deteriorate the quantum properties of
the signal. The fact that quantum information can-
not travel far enough constitutes one of the outstanding
challenges in the field [30], and it is the main focus of
our work.
Note that as optical modes are like harmonic oscil-
lators, photon loss is equivalent to the relaxation of
a harmonic oscillator towards equilibrium [20]. This
makes our findings applicable to any domain in which
one needs to fight relaxation (e.g. fields in a cavity,
nanoscale resonators, etc).
2 Preliminaries
We know from the theory of quantum information that
we can “stretch” the distance between two entangled
systems by a process that involves entanglement swap-
ping, which is a core component of the first genera-
tion of quantum repeaters [5]. The newest generation
of repeater architectures relies instead on quantum er-
ror correction, which allows for “one-way” functionality
by removing the need to broadcast measurement results
backwards and forwards [14, 23, 24, 25]. It is even pos-
sible to omit measurements altogether: just as one can
implement quantum error correction by measuring the
error syndrome and then applying the appropriate cor-
rection, it is also possible to do so coherently by using
an ancillary system initialized in a fiducial state |0i to
store the index i of each syndrome and of the corre-
sponding Kraus operator:
U
rep
=
syndromes
X
i=1
R
(i)
⊗ |iih0|. (1)
In this expression, R
(i)
= U
i
P
(i)
where P
(i)
is the pro-
jector onto the i-th syndrome space, and U
i
is the uni-
tary that brings the system from the i-th syndrome
space back into codespace. Note that expression (1)
is intended as an isometry. The need for an ancilla is
a simple consequence of the Stinespring dilation theo-
rem [28], according to which any CPTP channel can be
Accepted in Quantum 2018-06-27, click title to verify 1
arXiv:1710.03063v3 [quant-ph] 28 Jun 2018
described through a unitary interaction with a larger
system.
In this work we present two recipes to build families of
Hamiltonians that generate such unitary one-way quan-
tum repeaters.
In order to evaluate the performance of our repeaters,
we will compare the entanglement between two systems
(one of which is travelling in a lossy channel interspersed
with repeaters), to the repeaterless key rate bound
[33, 29], see also [36] for results in the non-asymptotic
regime. In particular, [29] found that the tightest upper
bound of the QKD rate per optical mode that can be
achieved using only a lossy channel with transmissivity
η is equal to:
R(η) = log
2
1
1 − η
. (2)
What we found is that we can beat this repeaterless
bound even if each repeater introduces a mild coupling
loss between the optical fibre and the repeater device.
Furthermore, the optimized separation between the re-
peater stations turns out to be rather reasonable, on
the order of a few kilometers.
The complexity of our repeater Hamiltonians depends
on the code space that they protect and on the fiducial
ancilla states. Consequently, if we pick them wisely, we
have the potential to obtain simple Hamiltonians that
can be implemented by combining a few known optical
elements, such as single photon sources, beamsplitters,
squeezers, four wave mixers, etc. Such a quest for sim-
plification is outside the scope of the present work, but
it is part of our ongoing research.
3 Working principles of a one-way quan-
tum repeater
Our task is to protect the quantum information that is
encoded in a photonic system. Although in this par-
ticular context we lack the typical qubit structure of
QEC codes, if we treat photon loss as an “error”, we
can still apply QEC successfully to our problem. Note
that the set of correctable errors cannot include all of
the possible loss operators (e.g. including the loss of all
photons), but rather a few of them. How many depends
on the particular code in question.
Furthermore, in practice the correction process is not
going to be perfect and it can itself introduce errors and
noise. This enforces an optimal separation between the
repeaters, as well as other important considerations re-
garding the noise parameters. In this work we describe
an ideal repeater, i.e. one that does not introduce any
additional noise. In this we consider the method of ex-
act error correction [32, 15] to illustrate our approach of
Hamiltonian Quantum Repeaters. It will be interesting
to extend this line of consideration also to approximate
error correction [34, 4, 27, 21] though we leave that to
upcoming work. For this reason it is sufficient for us
to consider exact and not approximate quantum error
correction.
In all that follows, we will seek the protection of a
two-dimensional subspace within each signal from the
loss of a single photon. If this is done frequently enough,
and if we separate the repeaters such that the single-
photon loss is the dominant error, we can witness the
benefits of a repeater action. All that we need from
QEC is encapsulated in the QEC condition in the form
P E
†
i
E
j
P = δ
ij
c
i
P where c
i
is the probability of the i-
th error, P is the projector onto the protected subspace
(called the “code space”) and {E
i
} are the correctable
effect operators of the undesired interaction, which map
P to the orthogonal spaces P
(i)
= E
i
P E
†
i
/
1
2
Tr(E
i
P E
†
i
)
[28]. The fact that such spaces are orthogonal to each
other and to P means that the quantum information
has not left the system, it is just in a different subspace
and it can be brought back.
To understand what the error operators are, we need
a description of the lossy channel. There are three main
ways to describe a single-mode lossy channel of trans-
missivity η:
(i) As a quantum channel in Kraus form (also e.g. in
[10]):
L
η
[ρ] =
∞
X
k=0
(1 − η)
k
k!
√
η
N
a
k
ρa
†
k
√
η
N
, (3)
where N = a
†
a is the number operator and a is the an-
nihilation operator. The Kraus operators are therefore
A
k
=
(1−η)
k/2
√
k!
√
η
N
a
k
.
(ii) As the action of a beamsplitter in each mode (ef-
fectively the Stinespring representation of the channel):
L
η
[ρ] = Tr
B
h
U
AB
(ρ
A
⊗ |0ih0|
B
)U
†
AB
i
, (4)
where U
AB
= exp[iφ(a
†
b + ab
†
)] is a beamsplitter be-
tween the signal and a local environment in the vacuum
state.
(iii) Through the master equation [20]:
˙ρ =
γ
2
2aρa
†
− {N, ρ}
. (5)
The connection between the three representations is
the relation η = cos(φ)
2
= exp(−γt). This chan-
nel generalizes to an M-modes lossy channel as a sim-
ple tensor product of M channels. We will indicate
the Kraus operator corresponding to losing k pho-
tons from mode i and none from the other modes as
A
(i)
k
=
(1−η)
k/2
√
k!
√
η
N
a
k
i
(where N =
P
i
a
†
i
a
i
). As
Accepted in Quantum 2018-06-27, click title to verify 2
we concentrate on correcting single photon losses, we
need the single-photon loss operator in the i-th mode:
E
i
:= A
(i)
1
=
√
1 − η
√
η
N
a
i
. If we replace these oper-
ators into the error-correction conditions and take the
trace of both sides, we obtain that the probability of
the i-th error (i ≥ 1) is:
c
i
=
1 − η
d
Tr
P a
†
i
a
i
η
N−1
(6)
where we used the operator identity f(N)a = af(N −1)
and d is the dimension of the protected subspace, which
comes from Tr(P ) = d. In our case, as we are protect-
ing a two-dimensional subspace, d = 2. Eq. (6) means
that, understandably, the more photons a code uses the
easier it is to lose one, which gives rise to an interesting
trade-off between the probability to lose a photon and
the ability to protect from photon loss. From inspec-
tion, it appears that a good choice is to make sure we
can correct for a single loss in each mode using as few
photons as possible in the code.
Note that in case of no loss, we have a non-trivial
evolution: the no-loss operator is E
0
= A
(i)
0
=
√
η
N
,
which is not the identity operator. This means that if
the code space P is not an eigenspace of the total photon
number operator, it will evolve non-trivially even if no
losses occur.
As anticipated, a unitary operation is sufficient to
rotate P
(i)
back to P , but there is a catch: the operation
that we should apply to the system depends on the error
syndrome. As we have more than one subspace that
should rotate back to P , this is not a unitary operation
unless we introduce an ancillary system, as in Eq. (1).
So the quantum repeater must produce a fresh local
ancilla, it must have the incoming system interact with
it, and then it can discard the used ancilla. The output
of the repeater is the recovered system (unless more
losses occurred than the code can handle). Note that
in principle we could measure the used ancilla to detect
the syndrome, but the repeater would work equally well
if we did not.
To summarize: we choose to put our quantum in-
formation in protected subspaces that do not become
entangled with the environment upon photon loss, but
rather they rotate to an orthogonal subspace. This en-
ables us to rotate them back without involving mea-
surements if we supply this operation with a ‘catalytic’
ancilla which we can discard at the end of the operation.
4 Repeater Hamiltonians
In this section we present two families of Hamiltonians
that generate a repeater unitary. Each Hamiltonian will
depend on the code space P as well as the initial ancilla
state, so one can view them as a resource, which can
be chosen wisely to yield simpler Hamiltonians in the
sense mentioned in the introduction. Note that a well-
designed repeater action will not let the uncorrectable
error spaces to eventually couple to code space (for in-
stance after a series of several losses and repeaters). So
the action of U
rep
on spaces that will never interfere
with code space can be thought of as another resource,
as we can use the freedom to choose this otherwise irrel-
evant action to simplify the Hamiltonian. We present
both architectures because although they are related by
a system-ancilla swap after the repeater, the Hamilto-
nians are rather different.
4.1 Direct unitary
U
direct
| i
|
(i)
i
|'i
|'
(i)
i
Figure 1: In the direct architecture, the system interacts with a
local ancilla, it “transfers the error” and then proceeds onwards.
Measuring the used ancilla only gives us information about the
loss, which does not influence the performance of the repeater.
The first architecture (Fig. 1) that we introduce is
that of a repeater which lets the input system “transfer”
the loss errors to the ancilla. Once this operation is
complete, we have essentially recovered the initial state.
In this construction the initial state of the ancilla is free
to live in a K-dimensional space, and one is free to pick
any value for K ≥ 1. The Hamiltonian is
H
direct
=
K
X
k=1
M
X
i=1
Ψ
(i)
0k
+ Ψ
(i)
1k
, (7)
where M is the number of modes, Ψ
(i)
jk
is a projec-
tor onto |Ψ
(i)
jk
i =
1
√
2
(|ji
s
|k
(i)
i
a
− |j
(i)
i
s
|ki
a
, and the
subscripts s and a indicate the system and the ancilla.
Here we must comply with the error spaces that are im-
posed by the loss operators, i.e. for the system we have
|j
(i)
i = E
i
|ji/
q
hj|E
†
i
E
i
|ji, however we have complete
freedom (even in terms of number of modes) to pick the
ancilla states |ki
a
and |k
(i)
i
a
as long as all together they
form an orthonormal set. As this Hamiltonian acts as
the identity operator on the space of non-correctable er-
rors (two or more photon lost), it automatically avoids
Accepted in Quantum 2018-06-27, click title to verify 3
these error spaces to couple back to the code space. It
is possible to simplify the direct Hamiltonian by min-
imizing the dimension of the initial ancilla space (i.e.
K = 1):
H
0
direct
=
M
X
i=1
Ψ
(i)
0
+ Ψ
(i)
1
, (8)
where Ψ
(i)
j
projects onto |Ψ
(i)
j
i =
1
√
2
(|ji
s
|0
(i)
i
a
−
|j
(i)
i
s
|0i
a
). Note that in the construction of H
direct
(as
opposed to the construction of H
swap
in the next sub-
section) we excluded the no-loss (i = 0) element, as the
repeater should act as the identity if no error occurs.
We remark that in case a code had a non-trivial evolu-
tion under zero loss, one might need to include a term
to act in this eventuality.
The fact that the Hamiltonian is a projector simpli-
fies enormously the computation of the unitary, which
implements the correction at time t = π:
U
direct
= exp (iπH
direct
) = −2H
direct
. (9)
It is a simple computation to verify that U
direct
indeed
corrects an error by “transferring” it to the ancilla:
U
direct
|ψ
(i)
i
s
⊗ |ϕi
a
= |ψi
s
⊗ |ϕ
(i)
i
a
, (10)
where |ψ
(i)
i
s
= α|0
(i)
i
s
+β|1
(i)
i
s
is the state of the sys-
tem after the error i, and |ϕi
a
=
P
k
γ
k
|ki
a
is the initial
ancilla state (|ϕi
a
is fixed, for example the vacuum).
4.2 SWAP unitary
U
| i
U
swap
|
(i)
i
|'i
|'
(i)
i
Figure 2: In the “swap” repeater configuration, the system and
the ancilla remain in their respective spaces (the error space
P
(i)
and code space), but they swap states: the system acquires
the state of the ancilla and the ancilla acquires the state of the
system and then proceeds onwards.
The second architecture (Fig. 2) is a repeater that
“swaps” the state of the input system with the state
of an ancilla prepared in code space, while leaving the
errors behind. More specifically, the SWAP operation
occurs between P (where the ancilla needs to be initial-
ized) and the space where the system currently lives,
which is P if no error occurred and it is P
(i)
if a single
photon loss from the i-th mode occurred. The Hamil-
tonian of this repeater is:
H
swap
=
M
X
i=0
Φ
(i)
(11)
where Φ
(i)
projects onto |Φ
(i)
i =
1
√
2
(|0
(i)
i
s
|1i
a
−
|1
(i)
i
s
|0i
a
). Similar to H
direct
, this Hamiltonian also
automatically avoids the uncorrectable error spaces to
couple back to the code space. Note that now in case of
no error the repeater still needs to swap, so we have to
include the i = 0 case. The correction unitary is again
given by
U
swap
= exp (iπH
swap
) . (12)
And it is again a simple computation to verify that the
unitary corrects by swapping (compare with Eq. (10)):
U
swap
|ψ
(i)
i
s
⊗ |ϕi
a
= |ϕ
(i)
i
s
⊗ |ψi
a
, (13)
where |ψ
(i)
i
s
= α|0
(i)
i
s
+ β|1
(i)
i
s
is the state of the
system after the error i, and |ϕi
a
= γ
0
|0i
a
+ γ
1
|1i
a
is
the initial ancilla state in code space.
5 Beating the repeaterless bound
In this section we evaluate how well our repeaters pro-
tect the quantum information along a lossy channel.
Our goal is to beat the repeaterless bound asymptot-
ically, which in terms of distance x with α dB/Km of
loss is R(x) = −log
2
(1−10
−αx/10
). For a telecom fibre,
α ≈ 0.2 dB/Km.
We present a single-mode code to show the effect
of the uncorrected no-loss operator
√
η
N
and we will
compare a four-photon two-mode code [8] with a three-
photon three-mode code [35] to show the advantage of
using fewer photons. We supply the logical codewords
that span P explicitly in terms of the number states in
Fock space:
1. single-mode code: Logical states |0
L
i = |1i and
|1
L
i = |3i.
2. two-mode code: Logical states |0
L
i =
|4,0i+|0,4i
√
2
and
|1
L
i = |2, 2i.
3. three-mode code: Logical states |0
L
i = |1, 1, 1i and
|1
L
i =
|0,0,3i+|0,3,0i+|3,0,0i
√
3
.
Accepted in Quantum 2018-06-27, click title to verify 4
These are just simple didactical examples, for an excel-
lent in-depth presentation of bosonic codes see [22, 16,
1].
We initialize an entangled state |ψi
AB
=
1
√
2
(|0, 0
L
i+
|1, 1
L
i) using the logical states in one of the example
codes presented above, and we propagate the B system
through a lossy channel with repeaters separated by a
distance L. For codes that are eigenstates of the to-
tal photon number, after one segment comprising loss
and repeater action, we find the original state in code
space with probability p
s
(defined below). Since the
uncorrectable error spaces do not couple back to the
code space, we can extract p
n
s
secret bits of key after n
segments, which means that the condition to beat the
repeaterless bound asymptotically is that the logarith-
mic slope of the key rate be larger than the logarithmic
slope of the repeaterless bound: p
s
> η
s
, where η
s
is the
transmissivity of a segment. The same condition also
appears from different arguments, see [26, 9].
We also investigate the stability if we have coupling
losses between an optical fibre and a repeater station.
To do this, we consider the total loss of a segment to
be composed of two parts: η = η
c
η
s
, where η
c
is the
coupling efficiency of the fibre into the repeater and
η
s
is the regular loss of the channel. As the success
probability p
s
depends on the code space P , on the
repeater separation L and on the coupling efficiency η
c
,
we can find for which values of these quantities we beat
the repeaterless bound (see Fig. 3). For the two-mode
code, we have p
s
= η
4
+ 4(1 −η)η
3
, for the three-mode
code we have p
s
= η
3
+3(1−η)η
2
, where η = η
c
η
s
. Here
we assume that a repeater does not couple the error
spaces with the code space in undesired ways (e.g. by
adding more photons than necessary, which may lead to
the system entering back into code space upon further
loss).
For codes that are not so well-behaved (such as the
single-mode code in the examples), things are more
complicated as the code spaces become distorted. In
the case of our single-mode code, we need to com-
pute explicitly the action of the lossy channel L
η
from
Eq. (3) (where the total loss η = η
s
η
c
is composed
of the segment loss and the coupling loss) followed by
the channel R that maps the signal from the input
to the output of the repeater, which we can describe
with M + 1 Kraus operators: R
(0)
= ( −
P
i
P
(i)
) and
R
(i)
= |0ih0
(i)
| + |1ih1
(i)
| for 1 ≤ i ≤ M (see Eq. (1)).
Note that we do not correct the distortion on the zero-
photon loss (for some codes it can be possible, especially
for small η [22]). We find:
R ◦ L
η
|1ih1| = |1ih1|
R ◦ L
η
|1ih3| = (η
2
+ (1 − η)
√
3η)|1ih3|
R ◦ L
η
|3ih1| = (η
2
+ (1 − η)
√
3η)|3ih1|
R ◦ L
η
|3ih3| = (3η
2
− 2η
3
)|3ih3| + (1 − 3η
2
+ 2η
3
)|1ih1|
This mapping allows us to compute the n-fold applica-
tion of R ◦ L
η
. After n segments, the state is
ψ
(n)
AB
=
1
2
|0, 1ih0, 1| +
1
2
(1 − (3η
2
+ 2η
3
)
n
)|1, 1ih1, 1|
1
2
(η
2
+ (1 − η)
√
3η)
n
(|0, 1ih1, 3| + |1, 3ih0, 1|)
+
1
2
(3η
2
− 2η
3
)
n
|1, 3ih1, 3| (14)
If we perform the six-states protocol [6, 18] with this
state, we can compute the key rate by following the
results in appendix A of [31]. We resort to the six-
state protocol because the distortion causes errors, and
thus the simple consideration of success probabilities
will not be sufficient to guarantee the cross-over. What
we find is that the region of parameter space that al-
lows us to beat the repeaterless bound at some distance
(note: not asymptotically) is very small, but it exists
(see Fig. 3). We remark that it might be possible to
increase the single-mode key rate by other means (cor-
recting the zero-loss effect, noisy pre-processing, two-
way codes, etc...).
We can finally plot for each code the parameter re-
gion that allows us to beat the repeaterless bound (non-
asymptotically for the single-mode code and asymptot-
ically for the two- and three-mode codes) in a telecom
fiber (Fig. 3).
6 Considerations
We can see that unitary one-way quantum repeaters
can beat the repeaterless bound asymptotically, which
means that the combination of loss and repeater sta-
tions effectively turns the lossy bosonic channel into a
different channel with lower effective loss, to the point
that in a one-way exchange of quantum states we can
extract more secret key than one could obtain from a
pure-loss channel. In Fig. 4 we include an example of
the key rate over distance obtained with the three-mode
code over a range of 600 Km including coupling loss, and
compare it to the repeaterless bound.
In this work we have assumed that the repeaters are
not introducing any additional noise, which is an unre-
alistic assumption. We address the limitations due to
added noise in our upcoming work [9].
Accepted in Quantum 2018-06-27, click title to verify 5
Single-mode
Two-mode
Three-mode
0 2 4 6 8 10 12 14
0.88
0.90
0.92
0.94
0.96
0.98
1.00
Repeater separation (km)
η
c
Figure 3: The shaded regions indicate for which coupling effi-
ciency η
c
and repeater separation L we can extract more key
than the repeaterless bound allows (non-asymptotically for the
single-mode code and asymptotically for the two- and the three-
mode code). Notably, the three-mode code can withstand up
to about 11% of additional coupling loss between a fibre and a
repeater station.
Repeaterless bound
η
c
= 99% (225 m separation)
η
c
= 95% (1.3 km separation)
η
c
= 90% (3.2 km separation)
0 100 200 300 400 500 600
10
-10
10
-8
10
-6
10
-4
0.01
1
10
-10
10
-8
10
-6
10
-4
0.01
1
Distance (km)
Key Rate (bits)
Figure 4: Example key rates per mode for the three-mode code.
For each curve we have optimized the repeater separation (in
parenthesis). Note that even with 10% of coupling loss between
fibres and repeaters, the three-mode code can still beat the
repeaterless bound starting around the 400 km mark.
This result shows that it is possible to build unsu-
pervised, one-way unitary stations that act as quantum
repeaters, which could be placed conveniently every few
kilometers along the optical fibre network, allowing us
to perform QKD and other distributed quantum infor-
mation tasks at much increased distances than what we
can currently achieve.
Furthermore, some codes show a remarkable re-
silience to the additional coupling loss between fibres
and repeater stations, which is encouraging from a tech-
nological point of view.
The actual physical realization of a repeater station
is a very challenging problem. There are many pos-
sible families of bosonic codes and within each family
there are many possible choices of specific codes. For
each code there is in turn a continuum of Hamiltoni-
ans that depend on the initial ancilla state. So there
are innumerable Hamiltonians that one could use, but
we do not yet understand how to determine if a given
Hamiltonian can be implemented (or approximated to a
satisfactory degree) by arranging a reasonable number
of optical components. Understanding thed connection
between a Hamiltonian and its implementation is part
of our ongoing work.
7 Conclusions
We introduced two families of Hamiltonians that gen-
erate a quantum repeater interaction which protects
one qubit of information propagating in a lossy bosonic
channel. We presented a few examples to show that it
is possible to overcome the repeaterless bound even in
the presence of a mild coupling loss between the lossy
channel and the repeater stations. Our construction is
general and it gives the freedom to choose a code space
and an initial ancilla state, which translates into the po-
tential to discover particularly convenient Hamiltonians
that can be decomposed into a feasible arrangement of
a few simple optical elements.
8 Acknowledgements
The authors thank Marco Piani, Ryo Namiki and John
Jeffers for helpful discussions. This work was supported
by NSERC Discovery Grant, Industry Canada and the
Army Research Lab.
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