Macroscopic superpositions require tremendous
measurement devices
M. Skotiniotis
†1
, W. Dür
2
, and P. Sekatski
†2
1
Física Teòrica: Informació i Fenòmens Quàntics, Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellatera (Barcelona)
Spain
2
Institut für Theoretische Physik, Universität Innsbruck, Technikerstr. 21a, A-6020 Innsbruck, Austria
November 15, 2017
We consider fundamental limits on the de-
tectable size of macroscopic quantum super-
positions. We argue that a full quantum me-
chanical treatment of system plus measure-
ment device is required, and that a (classical)
reference frame for phase or direction needs
to be established to certify the quantum state.
When taking the size of such a classical ref-
erence frame into account, we show that to
reliably distinguish a quantum superposition
state from an incoherent mixture requires a
measurement device that is quadratically big-
ger than the superposition state. Whereas for
moderate system sizes such as generated in
previous experiments this is not a stringent re-
striction, for macroscopic superpositions of the
size of a cat the required effort quickly becomes
intractable, requiring measurement devices of
the size of the Earth. We illustrate our re-
sults using macroscopic superposition states of
photons, spins, and position. Finally, we also
show how this limitation can be circumvented
by dealing with superpositions in relative de-
grees of freedom.
1 Introduction
While natural sciences exist in some form or an-
other for more than two millennia, quantum mechan-
ics was only formulated about a century ago. It took
about fifty years to go from the basic formulation of
quantum mechanics to its most radical philosophical
implication—the Bell theorem [1]. The reason both
things took so long is quite simple: the world we per-
ceive as human beings, outside our laboratories, is
classical. There is essentially nothing quantum about
it.
Today we know that there is nothing surprising
about the fact that quantum effects are hidden from
our everyday experience. Unavoidable interaction
with the environment (decoherence) makes things
†
These authors contributed equally to this work
look classical [2]. There is also growing evidence that
the usage of imperfect detectors (such as our sense-
organs) often forbids the observation of quantum phe-
nomena [3–6]. In fact, whenever a superposition state
is deemed more macroscopic (accordingly to some def-
inition) than another, then it is also more fragile with
respect to one of the above, see e.g., [7]. Neverthe-
less, decoherence and detector resolution are technical
problems that can be overcome by working harder and
harder in the lab, where we do observe quantum phe-
nomena and violation of Bell inequalities for bigger
and bigger systems [8–19]. But how far can we po-
tentially go? Is there hope to observe a macroscopic
superposition of an everyday classical object in some
future lab?
Here we attempt to answer this question, and pro-
vide fundamental limits on the resources required to
certify a macroscopic quantum superposition state.
We concentrate on the measurement, i.e., the cer-
tification of a macroscopic quantum superposition
state, although similar arguments may be used for
the preparation process of such states as well. Our
approach is based on the observation that the descrip-
tion of any physical system in quantum mechanics
in terms of a pure quantum state implicitly assumes
the existence of a suitable, ideal background reference
frame (RF). In reality such an RF is itself a quantum
system of some limited size and can only encode a
finite amount of “unspeakable" information [20]. In-
corporating the RF into the quantum formalism the
joint system of macroscopic superposition and RF can
be treated as a closed quantum system whose phys-
ical laws are invariant with respect to fundamental
symmetries of nature. We exploit this invariance to
show that, within some simple model, in order to ob-
serve a macroscopic superposition of the size of a cat
one requires an RF that is of the size of the whole
Earth. However, this restriction on the size of the RF
can be circumvented in the case of macroscopic su-
perpositions of relative degrees of freedom, for which
no fundamental conservation laws exist.
Accepted in Quantum 2017-10-31, click title to verify 1
arXiv:1705.07053v3 [quant-ph] 14 Nov 2017
2 Background and Motivation
We begin by introducing the macroscopic quantum
states that will occupy us for the remainder of this
work in Secs. 2.1 and 2.2, and in Sec. 2.3 we for-
mally introduce the task of discriminating a macro-
scopic superposition from the corresponding mixture.
Next, we consider constraints on measurements im-
posed by the symmetry of physical laws and exem-
plify how such constraints arise and are circumvented
in a familiar experimental situation from quantum
optics (Sec. 2.4). The general treatment of measure-
ments under symmetry, as well as how the constraints
imposed by symmetries can be relaxed by appropriate
systems that act as suitable RFs, is given in Sec. 2.5.
Finally, we provide a working definition of the mea-
surement devices used to certify macroscopic quan-
tum states in Sec. 2.6.
2.1 Macroscopic quantum superpositions
A definitive test of macroscopic quantum behaviour
is to demonstrate the trade-mark signature of quan-
tum mechanics—quantum interference on the macro-
scopic scale [17–19, 21]. This amounts to being able to
distinguish between coherent and incoherent superpo-
sitions of macroscopically distinct states [22, 23] and
forms the basis for the remainder of this work. Specif-
ically, our goal is to distinguish the macroscopic co-
herent superposition
|Ψi =
1
√
2
|ψ
0
i +
1
√
2
|ψ
1
i, (1)
from the corresponding incoherent mixture
Ψ =
1
2
|ψ
0
ihψ
0
| +
1
2
|ψ
1
ihψ
1
|. (2)
Here
ψ
0(1)
are macroscopically distinct eigenstates
of a classical observable, or narrow superpositions of
neighbouring eigenstates. By macroscopically distinct
we mean that the difference between the correspond-
ing eigenvalues of
ψ
0(1)
is large. In this work we
shall consider the following three types of macroscopic
superposition states:
Total photon number (or energy) : the relevant
macroscopic superposition can be taken to be
|Ψi =
1
√
2
(|0i + |Ni) , (3)
where |0i and |Ni are single-mode Fock states
that are eigenstates of the photon number oper-
ator
ˆ
N with corresponding eigenvalues 0 and N.
Total spin : the macroscopic superposition state
|Ψi =
1
√
2
|↑i
⊗N
+ |↓i
⊗N
, (4)
is the GHZ state of N spin-
1
2
particles (or qubits),
where |↑i, |↓i are eigenstates of σ
z
with eigenval-
ues ±
1
2
respectively.
Delocalized object : the superposition state in
question can be taken to be
|Ψi =
1
√
2
−
L
2
+
L
2
, (5)
i.e., a superposition of position eigenstates
ˆ
X
±
L
2
= ±
L
2
±
L
2
for a particle of mass m.
This can be a quasi-particle describing the center
of mass of a composite system. The states
±
L
2
need not be eigenstates of the position operation
but can be, for instance, Gaussian wave packets
of width σ. However, for our purposes the width
of an individual wave-packet σ can be ignored as
long as it is much smaller that the separation,
σ L.
2.2 Macroscopic quantum states
More generally, one is also interested in states that do
not have a two-component superposition structure of
Eq. (1). Such states
|Ψi =
X
n
ψ
n
|ni, (6)
with |ni being eigenstates of photon number, total
spin component or position of a particle, can also
exhibit macroscopic quantum coherence ψ
n
ψ
m
|nihm|
between far away components |n − m| ≈ N if the
variance of ψ
2
n
is large enough
1
. Notice that a large
variance and the presence of far-away coherence is a
feature that renders such states macroscopic accord-
ing to different measures, see [21]. In order to witness
such macroscopic coherences one has to be able to
distinguish |Ψi from any semi-classical state Ψ that
does not contain such coherences. An important class
of semi-classical states is given by
Ψ =
X
n,m
ψ
n
ψ
m
c
|n−m|
|nihm|, (7)
with lim
N→∞
c
N
β
0
= 0 for all β
0
> β. Such states
do not contain coherences on the scale |n −m| > N
β
.
Hence if there is no measurement that allows to dis-
tinguish |Ψi from Ψ, there is also no way to detect
coherences on the scale larger then N
β
. Note that for
β ≤ 1/2 the state is typically not considered to be
macroscopic quantum by many measures [21].
It turns out that the requirements on the measur-
ing device in order to witness macroscopic coherence
are the same whether one considers the macroscopic
superpositions of Eq. (1), or the more general macro-
scopic quantum states of Eq. (6). However, in the
interest of delivering the main message of our work as
clearly—and devoid of cumbersome technicalities—as
possible we will first concentrate on macroscopic su-
perpositions, and provide the details for the more gen-
eral macroscopic states in Sec. 3.4.
1
Without loss of generality we assume the coefficients ψ
n
to
be real.
Accepted in Quantum 2017-10-31, click title to verify 2
2.3 The Task
Our goal is to determine how the size of a macro-
scopic superposition state determines the correspond-
ing size of the classical measuring device required to
distinguish such states from their respective incoher-
ent mixtures. This problem corresponds to the fol-
lowing quantum hypothesis testing task [24–26]. An
experimentalist is given a single system known to be
in one of two possible states with equal prior probabil-
ity
2
; the superposition state, Eq. (1), or the state of
Eq. (2). The experimentalist’s goal is to discriminate
between the two states with the maximum probability
of success, given by P =
1
2
+ t, where
t ≡
1
2
max
{M}
Tr [M (|ΨihΨ| − Ψ)] =
1
2
|||ΨihΨ| − Ψ ||
(8)
is the trace distance between the states of Eqs. (1, 2),
and the maximization is over all possible measure-
ments. Formally speaking any resolution of the
identity operator constitutes a valid measurement in
quantum mechanics. However, measurement is a
physical process performed by a physical system that
should, in principle, also be treated quantum mechan-
ically.
In practice one is usually given a device that pre-
pares the hypothetical state rather then a single copy
of it. Hence one is able to subsequently repeat the
preparation and the measurement n times in order to
confirm if the device is preparing a superposition or
a mixture
3
. In the limit of asymptotically many rep-
etitions the probability to obtain a wrong conclusion
after n repetitions is known to follow the Chernoff
bound [27]
P
(n)
Err
∼ e
−nξ
CB
(9)
where ξ
CB
= −log
min
0≤s≤1
P
i
p
Ψ
(i)
s
p
|Ψi
(i)
(1−s)
and p
Ψ
(i) and p
|Ψi
(i) denote the conditional proba-
bilities of obtaining outcome i in a single measure-
ment for the mixture and the superposition state re-
spectively. This quantity can be related to the trace
distance via the quantum Chernoff bound [28]
P
(n)
Err
& (1 − t)
n
. (10)
Hence, the trace distance of Eq. (8) is also directly
useful for the case where one repeats the measure-
ment many times, and allows us to establish the rela-
tion between the size of the superpositions, the size of
the measurement device, and the required number of
repetitions. Note that the quantum Chernoff bound
can also be used to obtain an upper bound on the
2
We note that this represents the worst case scenario.
3
We assume that the RF resets to its initial state between
each run, or that one is provided with a fresh RF for each
measurement. In reality the repeated use of a single RF will
progressively degrade its state. The study of this degradation
is beyond the scope of this paper.
error P
(n)
Err
. (1 −t
2
)
n/2
. However, this upper bound
may require collective measurements on the n copies
of the states.
2.4 Experimental example
In order to illustrate how the notion of a RF enters
into the description of a measuring device we first
show how such a RF is used in a simple example from
quantum optics. This example will serve as a starting
point from which we will provide our working defini-
tion of what we mean by a classical measuring device
which we will explicitly state in Sec. 2.6
Suppose that a two level atom is prepared in the
state |ψi =
1
√
2
(|0i + |1i), and we wish to perform a
measurement in the x-basis, {|±i =
1
√
2
(|0i ± |1i)}.
What we have at our disposal are photon counting
measurements, i.e., z-basis measurements. Thus, in
order to perform an x-basis measurement we first need
to apply a π/4-rotation about the y-axis to the state of
the two-level atom followed by z-basis measurement.
Such a rotation can be achieved via the interaction
of the atom with a laser pulse in a coherent state |αi.
In the rotating wave approximation, the interaction is
described by the Jaynes-Cummings Hamiltonian [29]
and results in the unitary transformation
U = e
γ(σ
+
a−σ
−
a
†
)
, (11)
where σ
±
are the raising/lowering operators acting
on the two-level atom, a, a
†
are the usual annihila-
tion/creation operators of the field, and γ is the in-
teraction strength. It is important to note here that
U preserves the total energy of atom-plus-field, and
that in order to perform the required rotation γ ≈
π
4α
.
Remarkably, an x-measurement is never perfect be-
cause the underlying unitary is never a perfect Rabi
rotation about the y-axis and consequently always
leaves some residual entanglement between the atom
and the laser. As we show in App. B, the error we
make in implementing the ideal rotation is equivalent
to performing a noisy z-measurement, {E
0
, E
1
}, given
by
E
0
= hα|U
†
(|0ih0| ⊗
a
) U |αi
=
1 −
(−2+π)
2
64π
2
α
2
4−π
16π
2
α
2
4−π
16π
2
α
2
1
4
+
1
π
2
+
1
π
16α
2
!
(12)
and E
1
= − E
0
. In the limit |α|
2
1, the
average overlap of {E
0
, E
1
} with the ideal z-basis
measurement is
¯
F = 1 −
0.22
|α|
2
. One can treat the
general case of a spin-N/2 system with N + 1 en-
ergy eigenstates |N/2, mi, that are eigenstates of J
z
in a similar fashion (see App. B). The quality of
the overall measurement, as quantified by the av-
erage overlap of each element with the ideal one
¯
F =
1
N+1
P
N
m=0
tr(E
m
|N/2, mihN/2, m|
x
), is
¯
F = 1 − 0.072
N(N + 2)
|α|
2
+ O
N
3
|α|
3
. (13)
Accepted in Quantum 2017-10-31, click title to verify 3
The coherent state |αi serves as a RF for phase,
relative to which the requisite rotation U can be ap-
plied to the state of the system before performing the
z-basis measurement. By tuning the phase of the co-
herent state, as well as the interaction strength γ, we
can realise all possible measurements that can be per-
formed on the state of the atom. Note that the qual-
ity of the measurement in Eq. (13) depends on the
ratio between the square of the corresponding spin-
system
4
, N
2
, and the number of photons in the laser
pulse |α|
2
. This is a typical relation that we will en-
counter throughout the remainder of this work.
2.5 Measurements under symmetry
The previous example illustrates how the size of the
reference limits the quality of a measurement in a
particular setup. However, the same argument can
be made more generally. Any measurement device is
comprised of a state, |RFi, that encodes all pertinent
information regarding the device’s size, orientation,
etc., and a measurement process consisting of an in-
teraction, U, between the state of the system, |ψi, and
the state of device |RFi, followed by the von Neumann
measurement |kihk| on some part of the global system
that realizes outcome k, see Fig. 1. The probability
+
k
U
interaction
outcome k
Figure 1: Measuring process under symmetry. The measure-
ment device is described by the state |RFi. The measurement
process consists of interacting the device with the system
of interest via a globally invariant unitary interaction U
inter
followed by a globally invariant von Neumann measurement
|kihk|.
to observe the latter is given by
p
k
= Tr
(|kihk| ⊗ ) U |ψihψ| ⊗ |RFihRF|U
†
. (14)
Importantly, by explicitly incorporating the state of
the RF (the measurement device) within our quantum
mechanical description, the joint system-plus-device
can be treated as a closed system, and as such must
respect the fundamental symmetries of natural laws.
4
Here we use the Schwinger representation to map any ar-
bitrary N-photon state to the state of a spin-N/2 particle
The most general, non-relativistic, group of space-
time symmetries of physical laws is the Galilean group
of transformations—time translations, space transla-
tions, rotations, and boosts. Noether’s theorem [30]
then tells us that to each such symmetry corresponds
a conservation law for the closed system: conserva-
tion of energy, momentum, angular momentum etc.
In particular, the only interactions, U, that can be
performed are those that satisfy [U, V
g
] = 0, ∀g ∈ G
where G is the relevant symmetry group and G ⊃ g →
V
g
is its corresponding unitary representation. Sim-
ilarly, this also restricts the possible measurements
that can be preformed on the closed system after
the interaction; the POVM elements that can be per-
formed have to satisfy [|kihk|, V
g
] = 0, ∀g ∈ G [31–35].
The invariance of the measurement might, at first
sight, appear less intuitive than that of the interac-
tion. After all, the measurement performed by our
eyes, or other sense-organs, do not commute with the
group of spatial rotations. However, in this exam-
ple the retina of the eye, as well as the rest of the
observer’s body, encode information about the ob-
server’s spatial orientation. As such, they are in-
cluded in the description |RFi, and the assumption
[|kihk|, V
g
] = 0 only holds when all physical systems
that can encode the relevant information are prop-
erly accounted for in the state of the RF. The action
of the projector |kihk| ultimately stands for the mere
awareness of the outcome k by the observer, which is
invariant under the relevant group of symmetries.
The fact that both the measurement and the in-
teraction commute with V
g
ensure that the probabil-
ity p
k
in Eq. (14) is left unchanged when the global
state is transformed |ψi|RFi → V
g
(|ψi|RFi). As this
holds for any g ∈ G, it also holds if g ∈ G is taken
at random according to the invariant measure, dg, of
the group yielding
p
k
= Tr
(|kihk| ⊗ ) U (G [|ψihψ| ⊗|RFihRF|]) U
†
,
(15)
where
G[|ψihψ|⊗|RFihRF|] =
Z
dg V
g
(|ψihψ| ⊗ |RFihRF|) V
†
g
(16)
is called the G-twirling map. Eq. (15) shows that
the restrictions imposed by symmetry on operations
and measurements are operationally indistinguishable
from the situation where the joint state of system-
plus-device is described by G[|ψihψ| ⊗ |RFihRF|] in-
stead of |ψi ⊗ |RFi. This is precisely the state that
one would ascribe to the joint system-plus-device in
the absence of an additional external RF (see App. A
for a more detailed explanation and [36] for the com-
plete theory).
It is important to note that for moderately sized
systems a fully quantum treatment of the RF is not re-
quired as one can consider the latter to be sufficiently
large, e.g., by increasing the power of a laser beam
to obtain a proper phase reference (as shown in the
Accepted in Quantum 2017-10-31, click title to verify 4
example above), or consider the spatial orientation to
be given by the laboratory equipment. However, this
is no longer justified if one considers quantum objects
of macroscopic size. In this case, the explicit inclusion
of the RF into our description is necessary in order to
take into account the required resources.
2.6 Classical states of reference frames
The preceding discussion emphasizes how all perti-
nent information of the measuring device, such as
its size, orientation, phase etc., are described by the
quantum state |RFi. However, as the measurement
device is by all accounts a classical system, the quan-
tum state |RFi describing it must be as classical as
possible. We define states of any macroscopic quan-
tum system, comprised of N constituent parts, as
classical if their collective quantum effects can be
explained as the accumulation of O(N) microscopic
quantum effects. This definition includes the so-
called pointer states [2, 37], and generalized coherent
states [38]. In addition such states are known to be
robust fixed point of generic local decoherence pro-
cesses [38, 39]. Finally, one can also show that after
the action of generic local noise any state can be de-
composed as a mixture of pure states with the average
variance of any extensive quantity that only scales
linearly with the number of subsystems. Note also
that the minimum average variance of a state is equal
to its quantum Fisher information [40]—a measure of
the states metrological performance. Thus, for the re-
mainder of this work, classical measuring devices will
be described either by coherent states of light, spin-
coherent states, or products of localized wave packets,
corresponding to classical measuring devices for en-
ergy, angular momentum, and position respectively.
However, in Sec. 3.4 we will show that our results can
be formulated without recourse to an explicit state
for a RF: all we need to assume is how its average
variance scales with respect to its size.
3 Detecting Macroscopic Superposi-
tions
In this section we derive the size of the classical mea-
suring device needed in order to discriminate between
a macroscopic superposition and a classical mixture in
a single shot as explained in Sec. 2.3. Specifically, we
will show that the macroscopic superposition states
given in Eqs. (3, 4, 5) can be distinguished from their
corresponding classical counterparts with an exponen-
tially small probability of error if the classical measur-
ing devices used are quadratically larger in size. In
Sec. 3.4 we show that the same result holds for the
more general macroscopic quantum states of Eq. (6).
In order to provide as much physical insight into the
results as possible, we give a heuristic account of our
findings below, with the rigorous mathematical proof
deferred to Apps. C and E.
3.1 Macroscopic Superpositions of photon
number
We begin with the case of distinguishing the macro-
scopic superposition of Eq. (3) from the incoherent
mixture
ψ =
|0ih0| + |NihN|
2
. (17)
Observe that in the absence of an appropriate phase
reference Eq. (3) is operationally indistinguishable
from that of Eq. (17)
5
; there exists no phase-invariant
measurement capable of distinguishing between these
two states.
It follows that in order to have any chance to dis-
tinguish between the two states we need a phase ref-
erence. Hence, our classical measurement device is
described by a coherent state of the RF with |α|
2
pho-
tons in total
|RFi = |αi =
∞
X
n=0
√
q
n
|ni, (18)
with q
n
= |α|
2n
e
−|α|
2
/n!. The joint system-plus-
device is invariant under global phase shifts, i.e., for
ρ ∈ B(H
S
), V
θ
(|RFihRF| ⊗ ρ)V
†
θ
= |RFihRF| ⊗ ρ,
where V
θ
= exp(iθ(
ˆ
N
RF
+
ˆ
N
S
)) with
ˆ
N
RF(S)
the pho-
ton number operator for the RF (system) respectively.
In other words the total photon number (or equiva-
lently the total energy) of the joint RF-plus-system is
conserved.
Now recall that the constraints due to global phase
shifts can be equivalently described by replacing the
joint state of the RF-plus-system by its G-twirled
version Eq. (16). As the latter involves averaging
over all possible θ ∈ [0, 2π), its effect on an arbi-
trary state of system-plus-device is to erase all co-
herences between states with different total photon
number
ˆ
K ≡
ˆ
N
RF
+
ˆ
N
S
, resulting in a mixture (a
direct sum) of states with different fixed total photon
number (see App. A). Hence the G-twirling map reads
G
|RFihRF| ⊗ ρ
=
∞
M
K=0
Π
(K)
|RFihRF| ⊗ ρ
Π
(K)
| {z }
Q
K
%
(K)
,
(19)
with
Π
(K)
=
K
X
n=0
|nihn|
RF
⊗ |K −nihK − n|
S
. (20)
5
We remark that the distinguishability of the superposition
of coherent states |SCSi =
1
√
2
(|αi + |−αi) from the mixture
|αihα|+|−αih−α|
2
does not suffer from a lack of a RF for total
photon number. Instead the relevant group of transformations
here is the set of Bogoliubov transformations associated with
the Lorentz group of symmetries of special relativity.
Accepted in Quantum 2017-10-31, click title to verify 5
For K < N the state Q
K
%
(K)
for both the macro-
scopic superposition state of Eq. (3) and the cor-
responding mixture of Eq. (17), reads Q
k
%
(K)
=
q
K
|KihK|
RF
⊗ |0ih0|
S
. For K ≥ N however, one gets
Π
(K)
|RFihRF| ⊗ |ψihψ|
Π
(K)
=
1
2
q
K
√
q
K
q
K−N
√
q
K
q
K−N
q
K−N
(21)
and
Π
(K)
|RFihRF| ⊗ ψ
Π
(K)
=
1
2
q
K
0
0 q
K−N
, (22)
expressed in the basis {|Ki
RF
⊗ |0i
S
, |K − Ni
RF
⊗
|Ni
S
}. The trace distance in Eq. (8) between the two
states after the twirling is now very easy to compute
and yields
t =
∞
X
K=N
1
2
√
q
K
q
K+N
≈
1
2
e
−
N
2
8|α|
2
, (23)
where the approximation holds in the limit of large α
and N , which is the regime we are interested in.
For completeness, note that this trace distance and
the corresponding optimal probability to distinguish
the states is attained with the phase-invariant POVM
composed of two elements
M
±
=
∞
M
K=0
|±
K
ih±
K
|, (24)
where
|±
K
i =
1
√
2
(|Ki
RF
⊗|0i
S
±|K − Ni
RF
⊗|Ni
S
). (25)
Recalling that |αi describes our measurement de-
vice, Eq. (23) states that the probability of suc-
cessfully distinguishing the macroscopic superposi-
tion from the incoherent mixture in a single shot
approaches unity so long as our classical measure-
ment device is quadratically larger than the size of
the macroscopic superposition (|α|
2
> N
2
).
3.2 Macroscopic Superpositions of Spin
Next, consider using a classical measuring device in
order to distinguish between the macroscopic super-
position of Eq. (4) and the incoherent mixture
ψ =
|↑ih↑|
⊗N
+ |↓ih↓|
⊗N
2
. (26)
In the absence of a RF for orientation in space, Eq. (4)
is operationally indistinguishable from Eq. (26). The
underlying symmetry in this case is the invariance of
physics under rotations V
n
= exp(i n · (S
S
+ S
RF
)),
where S = (S
X
S
Y
S
Z
) are the total spin operators
for the system and RF respectively. The resulting
G-twirling map corresponds to averaging over all pos-
sible orientations of the joint system-plus-device and
results in the elimination of all coherences between
subspaces corresponding to different values of the to-
tal spin S
2
= (S
S
+ S
RF
)
2
, as well as the com-
plete elimination of coherences between various di-
rections within each subspace (see App. A for fur-
ther details). It follows that an orientational RF
state needs to have some spread for its total spin
S
2
RF
. A classical state of RF for orientation car-
ried by M spin-
1
2
subsystems is, for example, given
by |RF
n
i = |↑i
⊗M/3
|→i
⊗M/3
|i
⊗M/3
, with |↑i, |→i
and |i being the eigenstates of S
X
, S
Y
and S
Z
op-
erators for a spin-
1
2
.
We will consider the problem of a full orientation
RF in Sec. 5. However, for the purposes of distin-
guishing the superposition state of Eq. (4) from the
mixture of Eq. (26) a simpler argument involving only
a directional RF suffices. A random rotation can be
thought of as first choosing a direction
ˆ
n, followed by
a random rotation by an angle θ ∈ (0, 2π] about
ˆ
n.
Hence, the G-twirling map for an orientation RF in-
cludes the G-twirling map of a directional RF. Choos-
ing without loss of generality
ˆ
n =
ˆ
z, and averaging
over all possible rotations about the
ˆ
z axis results
in an isomorphism between a directional RF and the
phase reference RF of Sec. 3.1. Projections onto the
total photon number are now replaced by projecting
the joint state of system-plus-RF onto sectors with a
fixed value of S
Z
= S
Z,S
+ S
Z,RF
, and the state of the
corresponding classical RF, representing the magnetic
field direction of the magnet in the Stern-Gerlach ex-
periment, is given by a spin coherent state
|RF
Z
i = |→i
⊗M
=
J
X
m=−J
s
1
2
M
M
m +
M
2
|J, mi,
(27)
with S
Z,RF
|J, mi = m |J, mi and J =
M
2
. As the
distribution of the spin-coherent state is well approxi-
mated by a Gaussian, we obtain for the trace distance
t ≈
1
2
e
−
N
2
8M
. (28)
Again, one requires that the magnets be at least
quadratically larger than the size of the macroscopic
superposition in question.
3.3 Macroscopic superpositions of position
Let us now consider the situation where the task is
to distinguish a system of mass m, say an atom, in a
superposition at two spatial locations separated by a
distance L, i.e., Eq. (5) versus the incoherent mixture
Ψ =
−
L
2
−
L
2
+
L
2
L
2
2
. (29)
Accepted in Quantum 2017-10-31, click title to verify 6
This corresponds to the case where one lacks a RF
for the two remaining elements of the Galilean group;
spatial translations and boosts.
In the case of spatial translations the unitary repre-
sentation is given by V
s
= e
is
P
i
p
i
, where p
i
are the
momenta of the particles in the closed system. Invari-
ance under spatial translation implies that the joint
state of the system-plus-device, % = ρ⊗|RFihRF|, can
be replaced by its G-twirled version
G
P
[%] = lim
ω→∞
Z
ω
0
ds e
isP
% e
−isP
, (30)
which corresponds to a projection onto sectors with a
fixed total momentum P ≡
P
i
p
i
.
In the case of boosts the fundamental symmetry
is the invariance of physical laws for all observers in
relative motion, i.e., observers that move with a con-
stant velocity with respect to each other. A boost
at some initial time t = 0 corresponds to a coordi-
nate change t
0
= t and x
0
= x + vt, so at t = 0 an
observer in R
0
sees all the particles with shifted mo-
menta p
0
i
= p
i
+ m
i
v as compared to the momenta
p
i
of the particles observed in R. Hence, the trans-
formation between the states in the two frames at
this time is given by V
v
= e
iv
P
i
x
i
m
i
. Because the
relation between the two observers R and R
0
is ex-
plicitly time dependent the treatment of the boost in
the Hamiltonian formalism, where time plays a special
role, slightly differs from the rest of the symmetries of
the Galilean group. In particular, the boost transfor-
mation does not commute with the Hamiltonian of a
closed system,
H =
X
i
p
2
i
2m
i
+ V (x
1
, . . . , v
N
). (31)
Instead, one finds V
†
v
HV
v
= v
P
i
p
i
≡ vP . This term
ensures the proper relation between the coordinates
x
0
= x + vt of the two observers at all times.
For example, consider the situation where a boost
applied at time t = 0 is later on inverted at time
t = τ. The overall transformation (boost+free evo-
lution+inverted boost) is not identical with the free
evolution H in Eq. (31) as
V
†
v
e
iτH
V
v
= e
iτH+ivtP
6= e
iτH
. (32)
In fact, the additional term e
ivtP
exactly accounts
for the coordinate shift x
0
i
→ x
i
+ vτ acquired be-
tween the two frames during this time
6
. Hence,
the twirling map corresponding to the boost and
generated with the center of mass position operator
X
C
≡
1
M
P
i
x
i
m
i
(with the total mass M ≡
P
i
m
i
)
does not commute with the free evolution. However,
6
The requirement V
†
v
e
itH
V
v
= e
ivtP
enforces the depen-
dence of the Hamiltonian on the momenta to be given by the
well known form of the kinetic energy
considering the translation and boost symmetries to-
gether, the twirling map for the momentum and the
position of the center of mass,
G[%] =
Z
dvds e
i(vX
C
+sP )
% e
−i(vX
C
+sP ),
(33)
does commute with the free evolution
G[e
itH
%e
−itH
] = e
itH
G[%]e
−itH
. Indeed, one sees
that e
itH
e
i(vX
C
+sP )
e
−itH
= e
i(vX
C
+sP +vtP )
(up to
an irrelevant phase) can be brought to the initial
form e
i(vX
C
+sP )
by a simple change of integration
variables in Eq. (33).
The G-twirling map washes out the center of mass
degree of freedom, seen as a quasi-particle respecting
the canonical commutation relation [X
C
, P ] = i (see
e.g. [41]). The twirling map performs a random dis-
placement in the phase space of the center of mass
variables X
C
and P , ensuring that this degree of free-
dom is totally undefined. This simply means that
for a closed system (once the RF is internalized) only
the relative degrees of freedom make physical sense,
while the center of mass position and momentum are
a mathematical fiction.
Let us now focus on the effect of the G-twirling
on the delocalized mass of Eq. (5) and a suitable RF
state. For the purpose of upper bounding the trace
distance between the superposition and the mixture,
the twirling over the total momentum can be omitted,
and we focus only on the twirling of the the center of
mass position, corresponding to the integration of v
in Eq. (33). The latter projects the overall state
G
X
C
[%] =
M
X
C
Π
X
C
% Π
X
C
, (34)
onto orthogonal sectors with a fixed position for the
center of mass X
C
≡
mx+MX
m+M
, where M and X are
the mass and position of the center of mass of the
RF. We assume the latter to be a product state of K
particles of mass m
0
, whose individual wave-functions
|hx
i
|ψ
i
i|
2
are Gaussians centered around fixed posi-
tions with equal spread σ
0
. Hence, the center of mass
of the RF is a quasi-particle with mass M = Km
0
,
position operator X ≡
P
m
0
x
i
M
=
1
K
P
x
i
and wave-
function
|RF(X)|
2
= tr |XihX||RFihRF| =
1
p
2π/Kσ
0
e
−K
X
2
2σ
2
0
.
(35)
One can then directly compute the trace distance of
Eq. (8) between the states G
X
C
[|ΨihΨ|⊗|RFihRF|] and
G
X
C
[Ψ ⊗|RFihRF|], which results in (see App. C.2 for
more details)
t =
1
2
Z
RF(X)RF
∗
(X + L
m
M
)dX
=
1
2
exp
−
L
σ
0
2
m
m
0
2
8K
. (36)
Accepted in Quantum 2017-10-31, click title to verify 7
Yet again, the size of the RF, K, has to be quadrat-
ically larger then the product of the ratio of masses
multiplied by the ratio of spreads between the super-
posed object and the particles constituting the RF.
3.4 Macroscopic quantum states
In this section we present the extension of the results
above to general states |Ψi of Eq. (6), as well as mixed
states of the reference frame system ρ
RF
. We give a
simplified account of our findings; details of the cal-
culation can be found in App. E. Firstly, we need
to identify a semi-classical counterpart, Ψ, of Eq. (7)
which does not contain macroscopic coherences. This
is not as trivial as it might seem at first sight as an
arbitrary choice of the “masking” function c
|n−m|
does
not necessarily yield a valid density matrix. One pos-
sible choice that guarantees the positivity of Ψ is to
associate the latter as the outcome of a completely
positive, trace-preserving map acting on the original
state |Ψi. In particular we choose
Ψ =
X
n,m
ψ
n
ψ
m
exp
−
(n − m)
2
2N
2β
|nihm|, (37)
corresponding to the action of a Gaussian dephasing
channel Ψ ≡ E
β
(|Ψi) on the initial state. This chan-
nel corresponds to the application of a random uni-
tary U
λ
≡ e
iλ(
P
n
n|nihn|)
on the initial state, where λ
is a random variable with associated probability dis-
tribution p(λ) =
N
β
2
√
π
e
−
N
2β
λ
2
4
. Notice that this noise
process is solely considered to construct a classical
state (without long-ranged coherences) that has the
same diagonal elements as the state in question. It is
not that we assume that this (or any other kind of)
noise actually acts on states.
Accordingly, the trace distance between the state
|Ψi and Ψ after twirling with the RF state ρ
RF
is
t =
1
2
tr |G [(|ΨihΨ| − Ψ) ⊗ρ
RF
]|. (38)
The state of the RF can always be decomposed into
a convex sum of pure RF states |RF
i
i =
P
n
r
(i)
n
|ni,
i.e., ρ
RF
=
P
i
q
i
|RF
i
ihRF
i
|. As the trace distance is
jointly convex, it follows that t ≤
P
i
q
i
t
(i)
where
t
(i)
≡
1
2
tr |G [(|ΨihΨ| − Ψ) ⊗|RF
i
ihRF
i
|]|. (39)
After twirling the state |Ψi|RF
i
i decomposes into
block-diagonal form
G [|ΨihΨ| ⊗ |RF
i
ihRF
i
|] =
M
k
P
k,i
|Ψ
k,i
ihΨ
k,i
|, (40)
where
|Ψ
k,i
i =
ψ
k−n
r
(i)
n
p
P
k,i
|k − ni
S
|ni
RF
≡
X
n
λ
(k,i)
n
n
(k)
E
(41)
and P
k,i
≡
P
n
(ψ
k−n
r
(i)
n
)
2
. The corresponding semi-
classical state, after twirling with the RF, reads
G (Ψ⊗|RF
i
ihRF
i
|) =
M
k
P
k,i
ρ
k,i
(42)
where
ρ
k,i
=
X
n,m
λ
(k,i)
n
λ
(k,i)
m
e
−
(n−m)
2
N
2β
n
(k)
ED
m
(k)
. (43)
Using the bound t ≤
√
1 − F
2
between the trace dis-
tance and fidelity [42] one obtains
t
(i)
≤
X
k
P
k,i
q
1 − hΨ
k,i
|ρ
k,i
|Ψ
k,i
i. (44)
Using the inequality e
−d
2
≥ 1 − d
2
and substituting
Eqs. (41, 43) into Eq. (44) one obtains
t
(i)
≤
X
k
P
k,i
v
u
u
t
X
n,m
(λ
(k,i)
n
)
2
(λ
(k,i)
m
)
2
(n − m)
2
2N
2β
. (45)
Expanding the quadratic term inside the square root
of Eq. (45), we may express the entire sum as the
variance of the distribution (λ
(k,i)
n
)
2
, yielding
t
(i)
≤
X
k
P
k,i
v
u
u
t
Var
(λ
(k,i)
n
)
2
N
2β
. (46)
Finally, repeated use of Jensen’s inequality and the
normalization condition
P
n
ψ
2
k−n
= 1 (see App. E)
gives the desired result
t ≤
v
u
u
t
P
i
q
i
Var
(r
(i)
n
)
2
N
2β
. (47)
Thus the trace distance vanishes whenever the aver-
age variance of the RF state ρ
RF
scales slower than
N
2β
. For the classical states of the RF considered
above, and for any state after the action of generic lo-
cal noise [43], the average variance scales linearly with
the size of the RF,
P
i
q
i
Var
(r
(i)
n
)
2
≈ M . Hence,
we have shown that in order to certify the presence
of quantum coherence on the scale |n −m| > N
β
one
requires a classical RF of size M > N
2β
. In particu-
lar, genuine macroscopic coherences |n − m| = O(N )
require a quadratically larger RF.
We stress that our result holds only for RF states
whose quantum features can be explained as the accu-
mulation of O(N) macroscopic quantum effects. If the
measurement device is allowed to possess macroscopic
quantum features as well, then the requirements on its
size change. For example, in the photonic case, it can
be shown that an RF prepared in the so called sine
state [44]
|RFi =
r
2
N + 2
N
X
n=0
sin
(n + 1)π
N + 2
(48)
Accepted in Quantum 2017-10-31, click title to verify 8
only needs to scale linearly with the size of the macro-
scopic superposition in order to distinguish the latter
form the corresponding mixture in a single shot.
4 Discussion
We have shown how in order to distinguish a macro-
scopic superposition from an incoherent mixture in a
single-shot, using classical measuring devices, the size
of such devices needs to be quadratically larger than
the size of the macroscopic superposition in question.
In what follows we relax our requirement for single-
shot certification (Sec. 4.1), and classical measuring
devices (Sec. 5) and determine how relaxing these con-
ditions affects the successful certification of macro-
scopic superpositions, and/or the size of the corre-
sponding measuring device. In Sec. 4.2 we discuss
the ramifications of our findings for certifying macro-
scopic quantum states.
We note that a similar result was obtained in a
different context by Kofler and Brukner [45]. In par-
ticular, Kofler and Brukner show that in order to ob-
serve violations of local realism for bigger and bigger
spins, one requires measuring devices with larger and
larger accuracy (or finer and finer coarse-graining).
The authors then connect the resolution of measure-
ment devices with their size by applying Heisenberg’s
uncertainty relations at the macroscopic scale and us-
ing relativistic causality. Here, we adopted a more
direct approach by internalizing the states of classical
measurement devices in the quantum formalism and
invoking fundamental symmetries.
4.1 Repeating the measurement
Our analysis so far was centered on the assumption
that we are given only a single copy of the macro-
scopic state and thus had to ensure that we are able
to distinguish it from the incoherent mixture, with a
high probability of success, in a single experimental
run. All cases we considered exhibited the same be-
havior for the probability of success, namely that for
a classical RF of size M the trace distance between a
superposition state of size N and the corresponding
mixed state is generically given by
t =
1
2
exp
−
N
2
8M
. (49)
If we want the probability of success to be high, then
M > N
2
.
However, in practice one is of course not bound to
a single shot measurement and one can overcome the
limitation set by the the size of the RF by repeating
the experiment sufficiently many times. A legitimate
question then is how many repetitions are required in
order to confirm that a superposition was prepared
given our RF is of size M < N
2
. The answer to
this question is provided by the Chernoff bound of
Eq. (10).
Consider a scenario where one performs a series of
experiments with superposition states of larger and
larger N , and with a corresponding RF whose size
is given by M = cN
2−ε
, i.e., the RF also increases
in subsequent experiments as as function of N . We
ask how the number of repetitions n, necessary to
witness the superposition from the mixture, grows as
a function of the size of the macroscopic quantum
state N. First, note that if the size of the RF can grow
at least quadratically with N , i.e., ε ≤ 0, then the
trace distance in fact decreases with N , so that large
superpositions can be certified without the need to
increase the number of repetitions. But for practical
reasons we are more interested in the case ε > 0 (e.g.
ε = 2 if the size of the RF is limited), where the
resources are limited such that a quadratic growth of
the RF is not possible. In this case the number of
repetitions, n, has to increase in order to compensate
for a decreasing trace distance, and one can easily see
from Eq. (10) that for a given constant probability of
error, lim
n→∞
P
(n)
Err
≤ p, one requires that
n ≥ 2 log
1
p
exp
N
ε
8c
. (50)
In other words, the required number of repetitions
blows up exponentially with N, so the experiment
becomes infeasible quite fast.
4.2 Implications and examples
We have demonstrated that when describing macro-
scopic superposition states of several types—having
larger and larger size—one eventually has to aban-
don the abstraction of an idealized, classical, RF. In
particular, we have shown how the task of distinguish-
ing the superposition of a certain size from the corre-
sponding mixture leads to a requirement on the size
of the classical RF. We now quantitatively illustrate
this relation with two examples.
First, consider a simple model for the cat in
Schrödinger’s thought experiment consisting of N
spins in a superposition state of Eq. (4). For sim-
plicity let us assume that each spin is carried by a
water molecule. For a cat of mass ≈ 3 kg, and with
a single water molecule having mass m
0
= 18 N
A
=
3 ·10
−26
kg, our cat is comprised of roughly N = 10
26
water molecules. Our results imply that in order
to certify such a cat in the superposition state of
Eq. (4) in a single shot, would require a RF of size
M ≈ N
2
= 10
52
. Assuming that the RF is also made
entirely of water molecules implies
m
RF
≥ N
2
m
0
≈ 3 · 10
25
kg ≈ 50M
⊕
, (51)
where M
⊕
is the mass of the Earth. This clearly in-
dicates the practical impossibility to perform such an
experiment.
Accepted in Quantum 2017-10-31, click title to verify 9
Second, consider the case of delocalized mass and
let us again assume that planet Earth plays the role
of the RF. For simplicity we use a model similar to
the one above where the RF is a crystal of K water
molecules, each of which is localized in a Gaussian
wave-packet of width σ
0
≈ 10Å = 10
−9
m— roughly
the size of an ice crystal cell. Requiring the exponent
in Eq. (36) to be ≤ 1 shows that even an RF of such
astronomical size only allows to observe a massive su-
perposition state of Eq. (5) for
L m ≤ 10
−9
kg m. (52)
This model leads us to conclude that in order to
observe a single water molecule delocalized over one
meter L = 1 m the requisite “water crystal” RF must
have a mass M = Km
0
≈ 3 µg which is not a severe
practical limitation. But for an object of mass m =
1 g and the RF of the size of the Earth K = M
⊕
/m
0
≈
2 · 10
50
this gives a very harsh restriction L ≤ 1 µm.
5 Many copies and relative degrees of
freedom
We have seen how symmetry arguments lead to se-
vere restrictions on the size of the measurement de-
vice that allow to observe macroscopic superpositions
of the type of Eqs. (3, 4, 5). Here, we will show how
such limitations can be circumvented in two ways: by
manipulating two copies of the superposition states,
or by preparing superpositions in relative degrees of
freedom.
5.1 Two copies of macroscopic superposition
Let us now suppose that one is able to prepare and
coherently manipulate two copies of the macroscopic
superposition states simultaneously. In this case the
macroscopic superpositions play the role of a RF for
each other, and it becomes possible to distinguish two
copies of a superposition from two mixtures without
the need of an additional RF as we now show.
For the purposes of illustration let us first consider
the photonic superposition states of Eq. (3). The
task now is to distinguish the superpositions |Ψi|Ψi
from two mixtures Ψ ⊗ Ψ without a external RF
state. Recalling that the restrictions imposed by time-
translation symmetry are equivalent to replacing the
global states of a closed system by their G-twirled
version
G[|ΨihΨ| ⊗ |ΨihΨ|] =
|00ih00|
4
+
|NNihN N|
4
+
σ
N
2
G[Ψ ⊗ Ψ] =
|00ih00|
4
+
|NNihN N|
4
+
ρ
N
2
,
(53)
where
σ
N
=
1 1
1 1
ρ
N
=
1 0
0 1
, (54)
are expressed in the basis {|0i|Ni, |Ni|0i}. It is triv-
ial to show that the trace distance between the states
of Eqs. (53) is t = 1/4. This means that the probabil-
ity to correctly distinguish the two states in a single
shot is as high as 3/4, and it quickly converges to
one when the experiment is repeated. The situation
is similar for the case of spins and for massive super-
positions, although for the case of spins care must be
taken as one has to consider the entire G-twirling map
over all possible orientations (see App. D).
5.2 Macroscopic superpositions in relative de-
grees of freedom
These observations are consequences of the fact that
superposition in relative (or interal) degrees of free-
dom are not affected by global symmetries. Hence,
for example a photonic two-mode superposition of
the form
1
√
2
(|N, 0i + |0, Ni) might suffer from various
technical limitations, but does not require an external
phase RF in order to be prepared.
A similar argument holds for the case of position
with two objects of masses m
1
and m
2
; the phase
space of the relative degree of freedom x = x
1
− x
2
and p =
m
1
p
1
−m
2
p
2
m
1
+m
2
is not affected by the twirling map
for the center of mass degree of freedom. These two
objects can be, for example, a heavy molecule and the
double slit in an interference experiment. However if
the mass of the molecule is big enough, this indicates
that the state of the double slit has to be involved in
the final measurement.
In the case of spins one can also construct a macro-
scopic superposition of two states both with J = 0
and thus invariant under twirling with respect to the
orientation group. To do so consider the J = 0
subspace for four spin-
1
2
particles. This subspace is
spanned by two states |♥i =
Ψ
−
12
⊗
Ψ
−
34
and |♦i =
1
√
2
Ψ
−
13
⊗
Ψ
−
24
+
Ψ
−
14
⊗
Ψ
−
23
, where the
Ψ
−
ij
is the singlet states carried by qubits i and j. These
states are orthogonal and are both invariant under
global rotations of al the qubits. Hence, the 4N par-
ticle superposition state
1
√
2
|♥i
⊗N
+ |♦i
⊗N
is in-
variant under rotations and does no require an exter-
nal RF for spatial orientation. As before, this can be
understood as a superposition in an internal (relative)
degree of freedom, as the two states |♥i and |♦i are
related by the action of the permutation group.
Accepted in Quantum 2017-10-31, click title to verify 10
6 Conclusion
We have shown that in order to distinguish macro-
scopic superpositions from their incoherent mixtures
in a single-shot measurement one requires classi-
cal measuring devices that are quadratically larger.
Specifically, we have highlighted how relevant degrees
of freedom of classical measuring devices, such as size,
orientation, position, velocity etc, play a pivotal role
acting as appropriate RFs. By incorporating the lat-
ter within the quantum formalism we are able to pro-
vide a quantification of the size of macroscopic quan-
tum states in terms of the size of the corresponding
classical measuring device needed to distinguish it.
In addition, we have shown how the requirement of
quadratically larger size for classical devices can be
relaxed if we are allowed to repeat the experiment.
However, the necessary number of repetition then in-
creases exponentially with the size of the superposi-
tion state, making this route practically unfeasible.
By using one macroscopic state as a RF for a second
copy of the state, we can significantly reduce the num-
ber of repetitions needed in order to distinguish the
states from their classical mixtures.
Whereas we have focused solely on certifying
macroscopic superpositions, we believe that the ar-
guments presented here should hold for the prepara-
tion of such states. A more direct connection of our
quadratically larger RF for the preparation of such
states as well is the subject of future work, along with
an extension to other symmetry groups, such as the
Lorentz and Poincare groups of special and general
relativity, as well as for more specific situations where
both the preparation and certification of macroscopic
states are performed by the same device.
Finally, we also showed how the limitations are cir-
cumvented by macroscopic superposition in relative
(or internal) degrees of freedom, that are invariant
under global symmetries and thus do not require an
external RF to be defined.
Acknowledgements
This work was supported by Spanish MINECO
FIS2013-40627-P, IJCI-2015-24643, and Generalitat
de Catalunya CIRIT 2014 SGR 966.GR 966. (MS),
the Austrian Science Fund (FWF): P28000- N27, SFB
F40-FoQus F4012-N16, and the Swiss National Sci-
ence Foundation grant P300P2_167749.
A Quantum mechanical treatment of
reference frames
In this appendix we briefly review the connection be-
tween symmetry with respect to a group of transfor-
mations and the lack of a RF. We then show how
to incorporate RFs within the quantum mechanical
formalism. We will only focus on the relevant ingre-
dients needed for this work; the detailed treatment of
the problem can be found in [36].
Let H
(S)
denote the state space of a quantum me-
chanical system. The action of a symmetry group
G on H
(S)
is represented by a set of transformations
{U(g) : g ∈ G}. We write U : G → H
(S)
, and
for ease of exposition we shall focus our discussion
to finite dimensional Hilbert spaces, and to unitary
representations of compact Lie groups. Now in the
presence of a symmetry associated with the group G
the only allowable operations, A, are those that sat-
isfy [A, U (g)] = 0, ∀g ∈ G. We will now show that
this restriction is formally equivalent to the lack of a
RF associated with G relative to which the states of
a physical system are defined.
Specifically consider performing the allowable
von Neumann measurement |kihk|. As |kihk| is
an allowable operation, it follows that |kihk| =
U(g)
†
|kihk|U(g). Moreover, as this is true for all
g ∈ G, it follows that
|kihk| =
Z
g∈G
dg U(g)
†
|kihk| U (g)
≡ G[|kihk|], (55)
where dg denotes the uniform Haar measure of the
group G. Now consider the corresponding probability
of obtaining the outcome associated to |kihk| given an
initial state |ψi, not necessarily obeying the symmetry
associated to G. One easily obtains
p
k
= Tr [|kihk| |ψihψ|]
= Tr [G[|kihk|] |ψihψ|]
= Tr [|kihk| G[|ψihψ|]] , (56)
where we have used the fact that G
†
= G. Eq. (56)
shows that the restrictions imposed by symmetry on
the allowable operations are operationally indistin-
guishable from the situation where the state of the
physical system is described by G[|ψihψ|] as opposed
to |ψi.
To see the connection with RFs, consider the fol-
lowing situation. Alice and Bob each hold their own
respective RFs associated with the group of transfor-
mations G. However, their RFs are not aligned, but
rather related by some g ∈ G. If Alice prepares the
state |ψi relative to her RF, then Bob’s description
of the state is U
†
(g) |ψi, to account for the relation
between his and Alice’s RFs. Now suppose that Alice
and Bob are completely ignorant as to which g ∈ G
relates their corresponding RFs. This is equivalent to
assuming that Bob does not possess the requisite RF
at all. In this case Bob’s description of the system is
given by the state
ρ =
Z
g∈G
dg U(g)
†
|ψihψ| U(g)
= G[|ψihψ|]. (57)
Accepted in Quantum 2017-10-31, click title to verify 11
Hence, the presence of symmetry associated with a
group of transformations G is operationally equiv-
alent to the lack of a RF associated with G. In-
deed, the restrictions can be partially lifted if Alice
and Bob perform RF alignment [46–49], where one
of the parties encodes their local RF in the state of
a physical system—known as a quantum token of a
RF—which the other party measures in order to ex-
tract as much information about the RF as possible.
Clearly a good token of a RF is one whose orbit un-
der the action of the group G is non-trivial. Denot-
ing by Γ : G → H
(RF)
the unitary representation
of G onto the state space of the RF token, it fol-
lows that the more distinguishable the set of states
{|RF(g)i, ∀g ∈ G} are from each other the more
asymmetric the RF token state |RFi ∈ H
(RF)
is with
respect to the symmetry group G and thus the better
its ability to act as a RF.
The above discussion highlights the crucial role
played by RFs in the description of physical systems.
Indeed, whenever one writes down a pure state of a
quantum system one implicitly assumes a RF; saying
that a system is in the +1 eigenstate of σ
z
assumes
the notion of a z-direction. More often than not, the
RF is assumed to be an idealized system external to,
and independent from, the theory used to describe the
systems in question. If, however, we adopt the point
of view that RFs are physical systems themselves then
we need to treat them on an equal footing as all other
systems.
To that end, let us now describe the situation where
we incorporate the requisite RF within our quantum
mechanical description. That is consider the state
|RFi ⊗ |ψi ∈ H
(R)
⊗ H
(S)
. Now consider a hypothet-
ical RF relative to which the joint state |RFi ⊗ |ψi
is defined. Changing this hypothetical RF by g ∈ G
is equivalent to performing the unitary transforma-
tion Γ(g)
†
⊗ U (g)
†
(|RFi ⊗ |ψi). However, as such a
global RF is not physically available, the joint state
of RF-plus-system is
G[|RFihRF| ⊗ |ψihψ|] =
Z
dg
Γ(g)
†
⊗ U
†
(g)
×
|RFihRF| ⊗ |ψihψ| (Γ(g) ⊗U(g)) .
(58)
Using Schur’s lemmas [50] there exists an orthonor-
mal basis relative to which Γ ⊗U can be decomposed
into several inequivalent and irreducible representa-
tions (irreps), Λ
(λ)
, as follows
Γ(g) ⊗ U(g) ≡
M
λ
Λ
(λ)
(g) ⊗ I
(λ)
, (59)
where the identity operator I
(λ)
has dimension equal
to the number of times the irrep U
(λ)
appears in the
decomposition of Γ ⊗ U. Consequently, the Hilbert
space H
(RF)
⊗ H
(S)
can be written in the same basis
as
H
(R)
⊗ H
(S)
≡
M
λ
H
(R+S)
λ
≡
M
λ
M
(R+S)
λ
⊗ N
(R+S)
λ
,
(60)
where H
(R+S)
λ
are the invariant subspaces of H
(R)
⊗
H
(S)
under the action of Γ ⊗U ; for any |ψi ∈ H
(R+S)
λ
,
Γ(g) ⊗ U (g) |ψi ∈ H
(R+S)
λ
, ∀g ∈ G. The state spaces
M
(R+S)
λ
(N
(R+S)
λ
) are the spaces upon which the ir-
reps Λ
(λ)
act non-trivially (trivially) and are known
as the carrier (multiplicity) spaces respectively. For
example, in the case where G = u(1)—the symme-
try group of time-translations that corresponds to a
RF for phase θ ∈ (0, 2π]—the carrier spaces are the
one dimensional spaces of total photon number
ˆ
N =
ˆ
N
RF
+
ˆ
N
S
, M
(RF+S)
n
≡ span{|ni ≡ |ni
RF
⊗|ni
S
}, and
the multiplicity spaces are simply all possible ways of
partitioning the total photon number into two parts.
In the case where G = su(2)—the symmetry group of
rotations that corresponds to an orientation RF—the
carrier spaces are given by M
(RF+S)
J
= span{|J, Mi}
where |J, Mi is the joint eigenstate of the total an-
gular momentum operator J
2
= J
2
x
+ J
2
y
+ J
2
z
, and
J
z
, and the multiplicity spaces are the various differ-
ent ways two quantum systems of angular momentum
j
RF
and j
S
can couple to give rise to a total angular
momentum J.
In the basis of Eq. (60) the G-twirling map of
Eq. (58) reads [36]
G[|RFihRF| ⊗ |ψihψ|] =
M
λ
D
M
(R+S)
λ
⊗ I
N
(R+S)
λ
◦
P
λ
[|RFihRF| ⊗ |ψihψ|], (61)
where P
λ
[ρ] ≡ Π
λ
ρΠ
λ
, ρ ∈ B(H
(R)
⊗ H
(S)
) with
Π
λ
the projector onto H
(R+S)
λ
, D
M
(R+S)
λ
[A] =
Tr(A)
|M
(R+S)
λ
|
M
(R+S)
λ
, ∀A ∈ B(M
(R+S)
λ
) with |M
(R+S)
λ
|
the dimension of the carrier space M
(R+S)
λ
, and
I
N
(R+S)
λ
[A] = A, ∀A ∈ B(N
(R+S)
λ
).
The intuition behind the particular decomposition
of the total space H
(R)
⊗ H
(S)
is as follows. The car-
rier space M
(R+S)
λ
corresponds to the global degrees
of freedom of the joint RF-plus-system, whereas the
multiplicity space N
(R+S)
λ
corresponds to the relative
degrees of freedom. It is in the latter that all relevant
properties of interest regarding the relationship of the
system to the RF preside.
B Measurements under conservation
laws
In this appendix we provide the detailed computations
of how well one can perform a measurement in the
eigenbasis of J
x
on a spin N/2 system under energy
conservation imposing that only projections onto the
Accepted in Quantum 2017-10-31, click title to verify 12
eigenbasis of J
z
are allowed. The measurement can be
achieved by utilizing a bright laser pulse in the coher-
ent state |αi, and the Jaynes-Cummings interaction
U = e
γ(J
+
a−J
−
a
†
)
, (62)
to perform a rotation by π/2 about the y-axis followed
by a measurement of J
z
. In Eq. (62), a and a
†
are the
annihilation and creation operators, γ is the interac-
tion strength, and J
±
are the corresponding raising
and lowering operators of a spin-N/2 system.
For ease of exposition, we will make use of the
Schwinger representation
J
x
=
1
2
b
†
c + bc
†
J
y
=
1
2i
b
†
c − bc
†
J
z
=
1
2
b
†
b − c
†
c
, (63)
of spin operators in terms of the operators of two
bosonic modes. The corresponding eigenstates of the
J
z
can be written in terms of the states of two bosonic
modes in the following way
|N/2, mi = |mi ⊗ |N −mi (64)
so that J
z
|N/2, mi =
2m−N
2
|N/2, mi.
Writing |αi = D(α) |0i, where D(α) is the displace-
ment operator, and noting that
e
γ(J
+
a−J
−
a
†
)
D(α) = D(α)e
i2αγJ
y
e
γ(J
+
a−J
−
a
†
)
, (65)
we see that upon tracing out the optical mode results
in the measurement
E[E
m
] = tr
RF
U(|αihα| ⊗E
m
)U
†
= tr
RF
h
D(α)e
γ(i2αJ
y
+(J
+
a−J
−
a
†
))
(|0ih0| ⊗ E
m
)
×e
−γ(i2αJ
y
+(J
+
a−J
−
a
†
))
D(−α)
i
= tr
RF
h
e
γ(i2αJ
y
+(J
+
a−J
−
a
†
))
(|0ih0| ⊗ E
m
)
×e
−γ(i2αJ
y
+(J
+
a−J
−
a
†
))
i
, (66)
where E
m
= |N/2, mihN/2, m|. Choosing γ =
π
4α
results in the desired rotation about the y-axis up to
a small perturbation that leads to noise described by
the CPTP map with Kraus operators
K
n
= hn|e
(i
π
2
J
y
+γ(J
+
a−J
−
a
†
))
|0i, (67)
and vanishes in the limit γN → 0.
In order to quantify the distance of the resulting
measurement from the ideal one we Taylor expand
Eq. (67) around γ = 0 yielding
K
n
= hn|
+ γO
1
+ γ
2
O
2
|0i + O(γ
3
O
3
), (68)
where
O
1
=
Z
1
0
dx e
ix
π
2
J
y
J
+
a − J
−
a
†
e
−ix
π
2
J
y
O
2
=
Z
1
0
dx
Z
x
0
dy e
iy
π
2
J
y
J
+
a − J
−
a
†
e
i(x−y)
π
2
J
y
×
J
+
a − J
−
a
†
e
−ix
π
2
J
y
. (69)
We note that, since ||[J
+
a − J
−
a
†
]|| ≤ N, kγ
3
O
3
k ≤
γ
3
N
3
and thus we only consider terms up to second
order in γ. As J
±
= J
x
± iJ
y
, using the property
e
iθJ
y
J
α
e
−iθJ
y
=
X
β∈(x,y,z)
R
y
(θ)
αβ
J
β
, (70)
and noting that, up to second order in γ, only the
Kraus operators K
0
and K
1
are of relevance, one ob-
tains
K
0
= −
2γ
2
π
2
L
†
A L
K
1
=
−2γ
π
L
†
· v, (71)
where L = (J
z
, J
+
, J
−
)
T
and
A =
1 2 −
3π
4
π
4
π−4
4
(π−2)
2
16
20−π(π+4))
16
π+4
4
−12−π(π−4)
16
(π+2)
2
16
,
v =
1
2−π
4
2+π
2
. (72)
We quantify the distance between the ideal mea-
surement and the noisy one given in Eq. (66), by
calculating the average overlap between the ideal pro-
jectors |N/2, mihN/2, m| and the corresponding noisy
ones E[|N/2, mihN/2, m|]. The latter is given by
¯
f ≡
1
N+1
P
N
n=0
tr[|N/2, mihN/2, m|, E[|N/2, mihN/2, m|]]
which reads
¯
f =
1
N + 1
N
X
m=0
N
2
, m
K
0
N
2
, m
2
+
N
2
, m
K
1
N
2
, m
2
. (73)
Observing that only terms involving J
2
z
, J
+
J
−
and
J
−
J
+
contribute in the first summand of Eq. (73) and
defining
z
m
=
2m − N
2
r
+
m
=
p
(m + 1)(N − m)
r
−
m
=
p
m(N − (m − 1)), (74)
Accepted in Quantum 2017-10-31, click title to verify 13
Eq. (73) reads
¯
f = 1 −
4γ
2
π
2
(N + 1)
N
X
m=0
(A
11
− v
2
1
)z
2
m
+A
22
r
−
(m+1)
r
+
m
+ A
33
r
+
(m−1)
r
−
m
= 1 −
γ
2
N(N + 2)(4 + π
2
)
12π
2
+ O(γ
3
N
3
) (75)
where A
ij
(v
i
) denotes the elements of matrix A (vec-
tor v) in Eq. (72). Recalling that γ = π/4α gives
¯
f = 1 − 0.07
N(N + 2)
α
2
+ O(γ
3
N
3
). (76)
Thus, the error we incur in performing the ideal mea-
surement along the x direction for a spin N/2 system
scales inversely proportional with the average num-
ber of photons present in our coherent laser beam.
For the case of a spin-1/2 system, the average error
reads
¯
f = 1 − 0.22α
−2
.
It is instructive to consider precisely the effect of
Eq. (66) on the projectors, |N/2, mihN/2, m|. The
resulting measurement is given by
E[|N/2, mihN/2, m|] = |N/2, mihN/2, m|
+
1
4α
2
v
T
L |N/2, mihN/2, m|L
†
v
−
1
16α
2
L
†
AL |N/2, mihN/2, m| + |N/2, mihN/2, m|L
†
AL
= |N/2, mihN/2, m| + ∆
m
+ ∆
T
m
, (77)
where the matrix ∆
m
has entries only within the
range (m −2, . . . , m + 2) and is explicitly given by
∆
m
=
−1
8α
2
0 0 0 0 0
0 −v
2
3
r
+2
m−1
0 0 0
A
23
r
+
m−2
r
+
m−1
(A
13
z
m−1
+ (A
21
− 2v
1
v
3
)z
m
) r
+
m−1
(A
11
− v
2
1
)z
2
m
+ A
33
r
+2
m−1
+ A
22
r
+
2
m
0 0
0 −2v
2
v
3
r
+
m−1
r
+
m
((A
31
− 2v
1
v
2
)z
m
+ A
12
z
m+1
) r
+
m
−v
2
2
r
+2
m
0
0 0 A
32
r
+
m
r
+
m+1
0 0
.
(78)
C Distinguishing macroscopic super-
positions
In this appendix we provide the detailed calculation
for discriminating macroscopic superpositions from
their incoherent mixtures. We first consider the pho-
tonic case (Appendix C.1), followed by the case of
macroscopic superpositions of positional eigenstates
(Appendix C.2).
C.1 Photonic macroscopic superpositions
Our goal is to distinguish the macroscopic superpo-
sition of Eq. (3), from the corresponding incoherent
mixture Eq. (17) using the coherent state |αi as a RF.
The joint RF-plus-system is invariant under phase
shifts, which in this instance implies that the total
photon number K = N
RF
+ N
S
is conserved. Hence,
superpositions of states of different total photon num-
ber are not allowed, as such states require a global
RF—a clock—relative to which the phases of such su-
perpositions are defined.
Operationally, this lack of knowledge, is described
by a group of transformations {V (θ) ≡ Γ
(RF)
(θ) ⊗
U
(S)
(θ); θ ∈ (0, 2π]}, that describe the relevant phase
reference. Here Γ
(RF)
(θ) describes the action of the
group transformation on the state space, H
(RF)
, of
the RF, whereas U
(S)
(θ) describes the corresponding
action on the state space, H
(S)
, of the system. It
is important to note that both system and RF are
transformed by the same θ.
The G-twirled version of the joint RF-plus-system
state (Eq. (58)) then reads
G[ρ
RF
⊗ σ
S
] =
1
2π
Z
2π
0
Γ
(RF)
(θ) ⊗ U
(S)
(θ) [ρ
RF
⊗ σ
S
]
Γ
(RF)
(θ)
†
⊗ U
(S)
(θ)
†
dθ, (79)
where ρ
RF
∈ B(H
(RF)
), σ
S
∈ B(H
(S)
), and 1/2π
corresponds to the invariant measure of the group
and represents our complete lack of knowledge of the
global background clock.
Decomposing H
(RF)
⊗H
(S)
into global and relative
photon number states (Eq. (60)) and using Eq. (61)
it follows that
G[ρ
RF
⊗ σ
S
] =
M
K
p
K
ω
K
, (80)
where p
k
= tr[ρ
RF
⊗ σ
S
Π
(K)
], ω
K
=
Π
(K)
ρ
RF
⊗σ
S
Π
(K)
p
K
and Π
K
=
P
N
S
|K − N
S
ihK − N
S
| ⊗ |N
S
ihN
S
|.
Now consider the case where the RF is given by
|αi =
∞
X
n=0
√
q
n
|ni (81)
where q
n
= |α|
2
ne
−|α|
2
/n!. Under the G-twirling
map, one can verify that
G[|RihR|⊗|ψihψ|] =
1
2
N−1
M
K=0
q
K
Π
(K)
∞
M
K=N
σ
K
G[|RihR|⊗ψ] =
1
2
N−1
M
K=0
q
K
Π
(K)
∞
M
K=N
ρ
K
, (82)
Accepted in Quantum 2017-10-31, click title to verify 14
with
σ
K
=
q
K
√
q
K
q
K−N
√
q
K
q
K−N
q
K−N
ρ
K
=
q
K
0
0 q
K−N
. (83)
We are now ready to compute the trace distance
(Eq. (8)) between the two state in Eq. (82). We note
that the trace distance between two states ρ and σ
can also be expressed as
t = ||ρ − σ|| =
1
2
Tr
q
(ρ − σ)
†
(ρ − σ)
. (84)
As the only non-trivial contributions come from good
quantum numbers with K ≥ N, and by making use
of the Gaussian approximation to the Poissonian dis-
tribution, the trace distance reads
t =
1
p
8π|α|
2
Z
∞
x=N
e
−(|α|
2
+N−x)
2
4|α|
2
e
−(|α|
2
−x)
2
4|α|
2
dx
=
e
−
N
2
8|α|
2
4
Erfc
N − |α|
2
p
2|α|
2
!
. (85)
Hence, if |α|
2
> O(N
2
), and noting that
Erfc
1−N
√
2
→ 2 for large N, the trace distance tends
to its maximum value of 1/2.
C.2 Positional macroscopic superpositions
We now consider the case of distinguishing the spa-
tial superposition of Eq. (5) from the corresponding
incoherent mixture of Eq. (29) using as a classical
measuring device comprised of K localized particles
each with mass m
0
and the same spread σ
0
and in the
state
|µ
i
i =
Z
∞
−∞
e
−(µ
i
−x)
2
4σ
2
0
(2πσ
2
0
)
1
4
|xidx. (86)
Here, |xi are the generalized eigenstates of the posi-
tion operator X, satisfying hx|x
0
i = δ(x − x
0
), µ
i
is
the mean position of particle i.
It is convenient to express the state of the device,
|RFi =
N
K
i=1
|µ
i
i, in terms of the center of mass po-
sition of the K systems, X =
1
K
P
K
i=1
(µ
i
− x
i
),
|RFi =
Z
∞
−∞
dX
R
dx
2
. . . dx
K
"
1
(2π
Q
K
i=1
σ
2
0
)
1
2
e
−
KX−
P
K
i=2
(µ
i
−x
i
)
2
2σ
2
0
K
Y
i=2
e
−
(
µ
i
−x
i
)
2
2σ
2
0
1
2
|Xi ⊗|x
2
. . . x
K
i.
(87)
As different states with the same center of mass posi-
tion are equivalent, defining
RF(X, x) = e
−
KX
R
−
P
K
i=2
(µ
i
−x
i
)
2
2σ
2
0
K
Y
i=2
e
−
(
µ
i
−x
i
)
2
2σ
2
0
(88)
and noting that the integrals over x
2
. . . x
K
are suc-
cessive convolutions of Gaussians the state of our clas-
sical RF can be written as
|RFi =
Z
∞
−∞
p
RF(X) |Xi dz, (89)
where
RF(X) =
e
−
KX
2
2σ
2
0
q
2πσ
2
0
K
(90)
Let us focus on the one-dimensional case, where the
lack of a momentum RF is described by the group
of boosts v ∈ (0, ∞). This implies that the cen-
ter of mass position of the system-plus-device X
C
=
mx+MX
m+M
, where the system is of mass m and the
RF has a total mass of M = Km
0
, is conserved.
Observe that unlike the case of a phase reference,
the group of co-linear boosts is not compact and
thus does not have a uniform Haar measure. De-
fine {Γ
(RF )
(v); v ∈ (0, ∞)} and {V
(S)
(v); v ∈ (0, ∞)}
the unbounded unitary representations of the one-
dimensional boosts acting on the state spaces, H
(RF)
,
and H
(S)
of the device and system respectively via
Γ
(RF)
(v) |RFi =
Z
∞
−∞
p
RF(X) e
ivX
|Xi dX ∈ H
(RF)
V
(S)
(v) |ψi =
1
√
2
x −
L
2
+ e
ivL
x +
L
2
∈ H
(S)
.
(91)
Then, the G-twirled version of the joint system-plus-
device state, ρ
RF
⊗σ
S
, that arises due to not knowing
v reads
G[ρ
RF
⊗ |ΨihΨ|] = lim
v→∞
1
v
Z
v
0
e
iv(X+x)
(ρ
RF
⊗|ΨihΨ|) e
−iv(X+x)
dv
=
∞
M
X
C
=−∞
RF(X)
2
√
RF(X)RF(X+
mL
M
)
2
√
RF(X)RF(X+
mL
M
)
2
RF(X+
mL
M
)
2
.
(92)
Here we have used the fact that the eigenspaces of
the total center of mass operator X
C
=
mx+MX
m+M
,
are all two-fold degenerate, namely
X
C
, ±
L
2
=
(
m
M
+ 1)X
C
∓
mL
2M
RF
⊗
±
mL
2M
S
7
.
7
We note that as only the relative distance between the two
states matters, when computing the overlap of the two gaussian
distributions. The relative distance is of course given by
mL
M
.
Accepted in Quantum 2017-10-31, click title to verify 15
A similar analysis gives for the joint system-plus-
device state |RFihRF|⊗ Ψ
G[ρ
RF
⊗ Ψ] =
∞
M
z=−∞
RF(X)
2
0
0
RF(X+
mL
M
)
2
!
. (93)
Computing the trace distance of the states in
Eqs. (92, 93) one obtains Eq. (36).
D Macroscopic RFs and relative orien-
tations
Here we consider two copies of a macroscopic super-
position for spin, Eq. (4), where one copy serves as the
required RF for the other, in order to certify macro-
scopic superpositions. Specifically, we are interested
in the situation where the total system is composed
of 2N spin-
1
2
particles and our goal is to distinguish
between the state
|Ψi|Ψi =
1
2
|↑i
⊗N
+ |↓i
⊗N
⊗2
(94)
and the corresponding mixture
Ψ ⊗ Ψ =
1
4
(|↑ih↑|)
⊗N
+ (|↓ih↓|)
⊗N
⊗2
. (95)
The difference between the two density operators ∆ ≡
|ΨihΨ| ⊗ |ΨihΨ|− Ψ ⊗ Ψ is given by
∆ =
1
2
(|↑ih↓|
⊗N
+ |↓ih↑|
⊗N
) ⊗ Ψ
| {z }
≡∆
1
+
1
2
Ψ ⊗ (|↑ih↓|
⊗N
+ |↓ih↑|
⊗N
)
| {z }
≡∆
2
+
1
4
|↑ih↓|
⊗N
⊗ |↓ih↑|
⊗N
+ |↓ih↑|
⊗N
⊗ |↑ih↓|
⊗N
| {z }
≡∆
12
(96)
As the relevant symmetry group is that of orienta-
tions, i.e., G = su(2), the G-twirling operation, acting
on an arbitrary state % ∈ B(H
⊗2N
), is given by
G[%] =
N
X
J=0
(D
M
J
⊗ I
N
J
) Π
J
%Π
J
, (97)
where the total state space H
⊗2N
decomposes ac-
cording to Eq. (60) with M
J
= span{|J, Mi}
J
M=−J
the spaces of spin-J systems (on which rotations
act non-trivially), and N
J
= span{|α
J,k
i}
n
J
k=1
, with
n
J
=
2J+1
N+J+1
2N
N+J
, the spaces on which permuta-
tions of qubits act non-trivially. Recall that the D
M
J
is the completely depolarizing map, I
N
J
is the identity
map and Π
J
=
P
J
M=−J
P
n
J
k=1
|J, M, α
J,k
ihJ, M, α
J,k
|.
Observe that as the G-twirling map destroys coher-
ences between states with a different number of spin-
up and spin-down components, G(∆
1
) = G(∆
2
) = 0.
The measurement that maximizes the trace dis-
tance of Eq. (8) corresponds to simply measuring the
total angular momentum J. It follows that for this
measurement the difference between the probabilities
∆P
j
≡ P
J
(|Ψi|Ψi) − P
J
(Ψ ⊗ Ψ) to observe the out-
come J for the two states is
∆P
J
=
J
X
M=−J
tr (Π
J,M
G[∆
12
]) =
J
X
M=−J
tr (G[Π
J,M
]∆
12
)
=
J
X
M=−J
tr (Π
J,M
∆
12
) = tr Π
J,0
∆
12
= n
j
n
J
X
k=1
hJ, 0, α
J,k
|∆
12
|J, 0, α
J,k
i. (98)
For ease of calculation it is preferable to use the
following overcomplete basis for the expansion of Π
J,0
in terms of qubits. Define
|J, 0, ki =
Ψ
−
(12)
. . .
Ψ
−
2(N−J−1),2(N−J )−1
|J, 0i
(99)
where k = (1, 2; . . . , 2(N −J −1), 2(N −J) −1) labels
the N − J singlets, and |J, 0i is a totally symmetric
state of the remaining 2J qubits with M = 0. Then
the projector Π
J,0
is proportional to the sum over all
inequivalent projectors |J, 0, ki of Eq. (99) obtained
by permuting all 2N qubits with the proportionality
constant given by the dimension n
J
[51]. In order to
compute Eq. (98) we need to determine which of these
projectors have a non-zero overlap with ∆
12
as well
as the value of this overlap.
To avoid counting each singlet twice we impose the
first qubit in each singlet to have a lower position
than the second. There are in total
2N
2N−2J
to chose
the 2N −2J particles to carry the singlet states, and
on top of this (2N − 2J − 1)!! inequivalent ways to
pair the particles in this group, yielding a total of
2N
2N−2J
(2N − 2J − 1)!! inequivalent states. From
the form of ∆
12
it is clear that a non-zero overlap
is achieved if and only if the first particle from each
singlet can be found in any of the first 0−N positions
(first half) and the second in any of the (N + 1) −2N
positions (second half). There in total
N
N−J
2
ways
to chose N − J spins from the first half and N − J
spins from the second. Finally there are (N −J)! ways
of pairing particles from the first group to those of the
second yielding that total number of state with non-
zero overlap with ∆
12
of
N
N−J
2
(N −J)!. Hence, the
probability that a random state from this overcom-
plete basis overlaps with ∆
12
is given by
β =
N
N−J
2
(N − J)!
2N
2N−2J
(2N − 2J −1)!!
. (100)
Using the fact that
↑↓ |Ψ
−
Ψ
−
| ↓↑
=
↓↑ |Ψ
−
Ψ
−
| ↑↓
= −
1
2
(101)
Accepted in Quantum 2017-10-31, click title to verify 16
Eq. (98) reads
∆P
J
= n
J
β
(−1)
N−J
2
N−J+1
tr(|J, 0ihJ, 0|((|↑ih↓|)
⊗J
))
= n
J
β
(−1)
N−J
2
N−J+1
2J
J
, (102)
and the trace distance yields
t =
1
2
N
X
j=0
|∆P
J
| =
1
4
. (103)
E General macroscopic quantum
states
We now address the demand on the size of the relevant
RF needed in order to distinguish a general macro-
scopic quantum state
|Ψi =
X
n
ψ
n
|ni, (104)
where we assume without loss of generality ψ
n
∈
R, ∀n, from the suitable semi-classical state given by
ρ
cl
(σ) = E
σ
(|Ψi) =
X
n,m
ψ
n
ψ
m
c
n−m
|nihm| (105)
with c
n−m
≡ e
−
(n−m)
2
2σ
2
. As mentioned in Sec. 3.4 the
matrix ρ
cl
(σ) is a valid density matrix, because it is
given by the action of a gaussian dephasing channel
E
σ
on the state |Ψi of Eq. (104).
As we are interested in detecting quantum coher-
ence between far away components |ni and |mi let
us introduce a size parameter N. Furthermore, we
shall assume a gaussian noise channel with σ = N
β
,
and consider the coherences of the semi-classical state
ρ
cl
(N
β
) for n − m = N
c
. In the limit of N → ∞
these coherence terms vanish if c > β meaning that
the state ρ
cl
(N
β
) exhibits no coherence between terms
separated by N
β
. We will now show that, if the size
of the measuring device is limited, for some choices
of β < 1, and after we apply the G-twirling map of
Eq. (58), the state ρ
cl
[N
β
] is indistinguishable from
|Ψi.
To that end, consider a RF whose size (given by
its variance with respect to the conserved quantity in
question) is M = N
2−ε
. In the large N limit the state
of the RF is well approximated by a Gaussian, which
we assume without loss of generality to be centred at
zero, i.e.,
|RFi =
∞
X
n=−∞
r
n
|ni (106)
After applying the G-twirling map of Eq. (58) we ob-
tain for the macroscopic state
G [|ΨihΨ| ⊗ |RFihRF|] =
X
k
P
k
Ψ
(k)
ED
Ψ
(k)
(107)
where
Ψ
(k)
E
=
X
n
ψ
n
r
k−n
√
P
k
|ni
S
|k − ni
RF
≡
X
n
λ
(k)
n
n
(k)
E
, (108)
and P
k
≡
P
n
(ψ
n
r
k−n
)
2
. Similarly the G-twirling of
the semi-classical state with the RF reads
G
ρ
cl
(N
β
) ⊗ |RFihRF|
=
X
k
P
k
ρ
(k)
cl
(N
β
), (109)
where
ρ
(k)
cl
(N
β
) =
X
n,m
λ
(k)
n
λ
(k)
m
e
−(n−m)
2
2N
2β
n
(k)
ED
m
(k)
.
(110)
Due to the block diagonal structure the trace dis-
tance is additive with respect to the blocks k
t =
X
k
P
k
1
2
Ψ
(k)
ED
Ψ
(k)
− ρ
(k)
cl
(N
β
)
≡
X
k
P
k
t
k
(111)
Using the relation t ≤
√
1 − F
2
, between the trace
distance and fidelity [42] we obtain
t
k
≤
q
1 −
Ψ
(k)
ρ
(k)
cl
(N
β
)
Ψ
(k)
=
v
u
u
t
X
n,m
(λ
(k)
n
)
2
(λ
(k)
m
)
2
1 − e
−
(n−m)
2
2N
2β
≤
v
u
u
t
X
n,m
(λ
(k)
n
)
2
(λ
(k)
m
)
2
(n − m)
2
2N
2β
=
1
N
β
v
u
u
t
X
n
n
2
(λ
(k)
n
)
2
−
X
n
n(λ
(k)
n
)
2
!
2
=
q
Var
(λ
(k)
)
2
N
β
, (112)
where we have used the inequality 1 − e
−x
2
≤ x
2
in
going from the second to the third line. It follows that
t ≤
P
k
P
k
q
Var
(λ
(k)
)
2
N
β
, (113)
and by Jensen’s inequality, E
√
X
k
≤
p
E (X
k
),
t ≤
s
P
k
P
k
Var
(λ
(k)
)
2
N
2β
, (114)
where Var
(λ
(k)
)
2
= hn
2
i
k
− hni
2
k
is the vari-
ance of the probability distribution {(λ
(k)
)
2
} in sec-
tor k. Employing Jensen’s inequality a second
Accepted in Quantum 2017-10-31, click title to verify 17
time,
P
k
P
k
hni
2
k
≥ (
P
k
P
k
hni
k
)
2
, and recalling that
(λ
(k)
)
2
=
ψ
2
n
r
2
n−k
P
k
, we arrive at
t ≤
P
n
n
2
(
P
k
ψ
2
n−k
)r
2
n
− (
P
n
n(
P
k
ψ
2
n−k
)r
2
n
)
2
N
2β
1/2
.
(115)
As
P
k
ψ
2
n−k
= 1, and recalling that r
2
n−k
has variance
N
2−
, the last expression becomes
t ≤
r
Var (r
2
n
)
N
2β
=
√
N
2−ε−2β
, (116)
and goes to zero so long as β > 1 −
ε
2
.
Thus, we have shown that with a RF of size N
2−ε
the superposition state |Ψi is indistinguishable from
a semi-classical state ρ
cl
, which does not exhibit any
quantum coherence on the scale n − m = N
c
for
c > 1 −
ε
2
. (117)
In the case where the RF state is itself a mix-
ture ρ
RF
=
P
i
q
i
|RF
i
ihRF
i
|, one obtains t ≤
P
i
q
i
s
Var
(r
(i)
n
)
2
N
2β
instead of Eq. (116). Another ap-
plication of Jensen’s inequality implies
t ≤
v
u
u
t
P
i
q
i
Var
(r
(i)
n
)
2
N
2β
(118)
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