
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