# Bipartite Leggett-Garg and macroscopic Bell inequality violations using cat
states: distinguishing weak and deterministic macroscopic realism

Manushan Thenabadu and M. D. Reid1 1 Centre for Quantum Science and Technology
Theory, Swinburne University of Technology, Melbourne 3122, Australia

###### Abstract

We consider tests of Leggett-Garg’s macrorealism and of macroscopic local
realism, where for spacelike separated measurements the assumption of
macroscopic noninvasive measurability is justified by that of macroscopic
locality. We give a mapping between the Bell and Leggett-Garg experiments for
microscopic qubits based on spin $1/2$ eigenstates and gedanken experiments
for macroscopic qubits based on two macroscopically distinct coherent states
(cat states). In this mapping, the unitary rotation of the Stern-Gerlach
analyzer is realized by an interaction $H=\Omega\hat{n}^{4}$ where $\hat{n}$
is the number of quanta. By adjusting the time of interaction, one alters the
measurement setting. We thus predict violations of Leggett-Garg and Bell
inequalities in a macroscopic regime where coarse-grained measurements
$\hat{M}$ need only discriminate between two macroscopically distinct coherent
states. To interpret the violations, we distinguish between different
definitions of macroscopic realism. Deterministic macroscopic local realism
(dMR) assumes a definite outcome for the measurement $\hat{M}$ prior to the
unitary rotation created by the analyser, and is negated by the violations.
Weak macroscopic realism (wMR) assumes a definite outcome for systems prepared
in a superposition $\psi_{pointer}$ of two macroscopically-distinct
eigenstates of $\hat{M}$, after the unitary rotation. We find that wMR can be
viewed as consistent with the violations. A model is presented, in which wMR
holds, and for which the macroscopic violations emerge over the course of the
unitary dynamics. Finally, we point out an EPR-type paradox, that a weak
macro-realistic description for the system prior to the measurement $\hat{M}$
is inconsistent with the completeness of quantum mechanics.

##  I Introduction

Bell’s theorem constrains the predictions of all local hidden variable
theories to values governed by inequalities bell-brunner-rmp ; CShim-review ;
Bell-2 ; det-bell . These inequalities can be violated by quantum mechanics,
giving a falsification of both local realism and local causality. Bell tests
involve measurements made on two separated particles, performed on timescales
such that the two measurement events are spacelike separated. At each
location, there is a choice between two measurement settings. To date, almost
all Bell tests have examined microscopic systems. Bell violations have been
predicted for large numbers of particles meso-bell-higher-spin-cat-states-bell
; cv-bell ; cat-bell ; mdr-mlr ; macro-bell-additional ; cv-bell-macro-ali ,
but, in almost all cases, the measurements for one of the settings require a
resolution of a few particles, or at the level of Planck’s constant.

By contrast, Leggett and Garg derived inequalities that do not require such
fine resolution of measurement legggarg-1 . Leggett-Garg inequalities are
designed to falsify the assumption of _macrorealism_ (M-R) for systems that at
certain times are in a superposition (e.g.
$\psi_{\alpha}\sim|a\rangle+|d\rangle$) of two macroscopically distinct states
s-cat-1 . M-R combines two premises: Macroscopic realism (MR) asserts that the
system must actually be in one or other of those two states at the given time.
The second premise, macroscopic noninvasive measurability (NIM), asserts the
existence of a measurement $\hat{M}$ that can distinguish between the two
macroscopic states, with negligible effect on the subsequent dynamics (at a
macroscopic level). Macroscopic realism implies a predetermined value
$\lambda_{M}$ for the outcome of the measurement $\hat{M}$ legggarg-1 . Since
the two states $|a\rangle$ and $|d\rangle$ can be distinguished by a
measurement $\hat{M}$ allowing a macroscopic uncertainty, the hidden variable
$\lambda_{M}$ need not specify the results of measurements to a resolution of
$\hbar$. This was recognised by Leggett and Garg who assigned the values
$\lambda_{M}=\pm 1$ for the two outcomes of $\hat{M}$, in order to derive
inequalities legggarg-1 .

Leggett-Garg inequalities have been reported violated for a wide range of
systems emary-review ; jordan_kickedqndlg2-1 ; weak-solid-state-qubits ;
NSTmunro-1 ; massiveosci-1-1 ; manushan-cat-lg ; weak-hybrid ; leggett-garg-
recent ; macro-bell-lg ; nst . A complication for the interpretation of the
violations has been the justification of the NIM premise for practical
measurements leggett-garg-recent . Different strategies are used, including
(in a recent macroscopic superconducting experiment) a control experiment
which quantifies the amount of disturbance induced by the measurement, if the
system is indeed in one of the states $|a\rangle$ or $|d\rangle$ NSTmunro-1 .
An alternative strategy (that does not assume the system to be in one of the
fully specified quantum states) is to violate bipartite Leggett-Garg or
macroscopic Bell inequalities macro-bell-lg ; weak-hybrid ; cv-bell-macro-ali
. Here, the NIM premise is replaced by that of macroscopic locality. Recent
work gives predictions for such violations using NOON and multi-component cat
states macro-bell-lg . The question becomes how to interpret such violations.
In particular, we ask whether the assumption of the macroscopic hidden
variable $\lambda_{M}$ for the coarse-grained measurement $\hat{M}$ can be
negated.

In this paper, we give full details of the results presented in the Letter
letter , which analyses this question. To facilitate the analysis, we give
predictions of violations of macroscopic Bell inequalities and of bipartite
Leggett-Garg inequalities for the conceptually simple case where a system at
certain times is found to be in a superposition of two macroscopically
distinct coherent states, $|\alpha\rangle$ and $|-\alpha\rangle$ (cat states),
as $\alpha\rightarrow\infty$. Similar to the results of Ref. macro-bell-lg ,
the violations are obtained using only measurements $\hat{M}$ which
discriminate between the two coherent states, thereby allowing macroscopic
uncertainties.

In order to interpret the violations, we are first careful to define
macroscopic realism in the weakest (i.e. most minimal) sense, as _weak
macroscopic realism_ (wMR), the premise that a macroscopic hidden variable
$\lambda_{M}$ specifies the outcome of the __ coarse-grained pointer
measurement $\hat{M}$. In contrast to many previous macrorealism tests, we
find it is not necessary to include in the definition of macroscopic realism
that the system be in one or other of two specific quantum states (e.g.
$|\alpha\rangle$ and $|-\alpha\rangle$), or even that the system be in one or
other of two unspecified quantum states. The violations given in this paper
therefore provide strong tests of both macrorealism (M-R) and macroscopic
local realism (MLR), and also of macroscopic local causality (MLC).

In this paper, we further weaken the definition of weak macroscopic realism,
by specifying $\hat{M}$ to be a pointer measurement, and the distinct states
$|a\rangle$ and $|d\rangle$ to be states with a definite outcome for the
(coarse-grained) $\hat{M}$. In other words, the assumption of wMR asserts the
validity of the macroscopic hidden variable $\lambda_{M}$ for the state
$\psi_{\alpha}$ prepared _after_ a unitary transformation $U$ that determines
the choice of measurement setting. We show that the violations of M-R and MLR/
MLC given in this paper can be viewed _consistently_ with wMR.

On the other hand, we show that the violations falsify _deterministic
macroscopic realism_ (dMR), which specifies well-defined values $\lambda_{M}$
_prior_ to such a unitary transformation. The assumption of dMR also includes
the assumption of macroscopic locality (ML), which states that the value of
$\lambda_{M}$ cannot be changed by spacelike separated measurement events at
another site. The premise of dMR implies the validity of _two_ hidden
variables, $\lambda_{M_{\theta}}$ and $\lambda_{M_{\theta^{\prime}}}$,
ascribed to the system _simultaneously_ (prior to $U$) to predetermine the
outcome of two pointer measurements, $\hat{M}_{\theta}$ and
$\hat{M}_{\theta^{\prime}}$, and is negated by the violation of the
macroscopic Bell inequalities.

We find that the consistency of the violations with wMR is possible because of
the unitary dynamics associated with the choice of measurement setting. The
unitary rotation $U=e^{-iHt/\hbar}$ that determines the measurement setting
transforms the system from one pointer superposition to another, over a time
interval $\Delta t$. The system is therefore not considered to be in a
superposition of eigenstates of both pointer measurements, $\hat{M}_{\theta}$
and $\hat{M}_{\theta^{\prime}}$, simultaneously. In our analysis, the
transformation $U$ that determines the measurement setting is given by a
nonlinear Hamiltonian $H=\Omega\hat{n}^{4}$ where $\hat{n}$ is the mode number
operator. The dynamics leading to the macroscopic violations can therefore be
observed over the interval $\Delta t$. Recent work predicts violations of Bell
inequalities at a microscopic level for the dynamical trajectories of two
entangled particles bell-trajectories . This is consistent with our conclusion
that deterministic macroscopic realism fails, since the choice of measurement
setting involves dynamics, as given by $U$.

There are open questions. While we show consistency with wMR, the results of
this paper neither falsify nor validate this premise. The validation of this
premise might be expected, in view of the known emergence of classicality with
coarse-grained measurements coarse-3 ; coarse-peres ; kofler-bruck-leggett-
garg-coarse . The concept might also be consistent with recent proposals to
interpret quantum mechanics using theories based on multiple interacting
classical worlds mhall , or with an epistemic restriction of order Planck’s
constant budiyono ; bartlett-gaussian ; macro-pointer ; q-trajectories-
drummond , and may also be consistent with recent interpretations of the
double-slit experiment double-slit-Aharonov .

However, it is clear from the violations of the Leggett-Garg inequalities
presented in this paper (and elsewhere) that if wMR holds, then the dynamics
associated with the macroscopic hidden variables $\lambda_{M}$ _cannot_ be
given by classical mechanics. _If_ wMR holds, we show for the tests of this
paper that the violations of the Leggett-Garg inequalities would therefore
arise from the breakdown of the NIM assumption. We explain how this breakdown
arises (in the bipartite model) from quantum nonlocality, the new feature
being that there is a _macroscopic_ nonlocality (associated with
macroscopically distinct __ qubits $|\alpha\rangle$ and $|-\alpha\rangle$),
which we show emerges over the timescales associated with the unitary
interactions $U$ at _both_ sites. While the distinction between predictions
for the superposition $\psi_{\alpha}$ and the classical mixture $\rho_{mix}$
of the two macroscopically distinct states is negligible (of order $\hbar
e^{-|\alpha|^{2}}$), a _macroscopic_ difference emerges over the course of the
dynamics corresponding to the unitary rotations.

From an alternative point of view, we summarise in this paper that if one does
postulate the validity of wMR, then inconsistencies arise. This can be shown
in the form of an EPR-type paradox epr-1 , similar to that given in earlier
papers macro-coherence-paradox ; eric_marg-1 ; irrealism-fringes ; macro-
pointer . If one assumes wMR, then the assumption that the system is in one of
two _quantum_ states, such as $|\alpha\rangle$ or $|-\alpha\rangle$, can be
negated. Hence, wMR is inconsistent with the completeness of quantum
mechanics, which gives predictions at the level of $\hbar$.

_Layout of paper_ : The paper is organised as follows. In Section II, we
review and extend previous work to give derivations of bipartite Leggett-Garg
and macroscopic Bell inequalities. __The macroscopic Bell inequalities are
derived assuming _deterministic macroscopic (local) realism_ or _macroscopic
local causality_ , both of which are negated by violations of the inequalities
given in this paper.__

In Section III, we demonstrate a mapping between the spin-$1/2$ Bell
experiments performed on microscopic qubits $|\uparrow\rangle$ and
$|\downarrow\rangle$, and a gedanken experiment performed on macroscopic
qubits, $|\alpha\rangle$ and $|-\alpha\rangle$. The unitary rotation of the
polariser or spin analyser is mapped to a transformation
$|\alpha\rangle\rightarrow\cos\theta|\alpha\rangle+i\sin\theta|-\alpha\rangle$,
which is realized for $\theta=0,\pi/8,\pi/4$ and $3\pi/8$ by an evolution
$U=e^{-iHt/\hbar}$ at certain times $t$ manushan-cat-lg . We thus show that
macroscopic Bell inequalities are violated for spacelike separated systems,
$A$ and $B$, prepared in an entangled two-mode cat state
$|\psi\rangle\sim|\alpha\rangle|-\beta\rangle-|-\alpha\rangle|\beta\rangle$.
The mapping relies on the orthogonality of the coherent states, which strictly
holds only for $\alpha$, $\beta\rightarrow\infty$. In our explicit model, we
give full calculations, and compute the values of $\alpha$ and $\beta$ for
which experiments could be performed.

In Section IV, we predict violations of Leggett-Garg’s macrorealism, for both
single and bipartite cat-state systems, following along the lines of Ref.
manushan-cat-lg . In the Bell test, the angle $\theta$ is the measurement
setting. In the Leggett-Garg test, the angle $\theta$ gives the time $t$ of
evolution between measurements $\hat{M}$ made at different times. The dynamics
associated with the unitary evolution $U=e^{-iHt/\hbar}$ can be visualised by
plotting the $Q$ function Husimi-Q-1 . We explain _weak macroscopic realism_
(wMR) in Sections III.E and also in Section VI. Evaluation of the $Q$ function
dynamics is given in Sections III.D and IV.B.2.

In Section V, we examine the measurement process more carefully. For the
bipartite Leggett-Garg test, whether the measurement is performed or not on
system $A$ is determined by the duration of unitary evolution, at the second
location $B$. By contrast, the timing of the final irreversible (“collapse”)
stage of the measurement at $B$ is unimportant. We show by calculation that
whether the collapse occurs before or after the subsequent evolution at $A$ is
irrelevant to the violations, making only a difference of order
$\mathcal{\hbar}e^{-|\alpha|^{2}}$ to the final probabilities. Here we point
out that although the results of this paper are consistent with weak
macroscopic realism (wMR), it is possible to establish inconsistencies with
quantum mechanics, similar to an EPR-type paradox. Finally, a conclusion is
given in Section VI.

##  II Macroscopic Bell and bipartite Leggett-Garg inequalities

We first outline and extend the Bell and bipartite Leggett-Garg inequalities
for a macroscopic system, as proposed in Refs. weak-hybrid ; macro-bell-lg .
In order to test macroscopic realism, we combine aspects of the original Bell
and Leggett-Garg proposals. As with Bell’s proposal, there are two spatially
separated systems $A$ and $B$ upon which local measurements are made. Similar
to the Leggett-Garg proposal, the systems at each location evolve dynamically
according to a local Hamiltonian $H^{(A)}$ (or $H^{(B)}$), so that
measurements can be made at different times. The measurements at each site $A$
and $B$ give only two possible outcomes, corresponding to two macroscopically
distinguishable states. The outcomes are designated as pseudo-spins,
$S_{A}=\pm 1$ and $S_{B}=\pm 1$, respectively.

In fact, we present two types of test of macroscopic realism. The first
(Section II.A) is a macroscopic Bell test, where for a given run of the
experiment, one makes single measurements at each site, $A$ and $B$. For each
measurement, there is a choice between two measurement settings, corresponding
to a choice between two different times of interaction with a measurement
device. The second type of test of macroscopic realism (Section II.B) is
similar to that proposed by Leggett and Garg, except there are two sites. In a
given run of the experiment, one makes sequential measurements at different
times $t_{1}<t_{2}<t_{3}$, on the systems at the separated sites, $A$ and $B$.

Figure 1: Bell test of macroscopic local realism using local dynamics:
Measurements $\hat{M}_{\theta}^{(A)}$ and $\hat{M}_{\phi}^{(B)}$ are made at
the sites $A$ and $B$ respectively. The measurements have two stages. At each
site, there is a choice of two measurement settings ($\theta$ and
$\theta^{\prime}$, and $\phi$ and $\phi^{\prime}$) corresponding to times
$t_{a}$ and $t^{\prime}_{a}$ at $A$, and $t_{b}$ and $t^{\prime}_{b}$ at $B$,
during which the system interacts with a local medium. The interactions at $A$
and $B$ correspond to unitary transformations $U_{A}(\theta)$ and
$U_{B}(\phi)$ respectively. After the unitary interactions, final pointer
measurements $\hat{M}^{(A)}$ and $\hat{M}^{(B)}$ are made each giving two
macroscopically distinct outcomes (denoted $\pm 1$).

###  II.1 Macroscopic Bell test

####  II.1.1 Deterministic macroscopic (local) realism

We consider the situation of Figure 1. One prepares the state of the system at
a time $t=0$. At each site there are two possible measurement settings
($\theta$ and $\theta^{\prime}$ at $A$, and $\phi$ and $\phi^{\prime}$ at $B$)
and for each of these measurements there are only two outcomes, corresponding
to states of the system that are macroscopically distinct. We refer to the
outcomes $S_{A}=\pm 1$ and $S_{B}=\pm 1$ as “spin” outcomes, and label the
measurements at sites $A$ and $B$ as $\hat{M}_{\theta}^{(A)}$ and
$\hat{M}_{\phi}^{(B)}$ respectively. Distinguishing only between
macroscopically distinct outcomes, the measurements $\hat{M}_{\theta}^{(A)}$
and $\hat{M}_{\phi}^{(B)}$ can allow a significant error, beyond the level of
$\hbar,$ as we will show by example in Sections III and IV. As such, we refer
to the measurements as coarse-grained, or macroscopic, measurements.

The assumption of _macroscopic realism_ (MR), as specified by Leggett and Garg
legggarg-1 , asserts that each system prior to the measurement
$\hat{M}_{\theta}^{(A)}$ and $\hat{M}_{\phi}^{(B)}$ is in one or other of the
two macroscopically distinguishable states, which have a definite outcome for
the spin. Here, we do not wish to restrict that the two macroscopically
distinct states are _specific_ states, $\psi_{1}$ and $\psi_{2}$, or even that
they are quantum states. The definition of MR is generalised to postulate that
prior to the measurement $\hat{M}_{\theta}^{(A)}$ and $\hat{M}_{\phi}^{(B)}$,
the system is in a state for which the outcome of the macroscopic, coarse-
grained measurement $\hat{M}_{\theta}^{(A)}$ or $\hat{M}_{\phi}^{(B)}$ is
definite, being either $+1$ or $-1$. Assuming MR, macroscopic hidden variables
$\lambda_{\theta}^{(A)}$ and $\lambda_{\phi}^{(B)}$ with values $+1$ or $-1$
can be ascribed to the systems $A$ and $B$ at the time immediately prior to
the measurement, the values giving the predetermined outcomes for the
measurements $\hat{M}_{\theta}^{(A)}$ and $\hat{M}_{\phi}^{(B)}$ respectively.
These variables, introduced by Leggett and Garg, are hidden variables, since
they are not explicitly given by quantum mechanics.

If there is a choice of measurement settings $\theta$ and $\theta^{\prime}$ at
a given site, then one assumes (based on the assumption of MR) that _two_
hidden variables $\lambda_{\theta}^{(A)}$ and
$\lambda_{\theta^{\prime}}^{(A)}$ describe the state of the system as it
exists prior to the time the choice is made, to predetermine the results of
_both_ measurements, $\hat{M}_{\theta}^{(A)}$ and
$\hat{M}_{\theta^{\prime}}^{(A)}$. Similar to the original Bell derivation
Bell-2 ; det-bell , we call the definition of MR as given here _deterministic
macroscopic realism_(dMR)_._

Assuming locality, that the value of the macroscopic hidden variable $\lambda$
for the system at one site cannot be changed by the local measurement at the
other site bell-brunner-rmp ; CShim-review ; Bell-2 ; det-bell , it is
straightforward to derive the Clauser-Horne-Shimony-Holt (CHSH) Bell
inequality, $|B|\leq 2$, where CShim-review

$\displaystyle B$ $\displaystyle=$ $\displaystyle
E(\theta,\phi)-E(\theta,\phi^{\prime})+E(\theta^{\prime},\phi)+E(\theta^{\prime},\phi^{\prime})$
(1)

Here $\theta$ and $\theta^{\prime}$ ($\phi$ and $\phi^{\prime}$) are two
measurement settings at sites A (and B) respectively, and $E(\theta,\phi)$ is
the expectation value for the product $S_{A}S_{B}$, for the choice of
settings, $\hat{M}_{\theta}^{(A)}$ and $\hat{M}_{\phi}^{(B)}$. The combined
assumptions of locality and deterministic macroscopic realism are referred to
as _deterministic macroscopic local realism_(MLR)_,_ or simply _deterministic
macroscopic realism,_ as the condition of locality is naturally implied by dMR
for two sites det-bell . The derivation of (1) follows from the assumption of
MLR, which implies $E(\theta,\phi)=\langle
S_{A}S_{B}\rangle=\langle\lambda_{\theta}^{(A)}\lambda_{\phi}^{(B)}\rangle$.

The local measurements $\hat{M}_{\theta}^{(A)}$ and $\hat{M}_{\phi}^{(B)}$
have two stages: a unitary stage $U$ for which there is an interaction with an
analyzer device, and a final stage $\hat{M}$ which includes an irreversible
coupling to a reservoir i.e. to a detector. The unitary stage determines the
measurement setting, and is analogous to the transmission of a photon or
spin-$1/2$ particle through a polarizing beam splitter or Stern-Gerlach
analyzer. Thus, in the traditional Bell tests, $\theta$ and $\phi$ are
polarizer or analyzer angles, determining the direction of polarisation or
spin that is to be measured at each site. The final pointer stage $\hat{M}$
corresponds to detection in one of the arms of the polarizing beam splitter.

For the examples given in this paper, following Refs. macro-bell-lg ;
manushan-cat-lg , the unitary stage of the measurement at each site comprises
an interaction with a nonlinear medium, and the measurement setting is
determined by the time duration of the interaction. This need not be a choice
of time itself, but can instead be a choice of the degree of nonlinearity of
the medium, the choice of measurement setting being made by a switch. We
identify at each site two choices of times for the measurements: $t_{a}$ and
$t_{a}^{\prime}$ ($t_{b}$ and $t_{b}^{\prime}$) at $A$ (and $B$) respectively.
The Bell inequality (1) becomes $|B|\leq 2$ where

$\displaystyle B$ $\displaystyle=$ $\displaystyle
E(t_{a},t_{b})-E(t^{\prime}_{a},t^{\prime}_{b})+E(t^{\prime}_{a},t_{b})+E(t^{\prime}_{a},t^{\prime}_{b})$
(2)

Modelling the measurement in this way as an interaction with a device
emphasises that the measurement will occur over a finite time interval. To
justify the locality assumption, it is assumed that the measurement events at
each site are spacelike separated.

####  II.1.2  Macroscopic local causality

When deriving (1) and (2), it is assumed that _two_ macroscopic hidden
variables describe each system at the time immediately prior to the
measurement ($t=0$) bell-brunner-rmp ; CShim-review ; Bell-2 ; det-bell .
These are $\lambda_{\theta}^{(A)}$ and $\lambda_{\theta^{\prime}}^{(A)}$ for
system $A$, and $\lambda_{\phi}^{(B)}$ and $\lambda_{\phi^{\prime}}^{(B)}$ for
system $B$. Each of these variables assumes values $+1$ or $-1$. Thus, the
system at $A$ is in a state with simultaneous predetermined values for the
measurements $\hat{M}_{\theta}$ and $\hat{M}_{\theta^{\prime}}$, and similarly
for $B$.

However, it is well known that the Bell inequality (1) can be derived with a
weaker assumption allowing for a stochastic interaction with a local
measurement device det-bell ; CShim-review . Here, one assumes that the system
at time $t=0$ is in a hidden variable state given by a set of hidden variables
$\\{\lambda\\}$ with probability density $\rho(\lambda)$. One defines the
probability $P_{A}(\pm 1|\lambda,\theta)$ for obtaining a spin outcome $+1$ or
$-1$ at site $A$, given the system prior to measurement is in the state
$\\{\lambda\\}$ and given the choice of measurement setting $\theta$ for the
local device at $A$. The probability $P_{B}(\pm 1|\lambda,\phi)$ is defined
similarly, for the spin outcome $\pm 1$ at $B$, given the measurement setting
$\phi$ for the local measurement device at $B$. The locality assumption is
that probability $P_{A}(\pm 1|\lambda,\theta)$ does not depend on $\phi$, and
$P_{B}(\pm 1|\lambda,\phi)$ does not depend on $\theta$. These assumptions can
be viewed as a single assumption, local causality det-bell .

Thus, the Bell inequalities (1) and (2) follow from a weaker assumption, that
we refer to as _macroscopic local causality_ (MLC). The assumption is similar
to local causality, except here one postulates that prior to the measurement,
the system is in a hidden variable state which gives a definite probability
for just two macroscopically distinct outcomes. In this paper, the Bell
inequalities can be derived from either MLR or MLC, and we will use MLR to
imply either definition.

###  II.2 Bipartite Leggett-Garg tests

The Leggett-Garg inequalities were derived for the situation where one
considers succcessive measurements on a single system, at times $t_{i}$. Here,
following macro-bell-lg ; weak-hybrid  and different to the original treatment
legggarg-1 , we consider two systems $A$ and $B$ (Figure 2). Measurements
$\hat{M_{i}}^{(A)}$ and $\hat{M}_{j}^{(B)}$ are made on systems $A$ and $B$ at
times $t_{i}$ and $t_{j}$ respectively. The measurements give two outcomes,
$+1$ and $-1$, corresponding to two macroscopically distinguishable states.
However, here there is no choice of measurement setting at a given time:
Rather, the unitary rotation stage $U_{A}$ (or $U_{B}$) of the measurement is
considered to be part of the dynamics. In the examples we consider in this
paper, the measurements $\hat{M_{i}}^{(A)}$ and $\hat{M}_{j}^{(B)}$ that occur
at the times $t_{i}$ and $t_{j}$ are analogous to the pointer measurements,
$\hat{M}^{(A)}$ and $\hat{M}^{(B)}$, of Figure 1.

The Leggett-Garg inequalities are derived based on the assumption of
macroscopic realism legggarg-1 . Hence one may identify specific macroscopic
hidden variables $\lambda_{i}^{(A)}$ and $\lambda_{i}^{(B)}$ to describe the
states immediately prior to the pointer measurements $\hat{M_{i}^{(A)}}$ and
$\hat{M}_{j}^{(B)}$. Different to the Bell derivation of Section II.A.1
however, one does not ascribe to the system simultaneously at a time $t_{i}$
predetermined results $\lambda_{i}^{(A)}$ and $\lambda_{j}^{(A)}$ for _both_
future measurements, $\hat{M}_{i}^{(A)}$ and $\hat{M}_{j}^{(A)}$. What was the
local unitary interaction ($U_{A}$ or $U_{B}$) with the measurement device in
the Bell test is now the local trajectory. One therefore assumes macroscopic
realism for a __ measurement with just a single setting, at the time $t_{i}$.
This means that the Leggett-Garg-Bell inequalities for the systems of interest
may be derived based on the assumption of _weak macroscopic realism_ (wMR):
that for a system $A$ prepared in a superposition of macroscopically distinct
pointer states, the system can be described by a single hidden variable
$\lambda_{i}$ at that time $t_{i}$ (and this is not affected by a space-like
separated measurement event at the different site $B$).

The traditional Leggett-Garg test invokes a second assumption, that it is in
principle possible to determine which of the two macrosopically distinct
states the system is in at any given time, by performing a _noninvasive
measurement_ that has a negligible effect on the subsequent dynamics
legggarg-1 . This assumption is necessary because the traditional Leggett-Garg
tests involve measurements at different times performed on the same system,
and there is the possibility of direct disturbance of the system due to
measurement. In this paper, we follow macro-bell-lg , and consider two systems
(Figure 2). One assumes validity of (weak) macroscopic realism, but the
additional assumption of noninvasive measureability is replaced by the
assumption of locality. Specifically, for measurements at $A$ and $B$ that are
spacelike separated events, one assumes the hidden variable value
$\lambda_{i}^{(A)}$ is not changed by whether the measurement at $B$ takes
place or not, and similarly for $\lambda_{j}^{(B)}$.

Figure 2: A Leggett-Garg test of macrorealism using bipartite systems. Two
systems $A$ and $B$ prepared in an entangled state at time $t_{1}$ undergo
dynamical evolution according to local unitary operators, $U_{A}(t)$ and
$U_{B}(t)$. At the later times $t_{2}$ and $t_{3}$, each system is found to be
in one or other of two macroscopically distinct states $-1$ or $+1$, and the
outcomes are anti-correlated. One can infer the outcome for $A$ by measuring
at $B$. A possible evolution is shown by the dashed lines.

We consider measurements made at times $t_{1}$ and $t_{3}$ at site $A$, and at
times $t_{2}$ and $t_{4}$ at site $B$, where $t_{1}<t_{2}<t_{3}<t_{4}$. Let us
denote the spin outcome $S_{A}$ at $A$ for the measurement made at time
$t_{i}$ by $S_{i}^{(A)}$. The spin $S_{j}^{(B)}$ is defined similarly, for
site $B$. With the assumptions of macroscopic realism and noninvasive
measurability (locality), the Leggett-Garg-Bell inequality

$\displaystyle-2$ $\displaystyle\leq$ $\displaystyle
E(t_{1},t_{2})-E(t_{1},t_{4})+E(t_{2},t_{3})+E(t_{3},t_{4})\leq 2$ (3)

where $E(t_{i},t_{j})=\langle S_{i}^{(A)}S_{j}^{(B)}\rangle$ can be shown to
hold legggarg-1 . This inequality was derived as a Leggett-Garg inequality in
Ref. legggarg-1 , and as a Leggett-Garg-Bell inequality, in Refs. macro-bell-
lg ; weak-hybrid . Since the expectation values $E(t_{i},t_{j})$ of the
inequality involve measurements made at diffferent sites, the inequality
follows based on the assumption of locality, which justifies that
$E(t_{i},t_{j})=\langle
S_{i}^{(A)}S_{j}^{(B)}\rangle=\langle\lambda_{i}^{(A)}\lambda_{j}^{(B)}\rangle$.
We note that while we assume the time order $t_{1}<t_{2}<t_{3}<t_{4}$, in fact
because we have two localised sites, we need only take $t_{1}<t_{3}$ and
$t_{2}<t_{4}$.

A three-time Leggett-Garg-Bell inequality

$\displaystyle-3$ $\displaystyle\leq$ $\displaystyle
E(t_{1},t_{2})-E(t_{1},t_{3})+E(t_{2},t_{3})\leq 1,$ (4)

where $E(t_{i},t_{j})=\langle S_{i}^{(B)}S_{j}^{(A)}\rangle$ is also useful.
This is based on the original three-time Leggett-Garg inequality, derived for
the single-system set-up, in Refs. weak-solid-state-qubits ;
jordan_kickedqndlg2-1 . Here, one assumes however that a measurement at time
$t_{1}$ at site $A$ does not affect the outcome at the later time $t_{3}$ at
the same site, which raises the issue of a disturbance due to the measurement
at $t_{1}$. However, for correlated initial states, one may use that the
result for $S_{1}^{(B)}$ can be inferred from a measurement of $S_{1}^{(A)}$.
Thus, assuming locality, one may deduce macro-bell-lg :

$E(t_{1},t_{3})=\langle S_{1}^{(A)}S_{3}^{(A)}\rangle=-\langle
S_{1}^{(B)}S_{3}^{(A)}\rangle$ (5)

We note that the no-signalling-in-time (NSIT) equality has been shown to give
a stronger test of Leggett-Garg’s macrorealism NSTmunro-1 ; nst . However, it
is more challenging to use this equality for a bipartite test.

###  II.3 Summary of assumptions

Since we are to show violation of the macroscopic Bell and Leggett-Garg-Bell
inequalities, it is important to summarise what is the weakest (i.e. minimal
_)_ assumption needed to derive the inequalities. The Bell inequalities (2)
can be derived from the minimal assumption of macroscopic local causality
(MLC), as in Section 1.A.2. In this paper, however, we use the terms
macroscopic local realism (MLR) and macroscopic local causality (MLC)
interchangeably. The Leggett-Garg-Bell inequalities (3) and (4) on the other
hand are derived assuming _weak macroscopic realism_ , combined with
macroscopic noninvasiveness (NIM) of the measurement, which for two sites is
justified by macroscopic locality (ML).

##  III Violating the macroscopic Bell inequalities

We consider the system of Figure 1 prepared in the initial “cat” state cat-
bell-wang-1

$|\psi_{Bell}\rangle=\mathcal{N}(|\alpha\rangle_{a}|-\beta\rangle_{b}-|-\alpha\rangle_{a}|\beta\rangle_{b})$
(6)

Here $|\alpha\rangle$ and $|\beta\rangle$ are coherent states for two modes,
labelled $a$ and $b$, and for convenience we assume $\alpha$, $\beta$ to be
real and positive. We will ultimately take $\alpha$, $\beta\rightarrow\infty$.
The normalisation constant is
$\mathcal{N}=\frac{1}{\sqrt{2}}\\{1-\exp(-2\left|\alpha\right|^{2}-2\left|\beta\right|^{2})\\}^{-1/2}$.
In the limit where $\alpha$, $\beta$ are large, we note that
$|\alpha\rangle_{a}$ and $|-\alpha\rangle_{a}$ become orthogonal, and
similarly $|\beta\rangle_{b}$ and $|-\beta\rangle_{b}$, so that
$\mathcal{N}\rightarrow\frac{1}{\sqrt{2}}$. In this limit, we may write the
cat state as a macroscopic two-qubit state

$|\psi_{Bell}\rangle=\frac{1}{\sqrt{2}}(|+\rangle_{a}|-\rangle_{b}-|-\rangle_{a}|+\rangle_{b})$
(7)

Here $|+\rangle_{a}$ and $|-\rangle_{a}$ are orthogonal states giving an
outcome of $+1$ and $-1$ respectively, for a measurement $\hat{M}^{(A)}$ of
the sign $S_{A}$ of the coherent amplitude, $\alpha$. Similarly,
$|+\rangle_{b}$ and $|-\rangle_{b}$ are orthogonal states, with outcome $+1$
and $-1$ for the measurement $\hat{M}^{(B)}$ of the sign $S_{B}$ of the
amplitude $\beta$. The two modes $a$ and $b$ are spatially separated, being
located at sites $A$ and $B$, respectively. The sign $S$ of the coherent
amplitude is taken to be the value of the “spin” for each mode (site).

###  III.1 Bell violations using unitary local transformations

At locations $A$ and $B$, we assume local unitary transformations
$U_{A}(t_{a})$ and $U_{B}(t_{b})$ take place, as in Figure 1. States
$|\pm\rangle_{a}$ and $|\pm\rangle_{b}$ are transformed after a time $t_{a}$
and $t_{b}$ according to (at site $A$)

$\displaystyle U_{A}(t{}_{a})|+\rangle_{a}$ $\displaystyle=$ $\displaystyle
a|+\rangle_{a}+i\sqrt{1-a^{2}}|-\rangle_{a}$ $\displaystyle
U_{A}(t{}_{a})|-\rangle_{a}$ $\displaystyle=$ $\displaystyle
a|-\rangle_{a}+i\sqrt{1-a^{2}}|+\rangle_{a}$ (8)

and (at site $B$)

$\displaystyle U_{B}(t{}_{b})|+\rangle_{b}$ $\displaystyle=$ $\displaystyle
b|+\rangle_{b}+i\sqrt{1-b^{2}}|-\rangle_{b}$ $\displaystyle
U_{B}(t{}_{b})|-\rangle_{b}$ $\displaystyle=$ $\displaystyle
b|-\rangle_{b}+i\sqrt{1-b^{2}}|+\rangle_{b}$ (9)

For a system prepared in the superposition $|\psi_{Bell}\rangle$ of eqn. (7),
the system will evolve into the final state __

$\displaystyle|\psi_{f}\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}(U_{A}|+\rangle_{a}U_{B}|-\rangle_{b}-U_{A}|-\rangle_{a}U_{B}|+\rangle_{b})$
$\displaystyle=\frac{1}{\sqrt{2}}\\{(ab+\bar{a}\bar{b})(|+\rangle_{a}|-\rangle_{b}-|-\rangle_{a}|+\rangle_{b})$
$\displaystyle\thinspace\thinspace{\color[rgb]{0,0,1}{\normalcolor+}}i(\bar{a}b-a\bar{b})(|+\rangle_{a}|+\rangle_{b}-|-\rangle_{a}|-\rangle_{b})\\}$
(10)

where we have abbreviated $\bar{a}=\sqrt{1-a^{2}}$ and
$\bar{b}=\sqrt{1-b^{2}}.$ Using that the states $|\pm\rangle$ are orthogonal,
and defining the measurement $\hat{S}$ which measures the sign $S$ (which we
label as its “spin”), according to
$\hat{S}^{(A)}|\pm\rangle_{a}=\pm|\pm\rangle_{a}$ and
$\hat{S}^{(B)}|\pm\rangle_{b}=\pm|\pm\rangle_{b}$, we find that the
expectation value $E(t_{a},t_{b})$ of the spin product
$\hat{S}^{(A)}\hat{S}^{(B)}$ is

$E(t_{a},t_{b})=|\bar{a}b-a\bar{b}|^{2}-|ab+\bar{a}\bar{b}|^{2}$ (11)

For the traditional Bell experiment where the transformation is achieved using
a polarising beam splitter, we have $a=\cos\theta$ and $b=\cos\phi$, and

$E(\theta,\phi)=-\cos 2(\theta-\phi)$ (12)

It is known that the choice of angles $\theta=0$, $\phi=\pi/8$,
$\theta^{\prime}=\pi/4$ and $\phi^{\prime}=3\pi/8$ in the Bell inequality (1)
will lead to $B=-2\sqrt{2}$ which violates the inequality.

###  III.2 Unitary transformations via nonlinear media

Now we consider how to realise the transformations (8) and (9). We seek a
unitary transformation $U_{\theta}^{(A)}\equiv U_{A}(\theta)$ such that at
site $A$ the transformation

$\displaystyle U_{A}(\theta)|\alpha\rangle$ $\displaystyle=$
$\displaystyle\cos\theta|\alpha\rangle+i\sin\theta|-\alpha\rangle$
$\displaystyle U_{A}(\theta)|-\alpha\rangle$ $\displaystyle=$
$\displaystyle\cos\theta|-\alpha\rangle+i\sin\theta|\alpha\rangle$ (13)

holds, for the particular angle choices $\theta=0$ and $\theta=\pi/4$. At site
$B$, we seek a transformation such that

$\displaystyle U_{B}(\phi)|\beta\rangle$ $\displaystyle=$
$\displaystyle\cos\phi|\beta\rangle+i\sin\phi|-\beta\rangle$ $\displaystyle
U_{B}(\phi)|-\beta\rangle$ $\displaystyle=$
$\displaystyle\cos\phi|-\beta\rangle+i\sin\phi|\beta\rangle$ (14)

for the angle choices $\phi=\pi/8$, and $\phi=3\pi/8$. We drop subscripts $a$
and $b$, where the meaning is clear.

These transformations can be achieved by interacting each single mode system
with a nonlinear medium at the given site, where the interaction Hamiltonian
is $H_{NL}=\Omega\hat{n}^{k}$ ($k>2$ and $k$ even) manushan-cat-lg ; yurke-
stoler-1 ; wrigth-walls-gar-1 ; macro-bell-lg . Here $\hat{n}$ is the mode
number operator and $\Omega$ is the nonlinear coefficient. We choose units
such that $\hbar=1$. The interaction takes place independently for each mode
$a$ and $b$, and we denote the respective Hamiltonians as $H_{NL}^{(A)}$ and
$H_{NL}^{(B)}$. Thus we define local interactions at each site

$H_{NL}^{(A)}=\Omega\hat{n}_{a}^{k},\ \ H_{Nl}^{(B)}=\Omega\hat{n}_{b}^{k}$
(15)

The boson destruction mode operators for modes $a$ and $b$ are denoted by
$\hat{a}$ and $\hat{b}$, respectively, and the number operators are
$\hat{n}_{a}=\hat{a}^{\dagger}\hat{a}$ and
$\hat{n}_{b}=\hat{b}^{\dagger}\hat{b}$.

Figure 3: An initial coherent state $|\alpha\rangle$ evolving according to the
Hamltonian $H_{NL}^{(A)}$, $k=4$, $\alpha=3$. The Figure shows contour plots
of the Q function $Q(x_{A},p_{A})$ for the states created at a time $t$. For
the times shown, a superposition of the states $|-\alpha\rangle$ and
$|\alpha\rangle$ is formed. Figure 4: The detailed dynamics of the creation of
the superposition state shown in Figure 4. The system is not a superposition
of states $|-\alpha\rangle$ and $|\alpha\rangle$ at all times $t$. Here,
$\alpha=4$.

One may solve for the dynamics given by $H_{NL}$. If the system $A$ is
prepared in a coherent state $|\alpha\rangle_{a}$, the state after time $t$ is
yurke-stoler-1

$\displaystyle\left|\alpha,t_{a}\right\rangle$ $\displaystyle=$ $\displaystyle
U_{A}(t_{a})|\alpha\rangle_{a}$ (16) $\displaystyle=$
$\displaystyle\exp[-\frac{\left|\alpha\right|^{2}}{2}]\underset{n=0}{\sum^{\infty}}\frac{\alpha^{n}\exp(-i\phi_{n})}{\sqrt{n!}}\left|n\right\rangle_{a}$

where $U_{A}(t_{a})=e^{-iH_{NL}^{(A)}t_{a}/\hbar}$ and $\phi_{n}=\varOmega
t_{a}n^{k}$. For an initial coherent state $|\beta\rangle_{b}$ at $B$, the
state after a time $t$ is given similarly, as
$\left|\beta,t_{b}\right\rangle=U_{B}(t_{b})|\beta\rangle_{b}$ where
$U_{B}(t_{b})=e^{-iH_{NL}^{(B)}t_{b}/\hbar}$. These solutions are well known
manushan-cat-lg ; yurke-stoler-1  and are depicted graphically in Figures 3
and 4 in terms of the $Q$ function, defined as
$Q(x_{A},p_{A})=\frac{1}{\pi}\langle\alpha_{0}|\rho|\alpha_{0}\rangle$
Husimi-Q-1 . Here $\alpha_{0}=x_{A}+ip_{A}$ and $|\alpha_{0}\rangle$ is a
coherent state. At certain intervals of time, the system evolves into a
superposition of the macroscopically distinct coherent states
$|-\alpha\rangle$ and $|\alpha\rangle$ (Figure 4).

Suppose at time $t_{a}=t_{b}=0$, the systems are in the initial states
$|\alpha\rangle$ and $|\beta\rangle$. After a time $t_{b}=\pi/4\Omega$, the
system at $B$ is in the state $U_{B}(\pi/4\Omega)|\beta\rangle$ (second
picture of Figure 3). This gives the transformation of eq. (14) with
$\phi=\pi/8$ manushan-cat-lg . We write the state at this time as

$\displaystyle U_{\pi/8}^{(B)}|\beta\rangle$ $\displaystyle=$ $\displaystyle
e^{-i\pi/8}\\{(\cos\pi/8|\beta\rangle+i\sin\pi/8|-\beta\rangle\\}$

After a time $t_{a}=\pi/2\Omega$, the state is
$U_{A}(\pi/2\Omega)|\alpha\rangle$ which corresponds to the transformation eq.
(13) with $\theta=\pi/4$. We write the state at $t_{a}=\pi/2\Omega$ as
manushan-cat-lg

$\displaystyle U_{\pi/4}^{(A)}|\alpha\rangle$ $\displaystyle=$
$\displaystyle\frac{e^{-i\pi/4}}{\sqrt{2}}\\{|\alpha\rangle+e^{i\pi/2}|-\alpha\rangle\\}$
(18)

To obtain the transformation (14) with $\phi=3\pi/8$, we consider the state at
time $t_{b}=3\pi/4\Omega$ manushan-cat-lg , which is given by
$U_{B}(3\pi/4\Omega)|\beta\rangle$. The state at the time $t_{b}=3\pi/4\Omega$
is

$\displaystyle U_{3\pi/8}^{(B)}|\beta\rangle$ $\displaystyle=$ $\displaystyle
e^{i\pi/8}\\{(\cos 3\pi/8|\beta\rangle+i\sin 3\pi/8|-\beta\rangle\\}$

These transformations are shown graphically in Figure 3. The transformations
at each site $A$ and $B$ are summarised as

$U_{A}(t_{a})=e^{-iH_{NL}^{(A)}t_{a}/\hbar},\ \
U_{B}(t_{b})=e^{-iH_{NL}^{(B)}t_{b}/\hbar}$ (20)

where $t_{a}=0$, $t_{b}=\pi/4\Omega$, $t^{\prime}_{a}=\pi/2\Omega$ and
$t_{b}^{\prime}=3\pi/4\Omega$. Noting from above,
$U_{B}(\pi/4\Omega)=U_{\pi/8}^{(B)}$, $U_{A}(\pi/2\Omega)=U_{\pi/4}^{(A)}$ and
$U_{B}(3\pi/4\Omega)=U_{3\pi/8}^{(B)}$.

The Bell test of Section III.A requires a measurement of the spin $\hat{S}$
for the states after the unitary rotations. As we have seen above, the unitary
parts of the measurements $\hat{M}_{\theta}^{(A)}$ and $\hat{M}_{\phi}^{(B)}$
are to be made by interacting with the nonlinear medium at each site. This
creates the superpositions (LABEL:eq:state3-20) of the coherent states
$|-\alpha\rangle$ and $|\alpha\rangle$ ($|-\beta\rangle$ and $|\beta\rangle$)
which are rotated relative to the initial state $|\alpha\rangle$
($|\beta\rangle$). The initial states determine the orientiation of real axis
(i.e. the phase direction). The coherent states $|-\alpha\rangle$ and
$|\alpha\rangle$ ($|-\beta\rangle$ and $|\beta\rangle$) are hence the basis
states for a particular pointer measurement, $\hat{M}^{(A)}$ ($\hat{M}^{(B)}$)
that determines the sign of the coherent amplitude i.e. the value of the
macroscopic qubit. The pointer measurements are hence measurements of the
quadrature phase amplitudes yurke-stoler-1

$\hat{X}_{A}={\color[rgb]{1,0,0}{\color[rgb]{0,0,1}{\color[rgb]{0,0,0}\frac{1}{\sqrt{2}}}}}(\hat{a}+\hat{a}^{\dagger}),\
\ \hat{X}_{B}=\frac{1}{\sqrt{2}}(\hat{b}+\hat{b}^{\dagger})$ (21)

on the evolved states at each site. The orientation of the axis that defines
quadratures is aligned with the orientation of the initial coherent states
$|\alpha\rangle$ ($|\beta\rangle$) in phase space. This is set in the
experiment by a local oscillator. Finally, such a measurement involves a
coupling to detectors. Let us denote the results of the pointer measurements
$\hat{X}_{A}$ and $\hat{X}_{B}$ by $X_{A}$ and $X_{B}$ respectively. The final
spin outcome of the measurement $\hat{M}_{\theta}^{(A)}$ is taken to be
$S_{A}=+1$ if $X_{A}>0$, and $-1$ otherwise. Similarly, the outcome of the
measurement $\hat{M}_{\phi}^{(B)}$ is $S_{B}=+1$ if $X_{B}>0$, and $-1$
otherwise.

###  III.3 Macroscopic Bell violations

At each site $A$ and $B$, the system of Figure 1 prepared in the cat state (6)
undergoes unitary transformations $U_{A}$ and $U_{B}$, given by (20). There is
the choice of two rotation angles, corresponding to two different times, at
each site. These give transformations of the form (13) and (14).

A violation of the Bell inequality (1) is obtained if we select angles
$\theta=0$ and $\theta=\pi/4$ at site $A$, and $\phi=\pi/8$ and $\phi=3\pi/8$
at site $B$ (Figure 1). Therefore, in order to violate the Bell inequality
(2), we select $t_{a}=0$ (which gives rotation $\theta=0$) and
$t^{\prime}_{a}=\pi/2\Omega$ (which gives rotation $\theta=\pi/4$). Similarly
at site B, we select $t_{b}=\pi/4\Omega$ (rotation $\phi=\pi/8$) and
$t^{\prime}_{b}=3\pi/(4\Omega)$ (rotation $\phi=3\pi/8$). Using the prediction
$E(t_{a},t_{b})=E(\theta,\phi)$ given by (12) for orthogonal states undergoing
the unitary transformations (8) and (9), we see that this choice of time-
settings will give a violation of the Bell inequality (2).

Figure 5: Distributions giving macroscopic Bell violations: Contour plots of
$P(X_{A},X_{B})$ for the system depicted in Figure 1 with an initial entangled
state $|\psi_{Bell}\rangle$ (6) and unitary transformations $U_{A}$ and
$U_{B}$ (20), for measurement settings $t_{a}$ and $t_{b}$. Here,
$\alpha=\beta=3$ and $\Omega=1$. For each setting $\theta$ and $\phi$ given by
_( $t_{a}$, $t_{b}$)_, the outcomes $S_{A}$ and $S_{B}$ for $\hat{M}^{(A)}$
and $\hat{M}^{(B)}$ are $+1$ or $-1$ if the measurement of $\hat{X}_{A}$ or
$\hat{X}_{B}$ yields a positive or negative value.

To account for the nonorthogonality due to finite amplitudes $\alpha$ and
$\beta$, we evaluate the distributions $P(X_{A},X_{B})$ for the outcomes of
the joint measurement of the amplitudes $\hat{X}_{A}$ and $\hat{X}_{B}$ (eqn
(21), for each of the time-settings. This constitutes the prediction for the
final pointer measurements. The final state after the unitary transformations
is

$\displaystyle|\psi_{f}\rangle$ $\displaystyle=$
$\displaystyle\mathcal{N}\thinspace
U_{A}(t_{a})U_{B}(t_{b})\Bigr{(}|\alpha\rangle|-\beta\rangle-|-\alpha\rangle|\beta\rangle\Bigl{)}$

where the unitary operators are given by (20) and $\mathcal{N}$ is the
normalisation constant. The probability is

$P(X_{A},X_{B})=|\langle X_{A}|\langle X_{B}|\psi_{f}\rangle|^{2}$ (23)

where $|X_{A}\rangle$, $|X_{B}\rangle$ are eigenstates of $X_{A}$ and $X_{B}$
respectively. Details of calculation are given in the Appendix. Integrating
gives the results needed for the evaluation of the Bell inequality term, $B$,
of eqn (2). We use $E(t_{a},t_{b})=P_{++}+P_{--}-P_{+-}-P_{-+}$ where

$\displaystyle P_{++}$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\int_{0}^{\infty}P(X_{A},X_{B})dX_{A}dX_{B}$
(24)

and $P_{--}$, $P_{-+}$ and $P_{+-}$ are defined similarly for the remaining
three quadrants.

The results for the four choices of time settings are shown in Figures 5 and
6. For $\alpha,\beta\geq 1.5$, the overlap of the coherent states
$|\alpha\rangle$ and $|-\alpha\rangle$ (and $|\beta\rangle$ and
$|-\beta\rangle$) is small and there is agreement with the calculations based
on the assumption of orthogonality of the states. For each time setting, the
results of the measurements $S^{(A)}$ and $S^{(B)}$ have two possible values,
$+1$ and $-1$, and there are four joint outcomes. In Figure 5, the plots of
$P(X_{A},X_{B})$ show four distinct Gaussian peaks corresponding to the
probabilities of these joint outcomes, the peaks becomes macroscopically
separated for large $\alpha$ and $\beta$. We thus verify that maximal
violations of the Bell inequality (2) ($|B|=2\sqrt{2}$) are indeed predicted
for the macroscopic qubits, where measurements distinguish macroscopically
distinct states, as $\alpha$, $\beta\rightarrow\infty$.

Figure 6: The violation of the macroscopic Bell inequality (2) for the system
depicted in Figure 1, as described in Figure 5. We show (top) $B$ versus
$\beta$ where $\alpha=\beta.$ In the lower plot, $\beta$ is varied for fixed
$\alpha$. A violation is obtained when $|B|>2$.

###  III.4 Dynamics of the unitary part of the measurement

In the set-up of Figure 1, two spacelike separated systems display failure of
macroscopic local realism. As the measurements take place, the systems evolve
dynamically over an interval of time. This is a realistic model because the
analyser measurements involve interactions over a finite time duration. In
order to gain insight into how the _macroscopic_ Bell violations arise over
the course of the dynamics, we present plots in terms of the Q function.

The Husimi Q function is defined uniquely as a positive function for a two-
mode quantum state $\rho$ by
$Q(\alpha_{0},\beta_{0})=\frac{1}{\pi^{2}}\langle\alpha_{0}|\langle\beta_{0}|\rho|\beta_{0}\rangle|\alpha_{0}\rangle$
where $|\alpha_{0}\rangle$ and $|\beta_{0}\rangle$ are coherent states for the
modes Husimi-Q-1 . It is a function of four real variables $x_{A}$, $p_{A}$,
$x_{B}$ and $p_{B}$, where $\alpha_{0}=x_{A}+ip_{A}$ and
$\beta_{0}=x_{B}+ip_{B}$. The Q function is a quasi-probability distribution,
corresponding to an anti-normal ordering of moments, and therefore (different
to a positive Wigner function) does not provide a probability distribution for
the actual outcomes $X_{A}$, $P_{A},$ $X_{B}$, $P_{B}$ of the measurements
$\hat{X}_{A}={\color[rgb]{1,0,0}{\color[rgb]{0,0,1}{\color[rgb]{0,0,0}\frac{1}{\sqrt{2}}}}}(\hat{a}+\hat{a}^{\dagger}),\hat{X}_{B}={\color[rgb]{1,0,0}{\color[rgb]{0,0,1}{\color[rgb]{0,0,0}\frac{1}{\sqrt{2}}}}}(\hat{b}+\hat{b}^{\dagger})$,
$\hat{P}_{A}={\color[rgb]{1,0,0}{\color[rgb]{0,0,1}{\color[rgb]{0,0,0}\frac{1}{i\sqrt{2}}}}}(\hat{a}-\hat{a}^{\dagger}),\hat{P}_{B}={\color[rgb]{1,0,0}{\color[rgb]{0,0,1}{\color[rgb]{0,0,0}\frac{1}{i\sqrt{2}}}}}(\hat{b}-\hat{b}^{\dagger})$.
However, the $Q$ function provides at any given time the correct probability
distribution for the outcomes of the _macroscopically separated_ spins
$S_{j}^{(B)}$ and $S_{i}^{(A)}$, as we confirm below.

Figure 7: Top: Contour plots of the $Q$ function marginal $Q(x_{A},x_{B})$ for
the states created at the times relevant for the violation of the macroscopic
Bell inequality (2), as in Figure 5\. The system is prepared at
$t_{a}=t_{b}=0$ in the entangled Bell cat state $|\psi_{Bell}\rangle$ (6).
Lower: The corresponding plots if the system is prepared in the non-entangled
mixture $\rho_{mix}$ (LABEL:eq:mixqstate), which does not violate the Bell
inequality. Here $\alpha=\beta=3$ and $\Omega=1$. Figure 8: The two-state
nature of the systems created at the specific times given in Figures 5 and 7
is evident on examining the contour plots of the marginal $Q(x_{A},p_{A})$.
The plots for the Bell cat state and the mixed state $\rho_{mix}$ are
indistinguishable.

Assuming the system is prepared in the cat state (6), the state of the system
after interaction times $t_{a}$ at site $A$ and $t_{b}$ at site $B$ is given
by eq. (LABEL:eq:234-1-1-1). The Q function at the time $t_{a}$ and $t_{b}$ is
$Q(x_{A},x_{B},p_{A},p_{B})=\frac{1}{\pi^{2}}|\langle\beta_{0}|\langle\alpha_{0}|\psi_{f}\rangle|^{2}$.
The marginal for the two $X$ quadratures, given by

$\displaystyle Q(x_{A},x_{B})$ $\displaystyle=$ $\displaystyle\int
dp_{A}dp_{B}Q(x_{A},x_{B},p_{A},p_{B})$ (25)

One can also evaluate the marginals for each system e.g. $Q(x_{A},p_{A}$) by
integrating over $x_{B}$ and $p_{B}$.

In Figures 7 and 8, we plot the marginals $Q(x_{A},x_{B})$ and
$Q(x_{A},p_{A})$ corresponding to the states created after the interaction
times $t_{a}$ and $t_{b}$, as in Figure 5, where there is a violation of the
Bell inequality (2). As expected, there is similarity in the macroscopic limit
of the Q function $Q(x_{A},x_{B})$ with the actual distributions
$P(X_{A},X_{B})$, of Figure 5. The additional noise of order $\hbar$ (scaled
to order $\sim 1$ in the plots) which is characteristic of the Q function does
not change the probabilities for the macroscopically distinct outcomes,
$S_{A}$ and $S_{B}$.

Figure 9: The entangled cat state $|\psi_{Bell}\rangle$ (6) evolves while
being measured at the different sites. We show the contour plots of
$Q(x_{A},x_{B})$ corresponding to the $4$ joint measurements leading to
violation of the Bell inequality (2), as in Figures 5 and 7\. The top sequence
shows the dynamics for the measurements with settings $(t_{a}=0,t_{b}=\pi/4)$
and $(t_{a}=0,t_{b}=3\pi/4)$. The centre and lower sequences correspond to
settings $(t_{a}=\pi/2,t_{b}=3\pi/4)$ and $(t_{a}=\pi/2,t_{b}=\pi/4)$,
respectively. Here $\alpha=\beta=3$ and $\Omega=1$. Figure 10: The plots are
as for Figure 9, but here the system evolves from a nonentangled state
$\rho_{mix}$. There is no violation of the Bell inequality. The plots with
$t_{a}=0$ (top) involving a unitary rotation at only one site are
indistinguishable from those of Figure 9. The centre and lower plots involving
transformations at two sites show a macroscopic difference at time
$t_{a}=\pi/(2\Omega)$.

To investigate the dynamics leading to the violation of macroscopic local
realism (MLR), we give a comparison with the predictions if the system is
initially prepared in the non-entangled mixed state

$\displaystyle\rho_{mix}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(|\alpha\rangle|-\beta\rangle\langle\alpha|\langle-\beta|+|-\alpha\rangle|\beta\rangle\langle-\alpha|\langle\beta|)$

The mixture evolves to

$\displaystyle\rho_{mix}(t_{a},t_{b})$ $\displaystyle=$ $\displaystyle
U_{B}(t_{b})U_{A}(t_{a})\rho_{mix}U_{A}^{\dagger}(t_{a})U_{B}^{\dagger}(t_{b})$

We plot the Q function of the mixed state $\rho_{mix}$, eqn.
(LABEL:eq:mixunit), in Figure 7 (lower) for the choice of time-settings that
for the state $|\psi_{Bell}\rangle$ lead to a violation of the Bell inequality
(top). The mixed state has the interpretation that the system is in one _or_
other of the quantum states with a definite outcome for $S_{A}$ and $S_{B}$ at
a time $t=0$, as $\alpha$, $\beta\rightarrow\infty$. Consistent with that
interpretation, the dynamics does not lead to a violation of the Bell
inequality. We see that the Q functions for the initial states
($|\psi_{Bell}\rangle$ and $\rho_{mix}$) at $t=0$ are indistinguishable. In
fact, calculation shows that there is a microscopic difference $\mathcal{C}$,
of order $e^{-|\alpha|^{2}-|\beta|^{2}}$, between these two Q functions. We
also see that the Q functions arising from initial states
($|\psi_{Bell}\rangle$ and $\rho_{mix}$) are indistinguishable if either
$t_{a}$ or $t_{b}$ is zero. However, for the time settings where there is
evolution (rotation) at _both_ the sites, the final outcomes are
macroscopically different.

The timescales of the dynamics leading to the violation of macroscopic local
realism (MLR) can be visualised if we calculate the full dynamics associated
with the unitary measurements (rotations) (Figures 9 and 10). The predictions
for the joint distributions $P(X_{A},X_{B})$ depend only on the absolute value
of the times $t_{a}$ and $t_{b}$ of interaction with the measurement apparatus
at each site. The same violation of the Bell inequalities can thus be achieved
in different ways relative to a shared clock. For example, one may measure
system $A$ first, and subsequently measure system $B$; or vice versa.
Alternatively, one may measure by evolving the systems simultaneously. In
Figures 9 and 10, we choose to evolve sequentially, with $B$ first. On
comparing the sequences of Q functions for the entangled state and for the
mixture $\rho_{mix}$ in Figures 9 and 10, we observe the transition from
indistinguishable Q functions at $t_{a}=t_{b}=0$ though to a small but
noticable difference at times $t_{a}=\pi/8\Omega$ and $t_{b}=\pi/4\Omega$.
Finally, at times $t_{a}=\pi/2\Omega$ and $t_{b}=\pi/4\Omega$, the difference
has become macroscopic.

To understand why there is little distinction between the results for
$|\psi_{Bell}\rangle$ and $\rho_{mix}$ if the unitary evolution acts on the
state at one site only ($t_{a}=0$), but not where for $t_{a}\neq 0$ and
$t_{b}\neq 0$, we calculate the probabilities $P(X_{A},X_{B})$ for each case,
in the Appendix. The difference arises from quantum interference terms damped
by $\exp(-x_{A}^{2}-2|\alpha|^{2})$ in the first case but which are not damped
for the entangled state for certain values of $x_{A}$, in the second case. The
effect for large $\alpha$ and $\beta$ is in direct analogy with the
calculations given for the two-qubit Bell experiment in Section III.A, as
expected since these calculations rely on the orthogonality of the states
forming the qubit.

The plots in Figure 8 show the marginals $Q(x_{A},p_{A})$ and $Q(x_{B},p_{B})$
at each site. These plots highlight the two-state nature of the systems
immediately after the unitary rotations $U_{A}(t_{a})$ and $U_{B}(t_{b})$ for
the choice of measurement settings required for the Bell violation. Identical
two-state plots are obtained for the coarsely selected times given in Figure 9
that show the dynamics during those measurements. However, as seen from Figure
4, over very much _shorter_ time scales, the unitary dynamics involves a
continuous transition, and the systems cannot be regarded as two-state systems
at all times.

###  III.5 Conclusions

It is clear from the summary given in Section II.C that the violations
predicted for the macroscopic Bell inequality (1) and (2) imply failure of
deterministic macroscopic (local) realism (dMR) and macroscopic local
causality (MLC), which we collectively refer to as macroscopic local realism
(MLR). We explain that there is no inconsistency however with the assumption
of weak macroscopic realism (wMR), as introduced in the Sections I and II.

The peaks of the Q function in the Figures give the values for the
probabilities of the macroscopically distinct outcomes ($+1$ and $-1$) for the
spins, $S_{A}$ and $S_{B}$. The description is of an entire ensemble, not of
an individual system. Therefore the negligible difference of order
$\mathcal{C}\sim e^{-|\alpha|^{2}}$ between the Q functions of the classical
mixture $\rho_{mix}$ (eqn (LABEL:eq:mixqstate)) and the superposition
$|\psi_{Bell}\rangle$ (eqn (6)) refers to the average over many identically
prepared systems, not to any individual system at any given time.

In what we refer to here as the weak macroscopic realism (wMR) model, an
interpretation is given that goes beyond this. Consider the cat-state
$|\psi_{Bell}\rangle$, which for large $\alpha,$$\beta$ is a superposition of
two states with definite and macroscopically distinct outcomes for the pointer
measurement $\hat{M}^{(A)}$ ($\hat{M}^{(B)}$) of the spins $S_{A}$ ($S_{B}$).
Similar superpositions (10) are created after the unitary rotations $U$ at
each site, as evident in Figures 7 and 9. In analogy with the polarising beam
splitter or analyser in a Bell experiment, $U$ brings about a change of basis,
in preparation for a pointer measurement. The Q function for such
superposition states $|\psi_{pointer}\rangle$ gives macroscopically distinct
peaks corresponding to outcomes $S_{A}$ ($S_{B}$) for $\hat{M}^{(A)}$
($\hat{M}^{(B)}$).

The wMR interpretation is that immediately prior to the measurement $\hat{M}$
at each site, the local system _was_ in one or other of two states (referred
to as $\varphi_{1}$ and $\varphi_{2}$) predetermined to give one or other of
the macroscopically distinguishable outcomes $+1$ or $-1$ respectively. This
assumption is defined in parallel with that of macroscopic locality of the
pointer (MLP): that the measurement at $B$ does not affect the (macroscopic)
spin outcome at $A$ (and vice versa). In this model, the violation of
macrorealism and of macroscopic local realism arises over the course of the
unitary dynamics. The distinction between predictions for the initial Bell
superposition and a classical mixture of the two pointer eigenstates is
quantified by terms $\mathcal{C}$ of order $\hbar e^{-|\alpha|^{2}}$. The
violations of macro-realism and MLR emerge with the amplification of the
effect of $\mathcal{C}$, over the course of the unitary dynamics corresponding
to the rotation of basis.

In this interpretation, it is _not_ assumed that the states $\varphi_{1}$ and
$\varphi_{2}$ correspond to $|\alpha\rangle$ and $|-\alpha\rangle$, nor that
$\varphi_{1}$ and $\varphi_{2}$ correspond to quantum states (indeed, they
cannot, as shown in Section V). Nonetheless, the wMR model asserts the
predetermination of the macroscopic prediction.

##  IV Leggett-Garg tests of macro-realism and macroscopic local realism

In this section, we propose tests of macrorealism and of macroscopic local
realism using cat states and the Leggett-Garg inequalities. For measurements
of spin $S_{j}^{(A)}$ made on a single system $A$ at consecutive times
$t_{1}<t_{2}<t_{3}$, macrorealism implies the Leggett-Garg inequality weak-
solid-state-qubits ; jordan_kickedqndlg2-1

${\color[rgb]{0,0,0}{\color[rgb]{0,0,0}\langle
S_{1}^{(A)}S_{2}^{(A)}\rangle+\langle S_{2}^{(A)}S_{3}^{(A)}\rangle-\langle
S_{1}^{(A)}S_{3}^{(A)}\rangle}\leq 1}$ (28)

###  IV.1 Leggett-Garg test with a single system

We first outline the approach using a single system, but as applied to the cat
states. Here, we follow an analysis similar to that given by Ref. manushan-
cat-lg . The system $A$ is prepared at time $t_{1}=0$ in a coherent state
$|\alpha\rangle$. The spin $S_{1}$ if measured at this time gives the result
$+1$. The result however is known deterministically, from the preparation.
Following preparation, there is evolution according to the Hamiltonian
$H_{NL}^{(A)}$ of eqn (15), for a time $t_{2}=\pi/4\Omega$. This gives the
state

$\displaystyle U_{\pi/8}|\alpha\rangle$ $\displaystyle=$ $\displaystyle
e^{-i\pi/8}\\{(\cos\pi/8|\alpha\rangle+i\sin\pi/8|-\alpha\rangle\\}$ (29)

At the time $t_{2}$, after the rotation, we may choose to measure the value of
the spin $S_{2}$. This gives $\langle S_{1}S_{2}\rangle=\cos(\pi/4)$. If _not_
measured, the system continues to evolve according to $H_{NL}$ until the time
$t_{3}=\pi/2\Omega$. The state of the system at $t_{3}$ is

$U_{\pi/4}|\alpha\rangle=\frac{e^{-i\pi/4}}{\sqrt{2}}\\{|\alpha\rangle+e^{i\pi/2}|-\alpha\rangle\\}$
(30)

which implies $\langle S_{1}S_{3}\rangle=0$.

To test the Leggett-Garg inequality, one requires to evaluate $\langle
S_{2}S_{3}\rangle$. Here, one assumes macrorealism. We follow the usual
approach, and suppose that an ideal measurement $\hat{M}$ of $S_{2}$ takes
place at time $t_{2}$, which would give a result of either $+1$ or $-1$. If
the result is $\pm 1$, then one _assumes_ the system was in the state
$|\pm\alpha\rangle$ at the time $t_{2}$, based on the assumption that the
measurement did not disturb the system, and noting that tomography of the
(average) state conditional on the result $\pm 1$ would be consistent with the
quantum state $|\pm\alpha\rangle.$ For either state $|\pm\alpha\rangle$, one
deduces $\langle S_{2}S_{3}\rangle=\cos(\pi/4)$, since the system evolves for
a further time $\pi/4\Omega$. Thus, the prediction is $\langle
S_{2}S_{3}\rangle=\cos(\pi/4)$. Overall, the value for the left-side of the
Leggett-Garg inequality (28) is $1.4$, which violates the Leggett-Garg
inequality.

A criticism of the above approach is that there _is_ a change due to the
measurement $\hat{M}$ at time $t_{2}$, and the state prior to the measurement
was different to $|\alpha\rangle$. However, macrorealism postulates the
existence of an ideal measurement $\hat{M}$ where any such effect will have
negligible impact on the future dynamics, as the system becomes macroscopic
($\alpha\rightarrow\infty$). This would not change the prediction of a
violation, if macrorealism is correct.

The dynamics as depicted in terms of the Q function is given for the sequence
of times $t_{1}<t_{2}<t_{3}$ in Figure 11. There are two cases to compare:
whether a measurement $\hat{M}$ is performed at time $t_{2}$, or not. Details
are given in the Appendix. The lower sequence plots where a measurement is
performed at time $t_{2}$, assuming that for the ideal measurement this is
done in such a way to instigate a collapse into one or other of the coherent
states, as above. The Q functions (LABEL:eq:q2-2) and (48) describe the states
for an ensemble of systems $A$ immediately after the time $t_{2}$, if a
measurement has or has not taken place at time $t_{2},$ respectively. The two
Q functions differ by the term

$\mathcal{C}=-\frac{e^{-p_{A}^{2}-x_{A}^{2}}}{\sqrt{2}\pi}e^{-\alpha^{2}}\sin(2p_{A}\alpha_{0})$
(31)

This term is negligible for even moderate $\alpha\sim 1.5$ and there is no
distinguishable difference between the functions at the time $t_{2}$ (refer
Figure 11). Despite the negligible difference between the (average) states at
time $t_{2}$, the states at time $t_{3}$ are macroscopically different. The Q
function model is consistent with the Leggett-Garg premise that a measurement
exists which has increasingly negligible effect on the immediate state of the
system as the system becomes larger legggarg-1 . This is because $\mathcal{C}$
is proportional to $e^{-\alpha^{2}}$. In this model, the Leggett-Garg
inequalities are violated because the small effect nonetheless impacts the
future dynamics at a macroscopic level.

?figurename? 11: The Q function $Q(x_{A},p_{A})$ for times $t_{1}$, $t_{2}$
and $t_{3}$ ($\alpha=3$). The top sequence assumes no measurement is made at
$t_{2}$. The lower sequence assumes a measurement is made at $t_{2}$, of the
type that collapses the system into one or other of two coherent states. While
the difference in the Q function at time $t_{2}$ due to measurement is
negligible, this leads to distinct dynamics, and a macroscopic difference at
time $t_{3}$.

A challenge for testing macrorealism is to overcome the criticism that the
ideal measurement cannot be realised. While arguments can be made that the
system at $t_{2}$ was in a state sufficiently close to $|\alpha\rangle$ or
$|-\alpha\rangle$, if macrorealism is correct NSTmunro-1 , it becomes
difficult to exclude that the system, in a theory alternative to quantum
mechanics, is described by some other state at time $t_{2}$, prior to
measurement. This motivates the next section.

###  IV.2 Bipartite Leggett-Garg tests

####  IV.2.1 Violation of inequalities: idealised measurement at $B$

We next analyse a different model for the ideal measurement that occurs at
time $t_{2}$, where strong justification can be given that the measurement at
$t_{2}$ is nondisturbing. We will see that the predictions of a violation of
the Leggett-Garg inequality (28) are unchanged. Consider the set-up of Figure
2 where the system is prepared in a Bell cat state

$|\psi_{Bell}\rangle=\mathcal{N}(|\alpha\rangle|-\beta\rangle-|-\alpha\rangle|\beta\rangle)$
(32)

as in eqn. (6). We propose a test of the Leggett-Garg inequality (28) for
system $A$, where the second system $B$ is used to perform measurements on
system $A$. After preparation, both systems $A$ and $B$ interact with the
nonlinear medium as given by (20) at the respective locations and evolve
according to the same shared clock. At the later time $t_{2}=\pi/4\Omega$, the
overall state is

$\displaystyle U_{\pi/8}^{(A)}U_{\pi/8}^{(B)}|\psi_{Bell}\rangle$
$\displaystyle=$
$\displaystyle\mathcal{N}e^{-i\pi/4}(|\alpha\rangle|-\beta\rangle-|-\alpha\rangle|\beta\rangle)$

The system remains in a Bell cat state and there is a perfect anticorrelation
between the spin results $S_{2}^{(A)}$ and $S_{2}^{(B)}$ at $A$ and $B$.
Hence, a measurement of $S_{i}^{(A)}$ can be made noninvasively, by performing
a measurement of the spin $S_{i}^{(B)}$, where $i=1,2$. The assumption of
noninvasiveness is justified by that of __ locality: It is assumed that the
outcome of the measurement on the spacelike separated system $A$ is
independent of the choice of duration of the unitary evolution at $B$.

One requires to measure $\langle S_{1}^{(A)}S_{2}^{(A)}\rangle$. One infers
the result for $S_{1}^{(A)}$ by measuring $S_{1}^{(B)}$. A measurement is then
made of $S_{2}^{(A)}$ at the time $t_{2}$, assuming such a measurement will
accurately reflect the macroscopic value of the spin $S_{2}^{(A)}$of system
$A$ immediately prior to the time $t_{2}$. Thus $\langle
S_{1}^{(A)}S_{2}^{(A)}\rangle=-\langle S_{1}^{(B)}S_{2}^{(A)}\rangle$. To
obtain the prediction for $\langle S_{1}^{(A)}S_{2}^{(A)}\rangle$, one argues
as follows. The result of the first measurement at time $t_{1}$ at $A$ (as
measured at $B$) is either $+1$ or $-1$. Assuming the measurement at $B$ is
such to cause a “collapse” so that the system at $A$ is then either in the
coherent state $|\alpha\rangle$ or $|-\alpha\rangle$, the system evolves as
given by eq. (29) and $\langle S_{1}^{(A)}S_{2}^{(A)}\rangle=\cos(\pi/4)$.
Similarly, $\langle S_{1}^{(A)}S_{3}^{(A)}\rangle=-\langle
S_{1}^{(B)}S_{3}^{(A)}\rangle=\cos(\pi/2)=0$.

One also requires to measure $\langle S_{2}^{(A)}S_{3}^{(A)}\rangle$. The
result for $S_{2}^{(A)}$ at time $t_{2}$ is inferred by measuring
$S_{2}^{(B)}$ at time $t_{2}$. However, to measure $S_{2}^{(A)}$ this way, we
use state (LABEL:eq:bellunitary), which means that _no_ measurement is made on
system $B$ (or $A$) at the earlier time $t_{1}$. The measurement at time
$t_{2}$ implies either $+1$ or $-1$, for the spin of $A$ at time $t_{2}$, and
hence we obtain the prediction for $\langle
S_{2}^{(A)}S_{3}^{(A)}\rangle=\cos(\pi/4)$. This gives the left side of the
Leggett-Garg inequality (28) as $1.4$ and a violation is predicted, as in
Section IV.A.

The analysis also implies violation of the Leggett-Garg-Bell inequality (4)
presented in Section II. We have taken the anti-correlated states (6) (and
(32)), so that the Leggett-Garg inequality (28) is written

${\color[rgb]{0,0,0}{\color[rgb]{0,0,0}-\langle
S_{1}^{(B)}S_{2}^{(A)}\rangle-\langle S_{2}^{(B)}S_{3}^{(A)}\rangle+\langle
S_{1}^{(B)}S_{3}^{(A)}\rangle}\leq 1}.$ (34)

It is convenient to consider the correlated Bell state (6) (and (32)) given by
$\beta\rightarrow-\beta$ so that the values of the spins $S_{i}^{(A)}$ and
$S_{i}^{(B)}$ are equal. The Leggett-Garg inequality (28) becomes

${\color[rgb]{0,0,0}{\color[rgb]{0,0,0}\langle
S_{1}^{(B)}S_{2}^{(A)}\rangle+\langle S_{2}^{(B)}S_{3}^{(A)}\rangle-\langle
S_{1}^{(B)}S_{3}^{(A)}\rangle}\leq 1}$ (35)

which may also be measured as

${\color[rgb]{0,0,0}{\color[rgb]{0,0,0}\langle
S_{1}^{(A)}S_{2}^{(B)}\rangle+\langle S_{2}^{(B)}S_{3}^{(A)}\rangle-\langle
S_{1}^{(B)}S_{3}^{(A)}\rangle}\leq 1}$ (36)

This reduces to the Leggett-Garg-Bell inequality (4) of Section I. Thus,
$E(t_{1},t_{2})=\langle S_{1}^{(A)}S_{2}^{(B)}\rangle$,
$E(t_{2},t_{3})=\langle S_{2}^{(B)}S_{3}^{(A)}\rangle$ and
$E(t_{1},t_{3})=\langle S_{1}^{(A)}S_{3}^{(A)}\rangle$.

####  IV.2.2 Predictions for the realistic measurement

The above calculation assumes that the measurement at $B$ “collapses” the
system at $A$ into one or other of the coherent states, $|\alpha\rangle$ or
$|-\alpha\rangle$, at the time of the measurement, $t_{2}$. The calculations
based on that assumption can now be rigorously validated, since we give a
specific proposal, that the measurement at site $B$ be performed as a
quadrature phase amplitude measurement $\hat{X}_{B}$. The spin $S_{j}^{(B)}$is
the sign of the outcome $X_{B}$. In this section, we carry out a complete
analysis by evaluating $P(X_{A},X_{B})$ as in Section III.C. The results shown
in Figures 12 and 13 confirm the prediction of the violation of the Leggett-
Garg-Bell inequality (28) and (4) for large $\alpha,$ $\beta$.

?figurename? 12: Distributions giving the violation of the bipartite Leggett-
Garg-Bell inequality (34). Two systems $A$ and $B$ are prepared in the
entangled cat state $|\psi_{Bell}\rangle$ (6) and then evolve according to the
unitary transformations $U_{A}(t_{a})$ and $U_{B}(t_{b})$ (20) as in Figure 2.
We show the contour plots of the joint probability of $P(X_{A},X_{B})$ for the
measurements $\hat{X}_{A}$ and $\hat{X}_{B}$ made at the times $t_{a}$ and
$t_{b}$, with $\alpha=\beta=3$. The last three distributions give the moments
$\langle S_{1}^{(B)}S_{2}^{(A)}\rangle$, $\langle
S_{1}^{(B)}S_{3}^{(A)}\rangle$ and $\langle S_{2}^{(B)}S_{3}^{(A)}\rangle$.
Here, $t_{1}=0$, $t_{2}=\pi/4\Omega$, and $t_{3}=\pi/2\Omega$. Similar
distributions and violations are obtained for the bipartite Leggett-Garg-Bell
inequality (4) for the correlated Bell state, where $X_{B}\rightarrow-X_{B}$.

?figurename? 13: The violation of the Leggett-Garg-Bell inequality (34) and
(4) for the system depicted in Figures 2 and 12. We plot $B_{lg}$ given by
(37) versus $\beta$ for the state $|\psi_{Bell}\rangle$ (6), where
$\alpha=\beta.$ Violation is obtained when $B_{lg}>1$. The verification of
$\langle S_{i}^{(A)}S_{j}^{(A)}\rangle=-\langle S_{i}^{(B)}S_{j}^{(A)}\rangle$
for $i=1,2$ is given by examination of the conditional distribution
$P(S_{i}^{(A)}=1|S_{i}^{(B)}=-1)$ as shown. The same results are obtained for
the Bell state (6) with $\alpha=-\beta$, on defining $B_{lg}=\langle
S_{1}^{(A)}S_{2}^{(B)}\rangle-\langle S_{1}^{(B)}S_{3}^{(A)}\rangle+\langle
S_{2}^{(B)}S_{3}^{(A)}\rangle$ and considering
$P(S_{i}^{(A)}=1|S_{i}^{(B)}=1)$.

The dynamics of the measurement and its disturbance is visualised, by plotting
sequences of the $Q$ function (Figure 14). In order to measure $\langle
S_{1}^{(A)}S_{2}^{(A)}\rangle$ or $\langle S_{1}^{(A)}S_{3}^{(A)}\rangle$, via
the measurable moments $\langle S_{1}^{(B)}S_{2}^{(A)}\rangle$ and $\langle
S_{1}^{(B)}S_{3}^{(A)}\rangle$, one measures the spin $B$ at $t_{1}=0$ (top
and centre sequences). In order to measure $\langle
S_{2}^{(A)}S_{3}^{(A)}\rangle$ via the measurable moment $\langle
S_{2}^{(B)}S_{3}^{(A)}\rangle$, one measures the spin $B$ at time $t_{2}$
(with no measurement at $t_{1}$). The measurements are made by _stopping_ the
evolution at $B$ at the time $t_{1}$, or $t_{2}$, respectively.

?figurename? 14: Q function dynamics as the entangled cat state
$|\psi_{Bell}\rangle$ evolves through the three measurement sequences of the
bipartite Leggett-Garg test. Plots show the marginal distribution
$Q(x_{A},x_{B})$ after evolving for times $t_{a}$ and $t_{b}$ at the sites $A$
and $B$. The subsystems evolve acording to the same shared clock, with
$t_{a}=t_{b}$, up to the time of measurement at the site $B$. Here, $t_{1}=0$,
$t_{2}=\pi/4\Omega$, and $t_{3}=\pi/2\Omega$. The top, centre and lower rows
give a sequence of snapshots corrresponding to the measurement of $\langle
S_{1}^{(B)}S_{2}^{(A)}\rangle$, $\langle S_{1}^{(B)}S_{3}^{(A)}\rangle$ and
$\langle S_{2}^{(B)}S_{3}^{(A)}\rangle$ respectively. The results agree with
those of Figure 11.

The top two sequences of Figure 14 show measurement of $\langle
S_{1}^{(A)}S_{2}^{(A)}\rangle$ or $\langle S_{1}^{(A)}S_{3}^{(A)}\rangle$,
where the system at $B$ is frozen at $t_{1}=t_{b}=0$, so that information
about the state of the system $A$ at time $t_{1}$ is stored at that site $B$,
and $A$ continues to evolve. We see from the plot of $Q(x_{A},x_{B})$ for
$t_{b}=0$ and $t_{a}=t_{3}$ (last plot of the centre sequence) that $\langle
S_{1}^{(B)}S_{3}^{(A)}\rangle=0$. Similarly, the plot for $t_{b}=0$ and
$t_{a}=t_{2}$ (last plot of the top sequence) shows $\langle
S_{1}^{(B)}S_{2}^{(A)}\rangle=-0.707$. The lower sequence of Figure 14
corresponds to a measurement at site $B$ made at time $t_{2}$, in order to
evaluate $\langle S_{2}^{(B)}S_{3}^{(A)}\rangle=-\langle
S_{2}^{(A)}S_{3}^{(A)}\rangle$. The evolution at $B$ ceases at $t_{2}$ so that
the value for $S_{2}^{(A)}$ can be measured. The system $A$ continues to
evolve, until $t_{3}$. The correlations giving $\langle
S_{2}^{(B)}S_{3}^{(A)}\rangle=-\langle S_{2}^{(A)}S_{3}^{(A)}\rangle$ are
indicated by the final plot of the lower sequence. We find $\langle
S_{2}^{(B)}S_{3}^{(A)}\rangle=-0.707$ and hence $\langle
S_{2}^{(A)}S_{3}^{(A)}\rangle=0.707$. The exact values for the correlations
are evaluated numerically by integration of the joint distributions
$P(X_{A},X_{B})$ at the specified times. The calculation given in the Appendix
leads to

$\displaystyle{\color[rgb]{1,0,0}{\color[rgb]{0,0,0}B{}_{lg}}}$
$\displaystyle=$ $\displaystyle-\\{\langle
S_{1}^{(B)}S_{2}^{(A)}\rangle-\langle S_{1}^{(B)}S_{3}^{(A)}\rangle+\langle
S_{2}^{(B)}S_{3}^{(A)}\rangle\\}$ (37) $\displaystyle=$
$\displaystyle\frac{\sqrt{2}erf(\sqrt{2}\alpha)erf(\sqrt{2}\beta)}{{\color[rgb]{0,0,1}{\color[rgb]{0,0,0}\\{1-e^{-2\left|\alpha\right|^{2}-2\left|\beta\right|^{2}}\\}}}}$

where $erf$ is the error function. Results are shown in Figure 13, in
agreement with the simple approach of the last section.

The pictures of Figure 11 are also justified for this measurement procedure.
The impact of whether the measurement takes place at time $t_{2}$, or not, is
seen to be minimal if we examine the marginal (reduced) state for the system
A, immediately after time $t_{2}$. This is consistent with the idealised model
of the measurement given in Figure 11. If there was no measurement at $t_{2}$,
and the measurement was made at time $t_{1}$, then the system $A$ at time
$t_{2}$ is in a superposition of the two coherent states. On the other hand,
if a measurement is made at time $t_{2}$, then the marginal (reduced) state
for $A$ is the mixture, since no preparation took place at $t_{1}$ and the
statistics for the system is given by (LABEL:eq:bellunitary). The marginal Q
functions in each case are (48) and (LABEL:eq:q2-2), given as the second plots
of the two sequences in Figure 11, which give indistinguishable predictions
for large $\alpha$, justifying the arguments that this is an (apparently) non-
disturbing measurement.

Despite the negligible difference in the marginal Q functions for system $A$
at the time $t_{2}$, a macroscopic difference in the _joint correlations_
measured between $A$ and $B$ emerges at the later time $t_{3}$ (and indeed,
has already emerged at the time $t_{2}$). The correlations shown in Figures 12
and 14 imply results for the two-time moments corresponding exactly to those
shown in the final plots of the two sequences of Figure 11. The macroscopic
difference between the final plots of the two sequences in Figure 11 _arises
over the timescale of the unitary evolution that determines the measurement_
$-$ this is the evolution seen in the first three plots of the centre and
lower sequence, of Figure 14. The subsequent dynamics at $A$ after $t_{2}$
leads to the macroscopic difference in correlations seen in the final plots of
the sequences of Figure 11. On very _short_ timescales (evident at
$t_{a}=\pi/16\Omega$ of Figure 14) as the unitary rotation takes place, we
note the system cannot be considered a two-state system.

##  V Delayed collapse

###  V.1 Delaying the collapse stage of the measurement

We now clarify a possible point of confusion about the timing of the final
readout (the “collapse” stage) of the measurement at $B$, for the Leggett-
Garg-Bell proposal of Section IV.B. The measurement of $S_{1}^{(A)}$ is done
by measuring $S_{1}^{(B)}$ at time $t_{1}=0$ and inferring from the
correlation between the two systems, $A$ and $B$. The measurement of
$S_{2}^{(B)}$ comes in two stages: First, it is necessary to _stop_ the
unitary interaction of system $B$ at the time $t_{1}=0$, so that information
from the correlation can be stored. The second stage of the measurement at $B$
constitutes the irreversible “collapse”, where there is a coupling to a
detector. While the timing of the first stage is crucial, the timing of the
“collapse” stage at $B$ is immaterial.

To clarify, we compare where the collapse at $B$ has, or has not, occurred
prior to the time of the (collapse) measurement of $X_{A}$ at $A$. First, let
us assume the collapse at $B$ has _not_ occurred. Here, the two systems $A$
and $B$ are prepared in the entangled Bell state at time $t_{1}=0$. System $B$
does not evolve further ($t_{b}=0$), while system $A$ evolves for a time
$t_{a}>0$. The state formed after the evolution at $A$ is the superposition

$U_{A}(t_{a})|\psi_{Bell}\rangle=\mathcal{N}U_{A}(t_{a})\\{|\alpha\rangle_{a}|-\beta\rangle_{b}-|-\alpha\rangle_{a}|\beta\rangle_{b}\\}$
(38)

The final joint distribution $P(X_{A},X_{B})_{sup}$ that describes the
statistics at the later time $t_{a}$, assuming there has been no prior
collapse at $B$, is calculated straightforwardly for this superposition state,
as shown in the Appendix.

?figurename? 15: The timing of the final collapse stage of the measurement
$S_{1}^{(B)}$ at time $t_{1}$ on $B$ is immaterial, for $\alpha,\beta>1$: The
plots show a sequence of snapshots of $Q(x_{A},x_{B})$ as the system described
in Figure 14 evolves for the evaluation of $\langle
S_{1}^{(A)}S_{2}^{(A)}\rangle$ and $\langle S_{1}^{(A)}S_{3}^{(A)}\rangle$.
(a) The top two sequences: Here, $\alpha=\beta=0.5$. The top sequence shows
the evolution if no collapse of the measurement at $B$ has taken place before
the time $t_{3}=\pi/2\Omega$. The lower sequence shows the evolution assuming
the collapse has taken place at time $t_{b}=t_{1}=0$. (b) The lower two
sequences: As for (a), but here $\alpha=\beta=1.5$. The states for the top and
lower sequences although different are visually indistinguishable.

Now let us assume the final stage of the measurement at $B$ _was_ made at the
time $t=0$, meaning that the pointer measurement consisting of a readout of
the value of spin $S_{1}^{(B)}$ occurred at this time. In this case, the
system $B$ has been coupled to a third system, so that a “collapse” occurs,
for the system $B$ (and also for $A$). The system immediately after $t=0$ is
given as the mixture $\rho_{mix}$ (eqn (LABEL:eq:mixqstate)). The subsequent
evolution is different to the evolution of the superposition (38). If a
measurement is made at the later time $t_{a}$, the joint probability
$P(X_{A},X_{B})_{mix}$ is readily calculated, as shown in the Appendix. The
two probabilities $P(X_{A},X_{B})_{sup}$ and $P(X_{A},X_{B})_{mix}$ are indeed
different, due to the interference terms present for the superposition state,
where the collapse does not occur. However, for $\alpha$ and $\beta$ large,
the two probabilities are indistinguishable.

The result is illustrated in Figure 15 where we consider measurements of
$\langle S_{1}^{(A)}S_{2}^{(A)}\rangle$ and $\langle
S_{1}^{(A)}S_{3}^{(A)}\rangle$, for both small and large $\alpha$ and $\beta$.
The Q functions depicted in the final plots of the top and lower sequences
directly reflect the probabilities $P(X_{A},X_{B})_{sup}$ and
$P(X_{A},X_{B})_{mix}$ for the two cases, where collapse has not, and has,
taken place, respectively, by the time of the detection at $A$. Although
numerically different, these plots are, in effect, indistinguishable, for
$\alpha,\beta>1$.

A similar result occurs if we consider the measurement of $\langle
S_{2}^{(A)}S_{3}^{(A)}\rangle$. It makes no effective difference to the joint
probabilities $P(X_{A},X_{B})$ whether or not the final collapse stage of the
measurement $S_{2}^{(B)}$ at $B$ is delayed until a later time. The details
are given in the Appendix.

###  V.2 Interpreting the delayed-collapse cat-state experiment

We ask what can be concluded from the Leggett-Garg-Bell experiments described
in Sections IV and V.A, where the collapse stage of the measurement can be
delayed? The latter will imply that one can perform the macroscopic Bell and
Leggett-Garg tests as delayed choice and quantum eraser experiments delayed-
choice-qubit . This follows naturally from the mapping from the microscopic to
macroscopic qubits, which is ideal in the limit of large $\alpha$, for the
angle choices needed for the delayed choice experiments. The new feature,
compared to former delayed-choice experiments, is that the results are at the
macroscopic level, where the qubit spin values reflect the macroscopically
distinct amplitudes.

An argument can be given for interpretating the delayed collapse gedanken
experiment in a way that is consistent with the premise of weak macroscopic
realism (wMR). Here, the definition of wMR includes the assumption of
macroscopic locality for the pointer (MLP) at A, as defined in Section III.E.
The argument is based on the macroscopic nature of the experiment. After the
time $t_{2}$, the final collapse stage of the measurement $S_{2}^{(B)}$ at $B$
gives information about the macroscopic value of the spin $S_{2}^{(A)}$ of
system $A$ (should it be measured). The delayed-collapse analysis of Sections
V.A confirms that this information can be revealed at system $B$ an infinite
time after $t_{2}$, and after further events at $A$. The system $B$ can be
separated from $A$ by an infinite distance. A strong argument can then be
given that carrying out the collapse stage of the measurement at $B$ does not
_macroscopically_ change the state-description of $A$ at the former time
$t_{2}$ i.e. does not change the outcome for the macroscopic spin
$S_{2}^{(A)}$ at $A$.

We note that the wMR interpretation does _not_ mean that prior to the
measurement of spin $S_{2}^{(B)}$ at $B$ the system is in the _quantum_ state
$|\alpha\rangle$ or $|-\alpha\rangle$, since these states are
_microscopically_ specified, giving predictions for all measurements that
might be performed on $A$. That the system in the cat state cannot be viewed
as a classical mixture can be negated in multiple ways, e.g. by observing
fringes in the distribution for the orthogonal quadrature $\hat{P}_{A}$ (for a
single system $A$) or by deducing an Einstein-Podolsky-Rosen (EPR) paradox for
the entangled state $|\psi_{Bell}\rangle$, along the lines given in Refs.
eric_marg-1 ; irrealism-fringes ; macro-coherence-paradox ; macro-pointer ;
Bohm-1 ; epr-1 ; bohm-eric ; epr-r2 . We give an example of this in the next
subsection.

The consistency with wMR is explained as follows. Suppose the systems $A$ and
$B$ are prepared at a time $t_{j}$ in a macroscopic superposition
$|\psi_{pointer}\rangle$ of states with definite outcomes for pointer
measurements $\hat{M}^{(A)}$ and $\hat{M}^{(B)}$, an example being (as
$\alpha,$$\beta\rightarrow\infty$)

$|\psi_{Bell}\rangle=\mathcal{N}(|\alpha\rangle|-\beta\rangle-|-\alpha\rangle|\beta\rangle)$
(39)

Here, the states with definite outcomes for $\hat{M}^{(A)}$ and
$\hat{M}^{(B)}$ are the states with definite outcomes for the spin
$S_{j}^{(A)}$. The assumption of wMR asserts that the system $A$ at the time
$t_{j}$ is in one or other of two macroscopically specified states
$\varphi_{+}$ and $\varphi_{-}$, for which the result of a measurement of spin
$S_{j}^{(A)}$ is deterministically predetermined i.e. the system $A$ at time
$t_{j}$ may be described by a macroscopic hidden variable $\lambda_{j}^{(A)}$.
For the bipartite system, wMR is to be consistent with MLP. MLP asserts that
the value of the macroscopic hidden variable $\lambda_{j}^{(A)}$ for the
system $A$ cannot be changed by any spacelike separated event or measurement
at the system $B$ that takes place at time $t\geq t_{j}$ e.g. it cannot be
changed by a future event at $B$. The interpretation is that the system $A$ at
each time $t_{1}$, $t_{2}$ and $t_{3}$ was in one or other of states
$\varphi_{j,+}$ or $\varphi_{j,-}$ with a definite value of spin
$S_{j}^{(A)}$, and that the failure of macrorealism arises from the initial
entanglement as the systems evolve dynamically at the intermediate times.

The assumption of MLP is to be distinguished from the stronger assumption,
macroscopic locality (ML), introduced earlier, as part of the premise of
macroscopic local realism (MLR). The premise of ML assumes locality to apply
for spacelike separations where the measurement on system $A$ can vary, so
that $A$ is _not necessarily prepared in the pointer basis_. The above wMR
interpretation is thus not contradicted by the violation of macroscopic Bell
inequalities. In fact, we have seen in Section III that, consistent with the
validity of wMR, there is no violation of these Bell inequalities when there
is no unitary transformation $U_{A}$ at $A$. This is because in that case the
system is in (immediately prior to the measurements) a state
$|\psi_{pointer}\rangle$ where wMR applies.

###  V.3 Einstein-Podolsky-Rosen-type paradox based on macroscopic realism

The EPR argument argues the incompleteness of quantum mechanics based on the
assumption of local realism, or local causality epr-1 . One may also argue an
EPR-type paradox for the cat states based on the validity of weak macroscopic
realism. Here, we apply the argument given in Ref. macro-coherence-paradox .

One considers the superposition state
$c_{1}|\alpha\rangle+ic_{2}|-\alpha\rangle$ (for $\alpha$ large), similar to
that prepared at the time $t_{2}$ in the Leggett-Garg-Bell tests. Here,
$|c_{1}|^{2}+|c_{2}|^{2}=1$. Macroscopic realism postulates that the system
$A$ in such a state is actually in one or other state $\varphi_{+}$ and
$\varphi_{-}$ for which the value of the macroscopic spin $S_{2}^{(A)}$ is
predetermined. The spin $S_{2}^{(A)}$ is measured by the sign of $X_{A}$: the
distribution $P(X_{A})$ gives two separated Gaussian peaks, each with variance
$(\Delta X_{A})^{2}=1/2$.

One may specify the degree of predetermination of $X_{A}$ for the macroscopic
states, $\varphi_{+}$ and $\varphi_{-}$, by postulating for each state
$\varphi_{+}$ and $\varphi_{-}$ that the variance be some specified value,
which we take as $(\Delta X_{A})_{+}^{2}$ and $(\Delta X_{A})_{-}^{2}$
respectively. If we assume each of $\varphi_{+}$ and $\varphi_{-}$ to be
_quantum_ states, then for each the Heisenberg uncertainty relation implies
$(\Delta X_{A})(\Delta P_{A})\geq 1/2$. We then argue, as was done in ref.
macro-coherence-paradox , that for the ensemble of systems in a classical
mixture of such states, $\varphi_{+}$ and $\varphi_{-}$, we require $(\Delta
X_{A})_{ave}(\Delta P_{A})\geq 1/2$, where $(\Delta
X_{A})_{ave}^{2}=P_{+}(\Delta X_{A})_{+}^{2}+P_{-}(\Delta X_{A})_{-}^{2}$ ,
$P_{+}=|c_{+}|^{2}$ and $P_{-}=|c_{-}|^{2}$. The observation of

$\varepsilon_{M}\equiv(\Delta X_{A})_{ave}(\Delta P_{A})<1/2$ (40)

implies incompatibility of macroscopic realism with the completeness of
quantum mechanics, since the localised states $\varphi_{+}$ and $\varphi_{-}$
cannot be given as quantum states.

If we take $(\Delta X_{A})_{+}^{2}=(\Delta X_{A})_{-}^{2}=1/2$, to match that
the states have the variances associated with the two Gaussian peaks (the
coherent states), we find the condition (40) is satisfied for $(\Delta
P_{A})^{2}<1/2$. This is indeed the case for the superposition above. The
distribution $P(P_{A})$ is

$P(P_{A})=\frac{e^{-P_{A}^{2}}}{\sqrt{\pi}}\\{1+\frac{1}{\sqrt{2}}\sin(2\sqrt{2}P_{A}|\alpha|)\\}$
(41)

which shows fringes, as given in ref. yurke-stoler-1 . The variance is
$(\Delta P_{A})^{2}=\frac{1}{2}-\alpha^{2}e^{-4\alpha^{2}}$ macro-pointer ,
indicating a paradox.

##  VI Conclusion

In this paper, we have provided ways to test quantum mechanics against
macroscopic realism using cat states. We consider two definitions of
macroscopic realism. The first, macrorealism (M-R), supplements the assumption
of macroscopic realism with the premise of macroscopic noninvasive
measurability. The second, macroscopic local realism (MLR), supplements with
the premise of macroscopic locality. In this paper, “macroscopically distinct”
refers to separations well beyond the quantum noise level $\hbar$.

In Sections II-IV, we show that quantum mechanics predicts failure of MLR (and
M-R) for two spacelike-separated systems prepared in entangled cat states, in
a way that is directly analogous to the original Bell and Leggett-Garg
proposals. To test MLR, we consider CHSH-type Bell inequalities and three-time
Leggett-Garg-Bell inequalities, where measurements are made at two spacelike
separated locations, at successive times. One may construct similar tests,
based on the Leggett-Garg-Bell inequality described in Section II, involving
four times.

The tests give a convincing strategy to demonstrate the non-invasiveness of a
measurement that is assumed in a Leggett-Garg test of macro-realism. For a
macroscopic system, the usual argument is that the disturbance (at least for
some ideal measurement) becomes vanishingly small, so as to have a negligible
affect on the later dynamics. In order to refute the criticism that there _is_
a disturbance, methods have been developed to quantify the effect of such a
disturbance, if macroscopic realism is satisfied. One approach performs a
control experiment that prepares two macroscopically distinguishable states
($\psi_{1}$ and $\psi_{2}$) and then quantifies the effect of disturbance on
the inequality, if the system is indeed in one of those states NSTmunro-1 . A
counterargument however is that the system at time $t_{2}$ is in a state
microscopically different to $\psi_{1}$ (or $\psi_{2}$), a state which was not
prepared in the laboratory. In the present paper, such criticisms are avoided,
because macroscopic realism is defined more broadly. One does not assume that
the macroscopically distinct states are specific __ states $\psi_{1}$ and
$\psi_{2}$, nor that the states are quantum states. A direct disturbance due
to measurement is ruled out, because the result for the measurement of $A$ is
inferred from the measurement on the separated system $B$. The non-
classicality is then explained by quantum nonlocality, but at a macroscopic
level.

In Section V, we have extended the bipartite Leggett-Garg analysis, to
demonstrate that a delay of the final irreversible stage of the measurement
makes no difference to the predictions, provided the amplitude of the coherent
states is large. This implies that one could perform delayed choice
experiments, similar to those examined for qubits in Refs. delayed-choice-
qubit ; delayed-choice-interpretation . The possibility of delayed choice
experiments is expected because of the mapping onto the qubit system, for
large $\alpha$, $\beta$ (as the coherent states become orthogonal), for the
rotation angles required for the qubit experiments. The new feature is the
observation at a macroscopic level, since the qubits correspond to
macroscopically distinct coherent states. As with the qubit experiments, this
does not however imply acausal effects delayed-choice-interpretation .

It is clear from the violations of the macroscopic inequalities presented in
this paper that MLR, macroscopic local causality (MLC), and M-R fail. We have
explained in Section II that the MLR is defined as deterministic macroscopic
(local) realism (dMR), that the system is predetermined to be in a state
giving a definite outcome for the pointer measurement ($\hat{M}_{\theta}$ or
$\hat{M}_{\theta^{\prime}}$) prior to the unitary rotation $U$ that determines
the measurement setting, $\theta$ or $\theta^{\prime}$. The violation of the
macroscopic Bell inequalities given in this paper show that dMR fails.

The results are however consistent with an interpretation in which a _weak
macroscopic realism_ (wMR) holds. The initial Bell state gives predictions for
joint probabilities $P(X_{A},P_{A},X_{B},P_{B})$ at time $t_{1}$ that are only
microscopically different (of order $\hbar e^{-|\alpha|^{2}}$) to those of a
non-entangled mixture. Yet, after the choice of measurement setting, the final
joint probabilities are _macroscopically_ different. In the wMR
interpretation, the macroscopic differences arise during the unitary dynamics,
corresponding in a Bell experiment to the choice of measurement setting. Such
dynamics shifts the system into a superposition $|\psi_{pointer}\rangle$ of
states with definite outcomes for the measurement of the macroscopic spin
$S_{i}^{(A)}$, at a given time $t_{i}$. The assumption of wMR postulates that
a system in a superposition $|\psi_{pointer}\rangle$ is describable by a
macroscopic “element of reality”, $\lambda_{M}$, which predetermines the
result of the pointer measurement for the macroscopic qubit value
$S_{i}^{(A)}$. The interpretation is that the system is in one or other states
$\varphi_{+}$ and $\varphi_{-}$ giving a definite value for the macroscopic
pointer measurement.

An EPR-type paradox arises from the assumption of weak macroscopic realism, if
correct. The element of reality “state”, $\varphi_{+}$ or $\varphi_{-}$, of
the system prior to the measurement cannot be a _quantum_ state. This is
evident from calculations given in macro-coherence-paradox  and explained in
section V.C, where the states $\varphi_{+}$ or $\varphi_{-}$ that would apply
to the cat state $|\alpha\rangle+|-\alpha\rangle$ can be shown to be
inconsistent with the uncertainty principle. Similar paradoxes have been
illustrated for the two-slit experiment using the concept of irrealism
irrealism-fringes , and for entangled cat states eric_marg-1 , based on the
logic of the original EPR paradoxes that reveal the inconsistency of the
completeness of quantum mechanics with local realism epr-1 ; Bohm-1 ; bohm-
eric ; epr-r2 .

Finally, we comment on the possibility of an experiment. Entangled cat states
have been generated cat-bell-wang-1 . The challenge is to realise the unitary
rotation, given by the nonlinear Hamiltonian which has a quartic dependence on
the field intensity. This is necessary for the full Bell experiment. However,
a macroscopic delayed choice/ quantum eraser experiment may be carried out
more straightforwardly, since this depends only on the creation of a simple
cat state superposition, which can be realised at times $t=\pi/2\Omega$ from
the nonlinear Kerr interaction $H_{NL}=\Omega\hat{n}^{2}$ yurke-stoler-1 ;
manushan-cat-lg . This interaction has been experimentally achieved collapse-
revival-bec-2 ; collapse-revival-super-circuit-1 .

## Acknowledgements

This research has been supported by the Australian Research Council Discovery
Project Grants schemes under Grant DP180102470.

## Appendix

###  VI.1 Calculation of Bell violations

The joint probability is $P(X_{A},X_{B})=|\langle X_{A}|\langle
X_{B}|\psi_{f}\rangle|^{2}$ where $|X_{A}\rangle$, $|X_{B}\rangle$ are
eigenstates of $X_{A}$ and $X_{B}$ respectively. Using the overlap $\langle
x|e^{i\theta}\alpha\rangle=\frac{1}{\pi^{1/4}}\exp(-\frac{x^{2}}{2}+\frac{2x|\alpha|e^{i\theta}}{\sqrt{2}}-\\{\frac{|\alpha|e^{i\theta}}{\sqrt{2}}\\}^{2}-\frac{|\alpha|^{2}}{2})$,
we find

$\displaystyle\langle X_{A}|\langle X_{B}|\psi_{f}\rangle$
$\displaystyle=\frac{\exp^{-(|\alpha|^{2}+|\beta|^{2})}}{\sqrt{2\pi}}\\{\exp^{-\frac{X_{A}^{2}}{2}+\sqrt{2}X_{A}|\alpha|}[b\exp^{-\frac{X_{B}^{2}}{2}-\sqrt{2}X_{B}|\beta|}-i\bar{b}\exp^{-\frac{X_{B}^{2}}{2}+\sqrt{2}X_{B}|\beta|}]$
$\displaystyle-\exp^{-\frac{X_{A}^{2}}{2}-\sqrt{2}X_{A}|\alpha|}[b\exp^{-\frac{X_{B}^{2}}{2}+\sqrt{2}X_{B}|\beta|}+i\bar{b}\exp^{-\frac{X_{B}^{2}}{2}-\sqrt{2}X_{B}|\beta|}]\\}$
(42)

###  VI.2 Interference effects

Where $t_{a}=0$, the joint probability $P(X_{A},X_{B})$ for outcomes $X_{A}$
and $X_{B}$ for the system prepared in the entangled superposition (32) is

$\displaystyle P(X_{A},X_{B})$ $\displaystyle=$
$\displaystyle\mathcal{N}^{2}\\{|\langle X_{A}|\alpha\rangle\langle
x_{B}|U^{(B)}(t_{b})|-\beta\rangle|^{2}+|\langle X_{A}|-\alpha\rangle\langle
X_{B}|U^{(B)}(t_{b})|\beta\rangle|^{2}$ (43) $\displaystyle+\langle
X_{A}|\alpha\rangle\langle X_{B}|U^{(B)}(t_{b})|-\beta\rangle\langle
X_{A}|-\alpha\rangle^{*}\langle X_{B}|U^{(B)}(t_{b})|\beta\rangle^{*}$
$\displaystyle+\langle X_{A}|\alpha\rangle^{*}\langle
X_{B}|U^{(B)}(t_{b})|-\beta\rangle^{*}\langle X_{A}|-\alpha\rangle\langle
X_{B}|U^{(B)}(t_{b})|\beta\rangle\\}$

This contains an interference term proportional to
$\exp(-X_{A}^{2}-2|\alpha|^{2})$ that vanishes with large $\alpha$. The
expression reduces to the result for the mixture $\rho_{mix}$, given by

$\displaystyle P(X_{A},X_{B})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\\{|\langle X_{A}|\alpha\rangle\langle
X_{B}|U_{2}(t_{b})|-\beta\rangle|^{2}+|\langle X_{A}|-\alpha\rangle\langle
X_{B}|U_{2}(t_{b})|\beta\rangle|^{2}\\}$ (44)

because the interference term contains the expression $\langle
X_{A}|\alpha\rangle\langle
X_{A}|-\alpha\rangle^{*}=\frac{1}{\pi^{1/2}}\exp(-X_{A}^{2}-2|\alpha|^{2})$
which is small for orthogonal state, with large $\alpha_{0}$.

By contrast, for $t_{a}\neq 0$ and $t_{b}\neq 0$, we compare the expressions
$P(X_{A},X_{B})$ for the evolved cat state given by $|\psi_{f}\rangle$ of eq.
(LABEL:eq:234-1-1-1), with that of the evolved mixture

$\displaystyle\rho_{mix}(t_{a},t_{b})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Bigl{(}U^{(B)}(t_{b})|-\beta\rangle
U^{(A)}|\alpha\rangle\langle\alpha|U^{(A)\dagger}\langle-\beta|U^{(B)\dagger}(t_{b})$
(45) $\displaystyle+U^{(B)}|\beta\rangle
U^{(A)}|-\alpha\rangle\langle-\alpha|U^{(A)\dagger}\langle\beta|U^{(B)\dagger}\Bigl{)}$

The expressions are the same apart from interference terms, such as

$\langle X_{A}|U^{(A)}(t_{a})|\alpha\rangle\langle
X_{B}|U^{(B)}(t_{b})|-\beta\rangle\times\langle
X_{A}|U^{(A)}(t_{a})|-\alpha\rangle^{*}\langle
X_{B}|U^{(B)}(t_{b})|\beta\rangle^{*}$ (46)

We see from the solutions given by eqs. (LABEL:eq:state3-20) that for the
suitable choices of $t_{a}$ , $U^{(A)}(t_{a})|-\alpha\rangle$ (for example)
may involve terms such as $|\alpha\rangle$, and $U^{(A)}(t_{a})|\alpha\rangle$
involves terms in $|\alpha\rangle$. This then leads to a contribution of type
$\langle X_{A}|\alpha\rangle\langle
X_{A}|\alpha\rangle^{*}=\frac{1}{\pi^{1/2}}\exp(-X_{A}^{2}+2\sqrt{2}X_{A}\alpha-2|\alpha|^{2})$
and therefore significant interference terms (when $X_{A}\sim\sqrt{2}\alpha$).
These terms are the origin of the macroscopic difference between the
probability distributions in Figures 6 for nonzero $t_{a}$ and $t_{b}$, which
leads to the macroscopic Bell violation.

###  VI.3 Leggett-Garg test for a single system: Q function dynamics

The Q function at time $t_{1}$ is that of a coherent state
$|\alpha_{0}\rangle$ where we take $\alpha_{0}$ to be real.

$Q(x_{A},p_{A},t_{1})=\frac{e^{-p_{A}^{2}}}{\pi}e^{-(x_{A}-\alpha_{0})^{2}}$
(47)

At time $t_{2}$, the system is in the superposition state (29) which has the Q
function

$\displaystyle Q(x_{A},p_{A},t_{2})=$
$\displaystyle\frac{e^{-p_{A}^{2}}}{2\sqrt{2}\pi}\\{c_{+}e^{-(x_{A}-\alpha_{0})^{2}}+c_{-}e^{-(x_{A}+\alpha_{0})^{2}}-2e^{-x_{A}^{2}}e^{-\alpha_{0}^{2}}\sin(2p_{A}\alpha_{0})\\}$
(48)

where $c_{\pm}=\sqrt{2}\pm 1$. __Assuming no measurement takes place at time
$t_{2}$, the system at time $t_{3}$ is in the superposition state (30) with

$\displaystyle Q(x_{A},p_{A},t_{3})=$
$\displaystyle\frac{e^{-p_{A}^{2}}}{2\pi}\\{e^{-(x_{A}-\alpha_{0})^{2}}+e^{-(x_{A}+\alpha_{0})^{2}}-2e^{-x_{A}^{2}}e^{-\alpha_{0}^{2}}\sin(2p_{A}\alpha_{0})\\}$
(49)

If the measurement is performed at $t_{2}$, the system “collapses” to either
$|\alpha_{0}\rangle$ (with probability $\cos^{2}\pi/8$) or
$|-\alpha_{0}\rangle$ (with probability $\sin^{2}\pi/8$) and then evolves at
time $t_{3}$ respectively to either $U_{\pi/8}|\alpha\rangle$ or
$U_{\pi/8}|-\alpha\rangle$ as given by eqn. (29).

There are two cases to compare: whether or not a measurement is performed at
time $t_{2}$. Figure 12 shows the sequence for the Q functions at the times
$t_{1}$ , $t_{2}$ and $t_{3}$ for these two cases. The top sequence, modelling
where a measurement is not performed, shows sequentially the three Q
functions, (47), (48) and (49). The lower sequence plots where a measurement
is performed at time $t_{2}$, assuming this is done in such a way to instigate
a collapse into one or other of the coherent states, as described above. The
second plot of the lower sequence is therefore the Q function

$\displaystyle Q(x_{A},p_{A},t_{2})$
$\displaystyle=\frac{e^{-p_{A}^{2}}}{2\sqrt{2}\pi}\\{c_{+}e^{-(x_{A}-\alpha_{0})^{2}}+c_{-}e^{-(x_{A}+\alpha_{0})^{2}}\\}$

representing the average state $\rho_{mix}(t_{2})$ of the system at time
$t_{2}$, immediately after the measurement at time $t_{2}$:
$\rho_{mix}(t_{2})=\cos^{2}\pi/8|\alpha_{0}\rangle\langle\alpha_{0}|+\sin^{2}\pi/8|-\alpha_{0}\rangle\langle-\alpha_{0}|$.
The Q function for the final state at time $t_{3}$, if the measurement has
taken place at time $t_{2}$, is given by the evolution of $\rho_{mix}(t_{2})$
for the time $\pi/4\Omega$. The final _average_ state is
$\rho_{mix}(t_{3})=\cos^{2}\pi/8|\psi_{+}\rangle\langle\psi_{+}|+\sin^{2}\pi/8|\psi_{-}\rangle\langle\psi_{-}|$
where $|\psi_{\pm}\rangle=U_{\pi/8}|\pm\alpha\rangle$. The Q function is __

$\displaystyle Q(x_{A},p_{A},t_{3})=$
$\displaystyle\frac{e^{-p_{A}^{2}}}{8\pi}\\{6e^{-(x_{A}-\alpha_{0})^{2}}+e^{-(x_{A}+\alpha_{0})^{2}}-4e^{-x_{A}^{2}}e^{-\alpha_{0}^{2}}\sin(2p_{A}\alpha_{0})\\}$
(51)

which is plotted as the third function of the lower sequence.

###  VI.4 Bipartite Leggett-Garg tests: correlation between the outcomes at
the different sites

Here, we evaluate $B_{lg}=-\\{\langle S_{1}^{(B)}S_{2}^{(A)}\rangle-\langle
S_{1}^{(B)}S_{3}^{(A)}\rangle+\langle S_{2}^{(B)}S_{3}^{(A)}\rangle\\}$ and
the value of the conditional probabilities $P(S_{i}^{(A)}=1|S_{i}^{(B)}=-1)$
where $i=1,2$ for the case $\alpha=\beta$. These quantities are worked out
exactly as one would measure them. We first evaluate $P(X_{A},X_{B})$ at time
$t_{2}$.

$\displaystyle P(X_{A},X_{B})$ $\displaystyle=\left|\langle X_{B}|\langle
X_{A}|U_{\pi/8}^{(A)}U_{\pi/8}^{(B)}|\psi_{Bell}\rangle\right|^{2}$
$\displaystyle=2\frac{e^{-X_{A}^{2}-X_{B}^{2}-2\left|\alpha\right|^{2}-2\left|\beta\right|^{2}}}{\pi{\color[rgb]{0,0,1}{\normalcolor\left(1-e^{-2\left|\alpha\right|^{2}-2\left|\beta\right|^{2}}\right)}}}\sinh^{2}(\sqrt{2}X_{A}\left|\alpha\right|-\sqrt{2}X_{B}\left|\beta\right|)$
(52)

where we use

$\displaystyle\langle X_{B}|\langle
X_{A}|U_{\pi/8}^{(A)}U_{\pi/8}^{(B)}|\psi_{Bell}\rangle$
$\displaystyle=\mathcal{N}e^{-i\pi/4}\left(\langle X_{A}|\alpha\rangle\langle
X_{B}|-\beta\rangle-\langle X_{A}|-\alpha\rangle\langle
X_{B}|\beta\rangle\right)$
$\displaystyle=2\mathcal{N}\frac{e^{-\frac{X_{A}^{2}}{2}-\frac{X_{B}^{2}}{2}}}{\pi^{1/2}}e^{-\left|\alpha\right|^{2}-\left|\beta\right|^{2}-i\pi/4}\sinh(\sqrt{2}X_{A}\left|\alpha\right|-\sqrt{2}X_{B}\left|\beta\right|)$
(53)

In fact, because the system at time $t_{2}$ remains in a Bell state
$|\psi_{Bell}\rangle$ (apart from a phase factor), this also represents
$P(X_{A},X_{B})$ for the time $t_{1}$. We note that

$\displaystyle P(X_{B})$ $\displaystyle=\int
P(X_{A},X_{B})dX_{A}=\frac{e^{-X_{B}^{2}-2\left|\alpha\right|^{2}-2\left|\beta\right|^{2}}}{\sqrt{\pi}{\color[rgb]{0,0,1}{\normalcolor\\{1-e^{-2\left|\alpha\right|^{2}-2\left|\beta\right|^{2}}\\}}}}\left(e^{2|\alpha|^{2}}\cosh(2\sqrt{2}|\beta|X_{B})-1\right)$
(54)

which on integration gives $P(X_{B}>0)=1/2$. For the value of the conditional
probability $P(S_{i}^{(A)}=1|S_{i}^{(B)}=-1)$, $i=1,2$, one measures

$\displaystyle{\color[rgb]{0,0,0}{\color[rgb]{0,0,1}}P(X_{A}>0|X_{B}\leq 0)}$
$\displaystyle{\color[rgb]{0,0,0}=\frac{\int_{-\infty}^{0}\int_{0}^{\infty}P(X_{A},X_{B})dX_{A}dX_{B}}{\int_{-\infty}^{0}P(X_{B})dX_{B}}{\color[rgb]{0,0,0}={\color[rgb]{0,0,0}\frac{1}{2}+\frac{erf(\sqrt{2}\alpha)\times
erf(\sqrt{2}\beta)}{{\color[rgb]{0,0,0}2\left(1-e^{-2\alpha^{2}-2\beta^{2}}\right)}}}}}$
(55)

Similarly, we evaluate

$\displaystyle{\color[rgb]{0,0,0}}{\normalcolor}{\normalcolor}{\normalcolor}\langle
S_{1}^{(B)}S_{2}^{(A)}\rangle$
$\displaystyle=-\frac{\sqrt{2}erf(\sqrt{2}\alpha)erf(\sqrt{2}\beta)}{2{\color[rgb]{0,0,0}{\color[rgb]{0,0,1}{\color[rgb]{0,0,0}\\{1-e^{-2\left|\alpha\right|^{2}-2\left|\beta\right|^{2}}\\}}}}}$
$\displaystyle\langle S_{1}^{(B)}S_{3}^{(A)}\rangle$ $\displaystyle=0$ (56)

where $\langle S_{2}^{(B)}S_{3}^{(A)}\rangle=\langle
S_{1}^{(B)}S_{2}^{(A)}\rangle$. This gives the result (37). The plots of
Figure 13 and 15 reveal that for $\alpha=\beta>1$, the positive and negative
values for the outcomes of $X_{A}$ correspond to distinct macroscopically
separated Gaussian peaks in the different quadrants. The conditional
probability goes to $1$ as $\alpha$ is larger, which justifies that the
measurement of $S_{i}^{(B)}$at $B$ indicates the value $S_{i}^{(A)}$at $A$.

###  VI.5 Delayed Collapse for measurement $\langle
S_{1}^{(A)}S_{2}^{(A)}\rangle$

The state formed after the evolution at $A$ is the superposition

$U_{A}(t_{a})|\psi_{Bell}\rangle=\mathcal{N}\\{U_{A}(t_{a})|\alpha\rangle_{a}|-\beta\rangle_{b}-U_{A}(t_{a})|-\alpha\rangle_{a}|\beta\rangle_{b}\\}$
(57)

The final joint distribution at time $t_{a}$ is (assuming there has been no
prior collapse at $B$)

$\displaystyle P(X_{A},X_{B})_{sup}$ $\displaystyle=$
$\displaystyle\mathcal{N}^{2}(\langle
X_{A}|U_{A}(t_{a})|\alpha\rangle_{a}\langle X_{B}|-\beta\rangle_{b}-\langle
X_{A}|U_{A}(t_{a})|-\alpha\rangle_{a}\langle X_{B}|\beta\rangle_{b})$ (58)
$\displaystyle\times(\langle\alpha|_{a}U_{A}^{\dagger}(t_{a})|X_{A}\rangle\langle-\beta|_{b}X_{b}\rangle-\langle-\alpha|_{a}U_{A}^{\dagger}(t_{a})|X_{A}\rangle\langle\beta|_{b}X_{b}\rangle)$

Now let us assume the final stage of the measurement at $B$ _was_ made at the
time $t=0$, meaning that the pointer measurement consisting of a readout of
the value of spin $S_{1}^{(B)}$ occurred at this time. In this case, the
system $B$ has been coupled to a third system, so that a “collapse” occurs,
for the system $B$ (and also for $A$). The system (or rather, an ensemble of
them) immediately after $t=0$ is given as the mixture $\rho_{mix}$ (eqn
(LABEL:eq:mixqstate)). The system then evolves from a state that is _either_
$|\alpha\rangle|-\beta\rangle$ or $|-\alpha\rangle|\beta\rangle$, which is
different to the evolution of the superposition (57). If a measurement is made
at the later time $t_{a}$, the joint probability is

$\displaystyle P(X_{A},X_{B})_{mix}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\langle X_{A}|U_{A}(t_{a})|\alpha\rangle_{a}\langle
X_{B}|-\beta\rangle_{b}\langle\alpha|_{a}U_{A}^{\dagger}(t_{a})|X_{A}\rangle\langle-\beta|_{b}X_{b}\rangle$
(59) $\displaystyle+\langle X_{A}|U_{A}(t_{a})|-\alpha\rangle_{a}\langle
X_{B}|\beta\rangle_{b})\langle-\alpha|_{a}U_{A}^{\dagger}(t_{a})|X_{A}\rangle\langle\beta|_{b}X_{b}\rangle)$

The two probabilities $P(X_{A},X_{B})$ are indeed different. However, for
$\alpha$ and $\beta$ large, examination shows that the two probabilities (58)
and (59) become indistinguishable _._

###  VI.6 Delayed collapse for measurement $\langle
S_{2}^{(A)}S_{3}^{(A)}\rangle$

We may also examine where a measurement is to be made at time $t_{2}$, as in
the evaluation of $\langle S_{2}^{(A)}S_{3}^{(A)}\rangle$. Again, it makes no
difference as to the evaluation, if the “collapse” stage is delayed until a
later time. Wishing to measure the spin of subsystem $B$ at time $t_{2}$ means
we then “stop” the evolution of system $B$ at $t_{b}=t_{2}$. If the system $A$
continues to evolve, then the state of the system is
($t^{\prime}_{a}=t_{A}-t_{2}$)

$\displaystyle U^{(A)}(t^{\prime}_{A})|\psi_{Bell}\rangle$ $\displaystyle=$
$\displaystyle
N_{\pm}e^{-i\pi/4}(U^{(A)}(t^{\prime}_{A})|\alpha\rangle|-\beta\rangle-U^{(A)}(t^{\prime}_{A})|-\alpha\rangle|\beta\rangle)$
(60)

The final distributions $P(X_{A},X_{B})_{sup}$ and $P(X_{A},X_{B})_{mix}$ are
similar to those above, replacing $t_{A}$ by $t^{\prime}_{A}$. The lower
sequence of Figure 15 shows the Q function dynamics for the evolution where
there has been no collapse at time $t_{2}$. When plotted, the sequence where a
collapse has occured shows, even for moderate $\beta$, no distinguishable
difference, in either the final measured probabilities or the Q function.

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