# Leggett-Garg tests for macrorealism in the quantum harmonic oscillator and
more general bound systems

C. Mawby<EMAIL_ADDRESS>J.J.Halliwell<EMAIL_ADDRESS>Blackett Laboratory
Imperial College
London SW7 2BZ
UK

###### Abstract

The Leggett-Garg (LG) inequalities were introduced to test for the possible
presence of macroscopic quantum coherence. Since such effects may be found in
various types of macroscopic oscillators, we consider the application of the
LG approach to the one-dimensional quantum harmonic oscillator and also to
more general bound systems, using a single dichotomic variable $Q$ given by
the sign of the oscillator position. We present a simple method to calculate
the temporal correlation functions appearing in the LG inequalities for any
bound system for which the eigenspectrum is (exactly or numerically) known. We
apply this result to the quantum harmonic oscillator for a variety of
experimentally accessible states, namely energy eigenstates, and
superpositions thereof. For the subspace of states spanned by only the ground
state and first excited state, we readily find substantial regions of
parameter space in which the LG inequalities at two, three and four times can
each be independently violated or satisfied. We also find that the violations
persist, although are reduced, when the sign function defining $Q$ is smeared
to reflect experimental imprecision. For higher energy eigenstates, we find
that LG violations diminish, showing the expected classicalization. With a $Q$
defined using a more general type of position coarse graining, we find two-
time LG violations even in the ground state. We also show that two-time LG
violations in a gaussian state are readily found if the dichotomic variable at
one of the times is taken to be the parity operator. To demonstrate the
versatility of the approach, we repeat much of the LG analysis for the Morse
potential, finding qualitatively similar physical results.

##  1 Introduction

Perhaps a great temptation within quantum mechanics is to ascribe the spookier
extrapolations of its mathematics to phenomena which, although beyond our
knowledge, are fundamentally classical in nature. This encourages us to try to
understand – and place constraints on – what the universe may be doing in
between the snapshots we take of it in the form of measurements. This
endeavour, although foundational in nature, is becoming ever more relevant –
quantum technologies rely upon non-classical behaviour, and so tests which can
verify the presence of truly quantum behaviour are very valuable.

One of the key open questions in this area is whether quantum coherence may
persist up to the macroscopic level, which led to the introduction of the
Leggett-Garg (LG) inequalities Leggett and Garg (1985); Leggett (1988, 2008);
Emary  _et al._ (2013). These inequalities test a precise realist world-view
known as macrorealism (MR), which is defined by the conjunction of three
realist tenets – that a system at all instants of time is definitely in one of
the states available to it, that said state may be measured without
influencing future dynamics of the system, and that future measurements do not
affect past ones. The violation of these inequalities imply a failure of MR,
and thus, loosely speaking a detection of non-classical behaviour.

The LG inequalities are typically formulated for a dichotomic observable $Q$,
with possible values $s_{i}=\pm 1$, which is measured at a sequence of times
$t_{1},t_{2},\ldots,t_{i}$ . From these measurements, we can construct a data
set consisting of the single time averages $\expectationvalue{Q_{i}}$, and the
correlators $C_{ij}$, defined through

$C_{ij}=\expectationvalue{Q_{i}Q_{j}}=\sum_{i,j}s_{i}s_{j}p(s_{i},s_{j}),$
(1.1)

where $Q_{i}$ denotes $Q(t_{i})$. In keeping with the tenets of MR, the
correlator must be measured in a non-invasive way, typically using ideal
negative measurements Knee  _et al._ (2012) but other methods are also used
Halliwell (2016a); Majidy  _et al._ (2019); Majidy (2019). In Eq.(1.1),
$p(s_{i},s_{j})$ is the two-time probability describing the likelihood of
obtaining results $s_{i}$ and $s_{j}$ and times $t_{i}$ and $t_{j}$. We then
consider making measurements at times $(t_{1},t_{2})$, $(t_{2},t_{3})$ and
$(t_{1},t_{3})$, which yields three two-time probabilities. For systems which
obey the assumptions of MR, these three probabilities can always be matched as
marginal probabilities of an underlying joint probability
$p(s_{1},s_{2},s_{3})$ and the correlators will obey the three-time LG
inequalities ($\rm LG{3}$):

$\displaystyle L_{1}=1+C_{12}+C_{23}+C_{13}$ $\displaystyle\geq$
$\displaystyle 0,$ (1.2) $\displaystyle L_{2}=1-C_{12}-C_{23}+C_{13}$
$\displaystyle\geq$ $\displaystyle 0,$ (1.3) $\displaystyle
L_{3}=1+C_{12}-C_{23}-C_{13}$ $\displaystyle\geq$ $\displaystyle 0,$ (1.4)
$\displaystyle L_{4}=1-C_{12}+C_{23}-C_{13}$ $\displaystyle\geq$
$\displaystyle 0.$ (1.5)

The same arguments can be made for measurements at just two times
$t_{1},t_{2}$, which leads to the two-time LG inequalities ($\rm LG{2}$)
Halliwell (2016b, 2017, 2019a); Halliwell and Mawby (2019):

$1+s_{1}\expectationvalue{Q_{1}}+s_{2}\expectationvalue{Q_{2}}+s_{1}s_{2}C_{12}\geq
0.$ (1.6)

On their own, the $\rm LG{3}$ inequalities Eqs. (1.2-1.5) form a set of
necessary conditions for macrorealism – although their violation indicates a
failure of macrorealism, if they are satisfied, macrorealism does not
necessary hold. It is only by augmenting them with the set of twelve $\rm
LG{2}$ inequalities, that we reach the complete set of necessary and
sufficient conditions for macrorealism Halliwell (2016b, 2017, 2019a);
Halliwell and Mawby (2019). The implications of this are two-fold. Firstly, to
test if a given data set admits a macrorealistic description, the entire set
of $\rm LG{2}$ and $\rm LG{3}$ inequalities must be tested. Secondly, a
violation of just one $\rm LG{2}$ inequality is sufficient to show a failure
of MR, which involving just two measurement times, may well be logistically
simpler than testing the $\rm LG{3}$ inequalities.

Although originally proposed within the context of macroscopic quantum
coherence, experimental LG tests have been nearly exclusively conducted on the
discrete properties of microscopic systems; such as photons, silicon
impurities, nuclear magnetic resonance, neutrino oscillations, and position
superpositions in quantum random walks Majidy  _et al._ (2019); Knee  _et al._
(2012); Goggin  _et al._ (2011); Formaggio  _et al._ (2016); Robens  _et al._
(2015). For a thorough review of the LG inequalities, and experimental tests
thereof, see Ref. Emary  _et al._ (2013) and for a critique see Ref. Maroney
and Timpson (2014). Some LG tests have come close to macroscopicity Knee  _et
al._ (2016). There have also been some recent macroscopic non-classicality
tests based on interferometric experiments, with quantum interference observed
in $C_{60}$ and $C_{70}$ molecules Arndt  _et al._ (1999, 2001), and in large
organic molecules of masses up to 6910 amu Gerlich  _et al._ (2011). These are
not specifically LG tests, but proposals have been made to adapt them for this
purpose Pan (2020); Halliwell  _et al._ (2021).

Most macroscopic systems that have been investigated with a view to exhibiting
quantum coherence effects have the feature that they are described by
continuous variables. Hence to approach LG tests in the macroscopic domain, it
is natural to develop such tests for continuous variables, using, for example,
a dichotomic variable $Q$ defined by simple coarse graining of position. In
this work, we therefore pursue an investigation into one of the most
ubiquitous continuous variable systems – the quantum harmonic oscillator
(QHO), and bound systems more generally.

Our theoretical investigation is in part inspired by the LG experiment
proposed by Bose et alBose  _et al._ (2018). They used the dichotomic variable
$Q={\rm sgn}(x)$, and focused on a single coherent state of the QHO. By
contrast with other tests for non-classicality, this approach does not require
the fabrication of non-Gaussian states Romero-Isart  _et al._ (2011), nor the
coupling to an ancillary quantum system Scala  _et al._ (2013); Asadian  _et
al._ (2014). The work of Bose et al suggests that these quantum correlations
persist, and remain experimentally feasible to detect, at scales of up to
$10^{6}$-$10^{9}$ amu, meaning LG tests on the QHO are likely to push the
limits of our observations of macroscopic coherence.

To further this programme, we derive analytical results for the temporal
correlators for the QHO in a variety of experimentally accessible states,
primarily energy eigenstates and superpositions thereof. We also derive an
approximation for the temporal correlation functions which is applicable to
any quantum system for which the energy eigenspectrum is (exactly or
numerically) known. We determine substantial areas in the various parameter
spaces in which combinations of LG2, LG3 and LG4 inequalities are satisfied or
violated, thereby preparing the ground for experimental tests.

In Section 2, we set up the problem and calculate temporal correlators working
within the energy eigenbasis. We develop a powerful technique for studying the
partial overlap of energy eigenfunctions, which enables us to calculate the
temporal correlators for any system with known energy eigenspectrum. We apply
this to the QHO, which yields a simple and useful approximation for the
temporal correlator in the first excited state as a simple a cosine (just like
the simple spin model used in LG tests). We also demonstrate a procedure to
calculate the exact temporal correlators for any energy eigenstate of the QHO.

In Section 3, we explore the LG violations present in the QHO. We find
significant violations in the first excited state, and see that by creating
superpositions of the ground state and the first excited state, we can find
substantial regimes where the $\rm LG{2}$s and $\rm LG{3}$s are independently
satisfied or violated. We find significant LG violations persist even with
significant smoothing of projectors.

In Section 4, we analyze the LG inequalities in the higher excited states of
the QHO, and observe the expected classicalization, with the magnitude of LG
violations rapidly decreasing as energy increases.

In Section 5, we use a more general dichotomic observable, defined for
arbitrary regions of space. Using this more general variable we find LG
violations even in the ground state.

In Section 6, to demonstrate the versatility of the techniques developed in
Section 2, we calculate temporal correlators for the Morse potential, and
perform a brief LG analysis, finding significant LG3 and LG4 violations. We
summarize and conclude in Section 7.

The fine details of the complicated machinery required to calculate temporal
correlators in the QHO, are largely relegated to a series of technical
appendices.

##  2 Temporal correlators in bound systems

###  2.1 Conventions and Strategy

For most of this paper we work with the simplest choice of dichotomic variable
$Q=\text{sgn}(x)$, however the method we develop also gives the result for
more general observables involving arbitrary regions of space. This is
investigated further in Section 5. We will work with a general bound one-
dimensional system with Hamiltonian $\hat{H}$ and energy eigenstates $\ket{n}$
with corresponding eigenvalues $E_{n}$.

We now outline our strategy. We make use of the two-time quasi-probability,
introduced in Ref. Halliwell (2016b), given by,

$q(s_{1},s_{2})=\text{Re}{\rm Tr}(P_{s_{2}}(t_{2})P_{s_{1}}(t_{1})\rho),$
(2.1)

where $s=\pm 1$, and

$P_{s}=\frac{1}{2}(1+s\hat{Q})=\theta(s\hat{x}),$ (2.2)

where $\theta(x)$ is the Heaviside step function. Time dependence is then
handled in the Heisenberg picture.

We will use the quasi-probability in two ways. Firstly, it is proportional to
the two-time LG inequalities, so negativity signifies an LG2 violation
Halliwell (2016b). (The quasi-probability has a maximum negative value of
$-\tfrac{1}{8}$). Secondly, it yields a simple way to extract the temporal
correlators needed for the three-time LG inequalities, via its useful moment
expansion

$q(s_{1},s_{2})=\frac{1}{4}\left(1+s_{1}\expectationvalue{\hat{Q}_{1}}+s_{2}\expectationvalue{\hat{Q}_{2}}+s_{1}s_{2}C_{12}\right).$
(2.3)

We will hence first calculate the quasi-probability, and then go on to extract
the temporal correlators, which we can then use in the LG inequalities Eqs.
(1.2-1.5). In contrast to the computation of correlators in simple spin
models, these calculations are quite non-trivial.

More detailed properties of the quasi-probability may be found in
Ref.Halliwell (2016b), but we make a few brief comments here. As indicated, it
is used here purely as a mathematical tool and has no particular conceptual
role beyond the fact that it is proportional to the two-time LG inequalities.
It is clearly non-negative for commuting pairs of projections, so any
negativity is associated with incompatibility of the pairs of measurements. It
can usefully be expressed in terms of the Wigner-Weyl representationHalliwell
_et al._ (2019), from which one can see that it can be negative even when the
Wigner function of the initial state is non-negative, as is the case for
gaussian initial states. We shall see an example of this later in this paper,
and see also Refs.Halliwell  _et al._ (2019, 2021). The quasi-probability is
also very closely related to the two-time sequential measurement formula

$p(s_{1},s_{2})=\real{\rm Tr}\left(P_{s_{2}}(t_{2})P_{s_{1}}(t_{1})\rho
P_{s_{1}}(t_{1})\right),$ (2.4)

This is always non-negative and has a moment expansion similar in form to
Eq.(2.3), with the only difference that $\langle\hat{Q}_{2}\rangle$ is
replaced with $\langle\hat{Q}_{2}^{(1)}\rangle$, the average of $Q_{2}$ in the
presence of an earlier measurement at $t_{1}$ which has been summed out. (See
Ref.Halliwell (2016b)). Note in particular that both the quasi-probability and
sequential two-time measurement formula have the same correlator, which has
the quantum-mechanical form

$C_{12}=\frac{1}{2}\langle\hat{Q}_{1}\hat{Q}_{2}+\hat{Q}_{2}\hat{Q}_{1}\rangle.$
(2.5)

Physically, this corresponds to the fact that the same correlator is obtained
using a pair of measurements in which the first measurement is projective or
weak Halliwell (2016b); Fritz (2010), or more generally, any one of a family
of ambiguous measurements intermediate between these two options (as can be
seen in Ref.Halliwell  _et al._ (2019)). The relevant non-invasive measurement
protocols for LG tests of these quantities are described in more detail in
Refs.Halliwell (2016b, 2017, 2019a, 2019b).

###  2.2 Calculating the quasi-probability

Using the definition of the two-time quasi-probability Eq.(2.1), and Eq.(2.2),
we have

$q(s_{1},s_{2})=\text{Re}{\rm
Tr}(e^{\frac{iHt_{2}}{\hbar}}\theta(s_{1}\hat{x})e^{-\frac{iH(t_{2}-t_{1})}{\hbar}}\theta(s_{2}\hat{x})e^{\frac{-iHt_{1}}{\hbar}}\rho).$
(2.6)

We will often work with the difference between measurement times, and so
introduce the variable $\tau=t_{2}-t_{1}$.

Initially we calculate one element of the quasi-probability $q(+,+)$, where
the full quasi-probability $q(s_{1},s_{2})$ may often be reconstructed through
symmetry arguments. Working within the energy eigenbasis, we write Eq.(2.1) as

$q(+,+)=\real\sum_{m=0,n=0}^{\infty}\\!\\!\\!\innerproduct{\psi}{m}\\!\\!\\!\matrixelement{m}{P_{+}(t_{2})P_{+}(t_{1})}{n}\\!\\!\innerproduct{n}{\psi},$
(2.7)

which equals

$q(+,+)=\real\sum_{m=0,n=0}^{\infty}\innerproduct{\psi}{m}\innerproduct{n}{\psi}q_{mn},$
(2.8)

where

$q_{mn}=\matrixelement{m}{e^{\frac{iHt_{2}}{\hbar}}\theta(\hat{x})e^{-\frac{iH\tau}{\hbar}}\theta(\hat{x})e^{-\frac{iHt_{1}}{\hbar}}}{n}.$
(2.9)

Using the position representation for the step functions
$\theta(\hat{x})=\int^{\infty}_{0}|x\rangle\langle x|\mathop{dx}$, this may be
written explicitly as

$q_{mn}=e^{i\frac{E_{m}}{\hbar}t_{2}-i\frac{E_{n}}{\hbar}t_{1}}\int_{0}^{\infty}\int_{0}^{\infty}\mathop{dx}\mathop{dy}\innerproduct{m}{x}\\!\\!\matrixelement{x}{e^{-\frac{iH\tau}{\hbar}}}{y}\\!\innerproduct{y}{n},$
(2.10)

where from here the calculation forks, with two obvious ways to proceed. One
approach is to insert a resolution of unity in the energy eigenbasis, and
truncate the infinite sum to yield an approximation to the correlators, which
we will proceed with first. The second approach is to insert the expression
for the propagator, if known, which will allow us to calculate exact
expressions for the quasi-probability.

The single time averages are given by

$\expectationvalue*{\hat{Q}}=\expectationvalue{{\rm sgn}(\hat{x})}{n}.$ (2.11)

In the special case of symmetric potentials, since $\text{sgn}(\hat{x})$ flips
the parity of $\ket{n}$, this represents the overlap between an odd and an
even state. Hence in this case, the single time averages in the moment
expansion Eq.(2.3) are identically zero. This yields a simple relationship
between the correlator and the quasiprobability,

$q(+,+)=\frac{1}{4}(1+C_{12}).$ (2.12)

We adopt the notation $C_{12}^{\ket{n}}$, to denote the temporal correlator
between times $t_{1}$ and $t_{2}$, for a given energy eigenstate $\ket{n}$.

###  2.3 Quasi-probability for energy eigenstates

We use the approximation approach first, whereupon inserting the resolution of
unity, we have

$q_{mn}=e^{i\frac{E_{m}}{\hbar}t_{2}-i\frac{E_{n}}{\hbar}t_{1}}\sum_{k=0}^{\infty}e^{-i\frac{E_{k}}{\hbar}(t_{2}-t_{1})}\int_{0}^{\infty}\int_{0}^{\infty}\mathop{dx}\mathop{dy}\innerproduct{m}{x}\\!\innerproduct{x}{k}\\!\innerproduct{k}{y}\\!\innerproduct{y}{n}.$
(2.13)

The integration is now separable into two integrals, the partial overlap of
energy eigenstates,

$J_{k\ell}=\int_{0}^{\infty}\mathop{dx}\innerproduct{k}{x}\innerproduct{x}{\ell}.$
(2.14)

Surprisingly, as detailed in Appendix A.1, these integrals can be completed
for generic potentials over arbitrary boundaries $x_{1}$, $x_{2}$, with result

$J_{k\ell}(x_{1},x_{2})=\frac{1}{2(\varepsilon_{\ell}-\varepsilon_{k})}\left[\psi_{k}^{\prime}(x_{2})\psi_{\ell}(x_{2})-\psi_{\ell}^{\prime}(x_{2})\psi_{k}(x_{2})-\psi_{k}^{\prime}(x_{1})\psi_{\ell}(x_{1})+\psi_{\ell}^{\prime}(x_{1})\psi_{k}(x_{1})\right],$
(2.15)

where $\psi_{k}(x)=\innerproduct{x}{k}$. Hence the matrix elements of the
quasi-probability are

$q_{mn}=e^{i\frac{E_{m}}{\hbar}t_{2}-i\frac{E_{n}}{\hbar}t_{1}}\sum_{k=0}^{\infty}e^{-i\frac{E_{k}}{\hbar}(t_{2}-t_{1})}J_{mk}J_{nk}.$
(2.16)

The quasi-probability for a single energy eigenstate is the real part of the
diagonal elements,

$q_{n}(+,+)=\real
e^{i\frac{E_{n}}{\hbar}\tau}\sum_{k=0}^{\infty}e^{-i\frac{E_{k}}{\hbar}\tau}J_{nk}^{2}.$
(2.17)

By truncating this sum, we are able to calculate the quasi-probability for any
coarse-graining of space, and for any soluble bound system, purely in terms of
the spectrum of its Hamiltonian. We will primarily use this result with the
QHO in this work, however in Sec. 6, we consider its application to other
systems. Further details about implementation of this formula, including a
truncation error estimate $\Delta_{n}(m)$ defined in Eq.(A.12), are found in
Appendix A.2.

We also note in passing that Eq.(2.15) unlocks the result of several
indefinite integrals in the form of partial spatial overlaps of special
functions, which we have not been able to find in the literature. These
include; generalised Laguerre polynomials (Morse potential Dahl and Springborg
(1988)), Mathieu functions (quantum pendulum Galvez  _et al._ (2019)), Airy
functions (triangular potential Khonina  _et al._ (2013)), and Hypergeometric
functions (transformation technique Morales  _et al._ (2015)), with the
exactly soluble potential leading to the result bracketed.

###  2.4 Application to the QHO

The results so far have been general to any bound potential. We now apply them
to the systems defined exactly (or approximately) by the harmonic oscillator
Hamiltonian,

$\hat{H}=\frac{\hat{p}_{\rm{phys}}^{2}}{2m}+\frac{1}{2}m\omega^{2}\hat{x}_{\rm{phys}}^{2},$
(2.18)

with physical position and momentum $x_{\rm{phys}}$ and $p_{\rm{phys}}$. We
continue to denote energy eigenstates by $\ket{n}$, and use natural units, so
we work with dimensionless position and momentum $x$ and $p$, defined by
$\sqrt{\hbar/(m\omega)}x=x_{\rm{phys}}$ and
$p\sqrt{(m\omega)/\hbar}=p_{\rm{phys}}$. We will similarly write energy in
terms of $\hbar\omega$, with
$E_{n}=(n+\frac{1}{2})\hbar\omega=\varepsilon_{n}\hbar\omega$. The energy
eigenstates within the position basis are then given by

$\innerproduct{n}{x}=\frac{1}{\sqrt{2^{n}n!}}\pi^{-\frac{1}{4}}\exp(-\frac{1}{2}x^{2})H_{n}(x),$
(2.19)

where $H_{n}(x)$ are the Hermite polynomials.

For the ground state, the infinite series needs many terms to reach good
convergence, and so we instead turn to $q_{11}$, whereupon taking the real
part, we have the quasi-probability $q(+,+)$, for the first excited state. We
calculate the first few $J_{nm}$ using Eq.(A.9). We find
$J_{1,0}^{2}=\frac{1}{2\pi}$, $J_{1,1}^{2}=\frac{1}{4}$,
$J_{1,2}^{2}=\frac{1}{4\pi}$, $J_{1,3}^{2}=0$, and
$J_{1,4}^{2}=\frac{1}{48\pi}$. Using just the first three terms leads to a
very good approximation to the correlator, with $\Delta_{1}(2)=0.011$. The sum
may be explicitly written as
$q_{11}=\frac{1}{4}+\frac{1}{4\pi}e^{-i\omega\tau}+\frac{1}{2\pi}e^{i\omega\tau}$.
Taking the real part, we find

$q(+,+)=\frac{1}{4}+\frac{3}{4\pi}\cos\omega\tau.$ (2.20)

By comparing Eq.(2.20) to the expression for the quasi-probability of a pure
eigenstate Eq.(2.12), we can identify the correlator as

$C_{ij}^{\ket{1}}\approx\cos\omega\tau,$ (2.21)

where we have dropped the coefficient of $\tfrac{3}{\pi}\approx 0.955$, by the
requirement that $C_{ij}\to 1$ as $\tau\to 0$.

This approximation is interesting since it says that for the first excited
state, the QHO to a good approximation behaves just like the canonical simple
spin-$\frac{1}{2}$ example used in much LG research. We hence may borrow
intuition from this simpler system, and we know to expect LG violations. This
similarity can be understood by explicitly calculating the temporal correlator
using the three-term approximation to the post-measurement state. The temporal
correlator here is defined as

$C_{12}^{\ket{1}}=\real\expectationvalue{\hat{Q}_{2}\hat{Q}_{1}}{1}.$ (2.22)

From Eq.(2.2), we have $\hat{Q}=2\theta(\hat{x})-1$, so using the three-term
approximation and taking $t_{1}=0$, we find

$\hat{Q}_{1}\ket{1}\approx\frac{2}{\sqrt{2\pi}}\ket{0}+\frac{1}{\sqrt{\pi}}\ket{2}.$
(2.23)

Similarly, taking $t_{2}=\tau$, we find

$\bra{1}\hat{Q}_{2}\approx
e^{i\omega\tau}\left(\frac{2}{\sqrt{2\pi}}\bra{0}+e^{-2i\omega\tau}\frac{1}{\sqrt{\pi}}\bra{2}.\right).$
(2.24)

Hence like the spin-$\frac{1}{2}$ case, it boils down to just two states.
Contracting these result, and taking the real part yields

$C_{12}^{\ket{1}}=\frac{3}{\pi}\cos(\omega\tau)\approx\cos(\omega\tau)$ (2.25)

We note that taking the next order of approximation we have
$\Delta_{1}(4)=0.005$, and clearly a much better approximation. The expression
for the correlator (again normalized to 1 as $\tau\to 0$), is then

$C_{ij}^{\ket{1}}\approx\frac{36}{37}\cos\omega\tau+\frac{1}{37}\cos
3\omega\tau.$ (2.26)

###  2.5 Exact Correlators for the QHO

For the QHO, we are also able to calculate exact expressions for the
correlators, whereupon inserting the expression for the QHO propagator into
Eq.(2.10), we find

$q_{mn}=\mathcal{N}_{mn}(\tau)e^{i\frac{E_{m}}{\hbar}t_{2}-i\frac{E_{n}}{\hbar}t_{1}}\times\\\
\int^{\infty}_{0}\int^{\infty}_{0}\mathop{dr}\mathop{ds}H_{m}(r)H_{n}(s)e^{-\frac{r^{2}}{2}}e^{-\frac{s^{2}}{2}}\exp({i\frac{1}{2\tan(\omega\tau)}(r^{2}+s^{2})-i\frac{1}{\sin(\omega\tau)}rs}),$
(2.27)

where $\mathcal{N}_{mn}(\tau)$ is a dimensionless normalisation factor,
defined by

$\mathcal{N}_{mn}(\tau)=\frac{1}{\pi}\frac{1}{\sqrt{2^{n}n!}}\frac{1}{\sqrt{2^{m}m!}}\frac{1}{\sqrt{2i\sin(\omega\tau)}}$
(2.28)

This integral can be completed, via a generating integral approach, which is
detailed in Appendix B. Using this result, we are able to write the exact
results for the temporal correlators. To do so, it is helpful to introduce the
function

$f(\tau)=-ie^{-i\frac{\omega\tau}{2}}\sqrt{2i\sin\omega\tau}.$ (2.29)

The correlators for the groundstate and first excited state may then be
written as

$\displaystyle C_{ij}^{\ket{0}}$
$\displaystyle=\frac{2}{\pi}\real\arctan\left(\frac{1}{f(\tau)}\right),$
(2.30) $\displaystyle C_{ij}^{\ket{1}}$
$\displaystyle=\frac{2}{\pi}\real\left[\arctan\left(\frac{1}{f(\tau)}\right)+f(\tau)\right].$
(2.31)

These two rather non-intuitive expressions are plotted in Fig. 1, where they
are easily seen to have the broadly expected physical behaviours The
expressions for the first nine correlators are included in Appendix B.4.

###  2.6 Matrix Elements for a two-state oscillator

It will turn out that very much can be studied in the QHO, with initial
superposition states of just the ground state and first excited state. That is
working with states of the form

$\ket{\psi}=a\ket{0}+b\ket{1}.$ (2.32)

To determine the off-diagonal elements of $q_{mn}$ defined in Eq.(2.9), it is
simplest to do so term-by-term in the moment expansion Eq.(2.3).

The quantum mechanical temporal correlator is given by Eq.(2.5), with matrix
elements

$\matrixelement{m}{\hat{C}_{ij}}{n}=\frac{1}{2}\matrixelement*{m}{\hat{Q}_{1}\hat{Q}_{2}+\hat{Q}_{2}\hat{Q}_{1}}{n}.$
(2.33)

We now note that $\hat{Q}_{1}\hat{Q}_{2}+\hat{Q}_{2}\hat{Q}_{1}$ is invariant
under reflections. This means that
$(\hat{Q}_{1}\hat{Q}_{2}+\hat{Q}_{2}\hat{Q}_{1})\ket{n}$ must have the same
parity as $\ket{n}$. Hence in cases where $m$ is odd and $n$ even, or vice-
versa, the correlator represents the overlap of an even state with an odd
state, and is therefore zero in these cases. Hence for the general state
Eq.(2.32), the correlator is simply given as the mixture

$C_{ij}=\absolutevalue{a}^{2}C_{ij}^{\ket{0}}+\absolutevalue{b}^{2}C_{ij}^{\ket{1}}.$
(2.34)

We now determine the matrix elements of the quasi-probability. Owing to the
argument preceding Eq.(2.12), the diagonal elements take the simple form

$q_{nn}(s_{1},s_{2})=\frac{1}{4}\left(1+s_{1}s_{2}C_{ij}^{\ket{n}}\right).$
(2.35)

Similarly, with the vanishing of the correlators on the off-diagonals, we see
that

$q_{01}(s_{1},s_{2})=\frac{1}{4}\matrixelement{0}{s_{1}\hat{Q}_{1}+s_{2}\hat{Q}_{2}}{1}.$
(2.36)

As these averages involve only a single time each, the time dependence is
trivial, and we just require the value of

$\expectationvalue*{\hat{Q}}=\matrixelement{0}{\text{sgn}(\hat{x})}{1},$
(2.37)

which is readily calculated using Eq.(A.9) as
$\expectationvalue*{\hat{Q}}=\sqrt{\frac{2}{\pi}}$.

We hence find the two-time quasi-probability for the state Eq.(2.32) to be

$q(s_{1},s_{2})=\frac{1}{4}\left[1+s_{1}\left(2\sqrt{\frac{2}{\pi}}\real
a^{*}be^{i\omega t_{1}}\right)+s_{2}\left(2\sqrt{\frac{2}{\pi}}\real
a^{*}be^{i\omega
t_{2}}\right)+s_{1}s_{2}\left(\absolutevalue{a}^{2}C_{ij}^{\ket{0}}+\absolutevalue{b}^{2}C_{ij}^{\ket{1}}\right)\right].$
(2.38)

###  2.7 The classical analogue

(a)

(b)

Figure 1: In (a), the temporal correlator for the ground state of the QHO Eq.
(2.30) is plotted, alongside the classical analogue Eq. (2.41) (dashed). In
(b), the temporal correlator for the first excited state Eq. (2.31) is
plotted, alongside the correlator for the simple spin case, showing the close
similarity mentioned in the text. The classical analogue is again shown
(dashed).

It is useful in many of these calculations to compare the correlators obtained
with their classical analogue. This is readily found from the classical
analogue of the quasi-probability, namely
$\langle\theta(x)\theta(x(\tau)\rangle$, with phase-space initial state
$f(x^{2}+p^{2})$ which is normalised to
$2\pi\int_{0}^{\infty}\mathop{dr}rf(r)=1$. This choice covers fixed energy
states, and mixtures thereof. We then have,

$\mathbbm{q}(0,\tau)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\mathop{dx}\mathop{dp}f(x^{2}+p^{2})\theta(x)\theta(x\cos\omega\tau+p\sin\omega\tau).$
(2.39)

The step-functions are easiest handled in polar coordinates as

$\mathbbm{q}(0,\tau)=\int_{0}^{\absolutevalue{\pi-\omega\tau}}\mathop{d\theta}\int_{0}^{\infty}\mathop{dr}rf(r),$
(2.40)

where inserting the normalisation condition on $f(r)$ yields

$\mathbbm{q}(0,\tau)=\frac{\lvert{\pi-\omega\tau}\rvert}{2\pi},$ (2.41)

which has the corresponding classical correlator of
$\mathbbm{C}_{12}=-1+\frac{2}{\pi}\lvert{\pi-\omega\tau}\rvert$.

We now plot two of the QHO correlators, alongside the classical analogue, in
Fig. 1. We also plot the correlator for the first excited state, alongside the
approximation Eq.(2.21), which also serves to compare the behaviour of the
first excited state of the QHO with the canonical simple spin model typically
used in LG research.

##  3 Leggett-Garg Violations in the two-state QHO

###  3.1 Four Regimes

In this section, we study the LG inequalities in the scenario where we have
access only to the ground state and the first excited state, the two-state
oscillator. This allows us to take full advantage of the simplicity of the
expression for the correlator in this two-state superposition, Eq.(2.34). We
will find that even in this state space, we have access to four regimes of
interest, which are laid out in Table 1.

Regime | LG3s | LG2s
---|---|---
I |  |
II |  |
III |  |
IV |  |
Table 1: The four regimes testable with the $\rm LG{2}$ and $\rm LG{3}$
inequalities. A tick denotes that the complete set of inequalities is
satisfied, whereas a cross indicates that one or more of the inequalities in
that set is violated.

These four regimes take their importance from the fact that, for the data set
consisting of the three $\langle Q_{i}\rangle$ and three correlators, the LG
inequalities form necessary and sufficient conditions for MR only when all the
$\rm LG{2}$ and $\rm LG{3}$ inequalities are satisfied. This has a couple of
interesting consequences in cases where the $\rm LG{3}$ inequalities are
satisfied. From the sufficiency aspect; although typically just $\rm LG{3}$s
are experimentally tested, their being satisfied does not in fact indicate the
system behaves macrorealistically, since the $\rm LG{2}$s must also be
verified as satisfied. This means using solely the $\rm LG{3}$s to test for MR
leaves space open for false positives. From the necessity aspect; since all
conditions must be satisfied, it means the violation of one or more of the
$\rm LG{2}$s is enough to indicate the failure of MR, which since it involves
just two times, may be easier to implement experimentally. These four regimes
have previously been explored both experimentally and theoretically in simple
spin models Majidy  _et al._ (2021).

The QHO limited to the superpositions of the $\ket{0}$ and $\ket{1}$ states
makes a complete playground for the experimentalist, exhibiting all variants
of MR tested by the LG inequalities. We analyze and provide examples of each
of the regimes in Table 1. In what follows, we survey the parameter-space of
these two-state superpositions to determine where each of the four regimes
lie.

###  3.2 LG inequalities for pure eigenstates

We first consider initial states which are energy eigenstates. The LG2s are
trivially satisfied for these states since $\expectationvalue{Q_{i}}=0$, so we
can only access regimes I and II in Table 1. This is not the case for
superpositions.

Consider now the $\rm LG{3}$s, $L_{i}^{\ket{n}}$, where $i$ indexes one of the
four $\rm LG{3}$ kernels Eq.(1.2-1.5), with the initial state $\ket{n}$. For
example,

$L_{1}^{\ket{0}}=1+C_{12}^{\ket{0}}+C_{13}^{\ket{0}}+C_{23}^{\ket{0}}.$ (3.1)

These four inequalities are plotted in Fig. 2, (although two of them
coincide). For the ground state, the $\rm LG{3}$ inequalities are satisfied
always. Since the $\rm LG{2}$ inequalities are trivially satisfied as well,
this means that for the ground state lies in regime I of Table 1. For the
first excited state, we see the $\rm LG{3}$ are violated always (except at
points of measure zero). The magnitude of the violation is significant,
reaching approximately 73% of the Lüders bound of $-\frac{1}{2}$. This means
that the first excited state corresponds to regime II.

(a)

(b)

Figure 2: In (a), the three-time LG inequalities are plotted for the ground
state of the QHO, where they are satisfied at all times. In (b), the same
inequalities are shown for the first excited state, where they are violated
everywhere, except for points of measure zero. These violations come close to
the maximal violation, the Lüders bound of $-\frac{1}{2}$.

###  3.3 LG inequalities for superposition states

Since the matrix elements of the correlator are diagonal for superpositions of
$\ket{0}$ and $\ket{1}$, the LG3s will simply be a convex sum of
$L_{i}^{\ket{0}}$ and $L_{i}^{\ket{1}}$. We can hence continuously transition
between states with no $\rm LG{3}$ violation, and those with $\rm LG{3}$
violation everywhere. We begin by parametrising the two-state superpositions
as

$\ket{\psi}=\cos\tfrac{\theta}{2}\ket{0}+e^{i\phi}\sin\tfrac{\theta}{2}\ket{1},$
(3.2)

where $0\leq\theta\leq\pi$, and $0\leq\phi\leq 2\pi$.

Since the $\rm LG{3}$ inequalities are constructed purely from correlators,
using Eq.(2.34), we find

$L_{i}(\theta)=\cos^{2}\tfrac{\theta}{2}~{}L_{i}^{\ket{0}}+\sin^{2}\tfrac{\theta}{2}~{}L_{i}^{\ket{1}},$
(3.3)

and so a two-dimensional space of $\theta$ and $\tau$ defines whether the $\rm
LG{3}$ inequalities for the superposition are satisfied. This space is plotted
in Fig. 3, where for a given $\theta$, there are three possible cases; the
$\rm LG{3}$ are violated for all $\tau$, the $\rm LG{3}$ are satisfied for all
$\tau$, or a non-trivial mixture of the two, where they are satisfied for some
ranges of $\tau$, and violated for others.

Figure 3: The regions where at least one of the $\rm LG{3}$ inequalities is
violated are shown as a function of time between measurements, and
superposition coefficients. The shading corresponds to the magnitude of the
$\rm LG{3}$ violation.

We now analyze the behaviour of the two-time LG inequalities. Due to
interference terms, the averages $\langle Q_{i}\rangle$ are non-zero for
superposition states, which can produce $\rm LG{2}$ violations. Using the
parametrisation Eq.(3.2) in the expression for the quasi-probability
Eq.(2.38), we find

$q(s_{1},s_{2})=\frac{1}{4}\Bigg{[}1+s_{1}\left(\sqrt{\frac{2}{\pi}}\sin\theta\cos(\phi+\omega
t_{1})\right)+s_{2}\left(\sqrt{\frac{2}{\pi}}\sin\theta\cos(\phi+\omega
t_{2})\right)\\\
+s_{1}s_{2}\left(\cos^{2}\tfrac{\theta}{2}~{}C_{ij}^{\ket{0}}+\sin^{2}\tfrac{\theta}{2}~{}C_{ij}^{\ket{1}}\right)\Bigg{]}.$
(3.4)

Without loss of generality, we set $t_{1}=0$, and in Fig. 4, we present the
full parameter-space for the two-state superpositions, showing where the $\rm
LG{3}$ and $\rm LG{2}$ inequalities are violated. Hence with superposition
states, we can reach regimes III and IV from Table 1.

(a)

(b)

Figure 4: In (a) the behaviour of the $\rm LG{2}$ and $\rm LG{3}$ inequalities
is plotted, for all possible superpositions of $\ket{0}$ and $\ket{1}$. Within
the orange (non-meshed) region, at least one of the four $\rm LG{3}$
inequalities are violated, and within the blue (meshed) region, at least one
of the twelve $\rm LG{2}$ inequalities are violated. In (b), a slice of this
parameter space at $\phi=\pi$ is presented.

In Fig. 5, the complete set of $\rm LG{2}$ and $\rm LG{3}$ inequalities are
plotted for the state $\theta=0.7$, and $\phi=\pi$. This state has been chosen
to represent the important regime III, where the only violation occurs in the
$\rm LG{2}$ inequalities between $t_{2}$ and $t_{3}$, despite the LG3s being
satisfied.

Figure 5: The complete set of two- and three-time LG inequalities are plotted
for the state $\theta=1.4$, $\phi=\pi$. For this state, the $\rm LG{3}$ (d)
inequalities are always satisfied, however there is still an LG violation
present, at the level of two times, seen by the violation of the $\rm LG{2}$
between $\tau$ and $2\tau$ in (b).

###  3.4 LG4s and higher

(a)

(b)

Figure 6: In (a) the $\rm LG{4}$ inequalities are plotted for the ground
state, where they are satisfied for all times. In (b), the inequalities are
plotted for the first excited state, showing some regions of violation. The
maximal violation is approximately $2.615$, which represents around $92\%$ of
the maximal violation of $2\sqrt{2}$.

We now consider the behaviour of the LG inequalities when more measurements
are made. We consider the case of the $\rm LG{4}$ inequalities, and higher
order $n$-time LG inequalities Halliwell and Mawby (2019). Constructed purely
from sums of correlators, it is apparent that the $\rm LG{n}$ for two-state
superpositions will again just be mixtures of the $\ket{0}$ and $\ket{1}$
cases, and for $n>2$, we have

$L_{i}(n)=\cos^{2}\tfrac{\theta}{2}~{}L_{i}^{\ket{0}}(n)+\sin^{2}\tfrac{\theta}{2}~{}L_{i}^{\ket{1}}(n).$
(3.5)

The $\rm LG{4}$ inequalities take the form

$-2\leq C_{12}+C_{23}+C_{34}-C_{14}\leq 2,$ (3.6)

together with the six more inequalities obtained by moving the minus sign to
the other three locations. The necessary and sufficient conditions for MR at
four times consist of these eight LG4 inequalities, together with the set of
sixteen LG2s for the four time pairs Halliwell (2016b); Halliwell and Mawby
(2019); Halliwell (2017, 2019a). We plot the $\rm LG{4}$ inequalities in Fig.
6. We again have the property that the LG4 inequalities are always satisfied
for the ground state, and are violated for the first excited state. However,
in contrast to the $\rm LG{3}$ inequalities, the $\rm LG{4}$ inequalities are
not violated everywhere, and have large regions of parameter space where they
are satisfied. They allow access to interesting combinations of MR violations,
whilst maintaining the experimental simplicity of working with a single energy
eigenstate.

For the LGn case, we found no violations for the groundstate. Since the
excited state correlator Eq.(2.21) is very similar to the simple spin case,
the asymptotic behaviour for large $n$ is the same as that of the example
analyzed in our earlier paper Halliwell and Mawby (2019).

###  3.5 Smoothed Projectors

A natural question that arises is whether the observed LG violations are an
artefact of using sharp projectors, that may fade when physically realisable
measurements are used. To provide an indication that this is not the case, in
Appendix C.1 we repeat the calculation of $C_{12}^{\ket{1}}$, but with a
spatially smoothed projector. We find violations persist while the projector
smoothing is less characteristic length-scale of the oscillator.

##  4 Higher states

(a)

(b)

Figure 7: In (a), the maximal violation for a given energy eigenstate is
plotted as a fraction of the Lüders bound. The odd eigenstates are represented
by the dashed line (circles), and always have significantly more violation
than the even states represented by the dotted line (squares). In (b), we plot
the average magnitude of difference between the quantum correlators and the
classical correlator, showing expected classicalization for large $n$.

Following Section 3, it is natural to ask whether the LG inequalities behave
significantly differently when including more of the energy eigenstate
spectrum. By two methods, we establish that for large $n$, the predictions of
the QHO tend towards classical statistics.

Firstly, by plotting the higher eigenstate correlators, we can visually see
that they tend toward the classical correlator Eq.(2.41). To make this exact,
we calculate the average distance from the classical correlator, as a function
of $n$. For this, we define

$\Delta(n)=\frac{1}{2\pi}\int_{0}^{\frac{2\pi}{\omega}}\mathop{dt}\lvert\mathbbm{C}_{12}-C_{12}^{\ket{n}}\rvert,$
(4.1)

which we plot up to $n=50$ in Fig. 7, where we see that the quantum temporal
correlators very rapidly match the classical correlator.

Secondly, with details in Appendix C.2, we perform an asymptotic analysis on
the infinite sum representation of the quasi-probability, and show that for
large $n$, this series tends towards the Fourier series of a triangle wave,
matching the classical result.

##  5 Different choices of Dichotomic Variables

We have so far worked with the simplest choice of dichotomic variable, $Q={\rm
sgn}(x)$. We now consider what new effects may be discovered using different
choices. We first consider a $Q$ defined using coarse grainings over more
general spatial regions. Secondly, we consider a more novel choice of $Q$,
namely the parity operator, which does not necessarily have a macrorealistic
description but reveals some interesting features.

###  5.1 Quasi-probability for arbitrary regions in the ground state

We now demonstrate how to calculate the quasi-probability where each
measurement is taken over an arbitrary coarse graining of space. In this work,
we consider just a pair of dichotomic variables, however the techniques
presented are readily applied to a full many-valued variable LG analysis as
introduced in Ref. Halliwell and Mawby (2020).

We define projectors over arbitrary regions of space as

$E(\alpha)=\int_{\Delta(\alpha)}\mathop{dx}\outerproduct{x}{x},$ (5.1)

where $\Delta(\alpha)$, where $\alpha=\pm$, is a set of intervals which
partition the real line, which may be chosen to be different for each
measurement time. The dichotomic variable $Q$ is then defined by $Q=2E(+)-1$,
and discerns whether a particle is within a given region of space ($+$), or
outside of it ($-$). The two-time quasi-probability is then given by

$q(+,+)=\real\expectationvalue{E_{2}(+)E_{1}(+)}$ (5.2)

and the LG2 inequalities are then simply $q(+,+)\geq 0$.

Remarkably, by using these more general dichotomic variables we can find $\rm
LG{2}$ violations in the ground state of the QHO. To demonstrate this, we
write the quasi-probability using the QHO propagator Eq.(2.27), with the
projectors Eq.(5.1) for the special case of the ground state,

$q(+,+)=\real\mathcal{N}_{00}(\tau)\int^{b}_{a}\int^{d}_{c}\mathop{dr}\mathop{ds}e^{-\frac{r^{2}}{2}}e^{-\frac{s^{2}}{2}}\exp({i\frac{1}{2\tan(\omega\tau)}(r^{2}+s^{2})-i\frac{1}{\sin(\omega\tau)}rs}).$
(5.3)

Since the real part of the integrand oscillates around zero, negative values
of $q(+,+)$ can clearly by achieved by suitable choice of the spatial
intervals $[a,b]$ and $[c,d]$.

In Fig. 8(a), we plot the result of the integration, with the first interval
$\Delta(+)=[0,\infty)$, and then plot the remaining integrand of the second
integral, in the regions where it is negative. This indicates the region in
which to make a second measurement, which will lead to negativity of the
quasi-probability, and hence $\rm LG{2}$ violations.

To calculate the quasi-probability with arbitrary coarse-grainings, it is most
efficient to use the techniques in Sec.2.3 since this approach already
encapsulates these more general projectors. The details of this calculation
are found in Appendix C.3.

(a)

(b)

Figure 8: In (a), we plot the result of the first integration in Eq.(5.3) over
$[0,\infty]$, which serves as a rough map to choosing a second measurement
which will lead to a negative quasiprobability. In (b), we plot the quasi-
probability, for $\omega t=2.77$, with the first measurement over
$[0,\infty]$, and the second over the interval $[c,d]$. The quasi-probability
reaches around $20\%$ of the maximal violation of $-0.125$.

In Fig. 8(b), we plot the quasi-probability for $\omega\tau=2.77$, over a
range of possible second measurement windows. From this, we observe that there
are a wide range of choices for the second measurement which lead to an LG2
violation, which can reach $20\%$ of the maximum possible violation. We also
observe that these measurements do not not need to be over particularly small
regions of space, and that simply shifting the axis of the second measurement
from $[0,\infty)$ to $[c,\infty)$, compared to the usual choice of dichotomic
variable, is sufficient for a good LG2 violation in the ground state.

The ground state wave function has positive Wigner function and from a
macrorealistic point of view, is sometimes thought of as a “classical” state
describing a particle localized at a phase space point, yet we see here that a
LG2 violation is possible. A similar phenomenon was noted in Ref.Halliwell
_et al._ (2021), which used a coherent state with $\langle x\rangle=0$. The
origin of this effect will be explored in a future paper. We have also looked
for LG3 and LG4 violations in the ground state but an extensive parameter
search yields no result.

###  5.2 LG violations using the parity operator

Our second choice of different dichotomic operator arises from the observation
that the variables $Q_{1}$, $Q_{2}$ etc used in LG inequalities do not
necessarily have to be the time evolution of a given dichotomic variable $Q$,
but could be a set of any dichotomic variables. The derivation of the LG
inequalities still holds so an LG test is still possible as long as the
variables can be measured. Such an approach has been used in some experiments
Goggin  _et al._ (2011). Note at this point that we are now departing from the
quantum harmonic oscillator, at least in terms of time evolution, as the
physical system of interest (although the QHO is still the natural arena in
which to create the gaussian state used below).

Two interesting variables to use in this context are the parity operator,

$\Pi=\int_{-\infty}^{\infty}dx\ |x\rangle\langle-x|,$ (5.4)

and the parity inversion operator,

$R=i\int_{-\infty}^{\infty}dx\ {\rm sgn}(x)\ |x\rangle\langle-x|.$ (5.5)

These two operators, taken together with the usual choice $Q={\rm sgn}(x)$
have the interesting property that they obey the same algebra as the Pauli
matrices $[Q,R]=2i\Pi$ etc and they also all anticommute with each other. By
comparison with the spin-$\frac{1}{2}$ model, this is a clear suggestion that
LG violations are readily possible with such variables. These variables were
previously used in Ref. Praxmeyer  _et al._ (2005) to demonstrate a Bell
inequality violation for a gaussian two-particle entangled state of the
Einstein-Podolsky-Rosen type.

We consider the LG2

$1-\langle Q_{1}\rangle-\langle Q_{2}\rangle+C_{12}\geq 0,$ (5.6)

and we take $Q_{1}=Q={\rm sgn}(\hat{x})$ and $Q_{2}=\Pi$. Since these
operators anticommute the correlator is zero and the LG2 is then

$1-\langle Q\rangle-\langle\Pi\rangle\geq 0.$ (5.7)

We take a gaussian state of width $\sigma$, $\langle x\rangle=q$ and $\langle
p\rangle=0$ and readily find that the LG2 inequality is

$1-{\rm
erf}\left(\frac{q}{\sqrt{2}\sigma}\right)-\exp(-\frac{q^{2}}{2\sigma^{2}})\geq
0.$ (5.8)

The left-hand side is readily shown to take its minimum value of approximately
$-0.3024$ at $q/\sigma=\sqrt{2/\pi}$, hence there is a clear LG2 violation.

Although the parity operator is measurable for some systems and therefore the
above inequality can be tested experimentally, it has no macrorealistic
counterpart (unlike Q = $\textrm{sgn}(x)$). Still, the fact that LG2
violations are obtained so easily with these variables could give interesting
clues to understand violations in more physical cases

##  6 Leggett-Garg violations in the Morse potential

In Sec. 2.3, we laid out a general technique to calculate temporal
correlators, when the energy eigenspectrum is known. As a demonstration, we
now apply this result to a different exactly soluble potential, the Morse
potential Morse (1929), an asymmetric potential with minimum at $r_{c}$, which
combines a short-range repulsion, with a long-range attraction. The potential
is defined by

$V(r)=D_{e}\left(e^{-2a(r-r_{e})}-2e^{-a(r-r_{e})}\right),$ (6.1)

where $D_{e}$ corresponds to the well depth, and $a$ to its width.

The details and results of applying our method to the Morse potential may be
found in Appendix D, where we find qualitatively similar behaviour to the QHO,
and observe significant $\rm LG{3}$ and $\rm LG{4}$ violations in the
$\ket{1}$ state. There are several other systems for which the Schrödinger
equation may be solved exactly, including the Hydrogen-like atom Landau and
Lifshitz (1977), the linear potential Khonina  _et al._ (2013), and the
quantum pendulum Galvez  _et al._ (2019). The result presented in Sec. 2.3
hence allows future exploration of the behaviour of the LG inequalities in
several bound systems.

##  7 Summary and Conclusion

We have presented an analytic investigation of the LG approach to macrorealism
in bound systems with an in-depth study of the QHO. For a dichotomic variable
given by the sign of the oscillator position, we developed a method to
calculate the temporal correlators for energy eigenstates of any bound system
with a known energy eigenspectrum.

We found that all the interesting variants of MR conditions tested for by
combinations of the LG2, LG3 and LG4 inequalities can be observed in states of
the QHO involving superpositions of only the $\ket{0}$ and $\ket{1}$ states.
Intriguingly, the temporal correlator of the $\ket{1}$ state was shown to be,
to a very good approximation, a cosine, like the temporal correlators in the
canonical simple spin systems used in much LG research. We also found that the
higher energy eigenstates rapidly begin to exhibit statistics in line with a
classical model.

We found that LG violations persist even with a significant amount of
smoothing of the projectors. This is important, since it means the LG
violations are not just an artefact of non-physical, sharp projectors, and
therefore the LG violations described here should be directly observable in
experiment.

Although the bulk of the work in this paper has been conducted using the
simplest dichotomic variable $Q=\text{sgn}(x)$, all the results presented are
general enough to also allow for dichotomic variables defined on arbitrary
regions of space. We briefly investigated the effect of using a different
variable, and found it provides a richer and more powerful test of MR, with
two-time LG violations even in the $\ket{0}$ state, despite being a state with
positive Wigner function. (See also Ref. Halliwell  _et al._ (2021)). In
addition, we found LG2 violations if one of the dichotomic variables at one of
times is taken to be the parity operator.

To demonstrate the method, we calculated the temporal correlators for another
exactly soluble system, the Morse potential, where we found significant LG3
and LG4 violations for the first excited state.

In conclusion, we have derived a number of results on LG inequalities for
coarse-grained position measurements in bound systems. These results provide
indications as to what sort of states to create in experiments on macroscopic
systems in order to find evidence of macroscopic coherence.

Future work will explore in more depth the phenomenon of LG violations in
these systems for initial states with positive Wigner function. We also
propose to develop non-invasive measurement protocols suitable for continuous
variable systems by adapting the continuous in time velocity measurement
scheme proposed and utilized in Refs. Halliwell (2016a); Majidy  _et al._
(2019).

## Acknowledgements

We are grateful to Sougato Bose and Dipankar Home for useful discussions. We
also thank Michael Vanner for discussions on the experimental feasibility of
the tests discussed in this paper. CM thanks Prof. Adams for choosing to
include the rather arcane Node theorem within MIT OCW’s free Quantum Physics
lectures, which many years ago supplemented my learning and helped to build my
foundations in the subject.

##  Appendix A Quasi-probability for generic bound potentials

###  A.1 Partial overlap integrals

To calculate the quasi-probability most generally, we need the integrals

$J_{k\ell}(x_{1},x_{2})=\int_{x_{1}}^{x_{2}}\mathop{dx}\innerproduct{k}{x}\innerproduct{x}{\ell},$
(A.1)

where we adopt the notation that when $J_{k\ell}$ appears without an argument,
$J_{k\ell}=J_{k\ell}(0,\infty)$. It is possible to compute these integrals, by
expanding a result found in Ref. Moriconi (2007), which bears resemblance to
Abel’s identity. We first denote energy eigenstates in position space as

$\psi_{n}(x)=\innerproduct{x}{n}.$ (A.2)

We then construct the Wronskian between the two states $k$ and $\ell$,

$W(x)=\psi_{k}^{\prime}\psi_{\ell}-\psi_{\ell}^{\prime}\psi_{k}.$ (A.3)

Differentiating the Wronskian, we find

$W^{\prime}(x)=\psi_{k}^{\prime\prime}\psi_{\ell}-\psi_{\ell}^{\prime\prime}\psi_{k}.$
(A.4)

Since we are working with energy eigenstates, from the Schrödinger equation we
have

$\psi_{n}^{\prime\prime}=2(V-\varepsilon_{n})\psi_{n},$ (A.5)

where $\varepsilon_{n}$ are the dimensionless energy eigenvalues. Substituting
this into Eq.(A.4), we find

$W^{\prime}(x)=2(\varepsilon_{\ell}-\varepsilon_{k})\psi_{k}\psi_{\ell},$
(A.6)

where any dependence on the form of the potential has disappeared from this
equation, being contained implicitly in the spectrum of the Hamiltonian. We
are now free to integrate both sides over any given region of space, which
yields

$\frac{1}{2(\varepsilon_{\ell}-\varepsilon_{k})}\evaluated{W(x)}_{x_{1}}^{x_{2}}=\int_{x_{1}}^{x_{2}}\mathop{dx}\psi_{k}\psi_{\ell}.$
(A.7)

The right-hand side of the equation is exactly the matrix elements
$J_{k\ell}$, and so completing the integration, we find

$J_{k\ell}=\frac{1}{2(\varepsilon_{\ell}-\varepsilon_{k})}\left[\psi_{k}^{\prime}(x_{2})\psi_{\ell}(x_{2})-\psi_{\ell}^{\prime}(x_{2})\psi_{k}(x_{2})-\psi_{k}^{\prime}(x_{1})\psi_{\ell}(x_{1})+\psi_{\ell}^{\prime}(x_{1})\psi_{k}(x_{1})\right].$
(A.8)

###  A.2 Calculational Details

We continue the analysis coarse-graining onto the left and right hand side of
the axis, in the important special case of symmetric potentials. In this case,
we have

$J_{ij}=\int_{0}^{\infty}\mathop{dx}\psi_{i}\psi_{j}=\frac{1}{2(\varepsilon_{j}-\varepsilon_{i})}\left[\psi_{j}^{\prime}(0)\psi_{i}(0)-\psi^{\prime}_{i}(0)\psi_{j}(0)\right],$
(A.9)

since wavefunctions must vanish at infinity. With a symmetric potential, we
have for $n$ odd, $\psi_{n}(0)=0$, and for $n$ even, we have
$\psi_{n}^{\prime}(0)=0$. Eq.(A.4) is not defined for $i=j$, however for a
symmetric potential, we know $J_{ii}=\frac{1}{2}$, as so the $k=n$ term
contributes $\frac{1}{4}$ is to both results.

Hence, if $n$ is odd, we have

$q(n)=\frac{1}{4}+\real
e^{i\frac{E_{n}}{\hbar}\tau}\psi_{n}^{\prime}(0)^{2}\sum_{k=0,k\neq
n}^{\infty}e^{-i\frac{E_{k}}{\hbar}\tau}\frac{1}{4(\varepsilon_{k}-\varepsilon_{n})^{2}}\psi_{k}(0)\psi_{k}(0).$
(A.10)

If $n$ is even, we have

$q(n)=\frac{1}{4}+\real
e^{i\frac{E_{n}}{\hbar}\tau}\psi_{n}(0)^{2}\sum_{k=0,k\neq
n}^{\infty}e^{-i\frac{E_{k}}{\hbar}\tau}\frac{1}{4(\varepsilon_{k}-\varepsilon_{n})^{2}}\psi_{k}^{\prime}(0)\psi_{k}^{\prime}(0).$
(A.11)

To be computationally feasible, we must truncate this sum for some $m$. To
estimate the error involved with a given truncation, we note that $J_{k\ell}$
form the coefficients of the expansion of $\theta(\hat{x})\ket{k}$. This
represents the probability of finding the particle on either side of the axis.
In the case of symmetric potentials, these are (anti-)symmetric states, and so
have norm $\frac{1}{2}$. Hence this gives us the truncation

$\Delta_{n}(m)=\frac{1}{2}-\sum_{k=0}^{m}J_{nk}^{2}.$ (A.12)

##  Appendix B Exact Quasiprobability in the QHO

###  B.1 Strategy

To calculate the quasi-probability exactly, we begin by writing Eq.(2.27) as

$q_{mn}=\mathcal{N}e^{i\frac{E_{m}}{\hbar}t_{2}-i\frac{E_{n}}{\hbar}t_{1}}\int^{\infty}_{0}\int^{\infty}_{0}\mathop{dr}\mathop{ds}H_{m}(r)H_{n}(s)\exp(\alpha
r^{2}+\beta s^{2}+i\gamma rs),$ (B.1)

where we have introduced the variables

$\displaystyle\alpha=\beta$
$\displaystyle=\frac{i}{2\tan(\omega\tau)}-\frac{1}{2},$ (B.2)
$\displaystyle\gamma$ $\displaystyle=-\frac{1}{\sin(\omega\tau)}.$ (B.3)

We now note that as a product of Hermite polynomials, the term
$H_{m}(r)H_{n}(s)$ will itself just be a polynomial involving powers of $r$
and $s$. Hence $q_{mn}$ may be broken up into a sum of terms of the form

$X_{k\ell}=\int_{0}^{\infty}\int_{0}^{\infty}\mathop{dr}\mathop{ds}r^{k}s^{\ell}\exp(\alpha
r^{2}+\beta s^{2}+i\gamma rs).$ (B.4)

To calculate these terms, we consider the generating integrals

$\displaystyle I(\alpha,\beta,\gamma)$
$\displaystyle=\int^{\infty}_{0}\int^{\infty}_{0}\mathop{dr}\mathop{ds}\exp(\alpha
r^{2}+\beta s^{2}+i\gamma rs),$ (B.5) $\displaystyle J(\alpha,\beta,\gamma)$
$\displaystyle=\int^{\infty}_{0}\int^{\infty}_{0}\mathop{dr}\mathop{ds}r\exp(\alpha
r^{2}+\beta s^{2}+i\gamma rs).$ (B.6)

We may then calculate any of the $X_{k\ell}$ integrals through repeated use of
partial differentiation. In particular, if $k$ and $\ell$ are both even, we
have

$X_{k\ell}=\partialderivative{\alpha^{\frac{k}{2}}}\partialderivative{\beta^{\frac{\ell}{2}}}I(\alpha,\beta,\gamma).$
(B.7)

Similarly, if $k$ and $\ell$ are both odd, we have

$X_{k\ell}=-i\partialderivative{\gamma}\partialderivative{\alpha^{\frac{k-1}{2}}}\partialderivative{\beta^{\frac{\ell-1}{2}}}I(\alpha,\beta,\gamma).$
(B.8)

Finally, for the case where say $k$ is odd, and $\ell$ is even, we have

$X_{k\ell}=\partialderivative{\alpha^{\frac{k-1}{2}}}\partialderivative{\beta^{\frac{\ell}{2}}}J(\alpha,\beta,\gamma),$
(B.9)

and vice-versa for $\ell$ odd, $k$ even. This approach although a handful on
paper, is simple to implement in computer algebra software.

###  B.2 Generating Integrals

We now proceed with calculating $I(\alpha,\beta,\gamma)$ Eq.(B.5), by
completing the square in the exponential function. This yields

$I(\alpha,\beta,\gamma)=\int_{0}^{\infty}\int_{0}^{\infty}\mathop{dr}\mathop{ds}\exp\left(\frac{r^{2}(4\alpha\beta+\gamma^{2})}{4\beta}\right)\exp\left(\beta
s+i\frac{\gamma r}{2\sqrt{\beta}}\right).$ (B.10)

We introduce the short-hand $\delta=\frac{4\alpha\beta+\gamma^{2}}{4\beta}$.
The $s$ integral may now be completed in terms of the error function as

$I(\alpha,\beta,\gamma)=\frac{\sqrt{\pi}}{2\sqrt{-\beta}}\int_{0}^{\infty}\mathop{dr}\exp\left(\delta
r^{2}\right)\left(1+i\operatorname{erfi}\left(\frac{\gamma
r}{2\sqrt{-\beta}}\right)\right).$ (B.11)

For simplicity of presentation, we separate the integral into two parts,

$\displaystyle I_{1}(\alpha,\beta,\gamma)$
$\displaystyle=\frac{\sqrt{\pi}}{2\sqrt{-\beta}}\int_{0}^{\infty}\mathop{dr}\exp\left(\delta
r^{2}\right),$ (B.12) $\displaystyle I_{2}(\alpha,\beta,\gamma)$
$\displaystyle=\frac{i\sqrt{\pi}}{2\sqrt{-\beta}}\int_{0}^{\infty}\mathop{dr}\exp\left(\delta
r^{2}\right)\operatorname{erfi}\left(\frac{\gamma r}{2\sqrt{-\beta}}\right).$
(B.13)

The first integral is simply the Gaussian integral on the half-plane, which
has the result

$I_{1}(\alpha,\beta,\gamma)=\frac{\pi}{4\sqrt{-\beta}}\sqrt{-\frac{1}{\delta}}.$
(B.14)

To proceed with $I_{2}(\alpha,\beta,\gamma)$, we rescale the variables to
$u=\frac{\gamma r}{2\sqrt{-\beta}}$, leading to

$I_{2}(\alpha,\beta,\gamma)=\frac{i\sqrt{\pi}}{\gamma}\int_{0}^{\infty}\mathop{du}\exp(-u^{2}\frac{4\alpha\beta+\gamma^{2}}{\gamma^{2}})\erf(u).$
(B.15)

This integral takes the form

$\int_{0}^{\infty}\mathop{du}e^{-cu^{2}}\erf{u}=\frac{1}{\sqrt{\pi}\sqrt{c}}\arctan\left(\frac{1}{\sqrt{c}}\right),$
(B.16)

which is convergent for $\real(c)>0$, which may be confirmed for the case with
$c=\frac{4\alpha\beta+\gamma^{2}}{\gamma^{2}}$. Overall, this gives the result

$I_{2}(\alpha,\beta,\gamma)=\frac{1}{\sqrt{4\alpha\beta+\gamma^{2}}}\arctan\left(\frac{i\gamma}{\sqrt{4\alpha\beta+\gamma^{2}}}\right).$
(B.17)

Hence the complete result is

$I(\alpha,\beta,\gamma)=\frac{\pi}{4\sqrt{-\beta}}\sqrt{-\frac{1}{\delta}}\left(1+\frac{2}{\pi}\arctan\left(\frac{i\gamma}{\sqrt{4\alpha\beta+\gamma^{2}}}\right)\right).$
(B.18)

To calculate $J(\alpha,\beta,\gamma)$, we pick up the calculation from
Eq.(B.11), with

$J(\alpha,\beta,\gamma)=\frac{\sqrt{\pi}}{2\sqrt{-\beta}}\int_{0}^{\infty}\mathop{dr}r\exp\left(\delta
r^{2}\right)\left(1+i\operatorname{erfi}\left(\frac{\gamma
r}{2\sqrt{-\beta}}\right)\right),$ (B.19)

which for clarity, we again separate into two parts,

$\displaystyle J_{1}(\alpha,\beta,\gamma)$
$\displaystyle=\frac{\sqrt{\pi}}{2\sqrt{-\beta}}\int_{0}^{\infty}\mathop{dr}r\exp\left(\delta
r^{2}\right),$ (B.20) $\displaystyle J_{2}(\alpha,\beta,\gamma)$
$\displaystyle=\frac{i\sqrt{\pi}}{2\sqrt{-\beta}}\int_{0}^{\infty}\mathop{dr}r\exp\left(\delta
r^{2}\right)\operatorname{erfi}\left(\frac{\gamma r}{2\sqrt{-\beta}}\right).$
(B.21)

These integrals are all simpler to compute, owing to the presence of the
factor of $r$. The first integral is easily calculated using
$\partialderivative{r}e^{ar^{2}}=2are^{ar^{2}}$, and is

$J_{1}(\alpha,\beta,\gamma)=-\frac{\sqrt{\pi}}{4\sqrt{-\beta}}\frac{1}{\delta}.$
(B.22)

The second integral is amiable to integration by parts, where we find the
result

$J_{2}(\alpha,\beta,\gamma)=-\frac{i\sqrt{\pi}\gamma}{2\beta\sqrt{\frac{\alpha\beta}{4\alpha\beta+\gamma^{2}}}\left(-\frac{4\alpha\beta+\gamma^{2}}{\beta}\right)^{3/2}}.$
(B.23)

In total, this gives the result

$J(\alpha,\beta,\gamma)=\frac{\sqrt{\pi}}{2\sqrt{-\beta}}\left(-\frac{2\beta}{4\alpha\beta+\gamma^{2}}+i\frac{\gamma}{\sqrt{-\beta}\sqrt{\frac{\alpha\beta}{4\alpha\beta+\gamma^{2}}}\left(-\frac{4\alpha\beta+\gamma^{2}}{\beta}\right)^{\frac{3}{2}}}\right).$
(B.24)

###  B.3 Example Calculations

With the generating integrals completed, we can find the expressions for
correlators using the approach in Appendix B.1.

The final results are significantly simplified by the identity

$\frac{\beta}{4\alpha\beta+\gamma^{2}}=-\frac{1}{4},$ (B.25)

which is simple to prove, by substituting in the definitions for
$\alpha,\beta$ and $\gamma$ as

$\frac{\beta}{4\alpha\beta+\gamma^{2}}=\frac{1}{2}\frac{-1+i\cot\omega\tau}{1-2i\cot\omega\tau-\cot^{2}\omega\tau+\csc^{2}\omega\tau},$
(B.26)

where using the identity $\csc^{2}x-\cot^{2}x=1$ yields the required result.

It is important that this identity only be applied after completing all the
differentiation steps of the algorithm in Appendix B.1.

We also note the use of the factor of $\frac{1}{\sqrt{-\beta}}$

$\frac{1}{\sqrt{2i\sin\omega\tau}}=e^{-i\frac{\omega\tau}{2}}\sqrt{-\beta},$
(B.27)

which allows us to take care of the time-dependence of the propagator
prefactor.

To for example calculate the correlator for the ground state, we find

$q_{00}=\frac{1}{\pi}e^{-i\frac{\omega\tau}{2}}\sqrt{-\beta}e^{i\frac{\omega
t_{2}}{2}-i\frac{\omega t_{1}}{2}}I(\alpha,\beta,\gamma).$ (B.28)

Making the appropriate cancellations and substitutions we find

$q_{00}=\left(\sqrt{-\frac{\beta}{4\alpha\beta+\gamma^{2}}}+\frac{2}{\pi}\sqrt{-\frac{\beta}{4\alpha\beta+\gamma^{2}}}\arctan\left(\frac{i\gamma}{\sqrt{4\alpha\beta+\gamma^{2}}}\right)\right).$
(B.29)

Making use of the identity Eq.(B.25), we find

$\displaystyle q_{00}$
$\displaystyle=\frac{1}{4}\left(1+\frac{2}{\pi}\arctan(\frac{\gamma}{\sqrt{-4\alpha\beta-\gamma^{2}}})\right),$
(B.30)

whereupon back-substitution of the $\alpha,\beta$ and $\gamma$, and taking the
real part, we find

$\displaystyle
q^{\ket{0}}(+,+)=\frac{1}{4}\left(1+\frac{2}{\pi}\real\arctan(\frac{ie^{i\frac{\omega\tau}{2}}}{\sqrt{2i\sin\omega\tau}})\right).$
(B.31)

Care must be taken with the branch-cut of the square-root function, and the
results presented here are consistent with the choice of taking the branch cut
along the negative real axis.

###  B.4 Temporal Correlators

We tabulate the exact correlators for the first nine energy eigenstates of the
QHO. We again rely on the function
$f(\tau)=-ie^{-i\frac{\omega\tau}{2}}\sqrt{2i\sin\omega\tau}$.

$\displaystyle C_{ij}^{\ket{0}}$
$\displaystyle=\frac{2}{\pi}\real\left[\arctan\left(\frac{1}{f(\tau)}\right)\right],$
(B.32) $\displaystyle C_{ij}^{\ket{1}}$
$\displaystyle=\frac{2}{\pi}\real\left[\arctan\left(\frac{1}{f(\tau)}\right)+f(\tau)\right],$
(B.33) $\displaystyle C_{ij}^{\ket{2}}$
$\displaystyle=\frac{2}{\pi}\real\left[\arctan\left(\frac{1}{f(\tau)}\right)+\frac{1}{2}f(\tau)\right],$
(B.34) $\displaystyle C_{ij}^{\ket{3}}$
$\displaystyle=\frac{2}{\pi}\real\left[\arctan\left(\frac{1}{f(\tau)}\right)+\frac{5+e^{-2i\omega\tau}}{6}f(\tau)\right],$
(B.35) $\displaystyle C_{ij}^{\ket{4}}$
$\displaystyle=\frac{2}{\pi}\real\left[\arctan\left(\frac{1}{f(\tau)}\right)+\frac{14+e^{-2i\omega\tau}}{24}f(\tau)\right],$
(B.36) $\displaystyle C_{ij}^{\ket{5}}$
$\displaystyle=\frac{2}{\pi}\real\left[\arctan\left(\frac{1}{f(\tau)}\right)+\frac{94+17e^{-2i\omega\tau}+9e^{-4i\omega\tau}}{120}f(\tau)\right],$
(B.37) $\displaystyle C_{ij}^{\ket{6}}$
$\displaystyle=\frac{2}{\pi}\real\left[\arctan\left(\frac{1}{f(\tau)}\right)+\frac{148+14e^{-2i\omega\tau}+3e^{-4i\omega\tau}}{240}f(\tau)\right],$
(B.38) $\displaystyle C_{ij}^{\ket{7}}$
$\displaystyle=\frac{2}{\pi}\real\left[\arctan\left(\frac{1}{f(\tau)}\right)+\frac{1276+218^{-2i\omega\tau}+111e^{-4i\omega\tau}+75e^{-6i\omega\tau}}{1680}f(\tau)\right],$
(B.39) $\displaystyle C_{ij}^{\ket{8}}$
$\displaystyle=\frac{2}{\pi}\real\left[\arctan\left(\frac{1}{f(\tau)}\right)+\frac{8528+904e^{-2i\omega\tau}+258e^{-4i\omega\tau}+75e^{-6i\omega\tau}}{13440}f(\tau)\right].$
(B.40)

##  Appendix C QHO Results

###  C.1 Smoothed Projectors

It is simplest to calculate the effect of smoothed projectors, working with
the quasi-probability expressed as a truncated infinite sum. To do this, we
switch from using $\theta(\hat{x})$ as our projector, to the continuous
$\frac{1}{2}(1+\erf(\sqrt{\frac{m\omega}{\hbar}}\frac{\hat{x}}{a}))$. Where
$a$ is a dimensionless parameter characterising the degree of smoothing. For
small $a$, we expect to recover the sharp projector result, and $a=1$
corresponds to a smoothing on the characteristic length scale of the QHO,
$\sqrt{\frac{m\omega}{\hbar}}$. This adjusts the matrix elements to be

$J_{k\ell}=\frac{1}{\sqrt{\pi
2^{k+\ell}k!\ell!}}\int_{-\infty}^{\infty}\mathop{dr}H_{k}(r)H_{\ell}(r)e^{-r^{2}}\frac{1+\erf(\frac{r}{a})}{2}.$
(C.1)

We compute these integrals numerically, and then investigate the LG violations
possible with different values of $a$. In Fig. 9(a), we plot one of the $\rm
LG{3}$ inequalities, varying the value of $a$. The smoothed projectors result
in a similar (although not identical) shape, but with a reduced amplitude. In
Fig. 9(b), we plot the minimal value taken by the $\rm LG{3}$ inequalities for
each value of $a$, where we can see that once the smoothing reaches the
characteristic length-scale of the QHO, LG violations vanish.

(a)

(b)

Figure 9: In (a), we plot one of the $\rm LG{3}$s, for different values of
$a$, showing the qualitative effect of projector smoothing. In (b), we plot
the minimum value taken by the $\textrm{LG}_{3}$ inequalities, for a given
value of $a$.

###  C.2 Classicalization

To understand the classicalization of the QHO, we look at the large $n$
asymptotic behaviour of the quasi-probability, using the original sharp
projectors. Using Eq.(A.10) for the QHO, we find

$q(n)=\frac{1}{4}+\real\psi_{n}^{\prime}(0)^{2}\sum_{k=0}^{\infty}e^{-i(k-n)\tau}\frac{1}{4(k-n)^{2}}\psi_{k}(0)^{2},$
(C.2)

where the sum is over even $k$. We first relabel the sum to simplify the
denominator,

$q(n)=\frac{1}{4}+\real\psi_{n}^{\prime}(0)^{2}\sum_{k=-n}^{\infty}e^{-ik\tau}\frac{1}{4k^{2}}\psi_{k+n}(0)^{2},$
(C.3)

which also shifts it to a sum over only odd $k$. We now make the first of two
observations relevant in the large $n$ regime. Since the Hermite functions
evaluated at $0$ remain finite, and of order of magnitude $1$, the magnitude
of the terms of this sum will fit into the envelope given by
$\frac{1}{k^{2}}$, and so, for large $n$, we can safely extend the lower limit
of the sum to $-\infty$, yielding

$q(n)=\frac{1}{4}+\real\psi_{n}^{\prime}(0)^{2}\sum_{k=-\infty}^{\infty}e^{-ik\tau}\frac{1}{4k^{2}}\psi_{k+n}(0)^{2}.$
(C.4)

Using the standard recurrence relations of the Hermite functions, we find that
at the origin we have

$\frac{\psi_{n+1}(0)}{\psi_{n-1}(0)}=\sqrt{\frac{n}{n+1}}.$ (C.5)

This leads to the second observation, that in the large $n$ regime, the rate
at which $\psi_{n+k}(0)$ changes is negligible compared to the rate of change
of $\frac{1}{k^{2}}$, and hence may be considered approximately constant in
the sum, and so we have

$q(n)\approx\frac{1}{4}+\real\psi_{n}^{\prime}(0)^{2}\psi_{n+1}(0)^{2}\sum_{k=-\infty}^{\infty}e^{-ik\tau}\frac{1}{4k^{2}}.$
(C.6)

To proceed, we look at the generating function of the Hermite polynomials,

$e^{2xt-t^{2}}=\sum_{n=0}^{\infty}H_{n}(x)\frac{t^{n}}{n!}$ (C.7)

Evaluating at $x=0$, and using the Taylor series for the Gaussian, we find

$\sum_{k=0}^{\infty}(-1)^{k}\frac{t^{2k}}{k!}=\sum_{n=0}^{\infty}H_{n}(0)\frac{t^{n}}{n!},$
(C.8)

whereby comparing powers of $t$, we find for even $n$,
$\absolutevalue{H_{n}(0)}=\frac{n!}{(\frac{n}{2})!}$. Performing a similar
analysis for the derivative term, we find that for odd $n$,
$\absolutevalue{H^{\prime}_{n}(0)}=\frac{(n+1)!}{(\frac{n+1}{2})!}$. The
product term, with normalisation re-included is then

$\psi_{n}^{\prime}(0)\psi_{n+1}(0)=\frac{(n!)^{2}}{2^{n-1}\sqrt{2n}(n-1)!\left(\frac{n}{2}!\right)^{2}},$
(C.9)

which through computer algebra software is found to have the limit

$\lim\limits_{n\to\infty}\psi_{n}^{\prime}(0)\psi_{n+1}(0)=\frac{2}{\pi}.$
(C.10)

Using this in Eq.(C.6), we find

$q(n)\approx\frac{1}{4}+\real\sum_{k=-\infty}^{\infty}e^{-ik\tau}\frac{1}{\pi^{2}k^{2}}.$
(C.11)

We can now identify the sum as the exponential Fourier series for a symmetric
triangle wave, with amplitude $\frac{1}{4}$, which is exactly the classical
correlator. This represents the classicalization of the QHO, as $n$ becomes
large.

###  C.3 QHO quasi-probability with arbitrary coarse-grainings

To calculate the quasi-probability in the QHO with arbitrary coarse-grainings,
we apply the method in Appendix A, using the integrals $J_{k\ell}(a,b)$.

We first note Eq.(2.15) is undefined for $k=l$, and must be calculated by hand
as

$J_{kk}(a,b)=\int_{a}^{b}\mathop{dx}\psi_{k}(x)\psi_{k}(x).$ (C.12)

The quasi-probability is then calculated through Eq.(2.17) as

$q_{nn}(+,+)=e^{i\frac{E_{n}\tau}{\hbar}}\sum_{k=0}^{\infty}e^{-i\frac{E_{k}\tau}{\hbar}}J_{nk}(a,b)J_{nk}(c,d).$
(C.13)

We note again, that these $\psi_{k}(x)$ and $E_{k}$ are not yet the QHO
eigenstates and energies, and that this result is generic.

To make things concrete, we will now calculate this for the ground state of
the QHO. We take the first pair of intervals $\Delta(\alpha_{1})$ to be
$\Delta(+)=[0,\infty)$ and $\Delta(-)=(-\infty,0]$. The second pair of
intervals $\Delta(\alpha_{2})$ are taken to be $\Delta(+)=[c,d]$, and
$\Delta(-)$ its complement. The quasi-probability for the ground state is,

$q(+,+)=\frac{1}{2}J_{00}(c,d)+\real\sum_{k=0}^{\infty}e^{-i\omega\tau
k}J_{0k}(0,\infty)J_{0k}(c,d).$ (C.14)

Again, due to it being a symmetric potential, we have
$J_{00}(0,\infty)=\frac{1}{2}$ as before. Doing the integral manually for the
ground state, we find

$J_{00}(c,d)=\frac{1}{2}(\erf(d)-\erf(c)).$ (C.15)

Using this result and the QHO eigenspectrum in Eq.(A.8), we are able to
calculate the quasi-probability using Eq.(C.14).

##  Appendix D Morse Potential

(a)

(b)

Figure 10:  The temporal correlator in the $\ket{1}$ state for the Morse
potential over time interval $\omega_{0}\tau=2\pi$ is shown in (a). It is not
quite periodic over this time interval but becomes exactly periodic over a
much longer period of time, as shown in (b).

The Morse potential is an asymmetric potential with minimum at $r_{c}$, which
combines a short-range repulsion, with a long-range attraction. The potential
is defined by

$V(r)=D_{e}\left(e^{-2a(r-r_{e})}-2e^{-a(r-r_{e})}\right),$ (D.1)

where $D_{e}$ corresponds to the well depth, and $a$ to its width. The Morse
potential supports up to $\lfloor\lambda-\frac{1}{2}\rfloor$ bound states Dahl
and Springborg (1988), where $\lambda=\frac{\sqrt{2mD_{e}}}{a\hbar}$ and
$\lfloor x\rfloor$ is the floor function. The eigenstates and energies are

$\displaystyle\varepsilon_{n}$
$\displaystyle=-\frac{1}{2}\left(\lambda-n-\frac{1}{2}\right)^{2},$ (D.2)
$\displaystyle\psi_{n}(z)$
$\displaystyle=N_{n}z^{\lambda-n-\frac{1}{2}}e^{-\frac{1}{2}z}L_{n}^{2\lambda-n-1}(z),$
(D.3)

where $L_{n}^{(\alpha)}(z)$ are the generalized Laguerre polynomials, and $z$
is a scaled spatial coordinate defined as $z=2\lambda e^{-a(r-r_{c})}$. In the
standard non-dimensionalization, physical energy eigenvalues relate to
dimensionless as $E_{n}=\frac{2\hbar\omega_{0}}{\lambda}\varepsilon_{n}$,
where $\omega_{0}=a\sqrt{\frac{2D_{e}}{m}}$ is the frequency of small
oscillations in the potential. Normalisation is given by

$N_{n}=\left(\frac{n!(2\lambda-2n-1)}{\Gamma(2\lambda-n)}\right).$ (D.4)

In the Morse potential, states are constrained to $r>0$, however we can still
consider coarse-graining onto the left and right halves of the well. The
energy eigenstates similarly still vanish at $r=0$ and $r\to\infty$, so many
of the terms in Eq.(2.15) still vanish. Filling out Eq.(2.17), we find

$q_{nn}=J_{nn}^{2}+e^{i\frac{\varepsilon_{n}}{\lambda}\omega_{0}\tau}\sum_{k=0,k\neq
n}^{\lfloor\lambda-\frac{1}{2}\rfloor}e^{-i\frac{\varepsilon_{k}}{\lambda}\omega_{0}\tau}\frac{1}{(\varepsilon_{n}-\varepsilon_{k})^{2}}\left(\psi_{n}^{\prime}(x_{c})\psi_{k}(x_{c})-\psi_{k}^{\prime}(x_{c})\psi_{n}(x_{c})\right)^{2},$
(D.5)

Although this sum is finite, by choosing a large enough $\lambda$, and just
looking at low energy states we can approximate it well. Physically, this
corresponds to measurements in shallower wells being likely to eject the
particle from the well, where we would then have to consider the continuous
positive energy solutions as well. We can again estimate the error for a
particular $n$ and $\lambda$ as

$\Delta(n,\lambda)=J_{nn}-\sum_{k=0}^{\lfloor\lambda-\frac{1}{2}\rfloor}J_{nk}^{2}.$
(D.6)

(a)

(b)

Figure 11: In (a), the LG3 inequalities for the $\ket{1}$ state of the Morse
potential with $\lambda=50$ is plotted. In (b), the LG4 inequalities are
plotted. There are significant violations for both, reaching $70\%$ and $92\%$
of their maximal violations respectively.

Owing to the fact the Morse potential is non-symmetric, we must calculate the
terms $J_{nn}$ by hand, which for the low $n$ is a simple integration. The
truncation error for the first excited state reaches $0.01$ for $\lambda=15$,
and similar to the QHO, is remains higher for the ground state. For accuracy,
we hence calculate just the $\ket{1}$ state, and choose $\lambda=50$, yielding
$\Delta(n,\lambda)=0.001$.

We extract temporal correlators from Eq.(D.5) using the moment expansion
Eq.(2.3), which we plot in Fig. 10(a). At first glance, this correlator is
qualitatively very similar to the $\ket{1}$ correlator for the QHO, and hence
similar to the simple cosine correlator for spin-$\frac{1}{2}$ models. However
owing to the asymmetry of the Morse potential, the correlator never reaches
the value $-1$, and due to the anharmonicity, the correlator is periodic over
a much longer time scale, which is shown in Fig. 10(b).

In Fig. 11(a), we plot the LG3 inequalities for the first excited state of the
Morse potential, where there are significant violations, reaching $70\%$ of
the Lüders bound. The LG4 inequalities are plotted in Fig. 11(b), with
significant violations reaching $92\%$ of the Lüders bound. Violations
diminish in magnitude for the large-time behaviour, but remain present for
nearly all intervals of $\omega_{0}\tau$.

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