Fiducial cross-section measurements of top-quark pair production in
association with a photon at $\sqrt{s}$ = 13 TeV with the ATLAS detector

Joshua Wyatt Smith

on behalf of the ATLAS collaboration
II. Physikalisches Institut
Goerg-August Universität, Germany

> Top-quark pairs in association with final state particles are produced in
> large quantities at the LHC due to the high centre-of-mass energy available
> in proton-proton collisions. One such topology is that of a prompt photon
> radiated from the top quark in addition to the final state particles from
> the top quark decay. Using 36.1 fb-1 of data collected at $\sqrt{s}$ = 13
> TeV with the ATLAS detector, fiducial cross-section results are shown in the
> single-lepton and dilepton channels. Object-level and event-level neural
> networks are used to increase sensitivity.

> PRESENTED AT
>
>
>
>
> $11^{\mathrm{th}}$ International Workshop on Top Quark Physics
> Bad Neuenahr, Germany, September 16–21, 2018
>
> Copyright 2019 CERN for the benefit of the ATLAS Collaboration. Reproduction
> of this article or parts of it is allowed as specified in the CC-BY-4.0
> license.

##  1 Introduction

The production of a top-quark pair in association with a photon is a direct
probe of the electromagnetic coupling of the top quark. Possible anomalous
top-quark couplings could manifest as shape discrepancies in various kinematic
distributions or in cross-section measurements [1, 2]. The results can also be
interpreted in the framework of an effective field theory in the search for
new physics [3]. Photons can be radiated from all charged particles, including
those in top quark decay and incoming initial state quarks. Kinematic cuts are
applied to reduce the contribution of photons being radiated from top quark
decay.

Previous studies have measured the cross section in the single-lepton channels
[4, 5, 6, 7]. In the analysis [8] summarised here, the fiducial cross sections
are measured in the single-lepton channels and, for the first time, the
dilepton channels. No distinction is made between electrons and muons from tau
lepton decays. The measurements are performed using the ATLAS [9] detector at
the LHC and a total dataset corresponding to 36.1 fb-1, collected in 2015 and
2016 at a centre-of-mass energy of 13 TeV. Dedicated $k$-factors applied to
the leading-order signal sample allow to compare the extracted cross sections
to the next-to-leading order (NLO) theoretical calculations [10].

##  2 Signal region definition

Event selections are made to enhance the purity of the $t\overline{t}\gamma$
signal. Electrons are required to have $p_{\mathrm{T}}>25$ GeV
($p_{\mathrm{T}}>27.5$ GeV) for the 2015 (2016) data, while muons are required
to have $p_{\mathrm{T}}>27.5$ GeV for the 2015 and 2016 data. Only one
reconstructed lepton is required for the single-lepton channels, while exactly
two leptons of opposite sign are required for the dilepton channels. Electrons
and muons are both required to be isolated based on calorimeter and track
information. At least four (two) jets are required in the $e$+jets and
$\mu$+jets ($ee$, $e\mu$ and $\mu\mu$) channels. At least one of these
reconstructed jets should be tagged as a $b$-jet with a working point
efficiency of 77%. In all the dilepton channels a minimum invariant mass
between the two leptons is such that $m(l,l)>15$ GeV. In the $ee$ and $\mu\mu$
channels values of the invariant mass of the two leptons close to the mass of
the $Z$-boson are vetoed, such that $m(l,l)\notin$ [85,95] GeV. The same
requirement is applied to the invariant mass of the system of the lepton pair
and photon, $m(l,l,\gamma)$. In the $ee$ and $\mu\mu$ channels a cut on the
missing transverse energy of an event is required to be above a threshold,
$E_{\mathrm{T}}^{\mathrm{miss}}>30$ GeV. In the $e$+jets channel a veto is
placed on the invariant mass of the photon and the electron such that
$m(\gamma,e)\notin$ [85,95] GeV. In all channels, exactly one photon of
$p_{\mathrm{T}}>20$ GeV is required. This photon needs to be isolated. A final
cut is applied on the $\Delta R$ distance between the photon and the leptons,
$\Delta R(\gamma,l)>1.0$. This cut reduces the contribution of photons from
top quark decay products.

The fiducial phase space is defined at particle level such that it mimics the
above event selection, with the exception of the invariant mass veto cuts and
the $E_{\mathrm{T}}^{\mathrm{miss}}$ cuts that are only intended to suppress
the background.

##  3 Background processes

The main background in the single-lepton channels consists of photons which
are misidentified as electrons, the so-called “$e\to\gamma$ fake” background.
This occurs mainly from the $t\overline{t}$ dileptonic decays, specifically
from the $e\mu$ and $ee$ channels. This background is negligible in the
dilepton channels. Scale factors derived from a data-driven _tag-and-probe_
method are used to correct the number of fake photons predicted by the Monte
Carlo (MC) samples.

Photons from hadrons, or hadrons misidentified as photons make up the
“hadronic fake” background. The majority of hadronic fake photons comes from
$t\overline{t}$ events. Small contributions arise from $W/Z$+jets and single-
top quark processes. Scale factors to correct MC prediction to data are
derived in the single-lepton channel using the “ABCD” method. These are
extrapolated to the dilepton channels.

Events with non-prompt leptons as well as fake leptons may satisfy the event
selections for the signal region. These are categorised as “fake lepton”
background. The main contribution comes from the QCD driven multi-jet
processes in association with a photon. The estimation of the fake lepton
background in the single-lepton channels follows the fully data-driven Matrix
Method approach. In the dilepton channels this contribution is negligible.

Events in which a real prompt photon is radiated from anything except top
quarks (or incoming quarks) make up the “Other prompt” background. For the
single-lepton channel this is largely due to the $W\gamma$ process, whereas
for the dilepton channel the main contribution is from the $Z\gamma$ process.
Other processes such as single-top, diboson and $t\overline{t}V$, where each
has an additional prompt photon in the final state, have small or even
negligible contributions. In the single-lepton (dilepton) channels the
$W\gamma$ ($Z\gamma$) contribution is largest and is therefore separated from
the rest of the prompt photon backgrounds. The “Other prompt” background is
estimated using MC.

##  4 Analysis Strategy

Object-level and event-level neural networks (NNs) are used to improve the
sensitivity of the measurement in the single-lepton channels, while only an
event-level NN is used in the dilepton channels. All NNs are implemented based
on a feedforward architecture in the Keras [11] framework using the Theano
[12] backend. The lwtnn [13] library is used to inject the trained NNs back
into the ATLAS software framework. The object-level NN, called the Prompt
Photon Tagger (PPT), is trained on independent prompt photon and di-jet
samples, using information from the calorimeters. This NN serves to
distinguish between prompt photons and hadronic fake photons. In the case of
the single-lepton channel, the PPT distribution is then used as an input into
the event-level NN, called the Event-Level Discriminator (ELD). The single-
lepton (dilepton) ELD is trained using 15 (7) variables. For the single-lepton
ELD, information about the $b$-tagged jet and the PPT proves to provide the
most powerful variables. For the dilepton ELD, the $b$-tagged jet and the
invariant mass of the two leptons prove to be very powerful. Examples of other
variables used in the training include the $E_{\mathrm{T}}^{\mathrm{miss}}$,
$p_{\mathrm{T}}$ of various jets and $m_{\text{T}}(W)$ (for the single-lepton
channel).

The fiducial cross sections for each channel are extracted by performing a
maximum likelihood fit to the ELD distribution. The cross section
($\sigma_{\text{fid}}$) is related to the number of signal events by

$N^{s}_{i}=\mathcal{L}\times\sigma_{\text{fid}}\times C\times
f_{i}^{\text{ELD}},$ (1)

where $\mathcal{L}$ is the integrated luminosity, $C$ is the ratio of all
simulated events passing event selections to those only falling into the
fiducial phase space, and $f_{i}^{\text{ELD}}$ is the fraction of events
falling into bin $i$ of the ELD.

##  5 Results

Eight cross-section measurements are extracted from a maximum likelihood fit
in the single-lepton, dilepton and combined channels. The post-fit ELD
distributions for two of the measurements (the single-lepton and dilepton
channel) are shown in Figure 1, where good agreement can be seen between MC
and data points. The ELD distributions for other channels look similar. Table
1 shows the effects that groups of systematic uncertainties have on the
fiducial cross-section measurements. In the single-lepton channel, the largest
contributions come from the modelling of jets, the backgrounds, and the PPT.
In the dilepton channel, signal and background modelling have the largest
effect, but are generally still small.

The eight extracted cross sections are summarised in Table 2. In summary, all
measurements are in agreement with NLO predictions.

(a)  (b)

Figure 1: The post-fit ELD distributions for the (a) single-lepton and (b)
dilepton channels. All the systematic uncertainties are included [8].

Source | Single lepton (%) | Dilepton (%)
---|---|---
Signal modelling |  $\pm$ 1.6 |  $\pm$ 2.9
Background modelling |  $\pm$ 4.8 |  $\pm$ 2.9
Photon |  $\pm$ 1.1 |  $\pm$ 1.1
Prompt-photon tagger |  $\pm$ 4.0 | -
Leptons |  $\pm$ 0.3 |  $\pm$ 1.3
Jets |  $\pm$ 5.4 |  $\pm$ 2.0
$b$-tagging |  $\pm$ 0.9 |  $\pm$ 0.4
Pile-up |  $\pm$ 2.0 |  $\pm$ 2.3
Luminosity |  $\pm$ 2.3 |  $\pm$ 2.3
MC sample size |  $\pm$ 1.9 |  $\pm$ 1.7
Total systematic uncertainty |  $\pm$ 7.9 |  $\pm$ 5.8
Data sample size |  $\pm$ 1.5 |  $\pm$ 3.8
Total uncertainty |  $\pm$ 8.1 |  $\pm$ 7.0

Table 1:  Summary of the effects of the groups of systematic uncertainties on
the fiducial cross section in the single-lepton and dilepton channels [8].

|  $\sigma^{\text{fid}}_{\text{$t\overline{t}\gamma$}}$ [fb]
---|---
Channel | Measured $\pm$(stat.) $\pm$(syst.) | Theory $\pm$(total)
$e$+jets | $265\pm 6\pm 21$ | $247\pm 49$
$\mu$+jets | $250\pm 7\pm 22$ | $248\pm 50$
$ee$ | $16\pm 2\pm 2$ | $16\pm 2$
$\mu\mu$ | $18\pm 1\pm 2$ | $16\pm 2$
$e\mu$ | $34\pm 2\pm 2$ | $31\pm 5$
Single-lepton | $521\pm 9\pm 41$ | $495\pm 99$
Dilepton | $69\pm 3\pm 4$ | $63\pm 9$
Inclusive (5 channels) | $589\pm 10\pm 34$ | $558\pm 110$

Table 2: Fiducial cross-section summary for all channels, as well as
theoretical predictions [8].

## References

  * [1] U. Baur, A. Juste, L. H. Orr, and D. Rainwater, Phys. Rev. D71 (2005) 054013.
  * [2] A. O. Bouzas and F. Larios, Phys. Rev. D87 (2013) 074015.
  * [3] O. Bessidskaia Bylund et al., JHEP 05 (2016) 052\.
  * [4] CDF Collaboration, Phys. Rev. D84 (2011) 031104.
  * [5] ATLAS Collaboration, Phys. Rev. D91 (2015) 072007.
  * [6] CMS Collaboration, JHEP 10 (2017) 006\.
  * [7] ATLAS Collaboration, JHEP 11 (2017) 086.
  * [8] ATLAS Collaboration. ATLAS-CONF-2018-048, 2018, http://cds.cern.ch/record/2639675.
  * [9] ATLAS Collaboration, JINST 3 (2008) S08003.
  * [10] K. Melnikov, M. Schulze, and A. Scharf, Phys. Rev. D 83 (2011) 074013.
  * [11] F. Chollet et al., Keras, https://keras.io, 2015.
  * [12] Theano Development Team, ArXiv e-prints 1605.02688 (2016).
  * [13] D. H. Guest, et al., 2017.  https://doi.org/10.5281/zenodo.290682.


# Vacuum thermal effects in flat space-time from conformal quantum mechanics

Michele Arzano<EMAIL_ADDRESS>Dipartimento di Fisica “E. Pancini”,
Università di Napoli Federico II, I-80125 Napoli, Italy
INFN, Sezione di Napoli,
Complesso Universitario di Monte S. Angelo,
Via Cintia Edificio 6, 80126 Napoli, Italy

###### Abstract

The generators of radial conformal symmetries in Minkowski space-time can be
mapped to the generators of time evolution in conformal quantum mechanics.
Within this correspondence we show that in conformal quantum mechanics the
state associated to the inertial vacuum in Minkowski space-time has the
structure of a thermofield double. Such state is built from a bipartite
“vacuum state”, the ground state of the generators of hyperbolic time
evolution, which cover only part of the time domain. When time evolution is
restricted to a finite time domain one obtains the temperature perceived by
static diamond observers in the Minkowski vacuum. When time evolution is
determined by dilations, covering only half of the time line, the temperature
of the thermofield double corresponds to the non-vanishing temperature
perceived by Milne observers whose proper time evolution is confined to the
future cone (Milne universe) of Minkowski space-time. The two pictures are
related by a conformal transformation on the real line. Our result provides a
purely group theoretical derivation of the Milne and diamond temperature and
shows that the fundamental ingredient for vacuum thermal effects is the
presence of a horizon rather than acceleration.

##  I Introduction

The notion of vacuum state in quantum field theory is observer dependent
Davies:1984rk . The importance of this basic fact was only started to be
appreciated with the discovery of black hole quantum radiance Hawking:1974sw ;
Unruh:1976db , which is one of its main consequences. The existence of a
temperature and thermal features associated to the horizon of a black hole is
directly related to two of the major open issues in quantum gravity: the fate
of unitarity in the quantum evolution of black holes Hawking:1976ra ;
Mathur:2009hf ; Giddings:2019vvj  and the nature of their Bekenstein-Hawking
entropy Bekenstein:1973ur ; Carlip:2008rk , and it is at the basis of the
interpretation of gravity as a “thermodynamics of space-time” Jacobson:1995ab
.

The ambiguity in the choice of the vacuum state in curved space-time relies on
the possibility of having different choices of time-like Killing vectors that
can be used to decompose the space of solutions of the classical field
equation into positive and negative energy subspaces Ashtekar:1975zn . The
former will define, together with a positive definite inner product, the one-
particle Hilbert space for static observers with respect to the Killing vector
which generates evolution in their proper time. From such Hilbert space the
standard Fock space construction follows with the associated vacuum and multi-
particle states Wald:1995yp . The Hawking effect boils down to the fact that
the Hartle-Hawking vacuum state associated to freely falling observer is a
thermal state at the Hawking temperature for static observers with respect to
Schwarzschild time whose associated Killing vector becomes null at the
horizon.

In maximally symmetric spaces the quantization ambiguity associated to
different possible choices of time evolution is eliminated by the requirement
of invariance of the vacuum state under the full group of isometries. In
Minkowski space-time this singles out the standard vacuum state associated to
inertial time evolution under the ordinary time translation Killing vector.
However, as it is well known, one can also consider time evolution generated
by a boost Killing vector which is time-like in the Rindler wedges and whose
orbits correspond to worldlines of uniformly accelerating observers. For such
observers the inertial vacuum is populated by a thermal distribution of
particles at a temperature proportional to the magnitude of their four-
acceleration: the so-called Unruh temperature Unruh:1976db .

Quite interestingly, if one restricts the attention to scale-invariant fields
the range of possible candidates for generators of time evolution can be
extended to include conformal Killing vectors. In Minkowski space-time this
has recently been used Higuchi:2017gcd ; Wald:2019ygd  (see also
Crispino:2007eb ; Olson:2010jy ) to consider “Milne quantization” of a two-
dimensional massless scalar field in the future and past cones, analogous to
Rindler quantization in the Rindler wedges, where time evolution is generated
by the dilation conformal Killing vector. Observers along the radial orbits of
dilations are comoving observers in a Milne universe (which is just Minkowski
space-time described in expanding coordinates covering only the future cone)
which perceive the inertial vacuum as populated by a thermal distribution of
Milne particles.

These considerations suggest that in higher dimensions, and here we refer for
simplicity to the $3+1$ dimensional case, one could define time evolution in
terms of conformal Killing vectors which are time-like in certain regions of
Minkowski space-time and similar thermal effects should manifest. Of
particular interest are, to this extent, radial conformal symmetries, whose
full classification for Minkowski space-time has been given in RCM . Radial
conformal Killing vectors divide Minkowski space-time into causal domains
separated by light-like surfaces which are conformal Killing horizons, they
are analogous to stationary spherically symmetric Killing horizons of black
hole geometries and have similar thermodynamic properties DeLorenzo:2017tgx ;
DeLorenzo:2018ghq . Among such radial conformal Killing vectors are the ones
which map a causal diamond into itself and whose orbits describe worldlines of
observers confined within the diamond. The fact that these diamond observers
should perceive the inertial Minkowski vacuum as a thermal state was first
suggested in Martinetti:2002sz  where the claim was supported with arguments
which used the so-called “thermal time hypothesis”. Another attempt to provide
evidence of such diamond temperature was given in Su:2016wop  using a model of
Unruh-DeWitt detector moving along the diamond observer trajectory.

In Arzano:2020thh  the present author pointed out that radial conformal
symmetries are in correspondence with generators of time evolution in
conformal quantum mechanics, a $0+1$ dimensional conformal field theory
deAlfaro:1976vlx . Using this correspondence, for which the “wordline quantum
mechanics” of diamond observers at the origin111This result has analogies with
the worldline quantum mechanics description of the temperature perceived by a
static patch observer in de Sitter space Anninos:2011af ; Nakayama:2011qh  is
viewed as a conformal quantum mechanics, it was shown that the Minkowski
vacuum is perceived by diamond observers as a thermal state at a temperature
inversely proportional to the size of the diamond. The results of
Arzano:2020thh , besides providing a simple but powerful tool for studying the
thermodynamics of causal regions in Minkowski space-time, also show that the
possibility of selecting different generators of time evolution in conformal
quantum mechanics leads to thermal effects in complete analogy with higher
dimensional quantum field theory. Since the focus of Arzano:2020thh  was on
causal diamonds, and the corresponding hyperbolic time evolution restricted to
a finite time domain in conformal quantum mechanics, the natural question is
whether similar thermal effects can be associated to other choices of time
evolution like, for example, the one determined by the generator of dilations,
and what is the physical interpretation of the associated temperature in
Minkowski space-time.

In this work we show that, besides the “diamond temperature” associated to
time evolution restricted to a finite time domain of conformal quantum
mechanics, there is a natural temperature associated to time evolution
generated by dilations and restricted to positive (or negative) times. These
two thermal effects have their origin in the thermofield double structure of
the vacuum state associated to the “inertial” time evolution generator which
is here evidenced using purely group theoretical arguments. The thermal effect
associated to time evolution restricted to half of the time line indicates
that in Minkowski space-time static Milne observers, i.e. observers at the
origin whose proper time evolution is determined by dilations, perceive the
Minkowski vacuum as a thermal state. Such result provides evidence for the
existence of a “Milne temperature” for space-time dimensions higher than two
and exhibits its intimate connection with diamond temperature given that, as
it shown below, the two pictures are connected on the conformal quantum
mechanics side by a conformal transformation. Finally, since both the diamond
and Milne temperatures are non-vanishing along the worldlines of static
observers sitting at the origin with vanishing acceleration, our result
demonstrates that the fundamental ingredient for quantum vacuum thermal
effects is the presence of a horizon, in this case associated to the non-
eternal nature of the lifetime of the observer, rather than a non-vanishing
acceleration.

##  II Radial conformal Killing vectors in Minkowski space-time and time
evolution in conformal quantum mechanics

Let us start from the line element of Minkowski space-time written in
spherical coordinates

$ds^{2}=-dt^{2}+dr^{2}+r^{2}d\Omega^{2}\,,$ (1)

where $d\Omega^{2}=d\theta^{2}+\sin^{2}\theta\,d\phi^{2}$. The most general
radial conformal Killing vector $\xi$ for which
$\mathcal{L}_{\xi}\eta_{\mu\nu}\propto\eta_{\mu\nu}$ where $\mathcal{L}_{\xi}$
is the Lie derivative and $\eta_{\mu\nu}$ is the Minkowski, metric is given by
RCM

$\xi=aK_{0}+bD_{0}+cP_{0}\,,$ (2)

where $K_{0}$, $D_{0}$ and $P_{0}$ generate, respectively, special conformal
transformations, dilations and time translations and close the
$\mathfrak{sl}(2,\mathbb{R})$ Lie algebra

$[P_{0},D_{0}]=P_{0}\,,\qquad[K_{0},D_{0}]=-K_{0}\,,\qquad[P_{0},K_{0}]=2D_{0}\,.$
(3)

These generators can be written in terms of radial and time derivatives as

$\displaystyle P_{0}$ $\displaystyle=$ $\displaystyle\partial_{t}$ (4)
$\displaystyle D_{0}$ $\displaystyle=$ $\displaystyle
r\,\partial_{r}+t\,\partial_{t}$ (5) $\displaystyle K_{0}$ $\displaystyle=$
$\displaystyle 2tr\,\partial_{r}+(t^{2}+r^{2})\,\partial_{t}\,,$ (6)

so that (2) becomes

$\xi=\left(a(t^{2}+r^{2})+bt+c\right)\,\partial_{t}+r(2at+b)\,\partial_{r}\,,$
(7)

and in light-cone coordinate $u=t+r$ and $v=t-r$ we have

$\xi=(au^{2}+bu+c)\,\partial_{u}+(av^{2}+bv+c)\,\partial_{v}\,.$ (8)

In conformal quantum mechanics the generators $K_{0}$, $D_{0}$ and $P_{0}$
play the role of different time evolution operators. In terms of the inertial
time variable $\tau$ associated to the generator $P_{0}$ they can be written
as differential operators

$\displaystyle P_{0}$ $\displaystyle=$ $\displaystyle\partial_{\tau}$ (9)
$\displaystyle D_{0}$ $\displaystyle=$ $\displaystyle\tau\,\partial_{\tau}$
(10) $\displaystyle K_{0}$ $\displaystyle=$
$\displaystyle\tau^{2}\,\partial_{\tau}\,.$ (11)

The first important observation is that radial conformal Killing vectors for
static observers at the origin $r=0$ and on the light-cones $u=const.$ or
$v=const.$ coincide with the most general time evolution operator in conformal
quantum mechanics

$G=aK_{0}+bD_{0}+cP_{0}=(a\tau^{2}+b\tau+c)\,\partial_{\tau}\,.$ (12)

The causal structure of the Killing vector (7) in Minkowski space-time has
been studied in RCM . The quantity $\Delta=b^{2}-4ac$ characterizes the norm
of the Killing vector and is used to classify different types of generators.
The generator $P_{0}$ for which $\Delta<0$ is everywhere time-like and in
conformal quantum mechanics generates time evolution in terms of ordinary
translation spanning the whole real (time) line. The generator $D_{0}$ with
$\Delta>0$ and $a=0$ is time-like inside the light-cone emanating from the
origin, null on the light-cone and space-like outside. In conformal quantum
mechanics it describes time evolution in terms of dilations restricted to the
positive or negative real line. In Minkowski space-time it describes conformal
time evolution in a Milne universe222The Milne universe being flat can be
indeed mapped to the future (or past, inverting the time direction) inner
light cone of Minkowski space-time. It corresponds to a slicing of the latter
in terms constant time hyper-surfaces given by Euclidean hyperboloids, orbits
of boosts. RCM ; Olson:2010jy ; Wald:2019ygd . The other relevant generator is
the radial conformal Killing vector which maps a causal diamond, i.e. the
region $|t|+|r|<\alpha$, into itself given by

$S_{0}=\frac{1}{2}\left(\alpha
P_{0}-\frac{K_{0}}{\alpha}\right)=\frac{1}{2\alpha}\left((\alpha^{2}-t^{2}-r^{2})\partial_{t}-2tr\partial_{r}\right)\,.$
(13)

The causal structure of this Killing vector for which $\Delta>1$ and $a\neq 0$
is more articulated. It is null on the light-cones emanating from $r=0$ at
times $\alpha$ and $-\alpha$. It is time-like inside or outside both light-
cones, i.e. within the causal diamond of radius $\alpha$ and its causal
complement, and space-like everywhere else. In conformal quantum mechanics the
combination $S_{0}$ generates time evolution restricted to the finite domain
$t\in(-\alpha,\alpha)$, the analogue of the diamond on the real line.

Notice that the quantity $\Delta=b^{2}-4ac$ is proportional to the determinant
of $G$ seen as a $2\times 2$ matrix and thus it is invariant under the adjoint
action of $SL(2,\mathbb{R})$ on its Lie algebra. Such determinant can be used
to classify different types of transformations generated by $G$. In particular
while $P_{0}$, for which $\Delta<0$, generates parabolic transformations of
the Euclidean plane (null rotations if we think of the relation between
$SL(2,\mathbb{R})$ and the three dimensional Lorentz group $SO(2,1)$), $D_{0}$
and $S_{0}$, for which $\Delta>0$, generate $SL(2,\mathbb{R})$ elements
belonging to the class of hyperbolic transformations whose representatives on
the Lorentz group side are boost transformations. Thus we can map one into
another using an $SL(2,\mathbb{R})$ transformation.

In order to do so let us first introduce time coordinates adapted to the
evolution defined by the generators $D_{0}$ and $S_{0}$ i.e. such that

$\displaystyle\frac{D_{0}}{\alpha}$ $\displaystyle=$
$\displaystyle\partial_{\nu}$ (14) $\displaystyle\frac{S_{0}}{\alpha}$
$\displaystyle=$ $\displaystyle\partial_{\sigma}\,,$ (15)

defining the Milne and diamond time for static observers with semi-infinite
and finite lifetimes respectively. We can easily derive the transformation
between the inertial time $\tau$ and the Milne and diamond time coordinates
$\nu$ and $\sigma$ comparing (14) and (15) with the expressions (10) and (13).
One obtains

$\displaystyle\nu$ $\displaystyle=$ $\displaystyle\alpha\log{|\tau|}+C_{1}$
(16) $\displaystyle\sigma$ $\displaystyle=$
$\displaystyle\alpha\log{\frac{|\tau+\alpha|}{|\tau-\alpha|}}+C_{2}\,.$ (17)

where the integration constants $C_{1}$ and $C_{2}$ can be determined by
requiring dimensional consistency. We set

$\displaystyle C_{1}$ $\displaystyle=$ $\displaystyle-\log{2\alpha}$ (18)
$\displaystyle C_{2}$ $\displaystyle=$ $\displaystyle 0\,,$ (19)

the reason for the choice $2\alpha$ for the argument of the $\log$ will be
clear later. We thus have for the Milne time

$\displaystyle\tau=$ $\displaystyle 2\alpha\,\exp{\frac{\nu}{\alpha}}$
$\displaystyle\qquad\text{for}\,\,\tau>0$ (20) $\displaystyle\tau=$
$\displaystyle-2\alpha\,\exp{\frac{\nu}{\alpha}}$
$\displaystyle\qquad\text{for}\,\,t<0\,.$ (21)

We see that on the positive half-line $\tau>0$ as $\nu$ ranges from $-\infty$
to $\infty$ we go from $\tau=0$ to $\tau=\infty$. On the negative half line
the time flow is inverted and we go from $\tau=0$ to $\tau=-\infty$ as $\nu$
ranges from $-\infty$ to $\infty$. Notice that, even though there is no
natural time scale associated with the future cone, a constant with dimensions
of time is needed in order to define a time evolution operator in terms of the
generator of dilations $\tau\,\partial_{\tau}$. This time scale is the point
on the real line measured in time $\tau$ corresponding to the origin $\nu=0$
of Milne time.
For the $S$-time variable $\sigma$ we have that in the region $|\tau|<\alpha$,
i.e. within the diamond, i.e.

$\tau=\alpha\,\tanh{\sigma/2\alpha}$ (22)

while for $|\tau|>\alpha$ one has

$\tau=\alpha\coth{\sigma/2\alpha}\,.$ (23)

We see that within the diamond as the diamond time ranges from $-\infty$ to
$\infty$ we cover the region from $t=\alpha$ to $t=-\alpha$.

Let us now consider a new time coordinate $\tau^{\prime}(\tau)$ in terms of
which the dilation generator $D=\tau\,\partial_{\tau}$ acts as the generator
$S$ of conformal transformations which preserve the diamond, i.e.

$\tau\,\partial_{\tau}=\frac{1}{2\alpha}\left(\alpha^{2}-\tau^{\prime
2}\right)\partial_{\tau^{\prime}}\,.$ (24)

Such equation can be easily solved leading to

$\log{\frac{|\tau^{\prime}+\alpha|}{|\tau^{\prime}-\alpha|}}=\log{|\tau|}+C\,.$
(25)

For dimensional consistency we choose for the integration constant
$C=-\log{A}$, where $A$ has dimensions of time and thus must be proportional
to $\alpha$. Restricting to the positive half-line $\tau>0$ we have

$\tau=A\,\,\frac{\tau^{\prime}+\alpha}{\tau^{\prime}-\alpha}$ (26)

In order to determine the constant $A$ we first notice that the point
$\tau^{\prime}=-\alpha$ gets mapped to $\tau=0$ while $\tau=\alpha$ to
$\tau=\infty$. The origin of the diamond $\tau^{\prime}=0$ gets mapped to
$\tau=-A$ so we know that $A$ must be negative. Let us further notice that the
origin of the diamond proper time $\sigma=0$ corresponds to $\tau^{\prime}=0$.
Under the map (26) this is mapped to $\tau=-A$. Now we require that the
origins of the diamond and Milne proper times are “syncronized” i.e. the point
corresponding to $\sigma=0$ in $\tau^{\prime}$ coordinates is mapped to the
point corresponding to $\nu=0$ in $\tau$-coordinates and thus, from (20),

$2\alpha=-A\,.$ (27)

We thus have

$\tau=-2\alpha\,\frac{\tau^{\prime}+\alpha}{\tau^{\prime}-\alpha}$ (28)

which can be inverted to give

$\tau^{\prime}=\alpha\,\frac{\tau-2\alpha}{\tau+2\alpha}\,.$ (29)

Notice how this transformations correspond to the map from the causal diamond
to the Rindler wedge and viceversa Casini:2011kv ; Jacobson:2018ahi  when
written in radial light-cone coordinates, indeed the Rindler wedge and the
future cone meet on the light cone.

##  III The thermofield double of conformal quantum mechanics

As shown in Chamon:2011xk ; Jackiw:2012ur  conformal quantum mechanics can be
seen as a one-dimensional conformal field theory in which correlation
functions are built from eigenstates of the time translation operator. Let us
recall the main features of this construction. In conformal quantum mechanics
we work with the “quantum mechanical” counterparts of the generators
introduced in the previous section i.e. $H=iP_{0}$, $D=iD_{0}$ and $K=iK_{0}$
closing the algebra

$[H,D]=iH\,,\qquad[K,D]=-iK\,,\qquad[H,K]=2iD\,.$ (30)

Let us look for states $|\tau\rangle$ on which the $SL(2,\mathbb{R})$
generator $H$ acts as a $\tau$-derivative

$H|\tau\rangle=-i\,\partial_{\tau}\,|\tau\rangle\,.$ (31)

These states were first constructed in deAlfaro:1976vlx  starting from an
irreducible representation of the $\mathfrak{sl}(2,\mathbb{R})$ Lie algebra.
Upon defining the operators

$L_{\pm}\equiv\frac{1}{2}\,\left(\frac{K}{\alpha}-\alpha\,H\right)\,\pm i\,D,$
(32)

and

$L_{0}\equiv\frac{1}{2}\,\left(\frac{K}{\alpha}+\alpha\,H\right)\,,$ (33)

with commutation relations

$[L_{-},L_{+}]=2L_{0}\,,\quad[L_{0},L_{\pm}]=\pm L_{\pm}\,,$ (34)

such representation is realized in terms of the eigenstates
$\left|n\right\rangle$ of $L_{0}$

$\displaystyle L_{0}\left|n\right\rangle=r_{n}\,\left|n\right\rangle$ (35a)
$\displaystyle r_{n}=r_{0}+n,\ \ r_{0}>0,\ \ n=0,1\ldots$
$\displaystyle\left\langle n|n^{\prime}\right\rangle=\delta_{n\,n^{\prime}}$
$\displaystyle L_{\pm}\left|n\right\rangle=\sqrt{r_{n}\,(r_{n}\pm
1)-r_{0}\,(r_{0}-1)}\,\left|n\pm 1\right\rangle$ (35b)

where $r_{0}$ is related to the eigenvalue of the Casimir operator of the
$\mathfrak{sl}(2,\mathbb{R})$ algebra

$\mathcal{C}=\frac{1}{2}\left(KH+HK\right)-D^{2}=r_{0}(r_{0}-1)\,.$ (36)

Notice that dimensional consistency requires the introduction of a scale
$\alpha$ with the dimension of time in the definition of the operators $L_{+}$
and $L_{-}$ and, in light of the correspondence with conformal symmetries in
Minkowski space-time, we can give a geometric interpretation to such scale,
following the discussion in the previous section, in terms of the radius of a
causal diamond whose boundaries are invariant under the flow of the conformal
Killing vector $S_{0}$.

In Chamon:2011xk ; Jackiw:2012ur  it was found that the $|\tau\rangle$ states
can be obtained from the $\left|n=0\right\rangle$ “vacuum state” via

$|\tau\rangle=\mathcal{O}(\tau)\left|n=0\right\rangle=N(\tau)\exp(-\omega(\tau)L_{+})\left|n=0\right\rangle\,,$
(37)

where

$\displaystyle N(\tau)$ $\displaystyle=$
$\displaystyle[\Gamma(2r_{0})]^{1/2}\left(\frac{\omega(\tau)+1}{2}\right)^{2r_{0}}\
,$ $\displaystyle\omega(\tau)$ $\displaystyle=$
$\displaystyle\frac{a+i\tau}{a-i\tau}\ .$ (38)

The action of the other generators on such $\tau$-states are given by

$\displaystyle D\left|\tau\right\rangle$ $\displaystyle=$
$\displaystyle-i\,\left(\tau\,\frac{d}{d\tau}+r_{0}\right)\left|\tau\right\rangle,$
(39) $\displaystyle K\left|\tau\right\rangle$ $\displaystyle=$
$\displaystyle-i\left(\tau^{2}\,\frac{d}{d\tau}+2\,r_{0}\,\tau\right)\left|\tau\right\rangle\,.$
(40)

In Chamon:2011xk  the two point functions of the theory are identified with
the inner product between the $\tau$-states

$G_{2}(\tau_{1},\tau_{2})\equiv\left\langle\tau_{1}|\tau_{2}\right\rangle=\frac{\Gamma\,(2r_{0})\,\alpha^{2r_{0}}}{[2i\,(\tau_{1}-\tau_{2})]^{2r_{0}}}\,,$
(41)

where $r_{0}$ plays the role of the conformal weight. General $n$-point
functions with $n>2$ are obtained by inserting generic primary operators
between $\langle\tau_{1}|$ and $|\tau_{2}\rangle$. When $\tau=0$ one has
$\omega(\tau)=1$ and thus

$|\tau=0\rangle=\Gamma(2r_{0})^{1/2}\,\exp(-L_{+})\left|n=0\right\rangle$ (42)

One immediately sees from (41) that such $\tau$-vacuum is not normalizable due
to the divergence of the two-point function for coincident points.

In Arzano:2020thh  it was noted that for $r_{0}=1$ the two-point function (41)
is proportional to the two-point function of a free massless scalar field in
Minkowski space-time evaluated along the trajectory of a static inertial
observer at the origin. This fact was used to derive the diamond
temperature333It should be noted that $H$ and $D$ form a sub-algebra of
$\mathfrak{sl}(2,\mathbb{R})$ which generates the group of affine
transformations of the real line, also known as the $ax+b$ group. In
Arzano:2018oby  it was shown, using one-dimensional fields carrying a
representation of the $ax+b$ group and Bogoliubov transformations between
$H$-modes and $D$-modes, that the $H$-vacuum is seen as a thermal distribution
of $D$-particles when restricted to the positive half line. upon re-writing
(41) in terms of the time variable associated to the diamond time evolution
generator $S=iS_{0}$. Here we provide an alternative derivation of the same
temperature by looking directly at the structure of the $\tau$-vacuum. This
will lead to a unified description of the diamond and Milne temperatures.

Let us start from a simple observation which, nonetheless, is a key passage of
our derivation. It is known from the quantum optics literature (see e.g.
Ban:1992zz ; Chaturvedi:1991zzb ) that the operators $L_{\pm}$ and $L_{0}$
with commutation rules (34) can be realized in terms of two sets of creation
and annihilation operators $a_{L},a_{L}^{\dagger},a_{R},a_{R}^{\dagger}$ as

$L_{+}=a_{L}^{\dagger}a_{R}^{\dagger}\,,\quad L_{-}=a_{L}a_{R}\,,\quad
L_{0}=\frac{1}{2}\left(a_{L}^{\dagger}a_{L}+a_{R}^{\dagger}a_{R}+1\right)\,.$
(43)

It immediately follows from (42) that the $\tau$-vacuum can be written
as444From this point on we set $r_{0}=1$ as in Arzano:2020thh .

$|\tau=0\rangle=\exp\left[-a^{\dagger}_{L}a^{\dagger}_{R}\right]|0\rangle_{L}\otimes|0\rangle_{R}~{},$
(44)

where we made explicit the bipartite structure of the vacuum state
$\left|n=0\right\rangle$ emerging from the decomposition (44)

$\left|n=0\right\rangle=|0\rangle_{L}\otimes|0\rangle_{R}\,.$ (45)

With simple manipulations we have that

$\displaystyle|t=0\rangle=$
$\displaystyle\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}\,\left(a^{\dagger}_{L}a^{\dagger}_{R}\right)^{n}|0\rangle_{L}\otimes|0\rangle_{R}$
(46) $\displaystyle=$
$\displaystyle\,\sum_{n=0}^{\infty}(-1)^{n}|n\rangle_{L}\otimes|n\rangle_{R}\,,$

where $|n\rangle_{L}$ and $|n\rangle_{R}$ are eigenstates of the “left” and
“right” number operators $N_{L}=a^{\dagger}_{L}a_{L}$ and
$N_{R}=a^{\dagger}_{R}a_{R}$. Our second key observation is that the Lie
algebra

$[L_{-},L_{+}]=2L_{0}\,,\quad[L_{0},L_{\pm}]=\pm L_{\pm}\,,$ (47)

can be realized via another combination of $H$, $D$ and $K$, namely

$L_{0}=iS\,,\qquad L_{+}=\frac{1}{2}\left(D-R\right)\,,\qquad
L_{-}=2\left(D+R\right)\,,$ (48)

where $R=\frac{1}{2}\,\left(\frac{K}{\alpha}+\alpha\,H\right)$ is generator of
elliptic transformations corresponding to rotations in the three-dimensional
Lorentz group. Under the identification (48) we can now interpret the
$|n\rangle$ states as eigenstates of the diamond time evolution generator $S$
and, in particular, the bipartite vacuum
$\left|n=0\right\rangle=|0\rangle_{L}\otimes|0\rangle_{R}$ as the “ground
state” of such Hamiltonian. In light of the correspondence with radial
conformal symmetries in Minkowski space-time, we can think of the vacuum state
$\left|n=0\right\rangle$ as the conformal quantum mechanics counterpart of the
the vacuum state associated the generator of time evolution for static diamond
observers, i.e. as the diamond analogue of the Boulware vacuum for Rindler
observers.

Along the same lines we can think of the $\tau$-vacuum $|\tau=0\rangle$ as the
conformal quantum mechanics counterpart of the Hartle-Hawking vacuum of a
massless scalar field in Minkowski space-time. Such connection is a bit more
subtle and should be understood in terms of a state-operator correspondence
illustrated in Chamon:2011xk . As observed in Arzano:2020thh  the Minkowski
light-cone two-point function maps to the $CFT_{1}$ two-point function
constructed from the $|\tau\rangle$ states (41). The latter can be written as

$\langle\tau_{1}|\tau_{2}\rangle=\langle
n=0|\,\mathcal{O}(\tau_{1})\,\mathcal{O}(\tau_{2})\,|n=0\rangle\,,$ (49)

with $\mathcal{O}(\tau)$ defined in (37). The vacuum sate $|n=0\rangle$ is not
conformally invariant and the operators $\mathcal{O}(\tau)$ do not transform
as primary operators under conformal transformations Chamon:2011xk . However,
as shown in Chamon:2011xk , the combination

$|\tau\rangle=\mathcal{O}(\tau)|n=0\rangle$ (50)

satisfies a constraint which combines both the requests of the invariance of
the vacuum with that of the operators $\mathcal{O}_{r_{0}}(t)$ to be primary.
In other words the two point functions $\langle t_{1}|t_{2}\rangle$ behave as
if they were constructed from a conformal invariant vacuum in analogy with the
two-point function for massless scalar field in Minkowski space-time
constructed from the Hartle-Hawking vacuum. Indeed it turns out Chamon:2011xk
that such two-point functions can be expressed as

$\langle\tau_{1}|\tau_{2}\rangle=\langle\tau=0|e^{-i(\tau_{1}-\tau_{2})H}|\tau=0\rangle\,,$
(51)

i.e. a quantum mechanical counterpart of the Minkowski space-time two-point
function with $|\tau=0\rangle$ playing the role of the Hartle-Hawking state.

The analogy between the $\tau$-vacuum and the Hartle-Hawking state is actually
deeper than it what might appear at first glance. Indeed the state
$|\tau=0\rangle$, like the Hartle-Hawking vacuum, exhibits the structure of a
thermofield double. To see this let us recall the expression of $L_{0}$ in
terms of creation and annihilation operators (43). Using the identification
(48), from (46) we can write the $\tau$-vacuum as

$|\tau=0\rangle=-i\,\sum_{n=0}^{\infty}e^{i\pi
L_{0}}\,|n\rangle_{L}|n\rangle_{R}\,=-i\,\sum_{n=0}^{\infty}e^{-\pi
S}\,|n\rangle_{L}|n\rangle_{R}\,.$ (52)

Such superposition of states is the same as the one appearing in the
thermofield double state of a bosonic oscillator Khanna:2009zz ;
Chapman:2018hou . Indeed, starting from the $\tau$-vacuum density matrix
$\rho=|\tau=0\rangle\langle\tau=0|$ and tracing over one set of $L$ or $R$
degrees of freedom one obtains a thermal density matrix at temperature
$1/2\pi$ with respect to the modular Hamiltonian $S$. Now, of course, $S$ does
not have the dimensions of a Hamiltonian, rather one should see the operator
$\frac{S}{\alpha}$ as the modular Hamiltonian of the system so that the
temperature associated to the thermofield double (52) is given by

$T_{S}=\frac{1}{2\pi\alpha}\,,$ (53)

i.e. precisely the diamond temperature obtained in Arzano:2020thh .

An interesting upshot of the analysis above is that it can be used to also
give a group theoretical derivation of the Milne temperature. Indeed, under
the conformal map (29) the identification (48) leads to the following
realization of the algebra (47)

$L_{0}=iD\,,\qquad L_{+}=-\alpha H\,,\qquad L_{-}=\frac{K}{\alpha}\,.$ (54)

With such identification the vacuum state $\left|n=0\right\rangle$ is now seen
as the conformal quantum mechanics analogue of the vacuum state associated to
$D$, the generator of conformal time translations in the future cone, and the
temperature

$T_{D}=\frac{1}{2\pi\alpha}\,,$ (55)

is simply the temperature perceived by a Milne observer sitting at the origin
of Minkowski space-time in the inertial vacuum. In two space-time dimensions,
where null planes and null cones coincide and Milne and Rindler particles can
de identified on the the light cone Wald:2019ygd , our arguments give a
unified, group theoretical derivation of the diamond, Milne and Unruh
temperatures.

##  IV Conclusions

The considerations above show that the structure of conformal quantum
mechanics is rich enough to reproduce, in the extremely simplified context of
a one-dimensional model, the basic features which characterize vacuum thermal
effects in quantum field theory, i.e. the appearance of the global vacuum as a
thermal state populated of excitations associated to Hamiltonians whose
evolution does not cover the entire time domain of the theory. Remarkably,
using the correspondence between time evolution in conformal quantum mechanics
and radial conformal symmetries in Minkowski space-time seen as generators of
proper time, this result provides direct evidence for the fact that the
inertial vacuum appears as a thermal state to observer whose time evolution is
not eternal. The results presented suggest that the thermodynamic properties
of causal diamonds and the Milne “patch” of Minkowski space are intimately
connected. It is tempting to speculate that the tools of conformal quantum
mechanics which are used to provide evidence for such connection might provide
an alternative and perhaps simpler way to describe entanglement features of
quantum fields across the causal boundaries of such regions. A thorough
exploration of such hypothesis is left to future studies.

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Research

Quantum Physics, Quantum Optics, Vector Optics Natalia Korolkova

# An operational distinction between quantum entanglement and classical non-
separability

Natalia Korolkova1,3 Luis Sánchez-Soto2,3 Gerd Leuchs3,4,5 1School of Physics
and Astronomy, University of St. Andrews, North Haugh, St. Andrews, KY16 9SS,
UK
2Departamento de Óptica, Facultad de Física, Universidad Complutense, 28040
Madrid, Spain
3Max-Planck-Institut für die Physik des Lichts, 91058 Erlangen, Germany
4Institut für Optik, Information und Photonik, Universität Erlangen-Nürnberg,
91058 Erlangen, Germany
5Department of Physics, University of Ottawa, Ottawa, Ontario K1N 6N5, Canada
<EMAIL_ADDRESS>
###### Abstract

Quantum entanglement describes superposition states in multi-dimensional
systems – at least two partite – which cannot be factorized and are thus non-
separable. Non-separable states exist also in classical theories involving
vector spaces. In both cases, it is possible to violate a Bell-like
inequality. This has led to controversial discussions, which we resolve by
identifying an operational distinction between the classical and quantum
cases.

###### keywords:

quantum entanglement, classical entanglement, quantum measurement, non-
separability

##  1 Introduction

In 1997, Spreeuw first discussed “a classical analogy of entanglement” [1],
followed four years later by a paper elaborating on the classical analogy of
quantum information processing using wave optics [2]. Depriving quantum
entanglement of its purely quantum nature and exploring the new capabilities
of classical optics have sparked a range of intriguing theoretical
considerations and experiments (for reviews, see, e. g., [3, 4, 5, 6]).

This exploration has prompted a controversial discussion, the essence of which
we highlight with a few selected examples. In 2011, Qian and Eberly suggested
a new interpretation of light polarization based on “non-quantum
entanglement," stating that “polarization is a characterization of the
correlation between the vector nature and the statistical nature of the light
field." Subsequent elaborations on the emerging links between quantum and
classical optics [7, 8] led Eberly et al. to a strong statement: “…
entanglement is a vector space property, present in any theory with a vector
space framework. There is no distinction between quantum and classical
entanglements, as such" [8]. This assertion has elicited equally strong
opposing statements. Karimi and Boyd, for instance, wrote “Entanglement is a
property of the quantum world; classical systems need not apply," and, “We do
not endorse this new nomenclature," adding, “… this situation lacks the key
feature - nonlocality" [9].

Despite the wide range of opinions, the past decade has witnessed a plethora
of potentially useful experiments with vector optics and other classical
implementations of vector spaces, demonstrating the power of classical
entanglement (or rather non-separability) in high-precision measurements,
metrology applications, and in emulation of various quantum information
protocols. However, the controversy in interpreting these experiments and
discussions about the fundamental nature of entanglement, both quantum and
classical, remain ongoing [6]."

We address the controversy by recognizing that the term should rely solely on
the details of the experimental observation rather than any prior knowledge
about the system. As we will explain, concepts like non-locality or counting
the number of particles involved fail to offer a clear distinction between
quantum and classical entanglement.

Instead, there is a decisive difference: the characterization of quantum
entanglement always involves correlating statistical outcomes of multiple
measurements on a given system, as elaborated below. Conversely, classical
entanglement typically refers to deterministic correlations between a single
measurement and a filter or sorter operation, such as those imposed on a light
beam by a polarizing beam splitter or a quantum sorter [10, 11]. The settings
of such filters or sorters do not involve measurement (and is a unitary
operation). This provides a clear distinction between the two phenomena.

##  2 Entanglement revisited: questions to ask

The term entanglement refers to a unique form of correlation between two or
more different variables (for an excellent recent discussion of quantum
entanglement, see [12]). Mathematically, entangled states are described by
non-separable functions within the vector space known as Hilbert space. Their
global nature leads to the puzzling property that the outcome of a measurement
on one part of such a non-separable state at one location appears to have an
instantaneous effect on the other part of the state at another location,
resulting in perfect correlations between measurement results at both
locations. However, this "immediate action" cannot be detected solely by
measuring at one location.

The emergence of the ‘non-local effect’ from a local measurement stems from
quantum theory, giving rise to several conceptual questions. Opponents of this
difficult-to-grasp action at a distance brought up explanations rooted in
classical physics. This was used by John Bell to demonstrate that quantum
physics cannot be explained by classical stochastic theories relying on local
hidden variables, hence coining the term quantum entanglement [13].

The non-local nature of correlations in quantum entanglement stems from non-
separability. However, similar types of correlations can be found in a
classical description involving mathematically non-separable functions. Are we
discussing here correlations of a common nature, as suggested, for example, by
Eberly? The term classical entanglement was coined. Is such terminology
appropriate? Does any mathematical non-separability imply Bell-type non-
locality, and is it this non-locality that makes up entanglement?"

Figure 1: Right: Different possible subsets of a set of general non-separable
states. The ellipses represent the four decisive properties of systems under
consideration: consist of two objects; exhibit non-locality; allows for two
measurements; can be written as mathematically non-separable mode functions.
The examples of physical systems populating a particular set are described in
detail in the text: I - non-separable states involving two excitations (two
objects); IIa,b - non-separable states with only one single excitation
(object), but where two distinct measurements can still be performed on two
non-separable partitions of the Hilbert space. III - classically entangled
states described by non-separable mode functions. Left: Two paradigmatic
examples of the states of type I and IIb, both allowing for two distinct
independent measurements - see text for discussion.

John Bell emphasized that his analysis applies to situations involving two
distinctly different particles. This establishes a clear and naturally unique
partitioning of the Hilbert space of the composite quantum system, as well as
the measurements to be performed. However, the choice of partitioning becomes
less obvious when different degrees of freedom contribute to the dimensions of
the Hilbert space, particularly in discussions involving light and mode
functions. Extending these discussions to such scenarios has recently gained
attention, partly in relation to the debate on quantum versus classical
entanglement.

In a recent study by Paneru et al. [12], quantum contextuality has been used
as a tool to identify the genuinely quantum nature of correlations between
different degrees of freedom. Realism, locality, and non-contextuality for
single and multiple particles represent different aspects of physical
theories. Paneru et al. argue that classical states of light, whether
separable or non-separable, can be fully described using the wave picture
without invoking field quantization. Therefore, they cannot challenge these
fundamental concepts in classical physics (for an introduction to field
quantization and the quantum theory of light, see [14]).

We want to address the problem from a different, more hands-on viewpoint.
Realism, locality, and non-contextuality are all related to a measurement
process. Figure 1 visually highlights some of the nontrivial aspects of
partitioning and the role of measurement in analyzing correlations in non-
separable systems. For simplicity, we restrict ourselves to bi-partite non-
separable states. Different possible partitions of the _mathematical_ vector
space can be of quite different _physical_ nature. They can correspond to
objects of zero or non-zero rest mass, such as a photon or an atom,
respectively, or to mode functions, such as particular spatial modes.

To re-iterate, the complication stems from the following observation: non-
separability is a mathematical property and is independent of the physical
nature of the different degrees of freedom; i.e., the different parts of
Hilbert space, whereas the ability to perform one or more measurements does
depend on the physical nature of the degree of freedom. A measurement requires
some excitation, some energy above the level of the vacuum, which can be
associated either with a zero or non-zero rest mass particle or a combination
thereof. Mode functions in optics, for example, can have numerous degrees of
freedom occupying a larger Hilbert space, but they are merely frames for
excitations and by themselves they do not refer to any excitation. If the mode
is not excited, there is nothing specific to measure (other than non-specific
vacuum fluctuation). We, therefore, attempt to structure the types of non-
separable states with regard to their physical nature. We take the liberty
here to use the word ‘excitation" in a wider sense. We speak about excitation
not only when we have some non-zero energy state of the electromagnetic field,
but we can also view a massive particle as a de Broglie wave-packet. We can
then also think of “modes" which are “excited" or “not excited," meaning the
particle is there or not. Furthermore, there is a whole range of quasi-
particles, meaning a combined excitation from entities of different physical
natures: phonons, excitons, polaritons, to name a few.

We list below the properties defining principally different cases of non-
separability and refer to excitations to accommodate massive particles,
photons or “inter-species” excitations.

###  2.1 Defining properties of non-separable systems of different types

We want to single out four defining properties that lead to non-separable
states of principally different physical nature (see Figure 1). We will
elucidate their distinctive features with examples in the next subsection.

#### Two objects.

The dark-blue innermost set in Fig. 1 encompasses systems with two distinct
objects (i.e., excitations) living in the sub-spaces of a bi-partitioned
Hilbert space. These objects are in a state described by mathematically non-
separable mode functions. Here, the available ‘modes’ are populated by two
excitations, rendering two distinct separate measurements (each on one of the
excitations) possible. Examples of such system are two electrons in a singlet
state (spin up/down, $|\uparrow\rangle,|\downarrow\rangle$) or two photons of
orthogonal polarizations (polarization horizontal/vertical,
$|H\rangle,|V\rangle$) in different spatio-temporal modes:

$\displaystyle|\Psi_{\rm singlet}\rangle_{12}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(|\uparrow\rangle_{1}|\downarrow\rangle_{2}-|\downarrow\rangle_{1}|\uparrow\rangle_{2}\right),$
(1) $\displaystyle|\Psi_{\rm pol}\rangle_{12}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(|H\rangle_{1}|V\rangle_{2}-|V\rangle_{1}|H\rangle_{2}\right).$

The mathematically non-separable mode functions in Eq. (1) correspond to the
possible combinations of outcomes of two projective measurement defined by the
excitations present. This subset, depicted in Fig. 1 by the dark central
ellipse, marks the subset that we historically perceive as THE entanglement.
Elements of this subset possess all the four properties mentioned in Fig. 1:
(1) they consist of two distinctly different objects; (2) two measurements are
possible; i.e., each of the two subsystems into which the system is
partitioned can be measured separately; (3) they are non-local; (4) they are
non-separable. Correlations are manifested through series of two measurements
performed on the two subsystems involved and constitute statistical
correlations.

#### Non-local.

The next medium-blue shaded subset includes all states that exhibit the
property of non-locality (in its trivial sense: parts of a composite system
are at spatially separated locations). We briefly discuss non-locality in more
general sense in Sec. 4, but it is deliberately not the subject of this paper.

#### Two measurements.

The larger ellipse of the yet lighter blue includes all states characterized
by a common feature: the possibility to perform two distinct, independent
projective measurements. Remarkably, two of the three subsets contained within
the ellipse labelled ‘two measurements’ deprive us of the first traditional
ingredient: two objects. Here, in these two lighter-shaded subsets, there is
only one single excitation initially. However, importantly, it still clearly
defines two Hilbert space partitions. These partitions are determined by two
distinct excitations generated by one single object, examples in the next
section. The mathematically non-separable mode functions describe the possible
combination of measurement outcomes defined by these secondary excitations.

#### Non-separable.

For states in the outermost ellipse (but outside the other subsets), we now
relinquish the possibility of performing two measurements. In this class, the
only remaining property is mathematical non-separability, which is also common
to all cases listed above.

###  2.2 Paradigmatic examples of non-separable systems

In this subsection, we aim to explore specific physical implementations of the
non-separable states encompassed in different subsets in Fig. 1, emphasizing
the features that define their entanglement properties.

#### Subset I.

This is an example of prototypical quantum entanglement, the most familiar
among physical implementations of non-separable states. These states belong to
the type depicted in Fig. 1, top left, and presented in Eq. (1).

#### Subset IIa.

Consider a single excitation shared by two mode functions, involving non-
separability. The seminal example of such quantum entanglement is the
entanglement of a single photon (or atom) incident on a beam splitter. The
state at the output of the beam splitter can be written as
$|\psi_{12}\rangle\propto|1\rangle_{1}|0\rangle_{2}+|0\rangle_{1}|1\rangle_{2}$,
where $|1\rangle_{j}$ refers to a photon (or atom) in output $j$ and
$|0\rangle_{j}$ represents no photon (or atom), with $j=1,2$.

In optics, there has been a debate on whether this scenario truly represents
quantum entanglement, a controversy that was resolved in 2004 [15]. This
debate is relevant to our current discussion, as the argument against
entanglement was based on only one excitation being present and measurable at
the output with on-off single photon detectors. However, as it is well-known,
an optical beamsplitter always has two inputs and two outputs. The incident
light field excitation determines the set of modes we are looking at and,
consequently, the two output modes of the beamsplitter.

In the case of an ‘undivisible’ single photon, one output mode contains a
single photon while the other exhibits vacuum fluctuations in the mode defined
by this excitation. Both of these secondary excitations (with one
corresponding to the ground state) can be measured using homodyne detection.
Thus, although only one physical excitation is present (i.e., one photon), two
measurements can still be performed at the output using homodyning, providing
evidence of entanglement (see also [16]). This example IIa, also demonstrates
the property of non-locality (see Fig. 1).

#### Subset IIb.

In this scenario, we consider a single massive object that undergoes further
excitations in two ways; e.g., motion in two dimensions. A seminal example of
this is a single two-dimensional harmonic oscillator with possible excitations
in both the $x$\- and $y$-directions. An intriguing observation made by Asher
Peres in his textbook [17] nearly 30 years ago (and seldom cited) clarifies
this example and underscores the significance of two measurements defined by
two excitations in discussions about quantum entanglement. The right part of
Fig. 1 displays the two systems Peres discusses. Two one-dimensional (1D)
harmonic oscillators coupled in a non-separable way (Fig. 1, top right) is a
generic example of quantum entanglement, ticking all four boxes as in dark
blue subset. However, mathematically this system is completely equivalent to a
single mass on two springs, a single two-dimensional (2D) oscillator (Fig. 1,
bottom right) [17]. Consequently, the wave functions of the energy eigenstates
are completely equivalent in both cases:

$\displaystyle|\Psi_{\rm 2\times 1D}\rangle_{12}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(|n\rangle_{1}|m\rangle_{2}-|m\rangle_{1}|n\rangle_{2}\right),$
(2) $\displaystyle|\Psi_{\rm 1\times 2D}\rangle_{xy}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(|n\rangle_{x}|m\rangle_{y}-|m\rangle_{x}|n\rangle_{y}\right).$

Here, $\\{|n,m\rangle\\}$ denotes two possible energy levels associated with
two excitations, either in the two oscillators $1$ and $2$, for two 1D
systems, or with two spatial directions $x$ and $y$ of a 2D system.

Crucially, similar to example IIa, there are two distinct secondary
excitations. Hence, despite containing only one mass, this system allows for
two distinct measurements to be performed. thus decisively manifesting the
entanglement property.111In this example we view the quantum harmonic
oscillator as a discrete-variable (DV) system due to its discrete energy
levels. Alternatively, one can view it as a continuous-variable (CV) system by
looking at position and momentum, which requires a different framework. Note
that a light mode in quantum optics can be represented as such a CV quantum
harmonic oscillator [14], and in example IIa, the two output modes can be
viewed as such two coupled harmonic oscillators.

A 2D oscillator provides a prototypic example of such states, but the subset
is not restricted to 2D systems. Non-separable states involving quantum
entanglement between two different degrees of freedom exhibit the same
property. A vivid example of locally entangled state is a mesoscopic
Schorödinger cat-like state of cold atoms realized with trapped Be ions in
1996 in the group of Wineland [18]. They created the following state of a
single ${}^{9}{\rm Be}^{+}$ laser-cooled ion:

$|\psi\rangle=\frac{1}{\sqrt{2}}(|x_{1}\rangle|\uparrow\rangle+|x_{2}\rangle|\downarrow\rangle)\,.$
(3)

Here $|x_{j}\rangle$ denotes localized wave-packet states corresponding to two
spatial positions of the atom, while $|\uparrow\rangle,|\downarrow\rangle$ are
two distinct internal electronic quantum states of the atom (hyperfine ground
states). The secondary excitations present for this single atom can be
modelled as a 2D harmonic oscillator or even interpreted as a Schrödinger cat-
like state of two spatially separated coherent harmonic oscillator states,
$|x_{j}\rangle$, $j=1,2$ (as suggested in [18]). The key point here is that
two measurements on two different degrees of freedom, $|x_{j}\rangle$ and
$|\uparrow\rangle,|\downarrow\rangle$, are possible for the state (3).
Therefore, the state (3) is prototypic for “local entanglement” of states
which cannot be separated spatially. In other words, such states not only lack
the “two-objects” property, but also lack another traditional feature, non-
locality.

#### Subset III.

All the examples discussed are characterized by mathematically non-separable
functions, as shown in Fig. 1. Now, consider the region of the full set
outside the three inner subsets. In contrast to the previous examples, here
the partition of the Hilbert space is not defined by two excitations,
representing a case of classical entanglement.

For classically entangled non-separable states, the Hilbert space is spanned
by the available mode functions, whereas a single existing excitation can, in
principle, be assigned to (included in) any of the partitions. This assignment
is guided solely by experimental convenience or our choice of which
measurements to perform.

Let us illustrate this with an example. Enhancement in a number of metrology
tasks has been demonstrated using a classically entangled state with
correlations between the transverse spatial mode and polarization (see, e.g.,
[4]):

$|\Psi\rangle\propto|\psi_{01}(\mathbf{r})\rangle_{1}|H\rangle_{2}-|\psi_{10}(\mathbf{r})\rangle_{1}|V\rangle_{2}.$
(4)

As we mentioned at the beginning of our discussion on Fig. 1, quantum
mechanics stems from observations, namely measurements. In the case of Eq.
(4), we are dealing with the non-separability of a vector space described
entirely by available mathematical mode functions connected to different
degrees of freedom. Initially, there is nothing specific to measure.

Contrast this with all the previous cases: there, the partitioning of the
Hilbert space has been based on physical excitations. The Dirac kets
corresponded to the chosen bases for the respective vector sub-spaces, and
their eigenvalues were the possible measurement outcomes of the two projective
measurements, one on each partition.

For Eq. (4), the partitioning initially is not connected to any excitations
and the eigenvalues of the basis kets are mode-functions, possible
decomposition of the initial state in some basis. As we will discuss in detail
later, one of the eigenvalues is the conditional measurement outcome of one
single projective measurement on a chosen partition, conditioned on the
deterministic choice of the value of the other variable. That is, to unlock
the physical nature of the state Eq. (4) and to connect it to observations, we
rather need to view Eq. (4) differently.

The Hilbert space has a partition 1 spanned by two mode functions $k$ and
$k^{\prime}$ corresponding to the first degree of freedom (spatial mode), and
partition 2 spanned by two mode functions $l$ and $l^{\prime}$ corresponding
to the second degree of freedom (polarization mode). However, the excitation
present is independent of these partitions and corresponds to the
electromagnetic wave, which can be written in the basis of various modes.
Hence, the physics behind Eq. (4) is better captured by rewriting it as

$\displaystyle|\Psi\rangle\propto[|\psi_{k}\rangle_{1}|\psi_{l}\rangle_{2}-|\psi_{k^{\prime}}\rangle_{1}|\psi_{l^{\prime}}\rangle_{2}]\;|{E(\mathbf{r},t)}\rangle,$
(5)

where $E(\mathbf{r},t)$ describes the excitation of the electromagnetic field.
Put simply, we have decomposed the electromagnetic wave into two spatial
modes: $\psi_{01}$ horizontally polarized and $\psi_{10}$ vertically
polarized. In this system, we cannot violate realism, a cornerstone of
classical physics absent in quantum mechanics. Here, we are dealing with
measurement outcomes predetermined before measurement, evident in the
correlations observed.

To elucidate the relationship between the spatial mode and polarization, we
can opt for one of the two possible scenarios. First, assume we aim to measure
polarization. We assign excitation to partition 2 and conduct measurement
$\hat{M}_{2}$ in the basis $\\{H,V\\}$, where potential outcomes are denoted
${\cal M}^{j}_{2}$, with $j=H,V$. However, to discern the correlation between
spatial mode and polarization, one must first choose a specific spatial mode
to examine: either $\psi_{01}(\mathbf{r})$ or $\psi_{10}(\mathbf{r})$ from
partition 1. This is not a measurement but rather a unitary operation 222A
sorter or filter introduces a certain projection in the corresponding vector
space, resembling quantum measurement. However, in a sorter, the state is
projected and the complementary projection is in principle still available,
such as in a polarizing beam splitter in optics. This operation can be
reverted if both output beams are used. In a quantum measurement one is
necessarily ‘loosing’ information which is in principle there before the
measurement. The quantum measurement projects the state of the system onto an
eigen state of the operator describing the measurement. A measurement often
involves a sorting or filtering operation but it has to be followed by some
kind of read out. The readout causes the collapse or projection of the state
of the system to just one possibility. With sorting without the readout all
possibilities would be still open. (in contrast to the projections in the
Hilbert sub-space in previous examples being always non-unitary, projective
measurements). The projection onto one of the axes in partition 1 determines
the axis onto which the projection in partition 2 occurs, thus predetermining
the measurement outcome on 2 before any measurement is performed: a unitary
operation on 1 selecting $\psi_{01}(\mathbf{r})$ implies ${\cal M}^{H}_{2}$.

Similarly, if our aim is to resolve the type of transversal mode in the
measurement, we will assign excitation to partition 1, where potential
measurement outcomes are denoted ${\cal M}^{j}_{1}$, with $j=01,10$. The
projection onto one of the axes in partition 2 determines the axis onto which
the projection in partition 1 occurs: a unitary operation on 2 selecting $H$
implies ${\cal M}^{01}_{1}$ prior any measurement. Note that only one
measurement on one of the partitions is possible in this case. We will delve
into this case further in the next two sections.

Another property to consider, besides non-separability, is non-locality, which
one would attribute to (I) and (IIa), but not to (IIb) and not, in general, to
the entire set. It is worth noting that in cases (I), (IIa), and (IIb), two
separate independent measurements can be performed.

From the discussion and examples above, it is evident that there are
inherently different questions one needs to distinguish when seeking to assess
the nature of correlations:

  * •

Quantum-classical demarcation. What is a proper verification tool here?

  * •

Is entanglement entirely about non-separability of states, regardless of the
physical character of the mathematical partition of a vector space?

  * •

Is entanglement about the non-local character of correlations? What does ‘non-
local’ mean? What are appropriate means to verify it?

  * •

When employing a non-separable state in certain protocols, where does the
advantage come from?

##  3 Measurements to reveal quantumness

Figure 2: Correlations in quantum entanglement. Two distinct projective
measurements on subsystems $A,B$ are performed depicted by two boxes, with a
possibility of detection in two bases of choice, ${\cal F}_{\theta},{\cal
F}_{\theta+\frac{\pi}{2}}$, and with statistical outcomes ${\cal
M}_{A,B}^{j}$. There are statistical correlations between measurement
outcomes, $C\\{{\cal M}_{A}^{j},{\cal M}_{B}^{j}\\}$.

In this section, we address the first two previous points. We begin by
emphasizing that the Bell inequality provides answers to very specific
questions (as will be addressed in the next section) and requires extreme
caution when applied to verify the quantumness of correlations [19].

Figure 3: Correlations in classical entanglement. Only one projective
measurement described by mode function $B$ is performed, denoted by a box with
a possibility of detection in two bases of choice, ${\cal F}_{\theta},{\cal
F}_{\theta+\frac{\pi}{2}}$ and with outcome ${\cal M}^{j}$. The other mode
function, $A$, describes a unitary operation of “a switch”, determining the
filter setting $k$, ${\cal F}_{\theta}$ or ${\cal F}_{\theta+\frac{\pi}{2}}$.
There is a deterministic correlation between the filter setting $k$ and the
measurement outcome, $C\\{k,{\cal M}^{j}\\}$.

In assessing quantumness, a system cannot be considered separately from the
measurement process. The key lies in the interaction between a quantum system
and a measurement apparatus, with quantum mechanics providing predictions for
the corresponding outcomes. Additionally, the experimental context, including
the details of the apparatus used, is crucial, as it directly influences the
observation process.

To assess a quantum property, one has to be sensitive to the projection
imposed by a quantum measurement. This cannot be achieved by a single
measurement alone; rather, one needs to perform a first measurement to impose
the projection and then a second measurement to detect the effect of the
projection. This is precisely what occurs in the case of quantum entanglement.
A projective measurement is performed on one part of the non-separable state,
$A$. Subsequently, a second projective measurement is performed on the other
part, $B$, revealing the effect of the first measurement on the global state.
Overall, quantum correlations inherent to entangled states are manifested
through the statistically correlated results of subsequent projective
measurements. Mathematically, in the partitioning of the vector space, both
mode-functions $A$ and $B$ in the non-separable state are eigenstates of
projectors, with their eigenvalues (say, $0$ and $1$) describing the possible
outcomes of the projective measurements, denoted as ${\cal M}^{0,1}_{A}$ and
${\cal M}^{0,1}_{B}$, respectively, one on each of the partitions (see Fig.
2):

$|\Psi\rangle\propto|0\rangle_{A}|1\rangle_{B}+|1\rangle_{A}|0\rangle_{B}.$
(6)

We see that the correlations are revealed between two statistical measurement
outcomes, $C\\{{\cal M}_{A}^{j},{\cal M}_{B}^{j}\\}$.

In the case of classical entanglement, the vector space is partitioned in a
fundamentally different manner (as discussed in subsection 2.2). There is only
one excitation present, which can be assigned to either of the partitions
spanned by available mode functions. Consider the same prototypic example as
in Eq. (4):

$|\Psi\rangle\propto|\psi_{01}(\mathbf{r})\rangle_{A}|H\rangle_{B}-|\psi_{10}(\mathbf{r})\rangle_{A}|V\rangle_{B}.$
(7)

For concreteness, let us assign the excitation to the polarization mode
function, making the available measurement that of polarization. Then, the
projective measurement will only be performed on the $B$-part of the non-
separable functions, resulting in the projective measurement outcomes ${\cal
M}^{H,V}_{B}$. The projective measurement on sub-system $A$ is replaced by
some sort of mode selection, basis choice, or filter setting, and is a unitary
operation (see Fig. 3).

To facilitate comparison with Eq. (6), let us further reformulate Eqs.(4) and
(7), highlighting the different nature of kets. The unitary operation, which
distinguishes between two orthogonal basis states encoded via $\theta$ and
$\theta+\frac{\pi}{2}$, is denoted ${\cal F}_{\theta},{\cal
F}_{\theta+\frac{\pi}{2}}$. This is a filtering operation corresponding to the
choice of basis, or mode, or another unitary ‘sorter’ operation. Assigning $0$
and $1$ to the measurement outcomes in the $\\{H,V\\}$ basis, we obtain:

$|\Psi\rangle\propto|{\cal F}_{\theta}\rangle_{A}|0\rangle_{B}-|{\cal
F}_{\theta+\frac{\pi}{2}}\rangle_{A}|1\rangle_{B}.$ (8)

Comparing to Eq. (6), we clearly see that there is no statistical correlation
between two measurement outcomes here. Instead, we have a deterministic
projective measurement outcome ${\cal M}^{j}$, given the choice of basis.
There is a deterministic correlation between the ‘filter’ setting
$k=\theta,\theta+\frac{\pi}{2}$ and the measurement result, $C\\{k,{\cal
M}^{j}\\}$.

Entanglement is about non-separability, but the nature of non-separable
functions is decisive. However, the key point is not whether the partitioning
of the vector space, $A,B$, is related to spatially separated particles or to
two degrees of freedom of the same particle. What is important is whether non-
separable functions lead to statistically correlated measurement outcomes due
to the effect of two subsequent quantum measurements, one on each sub-system
(quantum entanglement, violation of realism), or whether mathematical non-
separability leads to a single measurement outcome conditioned on some basis
choice (classical entanglement, realism observed).

##  4 Locality and non-locality of non-separable states

In discussions comparing classical and quantum entanglement, distinctions are
often made using characterizations such as ‘local’ and ‘non-local.’ But what
do they mean? This question has sparked a controversial dispute that is still
ongoing [19, 6]. Is entanglement primarily about non-locality in the sense
that an action on one part of a global system seems to have an immediate
effect on the other part of the system? Or does ’non-local’ simply mean that
the system is distributed in space?

Regarding the latter, we have already shown in Sec. 2 that the existence of
two spatially separated objects is not a necessary pre-requisite for
entanglement. The former is more subtle. The principle of locality states that
any interaction propagates with finite velocity, and it takes time for a
change in one degree of freedom to have an effect on the other degree of
freedom. On the surface, any state involving mathematically non-separable
functions (equally describing either spatially separated systems or the
internal degrees of freedom of one and the same system) should violate this
principle of locality. Therefore, the concept of “spooky action at a distance"
stands as a characteristic feature of entanglement from the early Einstein-
Podolsky-Rosen (EPR) debates in 1935 onwards. Examples of the violation of
Bell-type inequalities with classically entangled light seem to confirm this
[20, 21]. This leaves anyone who attributes the term ’entanglement’ to a
system that is indisputably in the quantum domain quite uncomfortable. What,
then, is the decisive point that would help reliably demarcate classical
correlations from genuinely quantum phenomena?

Already in this discussion of locality and non-locality, an important point
emerges that enables the demarcation. First of all, recall that the violation
of the Bell inequality means that quantum mechanics is not a local realistic
theory. It is not just about locality, although we often use the term ‘non-
locality’ in connection with the Bell inequality. This raises the question:
Does quantum mechanics violate the locality principle, the principle of
reality, or both together? The discussion is still ongoing, especially when
different interpretations of quantum mechanics are debated. However, note: The
violation of local-realistic assumptions occurs in a non-separable state only
if a measurement is performed on one of the subsystems.

As strongly emphasized in [19], “the main deviation of classical light models
from quantum theory is not only in the states but also in the description of
measurement procedures." Khrennikov argues strongly against using quantum non-
locality to differentiate between quantum and classical correlations,
attributing the crucial role to measurement procedures [19], which aligns well
with our arguments. Importantly, the conventional view is that quantum theory
is local, but the assumption of realism is incorrect. It is quite
straightforward to argue for this. The principle of locality dictates that
interactions propagate with finite velocity. However, when the measurement
outcome on $A$ determines the measurement outcome on $B$ in a non-separable
state, it is not about interaction between these two subsystems. A system in a
non-separable state is in a superposition of two possibilities (in the
bipartite case): for the state of Eq. (6), the system is simultaneously in
state $|0\rangle_{A}|1\rangle_{B}$ and $|1\rangle_{A}|0\rangle_{B}$; it is not
in a definite, ‘pre-existing’ state. The distinct state emerges only upon
measurement and is ’decided’ at random in the process of measurement with a
given probability. This is a clear demonstration of the violation of the
principle of reality, which states that properties of objects are real and
exist independent of our observations of them.

Importantly, the conventional view is that quantum theory is local, yet the
assumption of realism is incorrect. This argument is quite straightforward.
The principle of locality stipulates that interactions propagate with finite
velocity. However, when the measurement outcome on $A$ determines the outcome
on $B$ in a non-separable states, it actually happens not due to the
interaction between these two subsystems. Instead, a system in a non-separable
state is in a superposition of two (in bi-partite systems) possibilities: For
the state of Eq. (6) the system is simultaneously in the state
$|0\rangle_{A}|1\rangle_{B}$ and $|1\rangle_{A}|0\rangle_{B}$, it is not in a
definite, ‘pre-existing’ state. The distinct state emerges only upon
measurement and is ‘decided’ at random in the process of measurement with a
given probability.

This clearly demonstrates a violation of the principle of reality, which
posits that the properties of objects are inherent and exist independently of
our observations. This once again underscores the importance of the quantum
measurement as a fundamental aspect of the quantum-classical boundary. The
manifestation of a specific ‘reality’ during the measurement process is the
source of correlations in case of quantum entanglement. The first projective
measurement, performed on $A$, collapses the global state in one of the
possibilities, at random, say $|0\rangle_{A}|1\rangle_{B}$, leading to random
‘0’ at $A$. The effect of this first measurement is verified by the second
measurement, which reveals the state of the second sub-system to be ‘1’,
confirming statistical correlation (Fig. 2).

In classical entanglement, as elucidated above, a unitary process of selecting
a particular basis or mode function replaces the measurement for certain
operations. This process, unlike a measurement, is unitary (at least in
principle), and we refer to it as “filtering" or “sorting". Due to the unitary
nature of the operation, there is no reduction of the wave function. Hence,
the mathematical non-separability in the framework of classical theory (e.g.,
in classical optics involving vector light fields) cannot result in any non-
local effects. In general, understanding the distinction between measurement
and certain unitary processes, which are sometimes confusingly interpreted as
measurements, provides insight into the controversial usage of the term
entanglement.

Figure 4: Principle experimental schemes for construction of Bell inequality
for verification of polarization correlations $f(\theta,\theta^{\prime})$.
Paramerization by $\theta$ happens either as part of the measurement process,
or as part of the state preparation process, as indicated by dashed boxes.
(Top Left) In case of LHV assumption, the polarization is predetermined at the
source and two measurements are performed parametrized by $\theta$; (Bottom
Left) In case of quantum entanglement, the polarization is not defined at the
source and is determined in the measurement process, at the detector, when two
measurements are performed parametrized by $\theta$; (Right) In case of
classical non-separability, light at the source is prepared in a certain
polarization state parameterized by $\theta$. A single, parametrized by
$\theta^{\prime}$ measurement is performed.

It is very tempting to exploit the analogy between wave-like properties in
quantum mechanics and all the wave phenomena in classical optics (wave
equation, superposition, interference, etc.). After all, in Erwin
Schrödinger’s early works, quantum mechanics went under the name wave
mechanics [22]. This viewpoint has been widely represented in the community,
strongly highlighted in the works of Eberly [7, 8]. This analogy has also been
extended to demonstrations of violations of Bell inequalities. Simulations of
Bell inequalities are discussed in a number of papers, for example [20, 21].

Let us depart for the moment from any specific experimental details on how it
has been implemented, but let us focus on the principal idea of an
experimental setting used to construct the inequality [13, 12]. The
statistical analysis required begins with performing two independent
measurements on two parts of the Hilbert space, these measurements being
parameterized by some parameter $\theta$. For example, in the case of
polarization entanglement of Eq. (1), the measurement process will include a
polarizer, set in front of a detector and rotated at a certain angle $\theta$.
In other words, the measurement will include a detector and a ‘filter’ set for
$\theta$, as discussed previously. The inequality is constructed by combining
probabilities of measurement outcomes obtained at detector 1 plus filter
$\theta$ and detector 2 plus filter $\theta^{\prime}$ for particular values of
$\theta$ and $\theta^{\prime}$ (for details, we refer the interested reader to
[13, 12]).

Consider now three relevant cases of polarization correlations
$f(\theta,\theta^{\prime})$: determined by local hidden variables (LHV);
stemming from quantum entanglement; originating from classical non-
separability. The three respective schemes are depicted in Fig. 4 with the
example of polarization measurements. The key difference can be easily
understood if we compare this figure with Figs. 2 and 3. The verification of
the local realism assumption has been derived by Bell by examining statistical
correlations of the results of two independent measurements on two partitions
of the Hilbert space (left part of Fig. 4). One can simulate this test using
classical non-separability, but it will not yield the same statistical
analysis as performed by Bell; instead, it exploits deterministic correlations
between state preparation and a single measurement to emulate Bell-like
correlations (right part of Fig. 4).

##  5 Conclusions

Quantum and classical optics constitute an amazing field for exploring the
interplay between the particle and wave nature of light [14]. The richness of
wave phenomena goes beyond usual applications, allowing for the emulation of
quantum features, as seen in classical entanglement, and even serving as a
platform for analogue gravity experiments [23]. Despite the beauty of both
quantum and classical theories of light, one should not be tempted to move the
quantum-classical boundary entirely on the ground of presence of wave
phenomena, or existence of non-separable vector spaces, or possibility for
enhancing metrology protocols. Quantum mechanics is first and foremost about
the interaction between a system and a measurement apparatus. Not the
properties of the system taken by itself, and not so much the microscopic
nature of the system itself are decisive.

Assessing the quantum nature of a non-separable state requires consideration
of its behavior in a measurement process. In simple, operational language, the
quantum nature of non-separable states manifests in statistical correlations
between the outcomes of two measurements performed on two partitions of the
corresponding Hilbert space. This distinctive feature transcends the the
details of state preparation taken by themselves and whether we are dealing
with single photons or intense light beams.

The scheme depicted at the bottom left part of Fig. 4 can be utilized not only
to test local realism for polarization entanglement of single photons, but
also for continuous variable polarization entanglement [24]. Such non-
separable states reveal quantum correlations in the measurement of Stokes
operators, quantum variables with a continuous spectrum, the counterparts of
classical Stokes parameters [25]. These states can be generated using intense
light beams that may appear classical.

The question arises: if classical non-separability is entirely a classical
phenomenon and not entanglement in its true sense, then where does the
advantage demonstrated in metrology, the possibility to simulate not only
Bell-like non-locality, but also a number of other quantum information
protocols, come from? This is not the subject of this paper, and we will only
briefly comment on it. In most cases, the advantage comes merely from the
increased parameter space. One can conceptualize it as the size and structure
of the Hilbert space, as seen in quantum cryptographic protocols based on
structured light [26], or as a form of classical parallelism, as observed in
polarization metrology [27, 5].

With great pleasure, we dedicate this article to Rodney Loudon. His legacy
continues to inspire scientists and pave the way for further exploration in
the realm of quantum optics. Over the years, we have benefited from questions,
suggestions, criticism, and advice from many colleagues. Particular gratitude
for help in various ways goes to Andrea Aiello, Gunnar Björk, Bob Boyd, Joe
Eberly, Robert Fickler, Elisabeth Giacobino, Aaron Goldberg, Ebrahim Karimi,
Antonio Z. Khoury, Andrei Khrennikov, and Andrei Klimov.
The work has been supported by the Scottish Universities Physics Alliance
(SUPA), the Alexander von Humboldt Foundation (NK), and by the Agencia
Española de Inbestigación (LLSS).

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