# Can single photon excitation of two spatially separated modes lead to a
violation of Bell inequality via homodyne measurements?

Tamoghna Das  International Centre for Theory of Quantum Technologies,
University of Gdańsk, 80-308 Gdańsk, Poland  Marcin Karczewski  International
Centre for Theory of Quantum Technologies, University of Gdańsk, 80-308
Gdańsk, Poland  Antonio Mandarino  International Centre for Theory of Quantum
Technologies, University of Gdańsk, 80-308 Gdańsk, Poland  Marcin Markiewicz
<EMAIL_ADDRESS>International Centre for Theory of Quantum
Technologies, University of Gdańsk, 80-308 Gdańsk, Poland  Bianka Woloncewicz
International Centre for Theory of Quantum Technologies, University of Gdańsk,
80-308 Gdańsk, Poland  Marek Żukowski  International Centre for Theory of
Quantum Technologies, University of Gdańsk, 80-308 Gdańsk, Poland

###### Abstract

We reconsider the all-optical homodyne-measurement based experimental schemes
that aim to reveal Bell nonclassicality of a single photon, often termed
‘nonlocality’. We focus on the schemes put forward by Tan, Walls and Collett
(TWC, 1991) and Hardy (1994). In the light of our previous work the Tan, Walls
and Collett setup can be described by a precise local hidden variable model,
hence the claimed nonclassicality of this proposal is apparent, whereas the
nonclassicality proof proposed by Hardy is impeccable. In this work we resolve
the following problem: which feature of the Hardy’s approach is crucial for
its successful confirmation of nonclassicality. The scheme of Hardy differs
from the Tan, Walls and Collett setup in two aspects. (i) It introduces a
superposition of a single photon excitation with vacuum as the initial state
of one of the input modes of a 50-50 beamsplitter, which creates the
superposition state of two separable (exit) modes under investigation. In the
TWC case we do not have the vacuum component. (ii) In the final measurements
Hardy’s proposal utilises a varying strengths of the local oscillator fields,
whereas in the TWC case they are constant. In fact the local oscillators in
Hardy’s scheme are either on or off (the local setting is specified by the
presence or absence of the local auxiliary field). We show that it is the
varying strength of the local oscillators, from setting to setting, which is
the crucial feature enabling violation of local realism in the Hardy setup,
whereas it is not necessary to use initial superposition of a single photon
excitation with vacuum as the initial state of the input mode. Neither one
needs to operate in the fully on/off detection scheme. This implies that the
confirmed Bell nonclassicality in the Hardy-like setup cannot be attributed to
the single-photon state alone, but should rather be considered a consequence
of its interference with the photons from auxiliary local fields. Neither can
it be attributed to the joint state of the single photon excitation and two
separate local oscillator modes, as in the experiment this state is
measurement setting dependent. Despite the failure of the Tan, Walls and
Collett scheme in proving Bell nonclassicality, we show that their scheme can
serve as an entanglement indicator. We reconsider the general problem of
transforming Bell inequalities for optical systems into indicators of mode
entanglement, and find a more efficient entanglement indicator based on a Bell
operator than theirs.

##  I Introduction

The ‘nonlocality of a single photon’, also known as ‘entanglement with
vacuum’, has long been a subject of controversy Tan  _et al._ (1991); Hardy
(1994); Vaidman (1995); Hessmo  _et al._ (2004); van Enk (2005); Dunningham
and Vedral (2007); Heaney  _et al._ (2011); Jones and Wiseman (2011); Brask
_et al._ (2013); Morin  _et al._ (2013); Fuwa  _et al._ (2015); Lee  _et al._
(2017). In its basic form, the problem concerns the nature of the state
$\frac{1}{\sqrt{2}}(|01\rangle_{b_{1}b_{2}}+|10\rangle_{b_{1}b_{2}})$,
obtained by casting a photon on a balanced beamsplitter. Here, the notation
$|10\rangle_{b_{1}b_{2}}$ indicates the presence of a photon in mode $b_{1}$
and its absence in $b_{2}$. Although the resulting state can be considered as
mode entangled Demkowicz-Dobrzański  _et al._ (2015), it can also be
interpreted as a mere superposition of mode excitations – the photon being
either here or there. This point of view is supported by writing down the
state as $\hat{a}^{\dagger}\ket{00}_{b_{1}b_{2}}$, where
$\hat{a}^{\dagger}=\frac{1}{\sqrt{2}}(\hat{b}^{\dagger}_{1}+\hat{b}^{\dagger}_{2})$,
and $\hat{b}^{\dagger}_{j}$ are photon creation operators for modes $b_{j}.$

Thus, one can question whether it can be used to demonstrate Bell
nonclassicality – both on its own, or with some additional resources like
local auxiliary optical fields.

Many experiments, some feasible, some gedanken, have been proposed to address
this fundamental problem. Let us briefly present their three major types.

The first one originates from a paper by Tan, Walls and Collett (TWC) Tan  _et
al._ (1991), in which homodyne-based coincidence intensity measurements with
weak coherent light as local auxiliary fields were used to violate the Bell-
like inequality of Reid and Walls (1986).

However, the Bell-like inequality in Reid and Walls (1986) does not rest
entirely on Bell’s assumptions. Because of that, one can question whether the
TWC scheme can be used to violate local realism. For instance, Santos Santos
(1992) provided an ad hoc local realistic model for the correlation functions
considered by TWC. Recently, this line of critique has been reinforced in Das
_et al._ (2021) by presenting a model that reproduces all detection events in
the TWC experimental proposal for the range of local oscillator strengths for
which the paper Tan  _et al._ (1991) reported ‘nonlocality’ of the single-
photon state. In the section III B of this work we show that optical Bell
inequalities Żukowski  _et al._ (2016), which must hold for any local
realistic description, are not violated for the TWC setup. Thus this case is
closed. Still, TWC correlations are interesting in themselves, and variants of
this scheme have been realized experimentally in Hessmo  _et al._ (2004) and
Fuwa  _et al._ (2015). In section V we show how to use the CHSH-like
inequality of Reid and Walls (1986) as an entanglement witness.

Another idea was put forward by Hardy Hardy (1994). He considered states
$q\ket{00}_{b_{1}b_{2}}+\frac{r}{\sqrt{2}}(|01\rangle_{b_{1}b_{2}}+|10\rangle_{b_{1}b_{2}})$,
with $q\neq 0$ which can be produced by sending a superposition of a vacuum
state and a single photon one on a 50-50 beamsplitter. Hardy investigated four
mutually complementary experimental situations and _proved_ that their joint
local realistic description would contradict the quantum predictions. His
setup relied on tunable amplitudes of the auxiliary coherent fields (local
oscillators). Its modification with mixed-state auxiliary states was proposed
in Dunningham and Vedral (2007), and a generalization to multimode initial
states in Heaney  _et al._ (2011).

Banaszek and Wódkiewicz suggested a measurement setup implementing the
displacement of the input field in the optical phase space for the single
photon superposition ($q=0$) Banaszek and Wódkiewicz (1999). The settings of
the Bell experiment they considered were defined by turning the displacements
on or off. Interestingly, their proof of a violation of the Clauser-Horne (CH)
inequality Clauser and Horne (1974) in their setup relied on the no-count
events. A combination of this approach with homodyne measurements was
investigated in Lee  _et al._ (2017), while an adaptation to multimode input
states was given in Brask  _et al._ (2013). Optical displacement was also used
in a scheme for a heralded distribution of a single-particle entanglement
Caspar  _et al._ (2020). We shall not analyze the approach of Banaszek and
Wódkiewicz (1999) as, first, their proof of violation of local realism is
impeccable, and second, despite the fact that their scheme involves only all-
optical measurements, the employed technique is essentially different from the
ones of TWC and Hardy.

Finally, van Enk van Enk (2005) observed that the single-photon superposition
can induce an entangled state of two atoms in spatially separated traps. In
such a case violation of local realism by the pair of entangled atoms is easy
to prove, and the entanglement is (in theory) easily detectable (see e.g., the
discussion in Ashhab  _et al._ (2007)). However, this puts us effectively back
to the two-qubit entanglement, and the problem is not interesting anymore
(except from being experimentally challenging). Moreover, note that the atoms
used in such schemes should be treated as auxiliary systems. Therefore, such
an approach will not be discussed here, as we want to discuss situations which
involve only quantum optical fields, passive optics and photon number
resolving (macroscopic) detectors.

In this work we shall concentrate on homodyne measurements which involve in
general arbitrary beamsplitters, and essentially weak local oscillator fields.
This includes not only the setups of TWC and Hardy, but also the intermediate
cases which have not been discussed yet. We start by analyzing the mechanism
behind the occurrence of the spurious nonclassicality, manifested by the
violation of the inequality of Reid and Walls (1986) in the TWC setup. Further
we shall test the ability of inequalities Żukowski  _et al._ (2016) to detect
violations of local realism in situations which are a kind of a hybrid of the
TWC configuration and the Hardy one. That is, we shall show that the
inequalities detect violation of local realism if one admits varying strengths
of the local oscillators for different settings in the Bell experiment. It
turns out that the approach allows to find genuine violations of local realism
for the initial state of the form of TWC (i.e, for $q=0$).

####  I.0.1 Detailed aims and analysis

In order to test the potentially nonclassical character of the correlations
appearing in the schemes of TWC and Hardy, we use the intensity rates of
optical fields as local observables. They were first introduced as a proposal
of normalized Stokes observables in He  _et al._ (2011, 2012) and rediscovered
in the context of Bell’s theorem in Żukowski  _et al._ (2016), further
developed in Refs Żukowski  _et al._ (2017); Ryu  _et al._ (2019).
Essentially, by an intensity rate we mean the ratio of measured intensity in
the given local detector to the total intensity measured in all local
detectors, in a given run of the experiment. In contrast to the CHSH Bell-like
inequalities of Reid and Walls (1986) used by TWC, this approach to analyzing
optical correlations does not lead to spurious violations.

The inequalities for intensities of Ref. Reid and Walls (1986) involve
additional constraints on local hidden variable description of intensities
passing beamsplitters. The additional constraints are a version of no-
enhancement assumption for CH-like inequalities (discussed in Clauser and
Horne (1974)), or a fair-sampling one in the case of CHSH-like inequalities.
Therefore the inequalities of Ref. Reid and Walls (1986) cannot be used to
test Bell nonclassicality, however, as we show further, they have an
interpretation of entanglement indicators. The problematic status of the
additional assumption in the inequalities of Reid and Walls (1986) in the
context of TWC setup has been pointed out by Hardy Hardy (1992) and Santos
Santos (1992). Nevertheless, because of the seemingly ‘innocent’ naturality of
this assumption, the inequalities of Reid and Walls (1986) can be found in
discussions of Bell’s theorem in some textbooks on quantum optics. e.g., Walls
and Milburn (2007). In this work, we show that Bell inequalities based on
rates of intensities Żukowski  _et al._ (2016), which do not invoke any
additional constraints, are not violated. This result is in perfect agreement
with the fact that a recently proposed local hidden variable model Das  _et
al._ (2021) reproduces all the probabilities of events considered ‘nonlocal’
by TWC in Tan  _et al._ (1991). It shows that the rate-based inequalities are
more reliable than the ones of Reid and Walls (1986), as they do not lead to
spurious violations of Bell nonclassicality. Interestingly, this is the first
found example supporting that claim. It complements the previous
investigations in which the rate-based inequalities were always more strongly
violated in case of genuine nonclassicality Żukowski  _et al._ (2016).

In contrast with the above, our analysis of the Hardy-like proposal using the
intensity rates approach indicates the presence of nonclassical correlations.
We have confirmed them in case of the initial state being just a single photon
($q=0$). Thus, the crucial change to the TWC setup introduced by Hardy
consists in the ability to vary the intensities of the coherent local
oscillators by turning them on or off.

We have also investigated a transition between the Hardy-like and TWC setups.
Our numerical calculations show that nonclassical correlations can still be
obtained when the amplitudes of auxiliary coherent fields are non-zero for
both local measurement settings. This means that turning the auxiliary
oscillators off, as proposed by Hardy, is not necessary. However, their
amplitudes must still be different for the alternative local settings of the
Bell-type experiment. This strongly suggests that the violations of local
realism in such an experiment cannot be associated with a unique state. In
other words, the full state
$\ket{\alpha_{1}}_{a_{1}}\frac{1}{\sqrt{2}}(|01\rangle_{b_{1}b_{2}}+|10\rangle_{b_{1}b_{2}})\ket{\alpha_{2}}_{a_{2}}$
includes local oscillators, which need to change from setting to setting. Thus
the experiment involves more than a single state, which is not consistent with
the Bell’s theorem.

This in turn suggests that the TWC-Hardy correlations result from an
interferometric effect based on bosonic indistinguishability between the
single photon in question, and the photons of the auxiliary local coherent
beams. Note that this feature of the investigated setup goes beyond the TWC
and Hardy cases. It is in fact characteristic to any nonclassicality test
relying solely on single-photon superposition, local oscillators, and passive
optical devices, including the scheme of Banaszek and Wódkiewicz Banaszek and
Wódkiewicz (1999). The necessity of introducing auxiliary systems in such
measurement schemes is related to the fact that it is impossible to perform a
projection onto a superposition between vacuum and a single photon using only
passive optical devices. This was already pointed out by Peres Peres (1995) in
a comment to the Hardy’s work.

An alternative interpretation is that the local oscillators of the homodynes
used to detect the violations of local realism induced by the state
$\frac{1}{\sqrt{2}}(|01\rangle_{b_{1}b_{2}}+|10\rangle_{b_{1}b_{2}})$ must be
treated as a part of the local measuring devices. In fact, in such all optical
scenario the local oscillators can be incorporated in the operational
definition of the measurements which are of the POVM class.

####  I.0.2 Controversies around the topic

The topic initiated by the TWC paper stirred up a lot of controversy. For
instance, Santos suggested that the intensity correlations in the Tan-Walls-
Collett scheme can be explained with local hidden variables (LHV) Santos
(1992), and cannot be used to convincingly demonstrate nonclassicality of a
single photon. However, his LHV model was not reproducing the quantum
probabilities of all photon count events. Greenberger, Horne and Zeilinger
argued Greenberger  _et al._ (1995) that Hardy’s setup is not single-particle
in nature, as analogous correlations can be observed in three-photon
interferometric experiments. Arriving to a similar conclusion, Vaidman pointed
out Vaidman (1995) that Hardy’s results are inconsistent with the local-
realistic Bohm’s description of a single-photon Bohm (1952). Hardy counter-
argued to both of these objections Hardy (1995), indicating that his proposal
uses on average less than a single photon and that the Bohm’s model for a
single photon is manifestly nonlocal, therefore no conclusion can be drawn
from it in the discussed context.

We review the above discussion of the proposals and objections in order to
indicate how unclear the subject issue has proved to be and how many
contradictory statements were made.

##  II Experimental Setup

Figure 1:  Most general schematic representation of the experimental setup for
testing single-photon correlational properties, which we consider here. In the
Tan-Walls-Collett scenario we have $q=0$, and
$|\alpha_{1}|=|\alpha_{2}|=const$ for all settings in Bell-like experiment. In
original Hardy’s scenario $q\neq 0$ and $|\alpha_{j}|=0$, or
$\alpha_{j}=\frac{i^{(1-j)}r}{q\sqrt{2}}$. Here we consider also intermediate
cases, including $\alpha_{j}$’s of varying absolute values and beamsplitters
$U_{BS_{j}}$ with transmissivity varying from setting to setting.

Here, we discuss the basic setup of the TWC and Hardy gedankenexperiments and
its variations which were studied in the literature, in greater detail. As
depicted in Figure 1, the experimental configuration consists of three
spatially separated beamsplitters $BS_{j}$ with ${j=0,1,2}$, whose action is
in general described by the unitary transformation:

$U_{BS}(\chi,\theta)=\begin{pmatrix}\cos\chi&e^{-i\theta}\sin\chi\\\
-e^{i\theta}\sin\chi&\cos\chi\end{pmatrix},$ (1)

where $\cos^{2}\chi$ is the transmission coefficient of the beamsplitter and
$\theta$ is the phase acquired by the reflected beam. As a beamsplitter is a
passive optical device this transformation links the photon annihilation
operators of the incoming beams with annihilation operators of the output
beams. We assume that all photons in the experiment have the same
polarization.

An initial state, $q\ket{0}_{s}+r\ket{1}_{s}$, a superposition of vacuum and
single-photon excitation in mode $s$, impinges on a symmetric beamsplitter BS0
(defined by $\chi=\frac{\pi}{4}$ and $\theta=-\frac{\pi}{2}$). Such will be
our notation, $\ket{n}_{m}$ is a Fock state of $n$ photons in (spatial) mode
$m$. If the initial state is a single photon, i.e., $q=0$, then it transforms
to:

$\ket{\psi}_{b_{1},b_{2}}=\frac{1}{\sqrt{2}}\left[\ket{01}_{b_{1},b_{2}}+i\ket{10}_{b_{1},b_{2}}\right],$
(2)

where $b_{1},b_{2}$, are the output modes of beamsplitter $BS_{0}$.

In the TWC scheme the quantum state, given in Eq. (2) is then shared between
two parties Alice and Bob, who perform a homodyne detection. We assume that
the homodyne measurement stations operated by local observers consist of a
$\theta_{j}$-dependent balanced beamsplitter $BS_{j},$ realizing the
transformation $U_{BS}(\frac{\pi}{4},\theta_{j})$, an auxiliary coherent beam
impinging on the remaining input mode of $BS_{j}$, and two photo-detectors
$D_{c_{j}}$ and $D_{d_{j}}$, placed in front of the output modes of $BS_{j}$,
for $j=1,2$, as shown in Figure 1. The amplitudes of the coherent beams
$\ket{\alpha}_{a_{1}}$ and $\ket{\alpha}_{a_{2}}$ are equal in the case of TWC
scheme. For simplicity we assume that $\alpha$ is real. After all that, the
total state at the inputs of the two beamsplitters $BS_{1}$ and $BS_{2}$, is:

$|\Psi(\alpha)\rangle=\frac{1}{\sqrt{2}}\,|\alpha\rangle_{a_{1}}(|01\rangle_{b_{1}b_{2}}+i\,|10\rangle_{b_{1}b_{2}})|\alpha\rangle_{a_{2}}.$
(3)

The photo-detectors $D_{c_{j}}$ and $D_{d_{j}}$ monitor output modes
$\hat{c}_{j}$ and $\hat{d}_{j}$ of $BS_{j}$, for $j=1,2$. These modes are
linked with modes $\hat{a}_{j}$ and $\hat{b}_{j}$ via symmetric beamsplitter
transformation $U_{BS}(\tfrac{\pi}{4},\theta_{j})$, that is we have:

$\displaystyle\hat{c}_{j}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(\hat{a}_{j}+e^{-i\theta_{j}}\,\hat{b}_{j}),$
$\displaystyle\hat{d}_{j}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(-e^{i\theta_{j}}\,\hat{a}_{j}+\hat{b}_{j}).$
(4)

It is assumed that the detectors measure photon numbers. The phases
$\theta_{1}$ and $\theta_{2}$ in the TWC case are tunable, and define the
local settings in the purported Bell experiment. Note that in the original TWC
setup the tunable phases are attached to the auxiliary coherent beams. While
both approaches describe physically equivalent situations, we decided to
attach the phases to the beamsplitters, which are part of the local
measurement apparatuses, in order to emphasize the fact, that in the TWC setup
one deals with _the same initial state_ for all measurement settings in the
Bell experiment. As we will see, this feature no longer holds in
generalisations of the setup.

####  II.0.1 Generalizations

In further discussions we will consider also a generalised scheme with
arbitrary tunable beamsplitters $BS_{1}$ and $BS_{2}$, for which the mode
transformation reads:

$\begin{pmatrix}\hat{c}_{j}\\\
\hat{d}_{j}\end{pmatrix}=U_{BS_{j}}(\chi_{j},\theta_{j})\begin{pmatrix}\hat{a}_{j}\\\
\hat{b}_{j}\end{pmatrix},$ (5)

where the unitary matrix is given in Eq. (1). Such devices have a realization
in the form of Mach-Zehnder interferometers. One can further generalize the
scheme, and assume that the local auxiliary fields may have different
amplitudes for both observers, therefore the overall state at the entry ports
of the final beamsplitters takes a more general form:

$|\Phi(\alpha_{1},\alpha_{2})\rangle=\frac{1}{\sqrt{2}}\,|\alpha_{1}\rangle_{a_{1}}(|01\rangle_{b_{1}b_{2}}+i\,|10\rangle_{b_{1}b_{2}})|\alpha_{2}\rangle_{a_{2}}.$
(6)

In this generalized case the dependence on the _real_ parameters $\alpha_{1}$
and $\alpha_{2}$ is local with respect to subsystems defined by modes
$\\{\hat{a}_{1},\hat{b}_{1}\\}$ and $\\{\hat{a}_{2},\hat{b}_{2}\\}$, so they
can be also used as local settings together with local phases $\theta_{1}$ and
$\theta_{2}$. However, in contrast to (3), the initial state is now explicitly
setting-dependent.

One can further generalize this to initial states of mode $\hat{s}$ being
superpositions of vacuum and single photon states (i.e $q\neq 0$). In such
case the state behind the first beamsplitter is given by:

$\ket{\psi^{\prime}}_{b_{1},b_{2}}=q\ket{00}_{b_{1},b_{2}}+\frac{r}{\sqrt{2}}\left[\ket{01}_{b_{1},b_{2}}+i\ket{10}_{b_{1},b_{2}}\right].$
(7)

##  III Intensities vs. intensity rates in analysis of TWC correlations

###  III.1 Intensity rate measurements allow loophole-free Bell inequalities.

For the analysis of nonclassicality the authors of Tan  _et al._ (1991) used
Bell-like inequalities derived in Reid and Walls (1986). The inequalities
involved correlation functions of intensities at pairs of spatially separated
detectors, one in Alice’s station one in Bob’s. They are applicable for the
following models:

$E_{T}(\theta_{1},\theta_{2})=\frac{\int
d\lambda\rho(\lambda)\prod_{j=1,2}\big{(}I_{c_{j}}(\theta_{j},\lambda)-I_{d_{j}}(\theta_{j},\lambda)\big{)}}{\int
d\lambda\rho(\lambda)I_{1}(\lambda)I_{2}(\lambda)}.$ (8)

In the above formula $\lambda$ symbolizes a hidden variable, $\rho(\lambda)$
is its distribution, $I_{x_{j}}(\theta_{j},\lambda)$ is the hidden variable
model of the local intensity (registered in a detector placed in front of mode
$x=c,d$,) at station $j=1,2$, i.e. respectively A and B of Fig. 1, for the
local (phase) setting $\theta_{j}$. Finally $I_{j}(\lambda)$ is the total
intensity at station $j=1,2$. However, the inequalities of Reid and Walls
(1986) cannot be used to refute altogether the possiblity of local realistic
description of an experiment, as in addition to local realism and ‘free will’,
they rely on an additional assumption (which holds in classical optics, but
not in e.g. stochastic electrodynamics Boyer (2019)). This assumption states
that the total local intensity of light $I_{j}(\lambda)$, for a given values
of a hidden variable $\lambda$ does not depend on the local setting of the
measuring device. Namely, the authors of Reid and Walls (1986) assumed that:

$I_{j}(\lambda)=I_{c_{j}}(\theta_{j},\lambda)+I_{d_{j}}(\theta_{j},\lambda),$
(9)

is independent of the local value of $\theta_{j}$. A possible violation of
inequalities of Reid and Walls (1986) indicated a failure of either local
realism, or ‘free will’, or of the assumption (9). However, in an earlier
paper Das  _et al._ (2021) we showed an explicit hidden variable model which
reproduces exactly the quantum predictions for the TWC experiment for the
whole range of $\alpha$ for which inequalities of Reid and Walls (1986) are
shown to be violated in Tan  _et al._ (1991). Therefore, the violation
reported in Tan  _et al._ (1991) cannot be attributed to Bell’s
nonclassicality of the state under consideration.

As mentioned above, the inequalities of Tan  _et al._ (1991) lead to
drastically wrong conclusions about the non-existence of local hidden variable
description of TWC experiment. To study this type of questions one must
therefore use an alternative approach, which uses Bell’s inequalities for
field intensities, since they rest only on the assumptions of local realism
and ‘free will’. The approach given in Żukowski  _et al._ (2016), allows to
construct a class of Bell’s inequalities having this trait. The Bell’s
inequalities of Żukowski  _et al._ (2016) involve correlations of functions of
the intensities that we will refer to as intensity _rates_. They are defined
as the ratios of the measured intensities in a given local mode to the total
intensity measured across all local modes:

$R_{x_{j}}(\theta_{j},\lambda)=\frac{I_{x_{j}}(\theta_{j},\lambda)}{I_{c_{j}}(\theta_{j},\lambda)+I_{d_{j}}(\theta_{j},\lambda)},$
(10)

where we assume that $R_{x_{j}}(\theta_{j},\lambda)$ _is assigned the value of
$0$ whenever the total intensity in the denominator is equal to $0$_.

Quantum optical observables that describe the rates can be defined as follows
Żukowski  _et al._ (2016). If one assumes, like in Tan  _et al._ (1991), that
the intensity observables of beams are modeled by photon number operators for
the given mode, the rate operator is defined by:

$\hat{\mathrm{R}}_{x_{j}}=\hat{\Pi}_{c_{j}d_{j}}\frac{\hat{n}_{x_{j}}}{\hat{n}_{c_{j}}+\hat{n}_{d_{j}}}\hat{\Pi}_{c_{j}d_{j}},$
(11)

where $\hat{n}_{x_{j}}=\hat{x}^{\dagger}_{j}\hat{x}_{j}$, represents the
photon number operator in the $x_{j}$-th mode of the optical field registered
in the detector $D_{x_{j}}$. Here, $\hat{x}^{\dagger}_{j},~{}\hat{x}_{j}$ are
the photon creation and annihilation operators of the local output mode
$\hat{x}_{j}$, $x=c,d$, of the beamsplitter for observer $j$. The operators
$\hat{\Pi}_{c_{j}d_{j}}=\mathbb{I}_{c_{j}d_{j}}-\outerproduct{\Omega_{c_{j}d_{j}}}{\Omega_{c_{j}d_{j}}},~{}j=1,2$,
are projectors onto the subspace spanned by the Fock basis of modes
$\hat{c}_{j},\hat{d}_{j}$, devoid of the vacuum state. It is worth mentioning
that the two mode vacuum $\ket{\Omega_{c_{j}d_{j}}}$ and the operators
$\hat{\Pi}_{c_{j}d_{j}}$ are invariant under the action of the local mode
transformations given in Eqs. (1) and (5). Since the operators
$\hat{\mathrm{R}}_{x_{j}}$ and
$\hat{\Pi}_{c_{j}d_{j}}\frac{1}{\hat{n}_{c_{j}}+\hat{n}_{d_{j}}}\hat{\Pi}_{c_{j}d_{j}}$
commute with each other, depending on needs one can put the rate operator also
as follows:

$\displaystyle\hat{\mathrm{R}}_{x_{j}}=\hat{n}_{x_{j}}\frac{1}{\hat{n}_{tot_{j}}}\hat{\Pi}_{c_{j}d_{j}}=\hat{\Pi}_{c_{j}d_{j}}\frac{1}{\hat{n}_{tot_{j}}}\hat{n}_{x_{j}}$
$\displaystyle=\hat{\Pi}_{c_{j}d_{j}}\frac{1}{\sqrt{\hat{n}_{tot_{j}}}}\hat{n}_{x_{j}}\frac{1}{\sqrt{\hat{n}_{tot_{j}}}}\hat{\Pi}_{c_{j}d_{j}},$
(12)

where $\hat{n}_{tot_{j}}=\hat{n}_{c_{j}}+\hat{n}_{d_{j}}$.

###  III.2 Homodyne detection with intensity rates and Bell CHSH inequality
for rates

The differences of intensity rates (10), when averaged over the probability
distribution of the LHV, can be used to construct the following correlation
functions:

$E_{R}(\theta_{1},\theta_{2})=\left\langle\prod_{j=1}^{2}\left(R_{c_{j}}(\theta_{j},\lambda)-R_{d_{j}}(\theta_{j},\lambda)\right)\right\rangle_{HV},$
(13)

depending on the value of the local settings defined by the variable
$\theta_{j}$ for $j=1,2$. This correlation function satisfies the CHSH
inequality for rates:

$|E_{R}(\theta_{1},\theta_{2})+E_{R}(\theta^{\prime}_{1},\theta_{2})+E_{R}(\theta_{1},\theta^{\prime}_{2})-E_{R}(\theta^{\prime}_{1},\theta^{\prime}_{2})|\leq
2.$ (14)

Note that the above inequality holds regardless of whether the condition (9)
is satisfied or not, which assures the loophole-free character of the Bell
test based on inequality (14). In order to check whether it can be violated in
the TWC setup, we introduce quantum implementation of the correlation function
(13) based on the theory of intensity measurements for optical fields with
rate operators outlined in Section III.1. It can be applied to write
explicitly the correct quantum expectation value of the homodyne photocurrent
difference measured by Alice and Bob:

$\displaystyle\hat{H}_{j}(\theta_{j})=\hat{R}_{c_{j}}-\hat{R}_{d_{j}}=\hat{\Pi}_{c_{j}d_{j}}\frac{\hat{n}_{c_{j}}-\hat{n}_{d_{j}}}{\hat{n}_{c_{j}}+\hat{n}_{d_{j}}}\hat{\Pi}_{c_{j}d_{j}},$
(15)

with $j=1,2$, and the implicit $\theta_{j}$-dependence is specified via mode
transformations (II). With abuse of notation, we also call
$E_{R}(\theta_{1},\,\theta_{2})$, the correlation coefficient of the joint
homodyne measurement on the initial state $\ket{\Psi(\alpha)}$ (3), performed
by both Alice and Bob:

$\displaystyle E_{R}(\theta_{1},\theta_{2})$ $\displaystyle=$
$\displaystyle\bra{\Psi(\alpha)}\hat{H}_{1}(\theta_{1})\hat{H}_{2}(\theta_{2})\ket{\Psi(\alpha)}$
(16) $\displaystyle=$ $\displaystyle
A_{R}\left(\alpha\right)\sin(\theta_{1}-\theta_{2}),$

where $\theta_{1}$ and $\theta_{2}$ are the tunable phases of the local
beamsplitters. The amplitude $A_{R}(\alpha)$ of the correlation function reads
explicitly:

$A_{R}(\alpha)=\frac{e^{-2\alpha^{2}}(e^{\alpha^{2}}-1)^{2}}{\alpha^{2}}.$
(17)

The correlation function (16) depends on the _fixed_ initial state parameter
$\alpha$ denoting the amplitude of the auxiliary local field. We can express
the correlation function (16) as arising from a POVM measurement performed
solely on the initial single-photon superposition (2) in modes $b_{1}$ and
$b_{2}$:

$\displaystyle E_{R}(\theta_{1},\theta_{2})$ $\displaystyle=$
$\displaystyle\bra{\psi}\hat{\mathcal{M}}_{b_{1}}(\alpha,\theta_{1})\hat{\mathcal{M}}_{b_{2}}(\alpha,\theta_{2})\ket{\psi}.$
(18)

The POVM operators $\hat{\mathcal{M}}_{b_{j}}(\alpha,\theta_{j})$ are
explicitly constructed in Appendix A. We note that a similar analysis of the
homodyne detection in terms of POVM operators has been already performed Tyc
and Sanders (2004), here we extend such an approach to homodyne detection with
intensity rate operators (15).

Our aim is to check whether the following CHSH inequality for rates,
constructed by substituting (16) into (14), is violated by the quantum state
given in (3):

$\displaystyle
A_{R}(\alpha)|\sin(\theta_{1}-\theta_{2})+\sin(\theta^{\prime}_{1}-\theta_{2})+\sin(\theta_{1}-\theta_{2}^{\prime})$
$\displaystyle-\sin(\theta^{\prime}_{1}-\theta^{\prime}_{2})|\leq 2.$ (19)

The term depending on the phases of the auxiliary coherent states can reach
the maximal value of $2\sqrt{2}$, hence (III.2) will be violated if
$A_{R}(\alpha)>\frac{\sqrt{2}}{2}$. The amplitude $A_{R}(\alpha)$ has been
plotted in the Figure 2, as the solid red line. The blue dotted straight line
depicts the value of $\frac{\sqrt{2}}{2}$. The position of the solid red line
below the dotted blue straight line, for all values of $\alpha$ shows that the
correlation function (16) does not lead to any violation of the loophole-free
CHSH inequality for rates (III.2).

Figure 2:  Comparison of the amplitudes of correlation functions
$E_{R}(\theta_{1},\theta_{2})$ and $E_{T}(\theta_{1},\theta_{2})$ for
different amplitudes $\alpha$ of the auxiliary coherent field. The solid red
line corresponds to the amplitude $A_{R}(\alpha)$ of
$E_{R}(\theta_{1},\theta_{2})$, and the dashed green line is for
$A_{T}(\alpha)$. For weak coherent light the amplitude $A_{T}(\alpha)$ exceeds
the threshold needed to violate the CHSH inequality.

We compare our results with the ones given in Tan  _et al._ (1991), where the
following Bell CHSH-like inequality has been considered:

$|E_{T}(\theta_{1},\,\theta_{2})+E_{T}(\theta^{\prime}_{1},\,\theta_{2})+E_{T}(\theta_{1},\,\theta^{\prime}_{2})-E_{T}(\theta^{\prime}_{1},\,\theta^{\prime}_{2})|\leq
2.$ (20)

The correlation functions $E_{T}(\theta_{1},\,\theta_{2})$ have been already
defined in Eq. (8). The quantum implementation of the correlators
$E_{T}(\theta_{1},\,\theta_{2})$ used in Tan  _et al._ (1991) reads as
follows:

$\displaystyle E_{T}(\theta_{1},\theta_{2})$ $\displaystyle=$
$\displaystyle\frac{\langle\Psi(\alpha)|(\hat{n}_{c_{1}}-\hat{n}_{d_{1}})(\hat{n}_{c_{2}}-\hat{n}_{d_{2}})|\Psi(\alpha)\rangle}{\langle\Psi(\alpha)|(\hat{n}_{c_{1}}+\hat{n}_{d_{1}})(\hat{n}_{c_{2}}+\hat{n}_{d_{2}})|\Psi(\alpha)\rangle}$
(21) $\displaystyle=$
$\displaystyle-\frac{\langle\Psi(\alpha)|(\hat{a}_{1}\hat{b}_{1}^{\dagger}-\hat{a}_{1}^{\dagger}\hat{b}_{1})(\hat{a}_{2}\hat{b}_{2}^{\dagger}-\hat{a}_{2}^{\dagger}\hat{b}_{2})|\Psi(\alpha)\rangle}{\langle\Psi(\alpha)|(\hat{a}_{1}^{\dagger}\hat{a}_{1}+\hat{b}_{1}^{\dagger}\hat{b}_{1})(\hat{a}_{2}^{\dagger}\hat{a}_{2}+\hat{b}_{2}^{\dagger}\hat{b}_{2})|\Psi(\alpha)\rangle}$
$\displaystyle=$ $\displaystyle A_{T}(\alpha)\sin(\theta_{1}-\theta_{2}),$

and the amplitude $A_{T}(\alpha)$ reads explicitly:

$A_{T}(\alpha)=\frac{1}{1+\alpha^{2}}.$ (22)

The coefficient $A_{T}(\alpha)$ as a function of the amplitude $\alpha$ of the
auxiliary coherent field has been plotted in Figure 2, as the dashed green
line. We observe a violation of the inequality (20) for the range
$0<\alpha^{2}<0.414$. This indication of Bell’s nonclassicality is apparent,
since in a recent work Das  _et al._ (2021) we have shown an LHV model
simulating the discussed correlations. The apparent nonclassicality occurs due
to the loophole connected with the additional assumption (9). Therefore the
inequality (20) is not suitable for Bell-type considerations concerning the
setup. Nevertheless it can be utilised as an entanglement witness, which is
discussed in more details in Section V.

Figure 3:  Phase distribution in the Pegg-Barnett formalism. The stronger the
coherent light, the narrower the peak of its phase distribution.

###  III.3 Classical approximation of the coherent light and the role of phase
uncertainty

In this subsection, we discuss a side but otherwise very interesting issue of
a potential trap that may occur when using a classical approximation for the
coherent light. Note that all the previous discussion has been done in an
approximation-free way. Below we will see that an interesting paradox occurs
in our intensity rate-based approach to the TWC setup if one treats the
auxiliary coherent field as a classical beam of light pulses. Such transition
can be obtained under the parametric approximation, that substitutes the
annihilation operator of the coherent light with a $c-$number, i.e., its
corresponding eigenvalue when acting on the coherent state, namely
$a_{j}\rightarrow\alpha_{j}$, and similar action when considering the creation
operator. Utilizing the same mode transformations as in (21) we obtain the
following form of the approximated correlation function (16):

$\displaystyle E^{C}_{R}(\theta_{1},\,\theta_{2})$ (23)
$\displaystyle=-{}_{b_{1},b_{2}}\langle\psi|\frac{(\alpha_{1}\hat{b}_{1}^{\dagger}-\alpha_{1}^{*}\hat{b}_{1})(\alpha_{2}\hat{b}_{2}^{\dagger}-\alpha_{2}^{*}\hat{b}_{2})}{(|\alpha_{1}|^{2}+\hat{b}_{1}^{\dagger}\hat{b}_{1})(|\alpha_{2}|^{2}+\hat{b}_{2}^{\dagger}\hat{b}_{2})}|\psi\rangle_{b_{1},b_{2}}$
$\displaystyle=\frac{1}{1+\alpha^{2}}\sin{(\theta_{1}-\theta_{2})}=A_{T}(\alpha)\sin(\theta_{1}-\theta_{2})$
.

This leads to a paradoxical situation in which the intensity rate-based
correlation function for a single photon and a classical auxiliary light
pulses seems to reveal correlations stronger than before the classical
approximation. One can therefore ask, what essential feature of coherent
states is lost in this approximation.

We will now show that the apparent increase of the strength of correlations
can be traced back to the phase uncertainty of the coherent states. There are
different approaches to defining phase in quantum optics and we use the one
introduced by Pegg and Barnett Pegg (1996). According to it, in the limit of
$\alpha\rightarrow 0$ the phase of a coherent state $|\alpha\rangle$ is
distributed as (see also Figure 3):

$P(\Delta\theta)=\frac{1+2\alpha e^{-\alpha^{2}}\cos{\Delta\theta}}{2\pi}.$
(24)

If one takes that into account, the correlation function for weak coherent
light taken under the parametric approximation reads as follows when averaged
over the phase distribution:

$\displaystyle\langle
E^{C}_{R}(\theta_{1},\theta_{2})\rangle_{\Delta\theta_{1}\Delta\theta_{2}}$
$\displaystyle=$
$\displaystyle\frac{1}{1+\alpha^{2}}\,\,\iint\limits_{[-\pi,\pi]\times[-\pi,\pi]}P(\Delta\theta_{1})P(\Delta\theta_{2})\sin{(\theta_{1}+\Delta\theta_{1}-\theta_{2}-\Delta\theta_{2})}d\Delta\theta_{1}d\Delta\theta_{2}$
(25) $\displaystyle=$
$\displaystyle\frac{\alpha^{2}e^{-2\alpha^{2}}}{1+\alpha^{2}}\sin{(\theta_{1}-\theta_{2})}=A^{C}_{R}(\alpha)\sin(\theta_{1}-\theta_{2}),$

where we introduced another amplitude of the correlation function:

$A_{R}^{C}(\alpha)=\frac{\alpha^{2}e^{-2\alpha^{2}}}{1+\alpha^{2}}.$ (26)

After the phase averaging the amplitude of the approximated correlation
function $A^{C}_{R}(\alpha)$ is strictly lower than $A_{R}(\alpha)$ and tends
to it in the limit of $\alpha\rightarrow 0$. All this discussion shows that
one has to be careful when applying the parametric approximation, as ignorance
of the phase uncertainty can lead to apparent increase in the strength of
correlations.

##  IV Violation of Clauser-Horne inequality based on intensity rates

In the previous section we have shown directly that the Bell-CHSH inequality
(14) based on intensity rates (10) is not violated in the TWC setup. This
result is concurrent with the existence of an LHV model for the TWC scenario
proposed by us in a recent work Das  _et al._ (2021). The crucial feature of
the TWC experiment is the fact that the local oscillators have constant
amplitudes. Hence, in order to search for a loophole-free Bell violation
within the general single-photon setup (Figure 1) we need to relax this
assumption, as it was done by Hardy Hardy (1994) in his version of the
_single-photon nonlocality_ test. Contrary to the original Hardy case we do
not use the single-photon-with-vacuum superposition as the initial state and
keep just the single photon one. On the other hand we introduce one more
modification with respect to both TWC and Hardy versions, namely the local
beamsplitters $BS_{1}$ and $BS_{2}$ in Figure 1 have now tunable
transmissivities, and are specified by formula (5). In his work Hardy (1994)
Hardy used Clauser-Horne (CH) inequality Clauser and Horne (1974), as it was
better suited to describe his _possibilistic_ paradox (meaning that the
contradiction occurs with a non-zero probability).

###  IV.1 Non-equivalence of CH and CHSH inequalities for optical fields

The CH inequality in its most general form reads as follows Clauser and Horne
(1974):

$\displaystyle-1\leq
P(A,B)+P(A,B^{\prime})+P(A^{\prime},B)-P(A^{\prime},B^{\prime})$
$\displaystyle-P(A)-P(B)\leq 0,$ (27)

where $A,A^{\prime}$ denote _fixed_ outcomes of arbitrary Alice’s
measurements, and $B,B^{\prime}$ refer to Bob’s ones. In the case of
dichotomic outcomes, namely $A,A^{\prime},B,B^{\prime}=\pm 1$, the CH
inequality (IV.1) can be written in four versions, each corresponding to a
definite choice of $\pm 1$ values for all the four outcomes
$A,A^{\prime},B,B^{\prime}$. Interestingly _each_ of these versions is
equivalent to the CHSH inequality. This equivalence rests on the following
property of dichotomic measurements:
$P(-1|\theta,\lambda)+P(+1|\theta,\lambda)=1.$

The equivalence persists even in the case of imperfect detection if one uses
the Garg-Mermin trick Garg and Mermin (1987). The trick simply leaves them
with dichotomic outcomes, despite possible three results, detection in channel
+1, in channel -1, or no detection at all. Simply the idea in Ref. Garg and
Mermin (1987) is to ascribe value $-1$ also for the non-detection event.

Note that for qubits the upper bound is often thought to be the CH inequality,
as it can be elegantly derived using geometric concepts Santos (1986). This
can be done by noticing that $S(A,B)=P(A)+P(B)-P(A,B)$ is non-negative,
$S(A,A)=0$, and it satisfies the triangle inequality. The CH inequality is
then the quadrangle inequality.

Any Bell inequality for two, or more qubits can be rewritten into an
inequality for intensity rates of fields, via a replacement:

$P(n_{x_{i}}=1|\theta_{i},\lambda)\rightarrow R_{x_{i}}(\theta_{i}|\lambda),$
(28)

where $P(n_{x_{i}}=1|\theta,\lambda)$ is an LHV probability for the qubit to
be registered at $x_{i}$, while the rates $R_{x_{i}}$ are defined in Eq. (10).
For a detailed presentation and justification of this method see Żukowski  _et
al._ (2016); Ryu  _et al._ (2019). In the case of the experimental setup
discussed throughout this paper the CH inequality for rates can be written in
four different versions, depending which pair of output modes we choose from
the set $\\{(c_{1},c_{2}),(c_{1},d_{2}),(d_{1},c_{2}),(d_{1},d_{2})\\}$.
Choosing the pair $(d_{1},d_{2})$, which coincides with the choice done in
Hardy’s work Hardy (1994), we obtain the following form:

$\displaystyle-1\leq\big{\langle}R_{d_{1}}R_{d_{2}}+R_{d_{1}}R^{\prime}_{d_{2}}+R^{\prime}_{d_{1}}R_{d_{2}}-R^{\prime}_{d_{1}}R^{\prime}_{d_{2}}$
$\displaystyle-R_{d_{1}}-R_{d_{2}}\rangle_{HV}\leq 0,$ (29)

in which the unprimed rates correspond to first setting, and the primed ones
to the second.

However, it turns out that the CH and CHSH inequalities for rates obtained by
application of the procedure (28) are no longer equivalent. As a particle can
hit only one detector we have $\sum_{x_{i}}P(n_{x_{i}}=1|\theta,\lambda)=1$.
The method (28) works because rates, just like probabilities, have values
between zero and one. However, rates of mutually exclusive events, do not have
to add up to $1$. We have $\sum_{x_{i}}R_{x_{i}}(\theta,\lambda)=1$ or $0.$
This prohibits one to show that the CHSH inequality (14) for rates is
equivalent to the CH one for rates (IV.1). Clearly the same conclusion holds
for other choices of pairs of output modes for the CH inequality for rates.

####  IV.1.1 The surprising case of CH inequalities for rates

In the case of dichotomic outcomes the lower and the upper bound of the CH
inequality (IV.1) are equivalent. Namely by a replacement of $P(A)$ by
$1-P(\tilde{A})$, where $\tilde{A}$ is an event opposite to $A$, i.e.
$P(A)=1-P(\tilde{A})$, and subsequent replacements of the form
$P(A,B^{(^{\prime})})=P(B^{(^{\prime})})-P(\tilde{A},B^{(^{\prime})})$, one
can transform the upper bound of the inequality (IV.1) into an inequality of
the form of the lower bound, for events $\\{\tilde{A},A^{\prime}\\}$ for Alice
and $\\{B,B^{\prime}\\}$ for Bob.

However by performing an analogous replacement of $R_{d_{1}}$ by
$R_{tot_{1}}-R_{c_{1}}$, where $R_{tot_{1}}=R_{c_{1}}+R_{d_{1}}$, which
follows the rule (28), utilising the upper bound and transforming it to the
lower bound by change of sign of both sides we obtain the following form of
the CH inequality for rates (IV.1):

$\displaystyle\big{\langle}(R_{tot_{1}}-1)(R_{d_{2}}+R^{\prime}_{d_{2}})-R_{tot_{1}}\rangle_{HV}$
$\displaystyle\leq\big{\langle}R_{c_{1}}R_{d_{2}}+R_{c_{1}}R^{\prime}_{d_{2}}+R^{\prime}_{d_{1}}R^{\prime}_{d_{2}}-R^{\prime}_{d_{1}}R_{d_{2}}$
$\displaystyle-R_{c_{1}}-R^{\prime}_{d_{2}}\rangle_{HV}.$ (30)

The above inequality is also a CH inequality for rates, where in contrast to
the original inequality in (IV.1) the measurement of Alice for the first
setting is now performed in mode $c_{1}$ and the role of primed and unprimed
settings for Bob’s measurement are swapped. A glance at the lower bound shows
that for $R_{tot_{1}}=0$ the algebraic expression which is averaged there,
reads
$(R_{tot_{1}}-1)(R_{d_{2}}+R^{\prime}_{d_{2}})-R_{tot_{1}}=-R_{d_{2}}-R^{\prime}_{d_{2}}$
and this can reach even the value of $-2$.

Thus the two sides of the CH inequality lead, via the procedure (28), to two
different CH inequalities for rates. This explains the phenomenon which we
shall see further down. In our analysis we get a violation of only the lower
bound of the CH inequality for rates (IV.1). Surprisingly this inequality is a
stronger bound on local realism in the case of rates, than the upper bound.

###  IV.2 CH violation in the extended setup

As already mentioned, in this section we consider measurement settings that
generalize the proposed measurement setup in Tan  _et al._ (1991). Now a
single photon ($q=0$ in our initial state), impinges on a balanced
beamsplitter, $BS_{0}$ and then the output modes interact with auxiliary
coherent beams $\ket{\alpha_{1}}$ and $\ket{\alpha_{2}}$ through additional
two $SU(2)$ beamsplitters $U_{BS_{1}}(\chi_{1},\theta_{1})$ and
$U_{BS_{2}}(\chi_{2},\theta_{2})$, given by the unitary (1). We assume that
the amplitudes of the local auxiliary fields $\alpha_{1}$ and $\alpha_{2}$ are
real. Following the idea of Hardy setup we treat amplitudes of local fields as
a part of measurement settings as well, therefore local settings are specified
by vectors:

$\displaystyle\vec{v}_{1}$ $\displaystyle=$
$\displaystyle(\chi_{1},\alpha_{1},\theta_{1}),$ $\displaystyle\vec{v}_{2}$
$\displaystyle=$ $\displaystyle(\chi_{2},\alpha_{2},\theta_{2}).$ (31)

The CH inequality for rates specified in the previous section (IV.1) can be
expressed in the following convenient form:

$\displaystyle-1\leq$ $\displaystyle
K(\vec{v}_{1},\vec{v}_{2})+K(\vec{v}_{1}^{\prime},\vec{v}_{2})+K(\vec{v}_{1},\vec{v}^{\prime}_{2})-K(\vec{v}^{\prime}_{1},\vec{v}^{\prime}_{2})$
(32) $\displaystyle-S_{1}(\vec{v}_{1})-S_{2}(\vec{v}_{2})\leq 0.$

The correlators $K(\vec{v}_{1},\vec{v}_{2})$ and the local averages
$S_{j}(\vec{v}_{j})$ are defined as follows:

$\displaystyle K(\vec{v}_{1},\vec{v}_{2})=\left\langle
R_{d_{1}}(\vec{v}_{1})R_{d_{2}}(\vec{v}_{2})\right\rangle_{HV},$
$\displaystyle S_{j}(\vec{v}_{j})=\left\langle
R_{d_{j}}(\vec{v}_{j})\right\rangle_{HV},$ (33)

where the rates $R_{d_{j}}(\vec{v}_{j})$ are defined as in (10), with the only
difference that now the hidden local intensities
$I_{d_{j}}(\vec{v}_{j},\lambda)$ explicitly depend on the new local settings
$\vec{v}_{j}$.

To study the Bell nonclassicality in this extended setup, we replace in (32)
the quantum-mechanical expressions for the two-mode correlators and the local
terms:

$\displaystyle K(\vec{v}_{1},\vec{v}_{2})$
$\displaystyle=\bra{\Phi(\alpha_{1},\alpha_{2})}\hat{R}_{d_{1}}(\chi_{1},\theta_{1})\hat{R}_{d_{2}}(\chi_{2},\theta_{2})\ket{\Phi(\alpha_{1},\alpha_{2})},$
$\displaystyle
S_{j}(\vec{v}_{j})=\bra{\Phi(\alpha_{1},\alpha_{2})}\hat{R}_{d_{j}}(\chi_{j},\theta_{j})\ket{\Phi(\alpha_{1},\alpha_{2})},$
(34)

where $\ket{\Phi(\alpha_{1},\alpha_{2})}$ is a generalised initial state
specified in (6). Note that we encounter here a peculiar feature, namely that
the correlation functions and local averages depend on the settings via
parameters of the initial state $\Phi$. If we would like to remove this
feature of the setup we have to use POVM operators, and calculate the
expressions (IV.2) as follows:

$\displaystyle
K(\vec{v}_{1},\vec{v}_{2})=\bra{\psi}\hat{\mathcal{M}}_{b_{1}}(\vec{v}_{1})\hat{\mathcal{M}}_{b_{2}}(\vec{v}_{2})\ket{\psi},$
$\displaystyle
S_{j}(\vec{v}_{j})=\bra{\psi}\hat{\mathcal{M}}_{b_{j}}(\vec{v}_{j})\ket{\psi},$
(35)

in which $\ket{\psi}$ denotes the initial single photon superposition (2), and
the POVM operators $\hat{\mathcal{M}}_{b_{j}}(\vec{v}_{j})$ are explicitly
constructed in Appendix B.

We choose to detect the transmitted beam $D_{d_{j}},~{}~{}j=1,2$, however we
stress that any other choice of the detectors is possible and equivalent
conclusions can be addressed. The terms read as follows:

$\begin{split}K(\vec{v}_{1},\vec{v}_{2})=\frac{e^{-\alpha_{1}^{2}}e^{-\alpha_{2}^{2}}}{2}&\Bigg{[}\Bigg{(}\frac{(e^{\alpha_{2}^{2}}-1)(1+e^{\alpha_{1}^{2}}(\alpha_{1}^{2}-1))}{\alpha_{1}^{2}}+\frac{(e^{\alpha_{1}^{2}}-1)(1+e^{\alpha_{2}^{2}}(\alpha_{2}^{2}-1))}{\alpha_{2}^{2}}\Bigg{)}\sin^{2}\chi_{1}\sin^{2}\chi_{2}\\\
&+\frac{(e^{\alpha_{1}^{2}}-1)(e^{\alpha_{2}^{2}}-1)}{\alpha_{2}^{2}}\sin^{2}\chi_{1}\cos^{2}\chi_{2}+\frac{(e^{\alpha_{1}^{2}}-1)(e^{\alpha_{2}^{2}}-1)}{|\alpha_{1}|^{2}}\cos^{2}\chi_{1}\sin^{2}\chi_{2}\\\
&+\frac{(e^{\alpha_{1}^{2}}-1)(e^{\alpha_{2}^{2}}-1)}{2\alpha_{1}\alpha_{2}}\sin
2\chi_{1}\sin 2\chi_{2}\sin(\theta_{1}-\theta_{2})\Bigg{]},\end{split}$ (36)

$\begin{split}S_{j}(\vec{v}_{j})=\frac{e^{-\alpha_{j}^{2}}}{2}\Bigg{[}\sin^{2}\chi_{j}\Big{(}(e^{\alpha_{j}^{2}}-1)&+\frac{1+e^{\alpha_{j}^{2}}(\alpha_{j}^{2}-1)}{\alpha_{j}^{2}}\Big{)}+\cos^{2}\chi_{j}\frac{e^{\alpha_{j}^{2}}-1}{\alpha_{j}^{2}}\Bigg{]}.\end{split}$
(37)

As mentioned we follow the Hardy-like pattern of settings, for which the first
setting of each observer corresponds to zero auxiliary field and local
beamsplitter acting as identity:

$\displaystyle\vec{v}_{1}$ $\displaystyle=$ $\displaystyle(0,0,0)$
$\displaystyle\vec{v}_{1}^{\prime}$ $\displaystyle=$
$\displaystyle(\chi_{1}^{\prime},\alpha_{1}^{\prime},\theta_{1}^{\prime})$
$\displaystyle\vec{v}_{2}$ $\displaystyle=$ $\displaystyle(0,0,0)$
$\displaystyle\vec{v}_{2}^{\prime}$ $\displaystyle=$
$\displaystyle(\chi_{2}^{\prime},\alpha_{2}^{\prime},\theta_{2}^{\prime}).$
(38)

We checked by numerical optimization that the double inequality (32) is
violated only on the left-hand side and the minimal achievable quantum value
reads $-1.0239$. The almost optimal settings for the violation read:

$\displaystyle\vec{v}_{1}^{\prime}$ $\displaystyle=$
$\displaystyle\left(\chi_{1}^{\prime}=\frac{3\pi}{20},\alpha_{1}^{\prime}=\frac{\sqrt{2}}{2},\theta_{1}^{\prime}=0\right),$
$\displaystyle\vec{v}_{2}^{\prime}$ $\displaystyle=$
$\displaystyle\left(\chi_{2}^{\prime}=\frac{3\pi}{20},\alpha_{2}^{\prime}=\frac{\sqrt{2}}{2},\theta_{2}^{\prime}=-\frac{\pi}{2}\right),$
(39)

which corresponds to local beamsplitters with transmissivity about $79\%$ and
local coherent fields with average photon number equal to $\tfrac{1}{2}$.

We have also investigated numerically the possibility of violating the CH
inequality for rates (32) for the case when for both local settings the
auxiliary field has non-zero amplitude, as opposed to (IV.2). We assumed
arbitrary parameters for both settings, with the constraint that the
amplitudes of auxiliary fields corresponding to a given setting are the same
for both observers, which allows for easy graphical representation:

$\displaystyle\vec{v}_{1}$ $\displaystyle=$
$\displaystyle(\chi_{1},\alpha,\theta_{1}),$
$\displaystyle\vec{v}_{1}^{\prime}$ $\displaystyle=$
$\displaystyle(\chi_{1}^{\prime},\alpha^{\prime},\theta_{1}^{\prime}),$
$\displaystyle\vec{v}_{2}$ $\displaystyle=$
$\displaystyle(\chi_{2},\alpha,\theta_{2}),$
$\displaystyle\vec{v}_{2}^{\prime}$ $\displaystyle=$
$\displaystyle(\chi_{2}^{\prime},\alpha^{\prime},\theta_{2}^{\prime}).$ (40)

The results of our findings are shown in the Fig. 4, in which we present a
plot of a CH-value as a function of the amplitudes $\alpha,\alpha^{\prime}$ of
auxiliary fields corresponding to two settings. The CH-values for each pair of
$\alpha$ and $\alpha^{\prime}$ are optimized over all the remaining parameters
in (IV.2). It turns out that a CH violation still exists for sufficiently
small but non-zero values of $\alpha$. On the other hand there is no violation
for any value $\alpha=\alpha^{\prime}$, which is consistent with our previous
findings that nonclassical correlations cannot be found for fixed-amplitude
auxiliary fields. The situation depicted in the Figure 4 shows an intermediate
case between the Hardy setup, in which we have on-off tuning of the auxiliary
field’s amplitudes and the TWC case in which the amplitudes are constant.

To complete our investigation, we have also examined the original Hardy
paradox from the point of view of the intensity rate-based approach. We found
that the Hardy setup with intensity rate operators does not lead to a
violation of the CH inequality for rates. Our result does not invalidate the
Hardy’s one, however it indicates a very different nature of the
nonclassicality tests based on aggregation of outcomes (like in rate approach)
and the ones based on very specific outcomes (like in Hardy case). All the
detailed considerations on this topic are presented in a forthcoming
publication Das  _et al._ .

Figure 4: Violation of the left-hand-side of the CH inequality (32) as a
function of amplitudes $\alpha,\alpha^{\prime}$ of the auxiliary fields
corresponding to two local settings. The CH-value is sign-flipped for the sake
of clarity of the plot. The amplitudes of auxiliary fields _for a given
setting_ ($\alpha$ or $\alpha^{\prime}$) are the same for both observers
(IV.2), while they are different for the first ($\alpha$) and second setting
($\alpha^{\prime}$). The black dashed line represents points for which
$\alpha=\alpha^{\prime}$, whereas the white dashed line is an envelope of the
violation region with the sign-flipped CH value strictly greater than $1$.
Notice that the black-line points lie strictly outside the violation region.
This means that it is impossible to obtain the CH violation if we just tune
the phases of the local oscillators. The amplitude tuning is necessary for the
violation.

##  V Bell operators as entanglement witnesses: intensities vs. rates

It is well known that Bell operators can be turned into entanglement
witnesses. Thus, the question arises if the inequality in Reid and Walls
(1986), used in the TWC paper, could also serve this purpose. As they are not
proper Bell inequalities the answer requires a careful analysis. Below we show
how to understand the additional condition (9) in this context. Next, we build
an optimized entanglement indicator based on the CHSH-like inequality of Reid
and Walls (1986). As the method which we employ to build entanglement
indicators for fields is also immediately applicable for Bell operators based
on intensity rates, we build an analogous entanglement indicator for this
case.

Since the CHSH-like inequality (20) seems to be more sensitive to the
correlations present in the TWC setup than the CHSH inequality for rates (14),
it suggests that the inequality (20) might be a stronger entanglement
indicator than (14). Surprisingly, it turns out the latter inequality leads to
an entanglement witness which is more efficient in the case of TWC
correlations. That is, it detects entanglement for a broader range of values
of $\alpha$.

###  V.1 The additional assumption (9) in the case of separable states

The original TWC correlations (with $|\alpha_{j}|=\alpha=const.$ for all
settings in the experiment), Tan  _et al._ (1991), do not violate local
realism, as shown by the local model presented in Das  _et al._ (2021). The
model covers the whole range of values of $\alpha$ for which TWC obtained
violations of the inequality of Reid and Walls (1986). As said earlier, the
inequality of Reid and Walls (1986) cannot be obtained without the additional
assumption (9), which states that for the given value of the hidden variable
$\lambda$ the total local intensity predicted by the local hidden variable
model does not vary with the measurement settings. Let us assume the model of
intensity, which was used above and leads to the predictions of Tan  _et al._
(1991). It is in the form of number of registered photons (in the given run of
the experiment). In such a case the additional assumption transforms into:

  * •

either

${n_{{tot}_{i}}(\lambda)}={n_{c_{i}}(\theta_{i},\lambda)}+{n_{d_{i}}(\theta_{i},\lambda)},$
(41)

where ${n_{x_{i}}(\theta_{i},\lambda)}$ is the number of photons predicted by
a LHV theory, for the given $\lambda$, to be detected at detector $x_{i}$,
under local setting $\theta_{i}$. The total number to be detected, in this
case, at both outputs ${n_{{tot}_{i}}(\lambda)}$ is assumed to be independent
of the local setting. Such must be the form of the condition (9), if we assume
the numbers of photons to be detected are predetermined by the model for each
value of $\lambda$.

  * •

or, for stochastic local hidden variable models, the condition (9) has to be
put as

$\langle{{n_{{tot}_{i}}}\rangle(\lambda)}=\langle{n_{c_{i}}(\theta_{i})\rangle(\lambda)}+\langle{n_{d_{i}}(\theta_{i})\rangle(\lambda)},$
(42)

where
$\langle{n_{x_{i}}(\theta_{i})\rangle(\lambda)}=\sum_{n_{x_{i}}=1}^{\infty}n_{x_{i}}P(n_{x_{i}}|\theta_{i},\lambda)$
for $x=c,d$.

Note that condition (42) does not imply (41).

We shall show below that in the case of hidden quantum product state model
(i.e., for separable states), which is a subclass of LHV models, the condition
of Reid and Walls (1986) should be understood as (42).

####  V.1.1 Condition (42) for separable states

As already discussed, assumption (41) imposes an additional constraint on the
structure of local hidden variable model. Still, a local hidden product
(quantum) state model is a form of local hidden variable model, which we get
when we consider a separable quantum state as the one that causes the
correlations. In such a case condition (42) definitely holds.

This suggests that the inequality of Reid and Walls (1986) can be an
entanglement witness.

Consider a density matrix describing separable states for optical fields, for
the operational scenario which we consider here. They are always a convex
combination of pure “product” states (inverted commas are for field theories,
see below). We shall use now $\lambda$ as an index related with any such
element of the convex combination, and the weights of the convex combination
we shall denote, as earlier, by $\rho_{\lambda}$. An arbitrary separable state
for the optical fields of the considered experiments reads

$\displaystyle\rho^{1,2}_{sep}$ $\displaystyle=\int
d\lambda\rho_{\lambda}f^{\dagger}_{\lambda}(\hat{a}_{1},\hat{b}_{1})g^{\dagger}_{\lambda}(\hat{a}_{2},\hat{b}_{2})\outerproduct{\Omega}{\Omega}f_{\lambda}(\hat{a}_{1},\hat{b}_{1})g_{\lambda}(\hat{a}_{2},\hat{b}_{2}),$
(43)

where $f^{\dagger}_{\lambda}(\hat{a}_{1},\hat{b}_{1})$ and
$g^{\dagger}_{\lambda}(\hat{a}_{2},\hat{b}_{2})$ are polynomial functions of
annihilation operators acting on modes $\hat{a}_{i}$ and $\hat{b}_{i}$ of
$i$-th party. Note that for arbitrary local observables
$\hat{O}_{i}(\hat{a}_{i},\hat{a}_{i}^{\dagger},\hat{b}_{i},\hat{b}_{i}^{\dagger})$,
and for any of the pure states
$f^{\dagger}_{\lambda}(\hat{a}_{1},\hat{b}_{1})g^{\dagger}_{\lambda}(\hat{a}_{2},\hat{b}_{2})\ket{\Omega}$
one has
$\langle\hat{O}_{1}\hat{O}_{2}\rangle_{\lambda}=\langle\hat{O}_{1}\rangle_{\lambda}\langle\hat{O}_{2}\rangle_{\lambda}$.
Thus, this is a proper description of separability for optical fields.

The probability of getting result $n_{x_{1}}$ for $1$-st party, with setting
$\theta_{1}$ for state indexed by $\lambda$ is given by:

$\displaystyle\text{Pr}(n_{x_{1}}|\theta_{1})_{\lambda}$
$\displaystyle=\Tr[\delta_{(n_{x_{1}},\hat{n}_{x_{1}}(\theta_{1}))}f^{\dagger}_{\lambda}(\hat{a}_{1},\hat{b}_{1})\outerproduct{\Omega}{\Omega}f_{\lambda}(\hat{a}_{1},\hat{b}_{1})\big{]}$
$\displaystyle=\langle{\delta_{(n_{x_{1}},\hat{n}_{x_{1}}(\theta_{1}))}}\rangle_{\lambda}$
(44)

where $\delta_{(n,k)}$ is the Kroenecker’s delta function.

A similar formula holds for 2-nd party for results $n_{x_{2}}$ under setting
$\theta_{2}$ Following formula (44) the joint probability for two parties goes
as follows:

$\displaystyle\text{Pr}(n_{x_{1}},n_{x_{2}}|\theta_{1},\theta_{2})_{sep}$
$\displaystyle=\Tr[\delta_{(n_{x_{1}},\hat{n}_{x_{1}}(\theta_{1}))}\delta_{(n_{x_{2}},\hat{n}_{x_{2}}(\theta_{2}))}\rho_{sep}^{1,2}\bigg{]}$
$\displaystyle=\int
d\lambda\rho_{\lambda}\text{Pr}(n_{x_{1}}|\theta_{1})_{\lambda}\text{Pr}(n_{2}|\theta_{2})_{\lambda}.$
(45)

Thus we have a typical structure of a local hidden variable model. Further,
assumption (42) is not in contradiction with the structure of separable
states. In fact we have an operator identity:

${\hat{n}}_{{tot}_{i}}=\hat{n}_{c_{i}}(\theta_{i})+\hat{n}_{d_{i}}(\theta_{i}),$
(46)

which holds for any settings $\theta_{i}$ and we have

$\langle{{\hat{n}_{{tot}_{i}}}\rangle(\lambda)}=\langle{\hat{n}_{c_{i}}(\theta_{i})\rangle(\lambda)}+\langle{\hat{n}_{d_{i}}(\theta_{i})\rangle(\lambda)},$
(47)

i.e., $\langle{\hat{n}_{{tot}_{i}}}\rangle_{\lambda}$ also does not depend on
the settings. Thus indeed the inequality of Reid and Walls (1986) is an
entanglement indicator.

### Entanglement indicators, a comparison

The inequality of Reid and Walls (1986) in the TWC version (20) can be put as
the following condition for separability

$\displaystyle\langle\delta\hat{n}_{1}(\theta_{1})\delta\hat{n}_{2}(\theta_{2})+\delta\hat{n}_{1}(\theta^{\prime}_{1})\delta\hat{n}_{2}(\theta_{2})$
$\displaystyle+\delta\hat{n}_{1}(\theta_{1})\delta\hat{n}_{2}(\theta^{\prime}_{2})-\delta\hat{n}_{1}(\theta^{\prime}_{1})\delta\hat{n}_{2}(\theta^{\prime}_{2})\rangle_{sep}$
$\displaystyle\leq
2\langle\hat{n}_{{tot}_{1}}\hat{n}_{{tot}_{2}}\rangle_{sep},$ (48)

where we introduced:
$\delta\hat{n}_{i}(\theta_{i})=\hat{n}_{c_{i}}(\theta_{i})-\hat{n}_{d_{i}}(\theta_{i})$,
and similarly for $\theta_{i}^{\prime}$. However, the right hand side of this
condition was derived following the additional assumption on local hidden
variable models. For separable states this can be lowered, as they form a
further constrained set of local hidden variable models.

We have to find its upper bound for separable states. This involves also
search for optimal settings. As in the usual case of two qubits optimal
violations of the CHSH inequality take place when Alice and Bob use fully
complementary settings, we shall give the separability conditions for such a
case. We are going to use an isomorphic mapping from entanglement witnesses
for qubits, to entanglement indicators for quantum optical fields Żukowski
_et al._ (2017); Ryu  _et al._ (2019). We derive a separability condition
based on Bell inequality for qubits and then map it to optical field
operators. As we need only two pairs of settings we reduce the Bloch sphere
for qubits to a circle and using the standard Pauli matrices we denote the
operator basis for each of them by
$\vec{\sigma}_{j}=(\sigma_{j}^{x},\sigma_{j}^{z})$, $j=1,2$. The optimal
separability condition based on the CHSH inequality can be obtained if we
choose Alice’s and Bob’s settings to be fully complementary. With each setting
we associate a unit Bloch vector. We can put: $\vec{a}=(1,0)$,
$\vec{a}^{\prime}=(0,1)$, $\vec{b}=\frac{1}{\sqrt{2}}(1,1)$ and
$\vec{b}^{\prime}=\frac{1}{\sqrt{2}}(1,-1)$. In this notation the LHS of the
CHSH inequality for any pure product state of two qubits gives the following:

$\displaystyle\left\langle\vec{a}\cdot\vec{\sigma_{1}}\otimes(\vec{b}+\vec{b}^{\prime})\cdot\vec{\sigma_{2}}+\vec{a}^{\prime}\cdot\vec{\sigma_{1}}\otimes(\vec{b}-\vec{b}^{\prime})\cdot\vec{\sigma_{2}}\right\rangle_{prod}$
$\displaystyle\leq\sqrt{2}\left(\left\langle\sigma_{1}^{x}\right\rangle_{prod}\left\langle\sigma_{2}^{x}\right\rangle_{prod}+\left\langle\sigma_{1}^{z}\right\rangle_{prod}\left\langle\sigma_{2}^{z}\right\rangle_{prod}\right)\leq\sqrt{2},$

because $\langle\sigma^{x}_{j}\rangle^{2}+\langle\sigma^{z}_{j}\rangle^{2}\leq
1$, where $j=1,2$, for any qubit state. Condition (V) holds as well for all
mixed separable states of two qubits as they are convex combinations of
product states. It holds also for arbitrary orthogonal pairs $\vec{a}$,
$\vec{a}^{\prime}$ and $\vec{b}$, $\vec{b}^{\prime}$. Its final form as an
entanglement witness thus reads:

$\displaystyle\langle\sqrt{2}\sigma_{1}^{0}\otimes\sigma_{2}^{0}-[\vec{a}\cdot\vec{\sigma}_{1}\otimes(\vec{b}+\vec{b}^{\prime})\cdot\vec{\sigma}_{2}$
$\displaystyle+\vec{a}^{\prime}\cdot\vec{\sigma}_{1}\otimes(\vec{b}-\vec{b}^{\prime})\cdot\vec{\sigma}_{2}]\rangle_{sep}\geq
0,$ (50)

where we have introduced identity matrices $\sigma^{0}$.

A method of finding an isomorphic entanglement witness given in Ryu  _et al._
(2019) is based on replacements $\sigma_{\mu}\rightarrow\hat{S}_{\mu}$, where
the latter is a quantum Stokes operator of optical field ($\mu=0,1,2,3$). In
the case of the TWC interferometric configuration $\hat{S}_{0}$ can be
replaced by the total intensity impinging on the detectors behind the local
interferometer, here modelled by $\hat{n}_{tot_{i}}$, and $\hat{S}_{1}$ with
$\hat{S}_{2}$ by respectively
$\delta\hat{n}_{i}(\theta_{i})=\hat{n}_{c_{i}}(\theta_{i})-\hat{n}_{d_{i}}(\theta_{i})$
and $\delta\hat{n}_{i}(\theta_{i}+\frac{\pi}{2})$. Thus the quantum optical
analog for the separability condition for two qubits is given, via the
isomorphism of Ryu  _et al._ (2019), in the following form:

$\displaystyle\big{\langle}\sqrt{2}\hat{n}_{tot_{1}}\hat{n}_{tot_{2}}-[\delta\hat{n}_{1}(\theta_{1})\delta\hat{n}_{2}(\theta_{2})+\delta\hat{n}_{1}(\theta_{1})\delta\hat{n}_{2}(\theta_{2}+\tfrac{\pi}{2})$
$\displaystyle+\delta\hat{n}_{1}(\theta_{1}+\tfrac{\pi}{2})\delta\hat{n}_{2}(\theta_{2})-\delta\hat{n}_{1}(\theta_{1}+\tfrac{\pi}{2})\delta\hat{n}_{2}(\theta_{2}+\tfrac{\pi}{2})]\big{\rangle}_{sep}$
$\displaystyle\geq 0.$ (51)

The above condition (V) can be also transformed into the one with rates
instead of intensities. This can be achieved using another isomorphism
presented in Ryu  _et al._ (2019):

$\displaystyle\big{\langle}\sqrt{2}\hat{\Pi}_{1}\hat{\Pi}_{2}-[\delta\hat{R}_{1}(\theta_{1})\delta\hat{R}_{2}(\theta_{2})+\delta\hat{R}_{1}(\theta_{1}+\tfrac{\pi}{2})\delta\hat{R}_{2}(\theta_{2})$
$\displaystyle+\delta\hat{R}_{1}(\theta_{1})\delta\hat{R}_{2}(\theta_{2}+\tfrac{\pi}{2})-\delta\hat{R}_{1}(\theta_{1}+\tfrac{\pi}{2})\delta\hat{R}_{2}(\theta_{2}+\tfrac{\pi}{2})]\big{\rangle}_{sep}$
$\displaystyle\geq 0,$ (52)

where
$\delta\hat{R}_{i}(\theta_{i})=\hat{R}_{c_{i}}(\theta_{i})-\hat{R}_{d_{i}}(\theta_{i})$,
and the intensity rate operators $\hat{R}_{x_{i}}(\theta_{i})$ are defined in
Eq. (11).

Witnesses (V) and (V) can reveal mode entanglement of the initial state
$\ket{\Psi(\alpha)}$ (3) in a certain range of the coherent field amplitudes
$\alpha$. Analogously to (16) we introduce $A_{R}^{EW}(\alpha)$ that stands
for the amplitude of the correlation function:

$\frac{\langle\Psi(\alpha)|\delta\hat{R}_{1}(\theta_{1})\delta\hat{R}_{2}(\theta_{2})|\Psi(\alpha)\rangle}{\langle\Psi(\alpha)|\hat{\Pi}_{1}\hat{\Pi}_{2}|\Psi(\alpha)\rangle}=A_{R}^{EW}(\alpha)\sin(\theta_{1}-\theta_{2}),$
(53)

where the amplitude is given by

$A_{R}^{EW}(\alpha)=\frac{e^{-\alpha^{2}}(e^{\alpha^{2}}-1)}{\alpha^{2}}.$
(54)

Condition (V) for $\ket{\Psi}$ can be put as follows:

$\displaystyle
A_{R}^{EW}(\alpha)[\sin(\theta_{1}-\theta_{2})+\sin(\theta^{\prime}_{1}-\theta_{2})$
$\displaystyle+\sin(\theta_{1}-\theta^{\prime}_{2})-\sin(\theta^{\prime}_{1}-\theta^{\prime}_{2})]\leq\sqrt{2},$
(55)

and its violation appears if $A_{R}^{EW}(\alpha)\geq\frac{1}{2}$. The same
holds for (V) with the difference that amplitude $A_{T}^{EW}(\alpha)\equiv
A_{T}(\alpha)=\frac{1}{(1+\alpha^{2})}$ replaces $A_{R}^{EW}(\alpha)$. We
present entanglement detection with (V) and (V) in terms of values of
$A_{R/T}^{EW}(\alpha)$ in Fig.5. The CHSH-based entanglement indicator
involving rates (V) is more effective than the one based on CHSH-like
inequalities of Reid and Walls (1986) involving intensities (V). This may seem
surprising, since the CHSH-like inequalities of Reid and Walls (1986) show
(spurious) violations of local realism for the TWC configuration, whereas
CHSH-Bell inequality involving rates is not violated for any $\alpha$. This
fact shows the advantage of the rate-based approach over the intensity-based
approach : it gives no spurious Bell violations and nevertheless it is more
effective as an entanglement indicator.

Figure 5:  Comparison of amplitudes $A(\alpha)$ in function of $\alpha$ for
CHSH based entanglement conditions for rates (V) and intensities (V). If
$A(\alpha)$ exceeds $\frac{1}{2}$ we observe violation.

Of course the presented entanglement indicators have two interpretations,
within the context of TWC correlations. Either they detect entanglement of the
state:

$\frac{1}{\sqrt{2}}\,|\alpha\rangle_{a_{1}}(|01\rangle_{b_{1}b_{2}}+i\,|10\rangle_{b_{1}b_{2}})|\alpha\rangle_{a_{2}}$
(56)

with von Neumann measurements, or entanglement of

$\frac{1}{\sqrt{2}}(|01\rangle_{b_{1}b_{2}}+i\,|10\rangle_{b_{1}b_{2}})$ (57)

with optical homodyne based POVM measurements. In the second case the settings
may depend also on varying moduli of values of local $\alpha$’s.

##  VI Discussion and Conclusions

In our work we reconsidered two prototypical experimental scenarios aiming at
demonstration of a single photon nonclassicality, namely the proposal of Tan,
Walls and Collett (TWC) Tan  _et al._ (1991), and the one of Hardy Hardy
(1994). We have shown that if one applies an intensity-rate-based approach,
without using additional assumption (9), the associated Bell-CHSH inequality
for rates is not violated in the case of TWC correlations. This finding is
concurrent with our previous work Das  _et al._ (2021), which shows a failure
of TWC setup in demonstrating nonclassicality by providing a precise local
hidden variable model for the setup. Nevertheless, we have shown that the
original TWC analysis can be still used as an entanglement indicator, however
a weaker one than the one based on intensity rate approach. These facts are a
strong premise towards treating the intensity rate-based approach to analysis
of correlations of the optical fields as more appropriate from both: Bell-type
perspective (as it is free of a certain loophole) and an entanglement
detection perspective (as it gives rise to more robust entanglement
indicators). As a side discussion we pointed out possible threats concerning
the application of a parametric approximation to a coherent light. Namely if
one applies such an approximation at the same time ignoring the phase
uncertainty, spurious increases of the strength of correlations may arise.

Our intensity rate-based analysis of Hardy-like scenario leads to several
conclusions. First, we point out that in this approach the CH inequality for
rates is strictly stronger than the CHSH one (for rates). We have shown a
violation of the CH inequality for rates in Hardy-like scenario, which
includes tunable amplitudes of the coherent local oscillator auxiliary fields,
but also (in contrast to the original Hardy analysis) local beamsplitters with
tunable transmissivities. Our approach involves modification of two further
features of the Hardy scheme, namely we use just a single photon initial state
as in the TWC approach ($q=0$ in the Figure 1) and do not limit the settings
specified by the local oscillator strengths to the on/off detection mode. This
indicates that the crucial aspect of the Hardy-like schemes which enables
violation of the local realism is the tunability of the strengths of the local
auxiliary fields, which need to differ from setting to setting.

Let us now discuss the physical interpretation of these observations. First,
notice that the Hardy-like scenario possesses a peculiar feature untypical for
Bell tests, namely that the entire initial state of the interferometric setup
is different for different measurement settings. To some extent one can
overcome this interpretational difficulty by treating presence of auxiliary
fields as an optical implementation of a positive-operator-value measurement
(POVM). We show analytical forms of such POVMs in Appendix A and Appendix B.
However even such a treatment does not dispel doubts whether nonclassical
correlations found in Hardy-like scenarios can be attributed solely to the
single photon (excitation) state. Despite several objections discussed earlier
majority of works devoted to this issue states that the sole source of
correlations in such experiments is the single photon superposition. Our
reconsideration of the mechanism of the nonclassical correlations in the setup
indicates that such a conclusion is unfounded due to a very subtle issue,
which goes beyond the duality between mode and particle entanglement in
interferometric setups (see e.g. discussion in Demkowicz-Dobrzański  _et al._
(2015)). Namely, the source of observed correlations is the quantum
interference, which occurs if and only if a given detected multiphoton event
could have been realised in two or more genuinely _indistinguishable_ ways.
The quantum interference observed in Hardy-like schemes is based on
indistinguishability of processes corresponding to detection of photons coming
from the signal state and the local oscillator fields. At the detection stage
due to mentioned interference all photons must be treated on equal footing,
hence the distinction between the input (signal) photons and the local
oscillator ones disappears.

Based on our findings, we argue that the resource behind observed nonclassical
correlations in the TWC-Hardy experiments is quantum interference due to the
indistinguishability of photons originating from different sources. This means
that the ‘nonlocality of a single photon’ shares the same interpretation as
the other profound interferometric nonclassical phenomena, like photonic
entanglement swapping Żukowski  _et al._ (1993), or nonclassical interference
for pairs of photons originating from independent sources Yurke and Stoler
(1992a, b); Kaltenbaek  _et al._ (2006); Blasiak and Markiewicz (2019).

## Acknowledgements

We thank Łukasz Rudnicki for comprehensive discussions on the topic of mode
entanglement. The work is part of the ICTQT IRAP (MAB) project of FNP, co-
financed by structural funds of EU. MK acknowledges support by the Foundation
for Polish Science through the START scholarship. AM acknowledges support by
National Research Center through the grant MINIATURA DEC-2020/04/X/ST2/01794.

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##  Appendix A Effective POVM operators for CHSH inequality rates

We obtained the correlation coefficient (see Eq. (16)) for local homodyne-type
measurements performed on the linear superposition of a single photon and
vacuum,
$\ket{\psi}_{b_{1},b_{2}}=\frac{1}{\sqrt{2}}(\ket{01}+i\ket{10})_{b_{1},b_{2}}$,
in terms of intensity rates in mode $\hat{c}_{j}$ or $\hat{d}_{j}$ in a
symmetric beamsplitter transformation $U_{BS_{j}}(\frac{\pi}{4},\theta_{j})$,
where an auxiliary coherent field $\ket{\alpha}_{a_{j}}$, impinges on the
remaining input of the beamsplitter for $j=1,2$. In this section our aim is to
figure out the form of the positive operators (POVMs) acting on the initial
state $\ket{\psi}_{b_{1},b_{2}}$, which give rise to an equivalent form of the
same correlation function (18). Suppose $\hat{\cal
M}_{b_{j}}(\alpha,\theta_{j})$ is the POVM operator in part of the mode
$b_{j}$, for $j=1,2$, then we obtain the defining condition for the POVM by
demanding equivalence of the two forms of the correlation coefficient (compare
Eqs (16) and (18)):

$\bra{\Psi(\alpha)}\hat{H}_{1}(\theta_{1})\hat{H}_{2}(\theta_{2})\ket{\Psi(\alpha)}={}_{b_{1}b_{2}}\langle\psi|\hat{\cal
M}_{b_{1}}(\alpha,\theta_{1})\hat{\cal
M}_{b_{2}}(\alpha,\theta_{2})\ket{\psi}_{b_{1}b_{2}},$ (58)

where the initial state $\ket{\Psi(\alpha)}$ is defined in Eq. (3). Hence, the
POVM operator is given by:

$\displaystyle\hat{\cal
M}_{b_{j}}(\alpha,\theta_{j})={}_{a_{j}}\langle\alpha|~{}\hat{H}_{j}(\theta_{j})\ket{\alpha}_{a_{j}}$
(59)
$\displaystyle={}_{a_{j}}\langle\alpha|~{}\hat{\Pi}_{c_{j}d_{j}}\frac{\hat{n}_{c_{j}}-\hat{n}_{d_{j}}}{\hat{n}_{c_{j}}+\hat{n}_{d_{j}}}\hat{\Pi}_{c_{j}d_{j}}\ket{\alpha}_{a_{j}}$
(60)
$\displaystyle={}_{a_{j}}\langle\alpha|~{}\hat{\Pi}_{a_{j}b_{j}}~{}\frac{e^{i\theta_{j}}\hat{a}_{j}\hat{b}_{j}^{\dagger}+e^{-i\theta_{j}}\hat{a}_{j}^{\dagger}\hat{b}_{j}}{\hat{a}_{j}^{\dagger}\hat{a}_{j}+\hat{b}^{\dagger}_{j}\hat{b}_{j}}~{}\hat{\Pi}_{a_{j}b_{j}}\ket{\alpha}_{a_{j}}$
(61)
$\displaystyle=e^{-\alpha^{2}}\sum_{n=0}^{\infty}\frac{\alpha^{2n+1}}{n!}\sum_{m=1}^{\infty}\frac{\sqrt{m}}{n+m}\left(e^{i\theta_{j}}\ket{m}\bra{m-1}_{b_{j}}+e^{-i\theta_{j}}\ket{m-1}\bra{m}_{b_{j}}\right).~{}~{}~{}~{}$
(62)

Figure 6: Schematic diagram of the effective POVM operators acting on modes
$\hat{b}_{1}$ and $\hat{b}_{2}$ to detect the nonclassicality of a single
photon. Average intensity rate $\langle\hat{R}_{x_{j}}\rangle$ detected in
detector $D_{x_{j}}$, for $x=c,d,~{}j=1,2$, or the correlation
$\langle\hat{R}_{x_{1}}\hat{R}_{x_{2}}\rangle$ detected in detectors
$D_{x_{1}}$ and $D_{x_{2}}$ for $x=c,d~{}~{}$ on the output modes of the
beamsplitters $U_{BSj},j=1,2$ along with an auxiliary coherent beam, can be
considered as an effective POVM $\hat{\cal{M}}_{b_{j}}$ acting on the modes
$b_{j},~{}j=1,2$. The semicircular shaded region shows that the beamsplitter
$U_{BSj}$, the coherent field $\ket{\alpha_{j}}$, and detection of the the
Homodyne $\hat{H}_{j}$ for $j=1,2$ are equivalent to a POVM
$\hat{\cal{M}}_{b_{j}}(\alpha_{j},\theta_{j})$, for $j=1,2$ is given in Eq.
(62) and the detection of $\hat{R}_{d_{j}}$, for $j=1,2$ are equivalent to a
POVM $\hat{\cal{M}}_{b_{j}}(\vec{v_{j}})$, for $j=1,2$ is given in Eq. (68).

##  Appendix B Effective POVM operators for CH inequality and tunable exit
beamsplitters

In the previous section, we have calculated the POVM for the balanced
beamsplitter and for the homodyne detectors $\hat{H}_{j}$. Here, we will try
to obtain the POVM for the correlation coefficients given in Eqs. (IV.2) and
(35) for arbitrary beamsplitters. The construction of the POVM is shown
schematically in the Figure 6.

For a beamsplitter with arbitrary transitivity, the output modes are given by
(see Eq. (5)):

$\displaystyle\hat{c}_{j}$ $\displaystyle=$
$\displaystyle\cos\chi_{j}\hat{a}_{j}+\sin\chi_{j}e^{-i\theta_{j}}\hat{b}_{j},$
(63) $\displaystyle\hat{d}_{j}$ $\displaystyle=$
$\displaystyle-\sin\chi_{j}e^{i\theta_{j}}\hat{a}_{j}+\cos\chi_{j}\hat{b}_{j}.$
(64)

Now we want to find out POVM elements ${\hat{\cal
M}}_{b_{j}}(\vec{v}_{j}),j=1,2$, such that the correlation coefficients and
the local terms from Eq. (35) read
$K(\vec{v}_{1},\vec{v}_{2})={}_{b_{1}b_{2}}\langle\psi|{\hat{\cal
M}}_{b_{1}}(\vec{v}_{1}){\hat{\cal
M}}_{b_{2}}(\vec{v}_{2})\ket{\psi}_{b_{1}b_{2}}$ and
$S_{j}(\vec{v}_{j})={}_{b_{1}b_{2}}\langle\psi|{\hat{\cal
M}}_{b_{j}}(\vec{v}_{j})\ket{\psi}_{b_{1}b_{2}}$, for $j=1,2$.

By demanding equivalence of the two forms of correlation coefficients in Eqs.
(IV.2) and (35) we obtain:

$\displaystyle{\hat{\cal
M}}_{b_{j}}(\vec{v}_{j})={}_{a_{j}}\langle\alpha_{j}|\hat{R}_{d_{j}}(\chi_{j},\theta_{j})\ket{\alpha_{j}}_{a_{j}}={}_{a_{j}}\langle\alpha_{j}|~{}\hat{\Pi}_{c_{j}d_{j}}\frac{\hat{n}_{d_{j}}}{\hat{n}_{c_{j}}+\hat{n}_{d_{j}}}\hat{\Pi}_{c_{j}d_{j}}\ket{\alpha_{j}}_{a_{j}}$
(65)
$\displaystyle={}_{a_{j}}\langle\alpha_{j}|~{}\hat{\Pi}_{a_{j}b_{j}}~{}\frac{\hat{d}_{j}^{\dagger}\hat{d}_{j}}{\hat{a}^{\dagger}_{j}\hat{a}_{j}+\hat{b}^{\dagger}_{j}\hat{b}_{j}}\hat{\Pi}_{a_{j}b_{j}}\ket{\alpha_{j}}_{a_{j}}$
(66)
$\displaystyle={}_{a_{j}}\langle\alpha_{j}|~{}\hat{\Pi}_{a_{j}b_{j}}\frac{\sin^{2}\chi_{j}\hat{a}_{j}^{\dagger}\hat{a}_{j}+\cos^{2}\chi_{j}\hat{b}^{\dagger}_{j}\hat{b}_{j}}{\hat{a}_{j}^{\dagger}\hat{a}_{j}+\hat{b}^{\dagger}_{j}\hat{b}_{j}}~{}\hat{\Pi}_{a_{j}b_{j}}\ket{\alpha_{j}}_{a_{j}}$
$\displaystyle\hskip 199.16928pt-\frac{1}{2}\sin
2\chi_{j}~{}{}_{a_{j}}\langle\alpha_{j}|~{}\hat{\Pi}_{a_{j}b_{j}}~{}\frac{e^{i\theta_{j}}\hat{a}_{j}\hat{b}_{j}^{\dagger}+e^{-i\theta_{j}}\hat{a}_{j}^{\dagger}\hat{b}_{j}}{\hat{a}_{j}^{\dagger}\hat{a}_{j}+\hat{b}^{\dagger}_{j}\hat{b}_{j}}~{}\hat{\Pi}_{a_{j}b_{j}}\ket{\alpha_{j}}_{a_{j}}$
(67)
$\displaystyle=e^{-\alpha_{j}^{2}}\bigg{(}\cos^{2}\chi_{j}\left(\mathbb{I}_{b_{j}}-\ket{0}\bra{0}_{b_{j}}\right)+\sum_{n=1}^{\infty}\frac{\alpha_{j}^{2n}}{n!}~{}\frac{\sin^{2}\chi_{j}~{}n+\cos^{2}\chi_{j}~{}b_{j}^{\dagger}b_{j}}{n+b_{j}^{\dagger}b_{j}}$
$\displaystyle\hskip 113.81102pt-\frac{1}{2}\sin
2\chi_{j}\sum_{n=0}^{\infty}\frac{\alpha_{j}^{2n+1}}{n!}\sum_{m=1}^{\infty}\frac{\sqrt{m}}{n+m}\left(e^{i\theta_{j}}\ket{m}\bra{m-1}_{b_{j}}+e^{-i\theta_{j}}\ket{m-1}\bra{m}_{b_{j}}\right)\bigg{)}.~{}~{}~{}~{}$
(68)

If beamsplitters $U_{BS1}$ and $U_{BS2}$ have $100\%$ transmissivity, then the
above POVM reduces to
$\hat{\Pi}_{b_{j}}=\mathbb{I}_{b_{j}}-\ket{0}\bra{0}_{b_{j}},~{}j=1,2$.