Photon distribution in the classical limit

Consider a single photon in a polarisation state \(\sqrt {p_{\mathrm{H}}} \left| H \right\rangle + \sqrt {p_{\mathrm{V}}} \left| V \right\rangle\), where H and V denote horizontal and vertical polarisations, respectively, and p H + p V = 1. When incident on a polarising beam splitter (PBS), the photon can either go through and become H polarised, or be reflected and become V polarised. These are two possibilities occuring randomly with probabilities p H and p V , if one decides to detect the photon after the PBS. Denote these two possible outcomes as {0, 1} and {1, 0}. This scenario is a physical implementation of a binary ±1 random variable X, where the outcome {1, 0} is associated with the value +1 and {0, 1} with −1.

Next, let us consider two indistinguishable photons in the above state entering the same PBS port. They are uncorrelated and therefore they scatter on the PBS independently.23 The photons cannot be distinguished and therefore there are only three exclusive outcomes: {2, 0}, {1, 1} and {0, 2}. These outcomes cannot be interpreted as products of two single-photon outcomes becasue of indistinguishability, i.e., events {1, 0} × {1, 0}, {1, 0} × {0, 1}, {0, 1} × {1, 0}, and {0, 1} × {0, 1} are meaningless. Moreover, unlike in the single-photon case, for two photons, statements “photon is detected on the left” and “photon is detected on the right” are not exclusive because there is a chance that photons can be detected on both sides. Interestingly, the average number of photons in each output is proportional to single-photon scattering probabilities, i.e., \(\bar n_{\mathrm{H}} = 2p_{\mathrm{H}}\) and \(\bar n_{\mathrm{V}} = 2p_{\mathrm{V}}\).

In general, for N photons scattering on the PBS one can observe N + 1 exclusive outcomes: {N, 0}, {N − 1, 1}, etc. Note, that although the scenario consists of only two distinguishable modes, the exclusivity structure depends on the number of photons, since the number of exclusive events scales linearly with N. This is drastically fewer than 2N outcomes observable for distinguishable particles. For N photons the single-photon random variable X is ill-defined because of indistinguishability. However, it is possible to define a random variable whose outcomes are given by \({\cal X} = \left( {n_{\mathrm{H}} - n_{\mathrm{V}}} \right){\mathrm{/}}N\), i.e., the difference between photon numbers in the output ports divided by the total number of photons. Note, that \(- 1 \le {\cal X} \le 1\) and \({\cal X} = X\) for N = 1. Additionally, since each photon is transformed independently and according to the same rule, the average number of photons in each polarisation mode is given by \(\bar n_{\mathrm{H}} = Np_{\mathrm{H}}\) and \(\bar n_{\mathrm{V}} = Np_{\mathrm{V}}\). Because of this \(\left\langle {\cal X} \right\rangle = p_{\mathrm{H}} - p_{\mathrm{V}}\) does not depend on N and the most probable outcome state is \(\left\{ {\bar n_{\mathrm{H}},\bar n_{\mathrm{V}}} \right\}\).

Finally, let us discuss the classical limit. In this case the total number of photons is undetermined but their average number is large \(\left( {\left\langle N \right\rangle \gg 1} \right)\). In quantum theory such situations are usually represented by a high amplitude coherent state.24 Once we go to the classical limit, it is quite natural to treat the beam of light as a continuous object that can be split into portions in an arbitrary way. The PBS transforms a single beam with intensity I into two beams, the H polarised beam with intensity I H and the V polarised one with I V . This is predicted by both, classical and quantum theories. In the classical limit the average value of random variable \({\cal X}\) becomes (I H − I V )/I. However, since the intensities of two beams are given by I H = Ip H and I V = Ip V , we get \(\left\langle {\cal X} \right\rangle = p_{\mathrm{H}} - p_{\mathrm{V}}\), as expected. Note, that for Poissonian light the fluctuations of \({\cal X}\) scale as \(1{\mathrm{/}}\sqrt N\), therefore in the classical limit \({\cal X}\) can be treated as a deterministic variable.

The above scenarios are schematically represented in Fig. 1. Although we considered only a single PBS, the similarity between classical intensities and probabilities generated by photonic distributions would also hold if one used an arbitrary number of linear optical devices (PBS, standard beam splitters (BS), phase shifters, etc). In this case the whole setup is equivalent to a multiport corresponding to a more complex random variable or a sequence of random variables.

Fig. 1 Schematic representation of the classical limit in an experiment with uncorrelated photons. Single photon on a polarising beam splitter (PBS) can either go through or reflect. There are two exclusive outcomes: the photon is either registered on the left with the polarisation V or on the right with the polarisation H. The corresponding probabilities are p V and p H , respectively (p H + p V = 1). Two photons on the PBS can produce three exclusive outcomes: both on the left with probability \(p_V^2\), one on the left and one on the right with probability 2p H p V , and both on the right with probability \(p_H^2\). N photons on the PBS can produce N + 1 exclusive outcomes, however the most probable events are those with approximately Np V photons on the left and Np H photons on the right. A classical beam of light of intensity I is split on the PBS into two beams. In principle there is a continuum of outcomes, however one always observes the one with the corresponding intensities I H and I V , where I H = Ip H and I V = Ip V Full size image

To summarise, we see that the same average value \(\left\langle {\cal X} \right\rangle\) is predicted by both, quantum and classical theories, since this value does not depend on N. Nevertheless, the underlying exclusivity structure of the outcomes of \({\cal X}\) strongly depends on N. This fact causes some bizarre interpretation difficulties in experimental Bell-type scenarios with classical light. We will discuss this problem in more details in the following sections.

Correlations in the classical limit

Next, we show that unlike photonic distribution, the correlations between spatially separated photons strongly depend on N and on the exclusivity structure of detection events. The problem of quantum correlations in the classical limit was discussed in details before (see for example ref. 5) so we provide here only a simple example.

Consider a pair of photons in an entangled polarisation state \(\sqrt {p_{\mathrm{H}}} \left| {HH} \right\rangle + \sqrt {p_{\mathrm{V}}} \left| {VV} \right\rangle\). These two photons are shared between two spatially separated parties, Alice and Bob, who measure their photons polarisations with respective PBSs. As before, we represent the two polarisation possibilities by {1, 0} and {0, 1}. Moreover, we can also use the ±1 random variables X A and X B , defined in the same way as X above, to represent the measurement of Alice and Bob. Alice and Bob register either {1, 0} × {1, 0} with probability p V , or {0, 1} × {0, 1} with probability p H . The average values and the corresponding correlations are 〈X A 〉 = 〈X B 〉 = p H − p V and 〈X A X B 〉 = 1.

Next, let us consider N such photonic pairs shared between Alice and Bob who measure polarisation on all pairs at the same time. As before, the local measurements are represented by the random variables \({\cal X}_{\mathrm{A}} = \left( {n_{{\mathrm{AH}}} - n_{{\mathrm{AV}}}} \right){\mathrm{/}}N\) and \({\cal X}_{\mathrm{B}} = \left( {n_{{\mathrm{BH}}} - n_{{\mathrm{BV}}}} \right){\mathrm{/}}N\). Again, there are correlations between the photonic pairs giving n AH = n BH = n H and n AV = n BV = n V , hence Alice registers the same photon distribution as Bob and \({\cal X}_{\mathrm{A}} = {\cal X}_{\mathrm{B}}\). However, for N entangled pairs the correlations \(\left\langle {{\cal X}_{\mathrm{A}}{\cal X}_{\mathrm{B}}} \right\rangle\) are much weaker. Note, that the outcome {n, N − n} × {n, N − n} happens with probability \({\textstyle{{N!} \over {n!(N \,-\, n)!}}}\,p_{\mathrm{V}}^{n}p_{\mathrm{H}}^{N - n}\), thus

$$\begin{array}{*{20}{l}} {\left\langle {{\cal X}_{\mathrm{A}}{\cal X}_{\mathrm{B}}} \right\rangle } \hfill & = \hfill & {\mathop {\sum}\limits_{n = 0}^N \left( {\frac{{2n \,-\, N}}{N}} \right)^2\frac{{N!}}{{n!(N \,-\, n)!}}\,p_{\mathrm{V}}^{n}p_{\mathrm{H}}^{N - n}} \hfill \\ {} \hfill & = \hfill & {\frac{{N \,-\, 4p_{\mathrm{V}}p_{\mathrm{H}}(N \,-\, 1)}}{N}.} \hfill \end{array}$$ (1)

For example, for p H = p V = 1/2 one gets \(\left\langle {{\cal X}_{\mathrm{A}}{\cal X}_{\mathrm{B}}} \right\rangle = 1{\mathrm{/}}N\). Interestingly, \(\left\langle {{\cal X}_{\mathrm{A}}} \right\rangle = \left\langle {{\cal X}_{\mathrm{B}}} \right\rangle = p_{\mathrm{H}} - p_{\mathrm{V}}\) and in the limit of the large number of photons

$$\mathop {{{\mathrm{lim}}}}\limits_{N \to \infty } \left\langle {{\cal X}_{\mathrm{A}}{\cal X}_{\mathrm{B}}} \right\rangle = 1 - 4p_{\mathrm{V}}p_{\mathrm{H}} = \left\langle {{\cal X}_{\mathrm{A}}} \right\rangle \left\langle {{\cal X}_{\mathrm{B}}} \right\rangle .$$ (2)

To conclude, the values \(\left\langle {{\cal X}_{\mathrm{A}}} \right\rangle\) and \(\left\langle {{\cal X}_{\mathrm{B}}} \right\rangle\) do not depend on N. However, the correlation between \({\cal X}_{\mathrm{A}}\) and \({\cal X}_{\mathrm{B}}\), \(\left\langle {{\cal X}_{\mathrm{A}}{\cal X}_{\mathrm{B}}} \right\rangle\), does. As a consequence, in the classical limit of large 〈N〉 the two random variables get practically uncorrelated. Therefore, the classical limit of Bell-type scenarios based on correlations between many particles can always be explained by a classical theory (for more details see ref. 5 and the methods). The idea of a classical simulation of quantum correlations using classical beams uses different approach, and in the next section we focus on correlations between random variables defined for the same particle.

Bell inequalities in the clasical limit

The idea of hidden variables goes back to the father founders of quantum theory. Some of them, like Einstein, could not accept the fact that the theory is fundamentally random. A programme to explain the quantum randomness as a lack of knowledge of the system’s state was developed. In particular, it was assumed that the quantum state provides only a partial information about the true state of the system. The remaining information exists, encoded in some parameters, but is inaccessible to an experimenter. These inaccessible parameters were called hidden variables. The standard hidden variable approach to quantum theory studies if it is possible to assign outcomes to all possible measurements in a consistent way, without violating commonly accepted features of nature, such as locality or non-contextuality. Bell inequalities11 serve as the most common tool to check whether a hidden-variable description is possible for a given experimental scenario.

Let us consider the CHSH scenario,9 which is the simplest Bell test involving four ±1 binary random variables A 0 , A 1 , B 0 , and B 1 . In a classical theory these four random variables are jointly distributed which implies that the following inequality must be satisfied

$$- 2 \le \left\langle {A_1B_1} \right\rangle + \left\langle {A_0B_1} \right\rangle + \left\langle {A_1B_0} \right\rangle - \left\langle {A_0B_0} \right\rangle \le 2,$$ (3)

where

$$\left\langle {A_iB_j} \right\rangle = \mathop {\sum}\limits_{a_i,b_j = \pm 1} {\kern 1pt} a_ib_jp\left( {A_i = a_i,B_j = b_j} \right).$$ (4)

In quantum theory it is possible to find a set of binary observables represented by Hermitian matrices, such that [A i , B j ] ≡ A i B j − B j A i = 0 (for i, j = 0, 1), but [A 0 , A 1 ] ≠ 0 and [B 0 , B 1 ] ≠ 0. This means that A i and B j can be jointly measured, but it is not possible to jointly measure A 0 and A 1 or B 0 and B 1 . Interestingly, for quantum correlations 〈A i B j 〉 the inequality (3) can be violated up to \(\pm 2\sqrt 2\) for an optimal choice of the state and observables. The violation implies that the measured correlations cannot be described by classical theories.

The simplest quantum system where such a scenario is possible has four levels. In the original Bell-type scenario we have two spatially separated systems, e.g., two polarisation entangled photons discussed in the previous section, see also Fig. 2a. In this case A 0 and A 1 correspond to the polarisation properties of the first photon, whereas B 0 and B 1 correspond to the polarisation properties of the second photon. However, using the arguments from the previous section, a large number of indistinguishable entangled pairs would produce 〈A i B j 〉 ≈ 〈A i 〉〈B j 〉 in the classical limit. Thus, the CHSH inequality (3) would not be violated.

Fig. 2 Clauser-Horne-Shimony-Holt (CHSH) Bell-type scenario. a Nonlocal setting in which a source S emits two correlated particles flying to Alice and Bob. Each of them performs one of the two measurements (denoted by 0 or 1). Each measurement produces a binary outcome (±1). Here, we present an instance in which Alice chooses to measure 0 and Bob chooses to measure 1. Alice’s outcome is + and Bob’s is −, hence they jointly register an event (+ − |01). b The same instance, but in a local scenario. Classical entanglement can only be tested in such scenarios. The measurement of A is performed before B. It is generally assumed that both properties are compatible (they commute in QM sense), therefore the order of measurement is irrelevant. c The exclusivity graph for the CHSH scenario. The vertices correspond to measurement events and the edges represent the exclusivity relations. Orange edges correspond to exclusivity of measurement outcomes for the same settings, e.g., (+ + |ij) and (− − |ij). Grey edges correspond to exclusivity of measurement outcomes in which the second measurement has different settings, e.g., (+ + |i0) and (− − |i1). This exclusivity can be tested in the setting represented in b, by choosing B 0 for the second left measuring device and B 1 for the second right measuring device—detailed discussion in the text Full size image

Let us now discuss another implementation of the CHSH scenario. This time the four-level system is made of a single photon which can occupy four modes, e.g., two polarisation modes (H and V) and two spatial modes (a and b). As a result the photon can be in one of four possible states a H , a V , b H and b V , or in an arbitrary superposition of them. The properties A 0 and A 1 can be associated with spatial modes, whereas B 0 and B 1 can be associated with polarisation. For example, A 0 can assign +1 to mode a and −1 to b. On the other hand, A 1 can assign ±1 to orthogonal superpositions of modes, like |a〉 ± |b〉. Similarly, B 0 can assign +1 to polarisation H and −1 to V, whereas B 1 can assign +1 to the right-handed circular polarisation and −1 to the left-handed one.

The Hilbert space of the system is a tensor product of two Hilbert spaces: the one corresponding to spatial modes and another one to polarisation. However, this time the system cannot be divided into parts that can be separated from each other. Still, it is possible to speak of entanglement between these two degrees of freedom, but this entanglement has nothing to do with nonlocality. Nevertheless, violation of the CHSH inequality with A i and B j confirms the presence of entanglement between spatial modes and polarisation. This entanglement gives non-classical correlations that can be attributed to contextuality rather than to nonlocality.

The properties A i and B j can be measured sequentially, as in the ref. 18 and the measurement of one property does not disturb the measurement of the other. More precisely, such a measurement can be implemented in a setup in which the system goes through the measuring device corresponding to A i and then through one of the two measuring devices corresponding to B j . The schematic representation of this setup is shown in Fig. 2b. Because A i and B j commute, the results of the measurements do not depend on their order, i.e., B j can be measured before A i . The measurements lead to four possible outcomes that we denote by (+ + |ij), (+ − |ij), (− + |ij) and (− − |ij). The result (+ − |ij) corresponds to A i = +1 and B j = −1.

A single run of the experiment makes one of the four detectors, placed after the outputs, click. The probabilities of these clicks are p(+ + |ij), p(+ − |ij), p(− + |ij) and p(− − |ij). They can be estimated after many experimental runs and used to evaluate correlations 〈A i B j 〉 = p(+ + |ij) − p(+ − |ij) − p(− + |ij) + p(− − |ij). One can observe violation of the CHSH inequality if in each experimental run the photon is prepared in the same special state and the measurements A i and B j are properly chosen. Although the setup is interpreted as a measurement of two random variables, it can also be viewed as a measurement of a single degenerate random variable X ij whose outcomes are products of the outcomes of A i and B j . Therefore, 〈A i B j 〉 = 〈X ij 〉.

What would happen if in a single experimental run one used many identical photons or a classical beam of light? From our initial discussion we know that the intensities at the outputs would be proportional to Np(+ + |ij), Np(+ − |ij), Np(− + |ij) and Np(− − |ij), where N is the number of photons. In the classical limit one would deal with a beam of light whose intensities would be I(+ + |ij), I(+ − |ij), I(− + |ij) and I(− − |ij). Moreover, I(+ + |ij)/I = p(+ + |ij), etc., where I is the input intensity.

In addition, one could consider a random variable

$${\cal X}_{ij} = \frac{{n\left( { + + |ij} \right) - n\left( { + - |ij} \right) - n\left( { - + |ij} \right) + n\left( { - - |ij} \right)}}{N},$$ (5)

where n(+ + |ij) is the number of photons in the output (+ + |ij), etc. For a single photon 〈A i B j 〉 = 〈X ij 〉 = \(\left\langle {{\cal X}_{ij}} \right\rangle\). For N > 1 it is impossible to assign definite values to A i , B j and to assign X ij to individual photons. However, \(\left\langle {{\cal X}_{ij}} \right\rangle\) can be evaluated and in the classical limit one gets

$$\left\langle {{\cal X}_{ij}} \right\rangle = \frac{{I\left( { + + |ij} \right) - I\left( { + - |ij} \right) - I\left( { - + |ij} \right) + I\left( { - - |ij} \right)}}{I}.$$ (6)

Thus, it is possible to prepare a classical state of light such that

$$\left\langle {{\cal X}_{11}} \right\rangle + \left\langle {{\cal X}_{01}} \right\rangle + \left\langle {{\cal X}_{10}} \right\rangle - \left\langle {{\cal X}_{00}} \right\rangle = \pm 2\sqrt 2 .$$ (7)

The above may lead to a discussion whether the classical light has some nonclassical properties.7,8,10,12,13,14,16,17,18 In the following sections we show that for more than one photon the classical bound is different than ±2. One needs to remember that although \(\left\langle {{\cal X}_{ij}} \right\rangle\) does not depend on N, the random variable \({\cal X}_{ij}\) and the corresponding exclusivity structure of events strongly depends on N, therefore in order to understand what is really going on it is better to examine the CHSH scenario from the point of view of events, not averages.

Exclusivity and classical bounds

The contextuality and non-locality scenarios can be studied within the graph theoretical model developed by Cabello, Severini, and Winter.22 This model is based on the exclusivity structure of measurable events and offers three different approaches to deriving bounds on sums of probabilities of measurable events. The first approach assigns logical truth/false values to each event and the resulting bound on the sum of probabilities, obtained by optimising over all possible assignments, is known as the classical bound. The second approach is equivalent to the quantum mechanical way of assigning probabilities to the events, which is based on the Born rule. The corresponding quantum bound is known as Lovasz theta function and is equal or greater than the classical bound. Finally, the third approach uses the minimal assumption, namely that the sum of probabilities of mutually pairwise exclusive events is bounded by one. This assumption is known as the E-principle25 and the corresponding bound is known as the non-signaling bound, since general non-signaling theories obey the E-principle. The non-signaling bound is equal or grater than both, classical and quantum bounds. It is worth to mention that there is a growing interest in finding ways to derive quantum bounds from non-signaling bounds using additional physical assumptions.25,26,27 In the following we discuss the CHSH scenario from the point of view of its exclusivity structure.

The CHSH inequality can be rewritten with probabilities of detection events. Since 〈A i B j 〉 = 1 − 2p(+ − |ij) − 2p(− + |ij) = 2p(+ + |ij) + 2p(− − |ij) − 1, the inequality (3) becomes

$$\begin{array}{r}p\left( { + - |11} \right) + p\left( { - + |11} \right) + p\left( { + - |01} \right) + \\ p\left( { - + |01} \right) + p\left( { + - |10} \right) + p\left( { - + |10} \right) + \\ p\left( { + + |00} \right) + p\left( { - - |00} \right) \le 3.\end{array}$$ (8)

This inequality can be derived in a completely different way. The upper bound equal to three comes from the exclusivity structure of events. Firstly, the events (+ + |ij), (− − |ij), (+ − |ij), and (−+ |ij) are pairwise exclusive. This is because they correspond to different outcomes of the same measurements. For example, (+ + |00) cannot happen together with (− − |00). In addition, two events are exclusive if they share the same measurement settings and the corresponding outcomes are different. This means that (+#|ij) is exclusive to (−#|ik) and (# + |ij) is exclusive to (# − |kj); Here # denotes an arbitrary outcome. For example, (+ − |10) is exclusive to (− + |11) and (− − |00) is exclusive to (− + |10). Such example can be realised in quantum theory by events corresponding to projections onto states |0〉 ⊗ |0〉 and |1〉 ⊗ (α|0〉 + β|1〉). Although |0〉 and α|0〉 + β|1〉 are in general nonorthogonal states, the exclusivity is provided by the orthogonality of |0〉 and |1〉 in the first Hilbert space. Verification of this type of exclusivity can be implemented in the sequential scenario represented in Fig. 2b in which the second left measuring device is set to B 0 and the second right to B 1 .

The exclusivity structure of the eight events can be represented with the exclusivity graph22 whose vertices correspond to events and edges to exclusivity between two events, see Fig. 2c. The upper bound of (8) is derived under assumption that the eight events are jointly distributed.28 The joined probability distribution (JPD) is constructed over all possible assignments of 1/0 (truth/false) values to these events. In principle there are 28 possible assignments, however the value 1 cannot be simultaneously assigned to two exclusive events. This significantly reduces the number of possible assignments. The maximum value of the sum of the eight probabilities is given by the maximal number of events that can be assigned the value 1. The problem of finding the maximal number of events that can be assigned 1 is equivalent to the graph theoretical problem known as maximum independent set.22 An independent set of a graph is a set of disconnected vertices. We are looking for a set with the largest possible number of vertices. In general, it is an NP-hard problem but it is solvable for our graph. Note, that the set of events that are assigned 1 must correspond to the independent set of the exclusivity graph, since two events from such set cannot be exclusive. It is easy to find that the maximum independent set of the graph from Fig. 2c contains three vertices. Therefore, the sum of the eight probabilities cannot be larger than three if these probabilities originate from some JPD. Not surprisingly, quantum theory can go as high as \(2 + \sqrt 2\) and it cannot be modelled with any JPD.

(Non-)contextuality of many indistinguishable particles and a proper classical bound

Hidden variable theories that aim to explain standard photonic experiments associate outcomes with detector clicks. However, here we show that this approach cannot be applied to experiments where more than one indistinguishable photon arrives at the detector. Instead, we propose an alternative hidden variable model, which provides a classical explanation of the Bell-like experiments done on electromagnetic waves. Within the entire discussion we adapt an operational approach in which a photon, or in general an indistinguishable particle, is identified with a detector click. The 1/0 assignment corresponds to a deterministic non-contextual (NC) model. The photon is assigned at most one event from each set of pairwise exclusive events. Such a set makes a measurement context—a set of events that can be jointly measured. If a context is complete, i.e., it consists of all possible measurement outcomes, the photon is assigned exactly one event. However, in the scenario considered here all contexts are not complete and contain exactly two events.

To properly discuss the problem of non-classicality of correlations in Bell-type scenarios for classical light, we need to redefine the introduced exclusivity graph model so that a transition from a single photon to a macroscopic electromagnetic wave is transparent. Instead of assigning events to a photon one should tie a photon to an event. This is a subtle difference but it leads to fundamental consequences once we deal with more than one photon. More precisely, a photon is assigned to at most one single event in each measurement context, where 1 corresponds to a photon and 0 to no photon event. In this new picture the events can be considered as modes and 1/0 as occupation numbers. The NC model assigns a well defined occupation number to each mode. The exclusivity of modes leads to conservation of the particle number—since there is a single photon in the system there could be at most a single photon in each context. If two exclusive events (modes) were assigned one, then there would exist a context containing two photons, which would contradict conservation of the particle number. The above interpretation was proposed for the first time in the ref. 29 This approach is discussed in details in the Methods section. It should be emphasised that the introduced model is very general and describes the single-photon-to-classical-wave transition in Bell-type scenarios irrespective of the direct physical implementation, which may be introduced in many different scenarios.7,10,14,16,17,18,19

In order to make our discussion as general as possible and not restricted to the case of classical and quantum optics let us consider the above described model for arbitrary indistinguishable particles. Our generalised model allows for arbitrary schemes of assigning particles to modes, therefore no physical constraints on the statistics (bosonic or fermionic behaviour), correlations and possible interactions are made. The only physical assumption we impose, is that there exists a macroscopic limit of a strong beam of the particles possessing two properties. The first one states that the occupation number ratios tend to intensity ratios of the macroscopic beam. The second one assumes that the ratio of the standard deviation of the particle number in a given mode to the average particle number tends to zero:

$$\frac{{\sigma _n}}{{\left\langle n \right\rangle }} \to 0.$$ (9)

The last assumption guarantees that the ratios of intensities of the macroscopic beam are fully deterministic. Note that the physical example of single-photon-to-classical-wave transition satisfies both conditions, since macroscopic EM wave can be treated as a limit of a strong beam of photons in a coherent state with particle number fluctuations following Poissonian statistics with standard deviation σ n of the form \(\sqrt {\left\langle n \right\rangle }\). In the introduced model the notion of exclusivity changes its implementation. Namely let us assume that at most N photons can be assigned to a single event. Then each experimental context, that is any set of mutually exclusive events, can be also filled with at most N photons.

Let us now discuss the bound for a CHSH inequality within the above model. One can rewrite the inequality (8) as

$$\begin{array}{r}\left\langle {n( + - |11)} \right\rangle + \left\langle {n( - + |11)} \right\rangle + \left\langle {n( + - |01)} \right\rangle + \\ \left\langle {n( - + |01)} \right\rangle + \left\langle {n( + - |10)} \right\rangle + \left\langle {n( - + |10)} \right\rangle + \\ \left\langle {n( + + |00)} \right\rangle + \left\langle {n( - - |00)} \right\rangle \le {\cal C},\end{array}$$ (10)

where n(+ + |ij), etc., are occupation numbers of the corresponding events and \({\cal C}\) is the NC bound on the sum of these numbers, which in the case of a single particle equals to three. A single particle violates this bound.

Now, consider the same CHSH scenario, but this time inject two indistinguishable particles to the system. The exclusivity and particle number conservation imply that there could be at most two particles per context. The possible occupation numbers are 0, 1, or 2. Since each context consists of only two events, one can assign a single particle to each event, see Fig. 3. Therefore, for two particles \({\cal C} = 8\), which is the maximal possible sum of non-contextually assigned occupation numbers over all events. We see that the bound depends on the number of particles. We divide (10) by N and rewrite it as

$$\begin{array}{r}\frac{{\left\langle {n( + - |11)} \right\rangle }}{N} + \frac{{\left\langle {n( - + |11)} \right\rangle }}{N} + \frac{{\left\langle {n( + - |01)} \right\rangle }}{N} + \\ \frac{{\left\langle {n( - + |01)} \right\rangle }}{N} + \frac{{\left\langle {n( + - |10)} \right\rangle }}{N} + \frac{{\left\langle {n( - + |10)} \right\rangle }}{N} + \\ \frac{{\left\langle {n( + + |00)} \right\rangle }}{N} + \frac{{\left\langle {n( - - |00)} \right\rangle }}{N} \le {\cal P}_C(N),\end{array}$$ (11)

where \({\cal P}_C(N) = C{\mathrm{/}}N\). For a single particle \({\cal P}_C(1) = 3\), whereas for two particles \({\cal P}_C(2) = 4\). Therefore, for N = 2 there is no violation and the measurements can be described by NC occupation number assignments.

Fig. 3 Examples of particle-assignments to events in the CHSH exclusivity graph. Left—one particle, right—two particles Full size image

As far as we know, the value \({\cal P}_C(2) = 4\) cannot be reached in any experimental setup, although it is allowed in our model. This is becasue the maximal quantum value of \(2 + \sqrt 2\) (attainable in the CHSH scenario) does not depend on the physical implementation of the experiment. In particular it does not depend on the dimension of the state space of the physical system, which in our case translates to independence on the particle number N.

Finally, let us consider the classical limit \(\left\langle N \right\rangle \gg 1\). This time the system is described by a classical beam of intensity I for which the inequality (11) reads

$$\begin{array}{r}\frac{{I( + - |11)}}{I} + \frac{{I( - + |11)}}{I} + \frac{{I( + - |01)}}{I} + \\ \frac{{I( - + |01)}}{I} + \frac{{I( + - |10)}}{I} + \frac{{I( - + |10)}}{I} + \\ \frac{{I( + + |00)}}{I} + \frac{{I( - - |00)}}{I} \le {\cal P}_{{\mathrm{cl}}},\end{array}$$ (12)

where N in the denominator of (11) was replaced by 〈N〉 due to the particle number uncertainty. The maximal experimentally attainable value of the left-hand side for an EM wave is still \(2 + \sqrt 2\) because the classical light beam in any linear optical setup behaves in the same way as a single-photon probability amplitude. The right-hand side can be evaluated in two ways. Firstly, in the classical limit the total intensityI that is distributed between the events can be treated as a continuous property. Therefore, in the NC model one can assign I/2 to each event and as a result \({\cal P}_{{\mathrm{cl}}} = 4\). This is the main result in this section: The corresponding CHSH inequality (12) cannot be violated by any macroscopic beam following assumptions of our model, like for example the classical light.

The other approach, which also confirms the above result, does not assume that the intensity is a continuous property. We consider two cases. First, let us take even N. The number of particles per context cannot be greater than N and since each context contains two events, one simply assigns N/2 particles per context. This leads to \({\cal P}_C(N) = 4\). Next, we consider odd N. In this case it is easy to show that one can assign (N − 1)/2 particles to five events and (N + 1)/2 particles to three events, such that there are at most N particles per each context, see Fig. 4. As a result one gets \({\cal P}_C(N) = 4 - {\textstyle{1 \over N}}\). In the classical limit N is undetermined, therefore \({\cal P}_{{\mathrm{cl}}} = \left\langle {{\cal P}_C(N)} \right\rangle\). However, since \(\left\langle N \right\rangle \gg 1\) the dominating terms in \(\left\langle {{\cal P}_C(N)} \right\rangle\) correspond to large values of N and hence \({\cal P}_{{\mathrm{cl}}} \approx 4\).

Fig. 4 Assignment of particles to events in the exclusivity graph of the CHSH scenario. Left—even number of particles, right—odd number of particles, where N + = (N + 1)/2 and N − = (N − 1)/2 Full size image

To conclude, we see that in the experiments discussed above light beams, as expected, do not exhibit any quantum behaviour. Quantum behaviour is only possible for a single photon and already for N > 1 one observes noncontextual (classical) behaviour. This is because for N ≥ 2 the non-contextual bound \({\cal P}_C(N)\) is either 4 or \(4 - {\textstyle{1 \over N}}\) and is always greater than the physically attainable value of \(2 + \sqrt 2 \approx 3.41\). We have only considered the CHSH scenario, however in the Methods section we show that for an arbitrary contextuality scenario the bound \({\cal P}_{cl}\) is always greater or equal than what can be achieved in the classical limit and therefore classical systems are always noncontextual and can never violate any Bell-type inequality.