Introduction

Bell nonlocality stands out as a hallmark of quantum theory, defying our classical conception of reality1,2,3,4. It boils down to strange correlations between outcomes of space-time separated measurements that cannot be explained in the usual causal manner implied by the experimental design, i.e. only as an effect of preparation at the source. Quantum theory predicts the appearance of such correlations when the system is prepared in an entangled state, which has been confirmed in a number of ingenious experiments5,6,7,8,9. Inherent to the notion of entanglement is the concept of a composite system described as the tensor product of subsystems, which presupposes that a full range of measurements on individual subsystems can be made10,11. This raises the subtle question of whether a given quantum state can actually demonstrate Bell non-locality. The answer will crucially depend on how the subsystems are defined and which measurements are allowed in the experiment.

The issue is especially relevant in the context of identical particles. According to the fundamental principle of indistinguishability, multi-particle systems are described by entangled states due to the symmetrisation or anti-symmetrisation of the wave function12,13. However, in the second-quantised theory, this fact is obscured by moving to the mode description of the system, where individual particles cannot be directly addressed, thereby preventing explicit probing of subsystems where this entanglement resides. A sensible criterion of entanglement for identical particles is therefore challenging14,15,16,17,18,19,20,21,22, which suggests a shift in focus towards operational notions of nonclassicality. One such important notion concerns whether and how states of identical particles can lead to non-local correlations that can be experimentally observed. This allows us to deliberately bypass the question of entanglement for a system of identical particles and instead ask directly about non-local correlations. Notably, the latter pertains to experimentally observed correlations rather than being a mathematically driven construct.

In this work, we treat passive linear optics (comprised of mirrors, phase shifters, beam splitters, and detectors) as a purely classical component of the experiment. From this perspective, any non-classical behaviour observed must have its origin in the remaining part of the experiment, and thus be attributed to the state that is fed into the classical optical setup. See Fig. 1 for illustration. This view provides a straightforward operational way to assess non-classical potential of a given quantum state with respect to passive linear optics.

Fig. 1: Passive linear optical experiment.

A quantum state (\vert \psi\rangle) of N identical particles (bosons or fermions) enters a classical optical setup built of paths, mirrors, phase shifters, beam splitters, and detectors. Arranged in various configurations, optical elements implement a unitary transformation ({\mathbb{U}}) on each particle27. Then, the modes/paths are distributed to different locations A, B, C…, where the particles can be further processed and finally detected. Each experiment has the potential to reveal non-local correlations in the observed statistics. Here, the state (\vert \psi \rangle) processed with the linear optical setup ({\mathbb{U}}) is interpreted as a source of particles for a Bell test that is performed at distant locations A, B, C, … .

Our research problem has a simple expression in the following question. For a given quantum state of identical particles, we ask:

Does there exist some passive linear optical experiment capable of demonstrating Bell non-locality?

A positive answer indicates that the state is a genuine non-local resource. Conversely, the lack of any such experiment means that the correlations observed in every optical setup can be explained by a generic local hidden variable model. Then, the state fails to provide adequate non-local resources. For example, it was shown that every single-particle state, for both bosons and fermions, can be locally simulated in a generic way23.

Note that the essence of such a formulated problem lies in the arbitrary nature of passive linear optical experiments, which can be composed in any possible manner. Furthermore, our query covers every single state within a sufficiently broad class of interest. In this article, the problem is fully solved for multi-particle states with a definite number of identical particles for both boson and fermion statistics.

We show that all fermion states and the vast majority of boson states provide a genuine non-local resource (with respect to passive linear optics), except for a narrow class of states that are reducible to a single mode.

See Fig. 2 for illustration. This result hints at an intriguing link between the basic concept of indistinguishability of particles and the puzzling feature of Bell non-locality within the framework of classical optical setups.

Fig. 2: Classification of states as a non-local resource (with respect to passive linear optics).

An illustration of fermion and boson states in Eq. (1)/(2). The extension of the bands depicts a wider range of boson states than fermion states (due to the Pauli exclusion principle), and the vertical wiggling line separates the single- and non single-mode type states described in Definition 1. Our main result Theorem 1 shows that, for states with a definite number of identical particles, the distinction between the states providing local vs non-local resource (with respect to passive linear optics) corresponds to a division into single-mode type vs non single-mode type states. More specifically, all fermion states provide a non-local resource, except for single-particle states (N = 1). While in the richer family of boson states, the only ones which are incapable of demonstrating non-locality are states reducible to a single mode.

Results

Informal statement of the problem

Suppose we lived in a purely classical world and someone presented us with a quantum state of N identical particles (either fermions or bosons). Would we be able to recognise this new quality with our classical tools? By classical tools, we mean standard passive linear optics, which includes mirrors, phase shifters, beam splitters, and detectors that can be arranged in various configurations. Arguably, these elements are found in any classical optical laboratory, and electromagnetic waves do not exhibit non-classical behaviour in experiments of this type24,25. What can be imagined is that such a quantum state is fed into an optical circuit, processed in any conceivable way, and then the particles are detected in different locations. See Fig. 1. The pattern of correlations observed in such an experiment will be deemed non-classical only if they cannot be explained by a local model with some ‘stuff’ (particles, fields and/or hidden variables) propagating along the paths as specified by the experimental design. Then, the state can be called a genuine non-local resource. Otherwise, there is an account of every optical experiment that does not require any non-local mechanisms, and therefore the state should not give rise to surprise.

The problem is formulated by thinking of the quantum state as a potential non-classical resource with respect to a fixed set of operations (i.e. passive linear optics and detectors). Specifically, we are interested in the following question:

Which quantum states of identical particles can lead to non-local behaviour in classical experiments composed of passive linear optical transformations and detectors?

For each particular state, it has two possible answers:

1. Yes:

The state is a genuine non-local resource. It means that the state can be used in some experimental procedure to demonstrate non-local correlations in space-time separated regions (by violating some Bell-type inequality in some specific experiment).

1. No:

The state does not provide a genuine non-local resource. That is, the correlations observed in any conceivable experiment can be explained in a local manner (via some generic model in which hidden variables propagate locally, following the pattern of connections in the experimental design).

This splits the space of quantum states into two disjoint classes. The first class can fuel classical optical setups to operate at a non-local level. The second class appears to be classical-like, i.e. simulable with local resources, and hence one should not expect a non-classical advantage in passive linear optics with those states.

Here, we are interested in the classification of all quantum states with a definite number of particles (N is fixed). Notably, for a single particle (N = 1), the question was resolved in ref. 23, where a generic local hidden variable model of arbitrary passive linear optical experiments was constructed (i.e. single-particle states do not provide a non-local resource). However, a far more complex and interesting problem is that of multi-particle systems (N ⩾ 2). We show that all fermion states and most boson states provide a genuine non-local resource.

Passive linear optical toolkit

In the occupation number representation (Fock space), a general state of N identical particles is described by

$$\vert \psi \rangle ,=\sum _{{n}_{1}+\ldots +{n}_{M}=N}{\psi }_{{n}_{1}\ldots, {n}_{M}}\vert {n}_{1}\ldots, {n}_{M}\rangle ,$$

(1)

which is as a (complex) combination of states (\vert {n}_{1}\ldots {n}_{M}\rangle) with a definite number of particles n**i in the respective modes i = 1, … , M. Those numbers are constrained by the particle statistics: for bosons ({n}_{i}\in {\mathbb{N}}), and for fermions n**i = 0, 1. In the following, we will consider situations where the total number of particles N is fixed and without any restriction on the number of modes M. Furthermore, since we are concerned with locality, we will focus on optical modes of the system, i.e. think of each mode as a different path along which particles can propagate.

It will be convenient to use the standard representation in terms of the creation/annihilation operators, which add/subtract a particle in the respective mode of the system. Then Eq. (1) can be written in the form

$$\vert \psi \rangle ,=\sum _{{n}_{1}+\ldots +{n}_{M}=N}{\psi }_{{n}_{1}\ldots ,{n}_{M}},\mathop{\prod }\limits_{i=1}^{{M},\frac{{{a}_{i}}{\dagger }}^{{n}_{i}}}{\sqrt{{n}_{i}!}},\vert 0\rangle ,$$

(2)

where (\left\vert 0\right\rangle) is the vacuum state and the creation and annihilation operators, ({a}_{i}^{\dagger }) and a**i, satisfy the usual commutation relations specific for the respective boson or fermion statistics12,13,[26](https://www.nature.com/articles/s41534-025-01086-x#ref-CR26 “For reference, we note that the conventional commutation relations are defined as follows:

$${[{a}{i},{a}{j}]}{\mp }={[{a}{i}^{{\dagger },{a}_{j}}{\dagger }]}{\mp }=0,,,{[{a}{i},{a}{j}^{\dagger }]}{\mp }={\delta }_{ij}$$

[

a

i

,

a

j

]

∓

=

[

a

i

†

,

a

j

†

]

∓

= 0

,

[

a

i

,

a

j

†

]

∓

=

δ

i j

, where the difference between the boson/fermion statistics boils down to the use of the commutation/anti-commutation relations (∓). We use their canonical representation whose action in a given mode for bosons is given by

$${a}{i}^{\dagger }\vert {n}{i}\rangle =\sqrt{{n}{i}+1},\vert {n}{i}\rangle$$

a

i

†

|

n

i

⟩

n

i

1

|

n

i

⟩

and

$${a}{i}\vert {n}{i}\rangle =\sqrt{{n}{i}},\vert {n}{i}-1\rangle$$

a

i

|

n

i

⟩

n

i

|

n

i

− 1 ⟩

, while for fermions we have

$${a}{i}^{\dagger }\vert {0}{i}\rangle =\vert {1}_{i}\rangle$$

a

i

†

|

0

i

⟩

|

1

i

⟩

,

$${a}{i}\vert {1}{i}\rangle =\vert {0}_{i}\rangle$$

{a}{i}\vert {1}{i}\rangle =\vert {0}_{i}\right\rangle

and

$${a}{i}^{\dagger }\vert {1}{i}\rangle ={a}{i}\vert {0}{i}\rangle =0$$

a

i

†

|

1

i

⟩

a

i

|

0

i

⟩

0

. This complies with the restriction on the occupation numbers for bosons

$${n}_{i}\in {\mathbb{N}}$$

n

i

∈ N

, and for fermions ni = 0, 1. Then, the number states can be written in a compact form

$$\vert {n}{1}\ldots {n}{M}\rangle =\frac{1}{\sqrt{{n}{1}!,\ldots ,{n}{M}!}},{a}{1}^{\dagger ,{n}{1}}\ldots ,{a}{M}^{\dagger ,{n}{M}}\vert 0\rangle$$

|

n

1

…

n

M

⟩

1

n

1

!

…

n

M

!

a

1

†

n

1

…

a

M

†

n

M

| 0 ⟩

. See refs. 12,13 for a full exposition.“).

In the following, we will be interested in transformations of the Fock states that can be implemented with passive linear optics, i.e. a sequence of single-mode and two-mode operations (or gates) that are conventionally realised by mirrors, phase shifters and beam splitters. Such transformations are described as follows

$${a}_{i}^{{\dagger }\ \mathop{\longrightarrow }\limits}{{\small \mathbb{U}}}\ {a}_{i}^{{{\prime} \dagger },=,\mathop{\sum }\limits_{j=1}}{M}{U}_{ij},{a}_{j}^{\dagger },$$

(3)

for each mode i = 1, … , M of the system, and ({\mathbb{U}}\equiv [{U}_{ij}]) is a unitary transformation on ({{\mathbb{C}}}^{{M}). Furthermore, it is known that any unitary ({\mathbb{U}}) on a single-particle space ({{\mathbb{C}}}}{M}) can be realised by passive linear optics (i.e. can be obtained as a sequence of single- and two-mode operations implemented by mirrors, phase shifters and beam splitters)27. However, we note that this is not enough to generate any transformation on the multi-particle Fock space since under passive linear optics, the state space splits into a myriad of non-equivalent continuous classes whose full classification is beyond reach, see ref. 28.

For further convenience, let us formally distinguish a special class of states that will play an important role in the following discussion.

Definition 1

A state is said to be of a single-mode type if it can be reduced to a single mode by passive linear optics. That is, for a state in Eq. (1)/(2), it means that there exists a unitary ({\mathbb{U}}) on ({{\mathbb{C}}}^{M}) (or a sequence of optical elements) such that

$$\vert \psi \rangle ,=,{\mathbb{U}},\left\vert N,0,\ldots ,0\right\rangle ,=,\frac{{a}_{1}^{{\prime} \dagger N}}{\sqrt{N!}},\vert 0\rangle .$$

(4)

Equivalently, those states can be thought of as having evolved from the initial state in which all the particles were in the same mode. A straightforward consequence of ref. 27 is that all single-particle states (N = 1) are of a single-mode type (for both bosons and fermions). Clearly, in the multi-particle case (N ⩾ 2), the class of single-mode type states is nontrivial only for bosons (it is empty for fermions due to the Pauli exclusion principle). We hasten to note that Definition 1 boils down to a few easy-to-check conditions; see Supplementary Information for a discussion.

Bell non-locality in optical experiments

Every experiment takes place in space and time. This is particularly manifest in optical setups that are composed of basic elements distributed in different positions and interconnected by optical paths into a larger circuit. In a typical optical experiment, an initial state (\left\vert \psi \right\rangle) is fed into a circuit that is responsible for evolving the system and registering the particles at specific locations. Here, without the loss of generality, the basic components of passive linear optical setups can be considered as single- and two-mode gates, such as mirrors, phase shifters, beam splitters and number-resolving detectors27.

The spatio-temporal arrangement of optical designs has implications for the possible interpretations of the experiment. Namely, it is common to think that the flow of information (or propagation of causes) follows the pattern of connections in the circuit. That is, its physical carriers propagate along the paths in the circuit (whether they are particles, fields or hidden variables), and the information is processed in a modular manner. This view stems from the principle of local causality, which is especially appealing for space-like separated arrangements. See Fig. 1. It was Bell’s fundamental insight that quantum correlations can be at odds with the causal structure implied by the design of an experiment1,2,3,4. In the optical framework, this idea can be expressed in the following manner.

Definition 2

An optical experiment demonstrates non-local behaviour (or Bell non-locality) if the observed correlations cannot be explained with the variables tracing the pattern of connections in the experimental setup.

In other words, there is no local hidden variable model explaining the observed experimental statistics, where locality requires that the variables propagate from one optical element to another along the paths determined by the circuit. In practice, demonstrating non-locality in a given experiment boils down to a violation of some Bell-type inequality derived from the corresponding causal structure. For example, in Fig. 1, one interprets the state (\vert \psi \rangle) together with the optical setup ({\mathbb{U}}) as a source of particles that are sent to distant locations A, B and C where a Bell test can be performed[29](https://www.nature.com/articles/s41534-025-01086-x#ref-CR29 “Let us remark that it is a simplified analysis that ignores the internal structure of the optical setup that is lumped into the transformation

$${\mathbb{U}}$$

U

. If this is not enough to demonstrate non-locality, one can look further into the causal pattern of connections in the experimental design, since it can impose much stronger constraints on local classical models, as demonstrated in the recently developed methodology of non-local networks53. However, for our purpose, there will be no need for the latter approach, as our proofs of non-locality will utilise the usual Bell test performed on appropriately prepared states.“). The opposite statement, regarding the apparent lack of non-locality, can only be justified by constructing an explicit local hidden variable model that is compatible with the experimental setup30.

Main result

An optical experiment can be seen as a test of certain properties of the state that is supplied to the circuit. In Definition 2, the question of non-locality is posed for a given experimental setup, which includes both the state and the optical design. In this paper, we are interested in the property of the state itself without restricting it to a particular implementation. We ask whether the state can demonstrate Bell non-locality by admitting a wider range of possible arrangements. This idea comes from thinking about the state as a resource of non-locality in a larger class of experimental designs. Here we focus on the use of passive linear optics, as defined below.

Definition 3

A state (\vert \psi \rangle) is a genuine non-local resource with respect to passive linear optics if it is capable of manifesting non-local behaviour in some passive linear optical experiment.

Note that to demonstrate non-locality of a given state, it is sufficient to provide a single example of a passive linear optical circuit in which Bell non-locality can be observed. However, showing the opposite for a given state requires constructing a generic local hidden variable model explaining correlations obtained in every conceivable experiment with passive linear optical setups.

Our main result takes the form of a simple criterion for states with a definite number of particles in Eq. (1)/(2).

Theorem 1

A state (\vert \psi \rangle) is a non-local resource with respect to passive linear optics if, and only if, it is not of the single-mode type.

This provides a straightforward classification of multi-particle states based on a simple condition Definition 1 and easy-to-understand interpretation. Interestingly, the only states that lack the non-local potential are those originating in the same mode. This leads to the following characteristics of states for both boson and fermion statistics; see illustration in Fig. 2.

Corollary 1

Neither fermions nor bosons provide a non-local resource for any single-particle state (N = 1). For the multi-particle case (N ⩾ 2), all fermion states are non-local resources, while some boson states are not (i.e., those of a single-mode type).

We refer to the Methods section for the detailed proof of Theorem 1. Let us conclude by unfolding a partial result which facilitates the full proof.

Yurke-Stoler test: A useful lemma

The key tool in our discussion of non-locality in the interferometric setups is the co-called Yurke-Stoler test. It is designed for the special case of two particles (N = 2) and two modes (M = 2), and asks about Bell non-locality of the following general state

$$\vert \phi \rangle ,=,\left(\alpha ,\frac{{{a}_{1}^{{\dagger }}}{2}}{\sqrt{2!}}+\beta ,{a}_{1}^{{\dagger },{a}_{2}}{\dagger }+\gamma ,\frac{{{a}_{2}^{{\dagger }}}{2}}{\sqrt{2!}},\right)\vert 0\rangle ,$$

(5)

where ∣α∣2 + ∣β∣2 + ∣γ∣2 = 1 in the boson case, whilst for fermions we have ∣β∣2 = 1, α = γ = 0. In the following, we show how the seminal idea in ref. 31 can be refined to serve the purpose at hand.

Yurke-Stoler test checks for a violation of Bell inequalities in the design depicted in Fig. 3. It takes the state Eq. (5) in two input modes 1 and 2, that are split into two dual-rail qubits {1,1’} and {2,2’}. Then the modes 1’ and 2’ get swapped, and the dual-rail qubits are sent out to Alice and Bob, who perform a typical Bell experiment. We note that the choice of settings on Alice and Bob’s side is unrestricted since any projective measurement on their dual-rail qubits can be implemented with passive linear optics. The protocol involves post-selection, which retains only those experimental runs in which a single particle is detected on Alice and Bob’s side (i.e. when the dual-rail qubits are well-defined). Notably, the specifics of post-selection used in the experiment do not compromise the conclusions from the violation of Bell inequalities when the number of particles is conserved, as shown in refs. 32,33. It means that the protocol provides a genuine test of non-local correlations in the system.

Fig. 3: Yurke-Stoler test.

A two-particle state, Eq. (5), in modes 1 & 2 is split by two beam splitters (Hadamard gates) into two dual-rail qubits {1,1’} and {2,2’}. Then the modes 1’ and 2’ get swapped and Alice & Bob make the usual Bell test on their (dual-rail) qubits when each one of them gets a single particle.

This scheme was originally proposed by Yurke and Stoler to prove Bell non-locality for two particles coming from independent sources31. For a recent discussion see ref. 34. Here, we build on this idea taking it as a test of Bell non-locality for a general two-particle state (\left\vert \phi \right\rangle) in Eq. (5), which via the setup in Fig. 3 evolves as follows

$$\vert \phi \rangle\ \mathop{\longrightarrow}\limits^{\small {H,{H}^{{-1}}}\ \left(\frac{\alpha ,}{2}\frac{{({a}_{1}}{\dagger }+{a}_{{1}^{{{\prime} }}}{\dagger })}^{{2}}{\sqrt{2!}}+\frac{\beta }{2},({a}_{1}}{\dagger }+{a}_{{1}^{{{\prime} }}}{\dagger }),({a}_{2}^{{\dagger }+{a}_{{2}}{{\prime} }}^{{\dagger })+\frac{\gamma }{2}\frac{{({a}_{2}}{\dagger }+{a}_{{2}^{{{\prime} }}}{\dagger })}^{2}}{\sqrt{2!}},\right)\vert 0\rangle$$

$$\mathop{\longrightarrow}\limits^{\small{{1}^{{{\prime} }\leftrightarrow ,{2}}{{\prime} }}}\ \left(\frac{\alpha ,}{2}\frac{{({a}_{1}^{{\dagger }+{a}_{{2}}{{\prime} }}^{{\dagger })}}{2}}{\sqrt{2!}}+\frac{\beta }{2},({a}_{1}^{{\dagger }+{a}_{{2}}{{\prime} }}^{{\dagger }),({a}_{2}}{\dagger }+{a}_{{1}^{{{\prime} }}}{\dagger })+\frac{\rm{\gamma }}{2}\frac{{({a}_{2}^{{\dagger }+{a}_{{1}}{{\prime} }}^{{\dagger })}}{2}}{\sqrt{2!}},\right)\vert 0\rangle$$

$$\mathop{\rightsquigarrow}\limits^{{post\hbox{-}select}}\ \left(\frac{\alpha}{\sqrt{2!}}, a_{1}^{{\dagger},a_{2}{\prime}}^{{\dagger} + \frac{\beta}{2}, a_{1}}{\dagger},a_{2}^{{\dagger} ,\pm,\frac{\beta}{2}, a_{1}{\prime}}^{{\dagger},a_{2}{\prime}}^{{\dagger} + \frac{\gamma}{\sqrt{2!}}, a_{1}{\prime}}^{{\dagger},a_{2}}{\dagger},\right),\vert 0\rangle,$$

(6)

where the “−” sign in “(\pm)” refers to fermions that anti-commute (recall in that case we also have ∣β∣2 = 1, α = γ = 0). Note that in the last line only the terms with a single particle in each dual-rail qubit {1,1’} and {2,2’} are retained. Clearly, this state is unnormalised due to post-selection, which succeeds with probability 1/2. As a result, Alice and Bob share a dual-rail encoded two-qubit state in the form

$$\vert \chi \rangle\ \sim\ \frac{\alpha }{\sqrt{2!}},\vert!! \uparrow \uparrow \rangle +\frac{\beta }{2},\vert!! \uparrow \downarrow \rangle \pm \frac{\beta }{2},\vert!! \downarrow \uparrow \rangle +\frac{\gamma }{\sqrt{2!}}\vert!! \downarrow \downarrow \rangle ,$$

(7)

where (\vert!!\uparrow\rangle \equiv \vert 10\rangle) and (\vert !! \downarrow \rangle \equiv \vert 01\rangle) is the computational basis for the respective dual-rail qubit {1,1’} and {2,2’}.

Alice and Bob then conduct the standard Bell test with an arbitrary choice of measurements on either side. It will only reveal non-local correlations for some choice of settings if, and only if, (\vert \chi \rangle) is an entangled state10. Conversely, the lack of non-local correlations observed in the Yurke-Stoler test requires that it must be a product state, i.e. we need to have β2 = 2 α**γ[35](https://www.nature.com/articles/s41534-025-01086-x#ref-CR35 “It is straightforward to see that the state

$$e\vert !! \uparrow \uparrow \rangle +f!\vert!! \uparrow \downarrow \rangle +g!\vert!! \downarrow \uparrow ! \rangle +h,\vert!! \downarrow \downarrow \rangle$$

e ∣

↑ ↑ ⟩ + f

∣

↑ ↓ ⟩ + g

∣

↓ ↑

⟩ + h

∣

↓ ↓ ⟩

is a product state if, and only if, eh = fg.“). Note that the latter condition implies that Eq. (5) simplifies to the single-mode type form (possible only for bosons)

$$\vert \phi \rangle ,=, \frac{1}{\sqrt{2!}}{\left(\sqrt{\alpha },{a}_{1}^{{\dagger }+\sqrt{\gamma },{a}_{2}}{\dagger }\right)}^{{2}\vert 0\rangle, =, \frac{{a}_{1}}{{\prime} \dagger 2}}{\sqrt{2!}},\vert 0\rangle .$$

(8)

This proves a helpful result that can be used to analyse more complex interferometric designs.

Lemma 1

The state (\vert \phi \rangle) in Eq. (5) does not manifest non-local correlations in the Yurke-Stoler test if, and only if, it is of a single-mode type, i.e., when we have β**2 = 2 αγ.

Note that for fermions, the Yurke-Stoler test is enough to demonstrate Bell non-locality for every two-particle (N = 2) and two-mode (M = 2) state. However, for the boson case, this setup is insufficient to prove non-locality for the single-mode type states. It will turn out that those states are locally simulable for any passive linear optical circuit. See Methods for the generic construction.

Discussion

The main interest of this work is to explore the boundary between the classical and quantum features of identical particles. We approach this task by drawing a line between the classical and quantum parts of experimental setups. This allows us to ask the question whether the state of the quantum part introduces a new quality that cannot be explained by classical means. In other words, can the quantum state itself be thought of as a genuine non-local resource? For this reason, in the classical part we include only standard optical means of processing the state, consisting of passive linear optics and single-mode detection. Note that if any additional quantum components were allowed, then the answer would refer to both the state and the extra quantum resources used in the experiment. However, this is a very different question. Therefore, seeking an unambiguous answer, we avoid using any additional resources, such as extra quantum particles or ancillary states. In particular, this excludes POVMs or homodyne detection from consideration—if these were included, the conclusion of non-classicality would refer jointly to the state and the auxiliary resource used in the experiment, cf. refs. 36,37,38,39,40,41. In short, we stand on the position that adding an ancilla introduces extra quantum particles being a potential source of non-local effects, which undermines the conclusion regarding the non-locality of the studied state itself in such extended experimental procedures. Therefore, we exclude them from considerations42.

Our result shows a fundamental difference between the system of particles that evolved from a single mode and those that evolved from separate modes. We coined the simple term single-mode type state, which turns out to fully characterise the class of states incapable of exhibiting non-local effects in any passive linear optical experiment. It was shown that, for all those states, there exists a generic local hidden variable model that explains the observed statistics. The constructed model can be viewed as the maximum achievable as regards local simulation of multi-particle quantum states in passive linear optics. On the other hand, we have also shown that for every non-single-mode type state, a passive linear optical experiment can be found that demonstrates Bell non-locality. As an aside, let us mention that the devised optical schemes can be unfolded in a non-trivial space-separated manner and, furthermore, fulfil the so-called ‘no-touching’ property; see refs. 31,43 and refs. 34,44 for a recent discussion. This completes the characterisation of non-local states with a definite number of identical particles, which boils down to a few easy-to-check conditions defining single-mode type states. We note that our result considerably extends the discussion in ref. 23, where only the single-particle case was discussed.

The derived classification of non-local resources applies to both bosons and fermions, as illustrated in Fig. 2. Interestingly, all multiparticle fermion states provide a non-local resource for passive linear optics, whereas for bosons, there is a non-trivial class of single-mode type states that lack this property. This difference is due to the Pauli exclusion principle, which for fermions rules out their single-mode origin. Notably, in the non-local sector, bosons admit a wider range of states which exhibit peculiar properties.

Note that the notion of non-classicality can have different meanings. Here, it is understood as the ability of a state to demonstrate non-local behaviour in passive optical setups. For a different computational perspective, take the boson sampling protocol that also works in optical designs45. It requires states being a non-local resource according to our classification, but for fermions, the computational advantage is lost. Another example is the difference in performance of bosons and fermions in various universal quantum computation schemes, e.g. see the KLM scheme46 with bosonic linear optics (for fermions it is not universal). Let it serve as an illustration of the intricate relationship and differences between the fundamental concept of Bell non-locality, complexity and quantum advantage in information processing with identical particles. Those aspects go beyond the scope of the present paper; see refs. 14,20,21,22 for a discussion of those topics.

We remark that our problem can also be phrased in the modern framework of resource theories, e.g. ref. 47. In that language, single-mode type states constitute free states, and passive linear optics with single-mode detection define the set of free transformations. Then, any other multi-particle state provides a non-local resource. Another interesting perspective is offered by the reference frames framework18. Notably, in our problem, we are handicapped by the lack of an external reference frame, since we limit ourselves to particle detection only (and the number of particles is preserved). From the reference frames viewpoint, this introduces certain constraints on observables that can be effectively measured (for example, see the discussion of non-locality of the superposition of a single particle in two modes in ref. 18). Those constraints predicted within the reference frames framework, via so-called superselection rules, crucially depend on the details of the physical implementation of the experimental setup, like the type of particles used (e.g. massive/massless, bosonic/fermionic) or the characterization of sources employed in the experiment (e.g. phase stability of used lasers). However, our approach is different as we do not utilise the reference frame approach, aiming at implementation-neutral results.

Finally, two general comments should be made about the physical focus of this research problem. Firstly, this work is not about the property of entanglement, which is a well-studied theoretical concept for distinguishable systems that is well-defined within the mathematical formalism of quantum theory11. It is about the property called Bell non-locality1,2,3,4, which concerns causal explanations of correlations observed in certain experimental setups that are crucially related to their spatial design (here, considered to be composed of passive linear optics and single-mode detection). This attitude is motivated by the fact that the definition of entanglement for a system of indistinguishable particles is challenging due to the mode description in the second-quantised theory15,16,17,18,19,20,[21](#ref-CR21 “Lo Franco, R. & Compagno, G. Indistinguishability of elementary systems as a resource for quantum information processing. Phys. Rev.