Local model of a qubit in the interferometric setup

We consider a typical realization of a qubit as a single particle in two-path interferometric circuits built from phase shifters, beam splitters and detectors. This framework is often taken as a standard example illustrating various paradoxes and quantum effects, including non-locality. In this paper we show that it is possible to simulate the behaviour of such circuits in a classical manner using stochastic gates and two kinds of particles, real ones and ghosts, which interact only locally. The model has built-in limited information gain and state disturbance in measurements which are blind to ghosts. We demonstrate that predictions of the model are operationally indistinguishable from the quantum case of a qubit, and allegedly 'non-local' effects arise only on the epistemic level of description by the agent whose knowledge is incomplete due to the restricted means of investigating the system.


I. INTRODUCTION
Quantum mechanics presents a challenge to many classical concepts that we hold about the world. In particular, it defies the very essence of particle ontology which says that a particle is localised only in one place at a time and interacts only with objects in its immediate vicinity. It was profound insight of John S. Bell [1,2] to point out that quantum mechanics admits correlations between particles which contradict the assumption of local realism. As a consequence, to recover quantum predictions in a realistic hidden-variable model one has to resort to spooky action at a distance and thus violate the paradigm of locality. In a similar manner it is often argued that in the case of a single particle a kind of non-local influence is also required to account for the effects associated with the collapse of the wave function. For illustration of this type of reasoning it is enough to consider simple interferometric setups, e.g. see the analysis of single-particle interference in the Mach-Zehnder interferometer [3,4] or the proposal of interaction-free measurements [5][6][7]. This sort of arguments exploit apparent difficulty in answering the question: How does the particle, being localised in a given path, knows what happens in the other path of the interferometer? Faced with a puzzle, conventional wisdom attributes this kind of behaviour to non-local effects -either of a particle itself or the wave function. However, it is unclear if this is enough to establish similar conclusions as in the reasoning of the Bell-type. In particular, does it imply impossibility of local hidden variable models simulating quantum behaviour in the considered interferometric setups? In this paper, we answer this question in the negative.
Clearly, in the Bell scenario one is concerned with correlations between a pair of quantum particles, whereas in the single particle case we are concerned with a single quantum particle interacting with classical apparatus. * Pawel. Blasiak@ifj.edu.pl Hence the question of hidden variable account is brought up in a different conceptual context. This makes interesting to ask if it is possible to simulate single-particle behaviour of simple interferometric circuits by replacing quantum gates with stochastic counterparts without violating the paradigm of locality. Note that argument of the Bell-type does not apply in this situation, and hence it should not be very surprising if a different conclusion is reached.
In this paper, we take a closer look at a single-particle framework for two-path interferometric setups built from phase shifters, beam splitters and detectors. It has a simple description which boils down to a qubit and, as such, is often taken as prototypical example illustrating various paradoxes and quantum effects, see e.g. [3][4][5][6][7][8][9][10][11][12][13]. We show that it is possible to simulate behaviour of such circuits in a classical manner using stochastic gates and particles which interact only locally. The crucial ingredient of the model is existence of two kinds of particles, real ones and ghosts, with the latter being invisible to detectors. This allows to construct a stochastic analogue of quantum circuits with built-in limited information gain and state disturbance. We show that operational description of the system -by an agent investing the system according to the rules of the model -parallels the quantum case, i.e. boils down to a qubit. At the same time, on the ontological level all gates and particles considered in the model conform to the paradigm of locality.
The paper is organised as follows. Sec. II gives a concise account of quantum-interferometric building blocks and their description in terms of a qubit. In Sec. III we explain ontology of the model, discuss the criterion of locality and define stochastic analogues of the interferometric gates. Then the model is carefully analysed in Sec. IV where we start by the ontic description of epistemic constraints, and then abstract away all unnecessary details to give purely operational account of the model as seen by the agent unaware of the underlying ontology. The latter will be shown equivalent to the quantum description of a qubit. We conclude with a brief discussion of results in Sec. V.

II. INTERFEROMETRIC SETUP FOR A QUBIT
A qubit is the simplest (i.e. two-state) quantum mechanical system which is described in the Hilbert space H = C 2 [14]. It is convenient to represent pure states of a qubit |ψ = cos θ 2 |0 + e iφ sin θ 2 |1 = cos θ 2 e iφ sin θ 2 (1) by points n = (θ, φ) on the so called Bloch sphere, where θ and φ are standard polar and azimuthal angles in 3D.
Mixed states have similar representation which extends to the Bloch ball via the parametrisation ρ = 1 2 (1 + n · σ) where |n| 1 (with pure states |ψ ψ| lying on the surface, i.e. |n| = 1). In this representation unitary transformations ρ → U ρ U † (or |ψ → U |ψ for pure states) correspond to rotations n → Rr(ϑ) n, where the axisr and the angle ϑ is determined from the parametrisation U = e iα e −iϑr·σ/2 (with α being irrelevant overall factor). According to the Born rule measurement in the computational basis {|0 , |1 } (or equivalently ±ẑ in Bloch repr.) on a system described by state ρ (equivalently n) yields outcome i = 0, 1 with probability and leaves the system in the corresponding pure state ρ (equiv. n) −→ |0 (equiv. +ẑ) , for i = 0 , Without conditioning on the outcomes after the measurement the system is described by the following mixture ρ = i P ρ (i) |i i| (equiv. (ẑ · n)ẑ ). These rules describe a qubit in a nutshell. In order to be physically meaningful this framework needs to be equipped with an interpretation. Here, we will be concerned with a typical realisation of a qubit associated with spatial degrees of freedom of a single particle, e.g. a photon traversing two arms of the Mach-Zehnder interferometer as shown in Fig. 1. In general, one can think of a particle which enters a system of two paths labelled i = 0, 1 arranged in a circuit built from optical gates and detectors [15,16]. Quantum description of such a system is equivalent to a qubit where states of the computational basis, |0 ≡ 1 0 and |1 ≡ 0 1 , correspond to the statement that the particle is present respectively in the 0-th or the 1-st path of the circuit. General state of the system is described by a mixed states ρ (equiv. n) and transformations are implemented by phase shifters and beam splitters which can be used to realise any unitary [16]. To be more specific, we have the following representation of the basic interferometric gates: FIG. 1. Mach-Zehnder interferometer. A particle enters a circuit which consists of two paths, the upper i = 0 and the lower i = 1 one, and undergoes a sequence of transformations via phase shifters Pi(ωi) and 50 : 50 beam splitters B = B( π 2 ). The initial pure state |0 = 1 0 is transformed to |ψ = ψ 0 ψ 1 and probability of registering the particle ('Click') in detector Di is given by
where P i (ω) denotes a phase shifter placed in the i-th path, and B(ξ) stands for a beam splitter characterised by reflectivity R = cos 2 ξ 2 and transitivity T = sin 2 ξ 2 . Note that phase shifters act locally on each path while beam splitters are non-local gates which require both path brought together. Finally, a measurement in the computational basis comes down to detection of presence/absence of the particle ('Click'/'No Click') via detectors D i placed in the paths, cf. Eqs. (2) and (3). 1 In summary, mathematical description of a single particle in two-path interferometric circuits boils down to a qubit. We note that this framework is often used for discussion of some typically quantum effects [3][4][5][6][7][8][9][10][11][12][13] which include demonstration of 'non-local' behaviour of quantum particles when the paths are taken to be spatially separated. In the following, we show that it is possible to simulate such interferometric setups in a classical manner using stochastic gates and two kinds of particles without violation of the paradigm of locality.

III. ONTOLOGY OF THE MODEL
In this section we construct a stochastic model which is based on the concept of particles propagating along two paths. We start by defining the ontic state space of the system and discuss its probabilistic account emphasising the distinction between local and non-local gates. Then we complete description of the model by postulating stochastic counterparts of the interferometric gates which define possible evolution of the system.

A. Ontic state space and stochastic gates
We will consider two kinds of particles with inner degrees of freedom: real ones described by a point on sphere S 2 in 3D, and ghosts described by a point on circle S 1 in 2D. That is a real particle carries a unit vector n = (θ, φ) where θ and φ are standard polar and azimuthal angles, and a ghost particle carries only a phase ϕ ∈ [0, 2π). Naming of the particles will become evident when we will introduce detectors which react only to real particles and remain blind to ghosts.
In the following we will consider evolution of such particles traversing a circuit which consist of two paths labeled i = 0, 1 and various types of gates which implement transformations. One may think of two spatially separated paths (or wires) through which particles are sent in direct analogy with quantum-interferometric circuits described in the previous section. Furthermore, we will restrict our model to situations with a single real particle present in one of the paths which is either accompanied by a ghost particle in the other path or the other path empty. See Fig. 2 for illustration. Hence, formal definition of the system boils down to the following description.
Definition 1 (Ontic state pace). At each time the system is described by a point in the ontic state space where S 1 * ≡ S 1 ∪ {∅}. The ontic state (i, n, ϕ) ∈ Ω corresponds to a situation in which there is a real particle in path i carrying vector n and the ghost particle is present in the other pathī ≡ i + 1 (mod 2) carrying phase ϕ. The case (i, n, ∅) ∈ Ω describes a situation in which the (other)ī-th path is empty.
In the following we will be interested in stochastic evolution of the system. This means that we admit incomplete knowledge which, in general, can be specified by a probability distribution over ontic states of the system.

Definition 2 (Probabilistic description).
At each time the system is described by a point in the probabilistic simplex Note that ontic states ω = (i, n, τ ) ∈ Ω, with τ = ϕ or ∅, correspond to extremal points of the simplex which are definite probability distributions δ i δ n δ τ : Ω −→ [0, 1]; note that the latter should be understood as a (tensor) product of delta functions . They form a basis in P(Ω) in a sense that each probability distribution p : Ω −→ [0, 1] can be uniquely written in the form A general stochastic transformation (or gate) is defined as a mapping which describes probability distribution T(ω) ∈ P(Ω) of final states given the system was in the ontic state ω ∈ Ω. This means that T ω,ω ≡ [T(ω)](ω ) is the transition probability from state ω to ω . In the following, we assume Markov property which entails that a sequence of transformations T 1 , T 2 , ..., T n defines a new transformation given by the product of matrices T = T 1 T 2 ... T n . Note that definition of stochastic gate in Eq. (6) is fully consistent with the assertion that at each time the system is in a well-defined state. It is only our incomplete knowledge about the actual ontic state (e.g. due to random disturbance of the system by transformations) which manifests by random results at each run of an experiment. Therefore, we may think of p ∈ P(Ω) as describing an ensemble of systems whose all elements are in definite ontic states which are distributed according to p. 2

B. Paradigm of locality
Our primary concern in building the model is to preserve the paradigm of locality which requires that particles (the real ones as well as the ghosts) interact only with objects in their immediate vicinity. This means that in a situation of spatial separation local gate affects only inner state of the particle present in the given path and does not depend on the inner state of the other particle nor the gate implemented in the other path. For the purpose of the present model it is enough to precise locality condition only for the case of deterministic gates, i.e. such that T(ω) = δ ω = δ i δ n δ τ for all ω ∈ Ω. We make the following formal definition. j : S 1 * −→ S 1 * , which define action of the gate on the given kind of particle (real or ghost/empty respectively) present in the j-th path and leave the other path unaffected. That is we have where the case i = j corresponds to the real particle being in the j-th path, while i = j means that it is the ghost or the path is empty.
A general deterministic gate T is said to be local if it can be decomposed into two local deterministic gates T j acting in the respective paths j = 0, 1, i.e. T = T 0 T 1 = T 1 T 0 . Otherwise the transformation is said to be non-local.
If we consider discrete time intervals, we may decompose evolution of the system and trace the character of consecutive steps. Usually it is presented in the graphical form of a circuit with spatially separated paths represented by lines and transformations depicted by blocks (or pictures). If the gate is known to be local then it is depicted by separate blocks attached to the respective lines, otherwise the lines 'meet' in the block indicating that particles are allowed to interact. Note that this is exactly the picture that we have for quantuminterferometric circuits discussed in Sec. II. It suggests interpretation of phase shifters P j (ω) and detectors D j as local gates, and beam splitters B(ξ) playing the role of non-local gates; cf. Fig. 1. In constructing our model we will pay special attention to get identical structure of circuits, i.e. the same character of locality/non-locality for the gates; cf. Fig. 2.
which transform entire probability distributions. We note that in the latter case account in terms of well-defined ontic states is no longer tenable. Since in this paper we seek for explanations in reference to the underlying ontological picture, we stick to the definition of Eq. (6).

C. Building blocks of the model
Now we are in position to complete the model by specifying selection of gates providing stochastic counterparts of the interferometric building blocks within our model.

Definition 4 (Limited set of stochastic gates).
We will consider stochastic circuits that are built from a few basic building blocks defined as follows.
(i ) Phase shifter P j (ω) is a local deterministic gate whose action is given by Note that phase shifter P j (ω) affects particles only in the j-th path and the inner states of real and ghost particles are rotated around theẑ axis in opposite directions. Clearly, if the path is empty, it remains so.
(ii ) Beam splitter B(ξ) is a non-local stochastic gate which requires information from both paths to effect the transformation. It is defined as follows In the case first case, the resulting state is a probabilistic mixture of two situations: particles remain in their respective paths (i → i) or particles get swapped (i →ī ).
Note that inner states of the ingoing particles change n → ± n and ϕ → 0 with the combination of two terms introducing non-trivial correlations between particles in the output. In the second case, action of the beam splitter can be seen as if before proceeding it complemented the lacking information in the empty path (∅) by creating there a ghost with phase ϕ = 0 and changing n →ẑ.
(iii ) Detector D j is a local deterministic gate which 'Clicks' if it finds real particle in the j-th path and remains silent otherwise ('No Click'). It has the following action and Note that the detector D j acts nontrivially on particles of either type. If it happens to be a real particle (i = j) then n →ẑ. If it is a ghost (i = j) then it gets absorbed and the path is left empty, i.e. ϕ → ∅. Clearly, detection is repeatable (subsequent detection gives the same result).
Notice that definition of the detection process via D j 's explains naming of the particles: the real ones are those 'observed' by detectors ('Click'), while the ghosts are particles invisible to detectors ('No Click').
It is instructive to remark that the beam splitter B(ξ) is the only place in the model where the ghosts can manifest their presence. When looked closer at the definition of n , it appears that the ghost particle provides a kind of 'reference frame' for the transformation effected in the beam splitter. On can think of this as information transfer between particles in different paths (it is allowed since beam splitter is a non-local gate where both paths 'meet'). If not this effect, predictions of the model would be insensitive to the existence of the ghosts. In the following we will show that their presence leads to 'observable' consequences in circuits composed of a few gates.
Clearly, we can replace quantum gates with their stochastic counterparts to imitate any two-path interferometric circuit (cf. Sec. II). So far this is only structural similarity and it is not clear why the analogy should go further. Although it might come as surprise in view of the fact that the model is built on the classical-like particle ontology with both kinds of particles as well action of the gates conforming to the paradigm of locality, we show in the following section that operational description of the model is equivalent to a qubit.

IV. OPERATIONAL DESCRIPTION OF THE MODEL
Imagine an agent without any prior notion of the model trying to make sense of how it works only on the basis of experiments she performs. Clearly, the agent acts under epistemic constraints which confine her perception of the system under study -she has a limited choice of gates for building circuits and her detectors 'see' only real particles (cf. Def. 4). That being the case the agent describing the model is legitimate to restrict herself only to situations within her reach. In other words, for all practical purposes it is sufficient to account for a limited set of preparation, transformation and measurement procedures that arise in any experimental circuit that she can build according to the rules of the model. Note that from this point of view any ontological commitment is superfluous and a minimal mathematical framework is just enough.
In the following, the full-blown ontological picture of the model is cut down to a minimal operational account adapted for the specific needs of the agent. We proceed in steps by answering the following questions: (a) Which distributions in P(Ω) can be prepared by the agent according the rules of the model?
(b) How do they transform and what information can be learned under the action of conceivable circuits?
We will see that the agent has access only to a limited range of distributions which form a well-structured set under possible transformations (Sec. IV A). A closer look will reveal that a detailed ontological account is in many respects irrelevant, which brings up the question: (c) What is the minimal description which is enough to predict behaviour of the system?
Our main result shows that operational account the system is equivalent to a qubit (Sec. IV B). It means that from the agent's perspective predictions of the model are indistinguishable from the behaviour of a single quantum particle in two-path interferometric circuits discussed in Sec. II.
A. Ontological account

Initialising the circuit
Before any analysis takes place we need to address the problem of initialisation, i.e. how the agent starts the circuit off by preparing a reliable ensemble of particles for which it is going to work. We know from Sec. III A that in order to meet the requirements posed by the model it has to be ensemble of particles with the property that at each time there is only a single real particle present in one of the paths and possibly a ghost in the other one, i.e. an ensemble described by a distribution in P(Ω).
It must be realised that without access to any particular source from the outside the agent has to find a method to prepare the initial ensemble only by herself. The least that can be assumed is that the agent is given access to an unknown (possibly random) source of particles. Then, to make it a reliable initial ensemble it has to be sieved in search for cases in which the circuit will certainly work by checking for presence of real particles in the paths. This can be done by filtering via detectors placed in both paths, D 0 & D 1 , and retaining only the cases when single detection occurred. 3 See Fig. 3 for illustration. It secures proper working of the circuit and provides the least framework to start investigation of the model. Thus, upon selection of events when a given detector D i 'Clicked' the agent prepares two initial ensembles described by the distributions (cf. Def. 4 (iii)) From the ontic point of view states in Eq. (8) correspond to the real particle with inner state n =ẑ being present in the i-th path and no ghost in the another path. ≡ δi δẑ δ∅ (on the right), which subsequently can be used to study behaviour of the model.
We note in advance that such a detailed description is not accessible to the agent. The only knowledge that she has is that the i-th detector 'Clicked' and immediate repetition of the test will necessarily yield the same outcome.
Certainly, the agent can go beyond the initial states in Eqs. (8) by processing them with the tools that she has at her disposal (phase shifters, beam splitters and detectors). Combining gates and measurements in various orders broadens the scope of preparation procedures leading to a considerable variety of distributions which will be systematically characterised in the following two subsections.

Some states of interest and action of the gates
For the purpose of analysis let us distinguish a few important classes of distributions in P(Ω). We will denote them by [ N ] and label by unit vectors N = (θ, ϕ) ∈ S 2 , where θ and ϕ are standard polar and azimuthal angles. See Fig. 4 for illustration.
For each N = ±ẑ we define the corresponding class [ N ] ⊂ P(Ω) as follows where All these distributions correspond to both real and ghost particle present in the system. Moreover, each distribution p  9) and (11). Classes associated to the poles ±ẑ contain special subclasses denoted by [±ẑ] * ⊂ [±ẑ] which describe situations in which one path is empty. All classes are disjoint, but do not exhaust all P(Ω).
with the real particle being present either in the upper path (i = 0) or the lower path (i = 1), with the respective probabilities P N (i = 0) = cos 2 θ 2 and P N (i = 1) = sin 2 θ 2 which depend only on the polar angle θ of the defining vector N = (θ, ϕ). Notice that in all distributions defined in Eq. (10) inner states of both particles are closely related, i.e. vector carried by the real particle is given by ± N (with sign depending on where it is i = 0, 1) rotated about theẑ axis by angle equal to the phase of the ghost (α or β respectively). It is important to realise that distributions defined in this way display correlations, which will play a crucial role in the following analysis. 4 For the north and south pole N = ±ẑ, we make the following definitions where [+ẑ] * ≡ δ 0 δ n δ ∅ : n ∈ S 2 , n = −ẑ , Every distribution in class [+ẑ] (resp. [−ẑ]) corresponds to a situation with the real particle being definitely in the 0-th (resp. 1-st) path of the circuit. Note that each class [±ẑ] is comprised of two subclasses distinguished by the presence or absence of the ghost. In the case when the ghost is present in the system, the inner state of the real particle always points north n =ẑ. Otherwise, when the other path is empty ∅, the vector carried by the real particle can point in any direction with the exception of the south pole n ∈ S 2 \{−ẑ}. We denote the latter subclasses by [±ẑ] * . Clearly, for the initial distributions in Eq. (8), we have p  In summary, we have associated to each point on sphere S 2 the corresponding class of distributions in P(Ω), i.e. we have a mapping S 2 N −→ [ N ] ⊂ P(Ω). See Fig. 4 for illustration. One readily checks that these classes are disjoint, i.e.
However, it is not a partition of P(Ω) since they do not exhaust all possible distributions in P(Ω), i.e.
In such a case, we say that the collection of all classes [ N ] : N ∈ S 2 defines partial equivalence relation on P(Ω). Clearly, in the restricted domain E it is a fullblown partition and equivalence relation with two distributions being equivalent when they belong to the same class.
Now, let us focus on the restricted set of distributions E ⊂ P(Ω) and characterise action of the stochastic gates defined in Sec. III C. We have the following two lemmas (see Appendix for the proofs).

Lemma 1 (Phase shifters & Beam splitters).
Action of phase shifters P j (ω) and beam splitters B(ξ) does not leave outside the set E, i.e. we have for any T = P j (ω) or B(ξ). Moreover, all elements of a given class are mapped (congruently) into the same class, For phase shifters P j (ω), we have the following rules [ N ] p For beam splitters B(ξ), we get In other words, classes of distributions [ N ] ⊂ E transform as a whole (congruently) under the action of phase shifters and beam splitters, and the mapping of classes boils down to appropriate rotation of the labelling vector N −→ Rr(γ) N as specified in Eqs. (16)- (18). See Fig. 5 (on the left) for illustration.
In the case of detectors situation is subtler since they provide additional information given by the outcome, i.e. 'Click' or 'No Click', which should be included in the description. In general, there are three different arrangements: with a single detector detector D j in one of the paths (j = 0, 1) or two detectors D 0 & D 1 placed in both paths at the same time. We get the following description.

Lemma 2 (Detectors).
Suppose that we have an ensemble which is described by distribution p ∈ [ N ] with N = (θ, ϕ), and there is a single detector D j placed in the j-th paths. For detector D 0 , its action is given by and afterwords the system is left in state described by the distribution For detector D 1 , we have − D 1 'Clicks' with probability and afterwords the system is left in state described by the distribution In the case of two detectors D 0 & D 1 placed in both paths at the same time their ' Clicks' are anti-correlated, but otherwise their behaviour follows the rules given above. This means that on the level of classes detection comes down to 'projection' of the labelling vector N −→ ±ẑ which depends on the outcome as specified in Eqs. (20) and (22). See Fig. 5 (on the right) for illustration. We note that without conditioning detection leads outside E, i.e. the whole ensemble is a mixture of two distributions in [±ẑ]. This aspect will be discussed in the following subsection.
It is important to remark that mappings in Eqs. (16)-(22) act nontrivially within each class, i.e. the resulting p depends on the input p ∈ [ N ]. We have skipped these details in the above statements since they are not essential for the following discussion. For explicit action of the gates see Appendix, where both Lemmas 1 & 2 are proved.

Complete description of accessible states
We have discussed above some restricted classes of distributions [ N ] ∈ P(Ω) labelled by vectors N ∈ S 2 . Now, we will use them to characterise the full range of states accessible to the agent who acts in accordance with the rules of the model. This means that we have to account for all distributions obtained from the initial ensemble in any kind of procedure that is allowed by the model, i.e.
(a) any sequence of gates (circuit), (b) possible conditioning on the outcomes, (c) probabilistic mixing of ensembles.
In the following we argue that the agent can not reach outside the set of distributions E ⊂ P(Ω) and probabilistic mixtures thereof.
Let us start by considering an ensemble described by one of the initial distributions p Then Lemma 1 guarantees that after processing through a circuit composed of a sequence of phase shifters and beam splitters the system ends in a state described by distribution within the restricted set E. Moreover, Eqs. (16)- (18) precise in which class [ N ] the resulting distribution will be contained in -it is specified by a sequence of rotations corresponding to the gates in the circuit acting on the vector labelling the respective initial class [±ẑ]. For example, if the circuit consists of a sequence of gates T 1 , T 2 , ... , T n , then we have where Rr l (γ l ) are rotations about axesr l =ẑ orx corresponding to the respective gates T l (cf. Lemma 1). If there are detectors placed along the circuit, then during evolution each element of the ensemble is tagged with the respective outcomes registered by the detectors. It follows from Lemma 2 that upon selection of cases corresponding to the same sequence of outcomes the associated subensamble is described by a well-defined distribution in the restricted set E. To put it differently, each instance of detection along the circuit, i.e. 'Click' or 'No Click' which discloses position of the real particle, splits the original ensemble into two groups in which all elements are described by the same distribution in the respective class [±ẑ], cf. Eqs. (20) and (22). In terms of classes this induces outcome dependent 'projection', i.e. we have [+ẑ] , real particle in path i = 0 , which happens with relative frequencies (cf. Eqs. (19) and (21) in Lemma 2) Then the selected subensemble evolves again in accord with Lemma 1 to a well-defined distribution in some other class [ N ] ⊂ E as specified by Eq. (24) until the next detection takes place and the procedure repeats. Hence, by conditioning on the readings of all detectors along the way description of the selected subensemble follows a path in the restricted set of distributions E ⊂ P(Ω).
In fact, we may generalise this analysis to ensembles described by any initial state p ∈ [ N ] ⊂ E that the agent might start with. By the same reasoning we conclude that processing via any conceivable circuit and conditioning on the outcomes does not lead outside the restricted set E ⊂ P(Ω). Moreover, evolution of classes [ N ] ⊂ E containing the corresponding distributions is given in the case of phase shifters P j (ω) and beam splitters B(ξ) by a sequence of appropriate rotations of the labelling vector (cf. Eq. (24)) and outcome dependent 'projections' of Eq. (25) in the case of detection D j . Let us summarise by the following observation (see Appendix for the proof of part (ii)).

Proposition 1.
(i) Processing distributions form the restricted set E ⊂ P(Ω) via any sequence of gates and conditioning on the outcomes does not leave outside E.
(ii) The agent can prepare ensembles described by any distribution in E by processing initial ensembles of Eq. (8) via an appropriate sequence of gates.
To exhaust all possibilities in which the agent can prepare new ensembles we need to include probabilistic mixing into the picture. This means that having a way to prepare ensembles described by distributions p 1 , ... , p K ∈ E the agent can also prepare probabilistic mixtures of the where p k 0 and k p k = 1. Clearly, if p k ∈ [ N k ], then we have We note that mixtures of this type arise when the agent does not condition on the outcomes. In such a case the entire ensemble after measurement in state p ∈ [ N ] is described by distribution p = P N (0) p 0 + P N (1) p 1 , with p 0 ∈ [+ẑ] and p 1 ∈ [−ẑ] and the respective probabilities P N (i) following the specification of Lemma 2. On the level of classes [ N ] ⊂ E this means that without conditioning on the outcomes we get with P N (i) = 1 2 1 + (−) iẑ · N ; cf. Eqs. (25) and (26). From linearity of stochastic gates we deduce that processing of such mixtures via any conceivable circuit does not lead to further extension of the set of accessible distributions. In this way, having considered all available possibilities for exploring the model, we conclude that the agent remains confined within distributions being probabilistic mixtures of states in the restricted set E. Hence the following result.

Proposition 2.
Most general distributions in P(Ω) which are accessible to the agent are probabilistic mixtures of states in E, i.e. the agent explores only the following set conv E = k p k p k : p k ∈ E ⊂ P(Ω) , (31) where p k 0 and k p k = 1, and the sums are finite.
We observe that having discussed action of the gates on distributions in the restricted set E, we can extend these results to trace evolution of ensembles described by any distribution in the set conv E. By virtue of linearity of the gates we get 5 One way of preparing such a probabilistic mixture is to use external source of randomness which allows for probabilistic choice of procedures which prepare the respective states p 1 , ... , p K ∈ E. Alternatively, the agent can draw on the internal source of randomness generated by the probabilistic nature of transformations effected by the beam splitters. For example, this can be implemented by preparing an ensemble described by a distribution in p ∈ [ N ], and then performing measurement D 0 & D 1 without registering the outcomes (i.e. no selection of subensembles corresponding to the respective outcomes). The resulting ensemble is described by distribution P N (0) p (in) 0 . This state can be processed again and in this way by appropriate repetition of non-selective measurements any convex combination of distributions in E can be obtained.
where the transformation p k → p k follows the prescriptions of Lemmas 1 & 2. By the same token we get evolution of mixtures on the level of classes, i.e. we have with the transformation rules of Eqs. (25), (27) and (30). This completes description of the model as experienced by the agent acting under epistemic constraints. Note that it is given form the position of an outside observer with full knowledge of the underlying ontology (it is the case of ourselves being acquainted with details of the model as laid out in Sec. III). However, it is important to realise that from the agent's perspective, who is unaware of the ontological aspects of the system, such a description is untenable. In the next section we show how to cut this ontological account down to size so that it complies with the specific needs and limitations of the agent.

B. Operational account and recovery of quantum description
Now, we come back to the problem of describing the model without commitment to the underlying ontology. This is situation of an agent trying to make sense of the system only by interacting with it through experiments. In such case ontic account of epistemic constraints in the form given in Sec. IV A is unsuitable and in many respects superfluous. For all practical purposes it suffice for the agent to focus on a minimal description which is just enough to properly predict behaviour of the system only in situations that she may come up with. With this in mind we will adopt results of the previous Sec. IV A to construct purely operational account of the model as seen by the agent being completely unaware of the underlying ontology given in Sec. III.
For the purpose of analysis we start by introducing an appropriate notion of equivalence on the set of accessible distributions conv E. Note that a general probabilistic mixture p ∈ conv E has the form (cf. Eqs. (31) and (14)) and N k ∈ S 2 . It follows that we can associate to each p ∈ conv E a vector defined as Clearly, this vector has length | N p | 1 and lays in the unit ball N p ∈ B 3 . Hence we get a mapping We note that this definition of vector N p is unambiguous (see Appendix for the proof).

Proposition 3.
For each p ∈ conv E the corresponding vector N p given in Eq. (35) is defined uniquely, i.e. does not depend on a particular decomposition in Eq. (34).
Observe that there are many distributions corresponding to the same vector N ∈ B 3 . For example, any distribution which obtains form Eq. (34) by a different choice of representatives p k ∈ [ N k ], or another decomposition N = k p k N k with N k ∈ S 2 and p k 0 and k p k = 1, are associated to the same vector given in Eq. (35). Now we are in position to define equivalence relation on conv E. We say that two distributions p, q ∈ conv E are equivalent if It is straightforward to see that each N ∈ B 3 defines equivalence class of distributions in conv E which has the form 6 Clearly, these classes are disjoint, i.e.
and define partition the set conv E, i.e.
It is important to realise that the agent collects information about the system only via measurements (i.e. 'Clicks'/'No Clicks' of the detectors). Therefore, distinguishing between two ensembles requires from the agent to point out a situation in which these ensembles give different predictions. By the rules of Eqs. (19) and (21) or Eq. (26) we may calculate probabilistic distribution of outcomes i = 0, 1 in a measurement for any distribution in the set conv E. For a general mixture of Eq. (34), we get This means that distributions in the same equivalence class [[ N ]] give identical probabilistic predictions which depend solely on the labelling vector N ∈ B 3 (actually only on its length | N | and polar angle θ). We conclude that a measurement does not differentiate between distributions in the same class The only way for the agent to distinguish between two distributions in the same class would be to process them via a circuit which makes experimental predictions different. Let us check the behaviour of an ensemble described by a general distribution p ∈ [[ N ]] as specified by Eq. (34). Clearly, we have N = N p . From the previous section we know that each gate in the circuit effects the corresponding transformation where p k ∈ [ N k ] are specified by Lemmas 1 & 2. For the purpose in hand it suffice to focus on the coarser level of classes which follow the rules of Eqs. (25), (27) and (30). We get the following description of the associated vector which obtains after processing through the respective gates. For phase shifters P j (ω) and beam splitters B(ξ) it is given by where Rr(γ) is rotation about axisr =ẑ orx corresponding to the respective gate (note that we have used linearity of rotations in R 3 ). For detectors D j by conditioning on the outcomes, we get N p where the ± signs correspond to the respective outcomes (i.e. real particle in path i = 0, 1) which happen with the relative frequencies P N (i) given in Eq. (41) (note that we have used the normalisation condition k p k = 1). Without conditioning on the outcomes, we have N p We conclude that whatever action taken by the agent distributions from the same class p, q ∈ [[ N ]] transform into distributions again in the same class p , q ∈ [[ N ]], i.e. we get N p = N q = N . As already has been noted this entails identical probabilistic predictions, cf. Eq. (41). In consequence, states in the same class [[ N ]] are operationally indistinguishable to the agent, which means that there is no way to make experimental predictions different by the means available to the agent. On the other hand, it is relatively easy to come up with a circuit which discriminates between distributions in different classes. 7 We may briefly summarise the foregoing discussion in the following theorem.
Theorem 1 (Ontic description of epistemic constraints). Agent subject to epistemic constraints explores the model defined in Sec. III only to a limited extent. Whatever arrangement of gates in the circuit, possible conditioning on the outcomes and probabilistic mixing of ensembles, the agent remains confined within a restricted set of distributions. Her processing the system via any conceivable circuit boils down to the following rules.
where N T depends only on N and the kind of action that is being performed T. In such a case, mapping of classes is conveniently represented on the labelling vectors as explained below. Phase shifters P j (ω) are described by rotations about theẑ axis Beam splitters B(ξ) correspond to rotations about thê x axis Detectors D j implement measurements which consist in registering ' Clicks' associated with the position of the real particle in the circuit (' No Click' in the case of a single detector D j is a 'negative' measurement result which indicates presence of the real particle in the other j-th path ). For a system described by a distribution in class [[ N ]] probability P N (i) of outcome i = 0, 1, i.e. real particle is located in the i-th path, is given by If outcomes are not registered, then the whole ensemble after the measurement is described by Observe that from the operational point of view large part of description in Theorem 1 is redundant. All information that is needed to make predictions about behaviour of the system processed by any conceivable circuit is fully specified by vector N ∈ B 3 which labels the class [[ N ]] in which the distribution is contained. Note that characteristics of classes in terms vector N is selfcontained, i.e. its use does not require further details concerning the underlying distribution. It is also minimal in a sense that different N 's can be distinguished by appropriately chosen experiment. In this way, we get a minimal operational framework which describes the model as seen from the perspective of the agent without any reference to the underlying ontological picture.

Corollary 1 (Operational account of the model).
From the operational point of view the system is fully specified by a vector in the unit ball N ∈ B 3 . Transformations effected by phase shifters P j (ω) and beam splitters B(ξ) correspond to rotations as given in Eqs. (48), (49) and (50). Clicks of detectors D j (measurements) are described by the probabilistic rule of Eq. (51) and conditioned on the outcomes post-measurement states follow the prescription of Eq. (52) (without conditioning we have Eq. (53)).
Note that in this formulation one is only concerned with 'Clicks' of detectors in experiments and how distributions of outcomes changes when processed via possible circuits. It does not even require the concept of particle to interpret the results. 8 As a matter of fact, this is the only information which is relevant for the agent who is unaware or indifferent to the underlying ontology.
From the mathematical point of view operational account of the model given in Corollary 1 should be compared with description of a qubit in Sec. II. It is straightforward to see that vector N plays the role of the Bloch vector n and follows the same rules describing transformations and measurements. This means that mathematical framework of Corollary 1 is identical with a qubit.
To conclude, recall that from the construction twopath quantum interferometric circuits (cf. Sec II) and circuits built in our stochastic model (cf. Sec. III) are structurally the same. In this section we have shown that operational descriptions are equivalent as well, i.e. in either case boil down to a qubit. This brings the final conclusion that from the agent's perspective quantuminterferometric circuits and their stochastic counterparts are indistinguishable.

V. DISCUSSION
We have considered the question whether behaviour of a single quantum particle in two-path interferometric setups is enough to declare non-locality. If it were the case, this would mean impossibility of a hidden-variable account based on particle ontology and the paradigm of locality. In this paper we express reservations towards statements to that effect. We have constructed explicit stochastic model which simulates behaviour of such quantum interferometric circuits with all particles and gates interacting strictly locally. It has been shown that from the agent's perspective predictions of the model are indistinguishable from the quantum case and its operational account is equivalent to a qubit. Thus, allegedly 'nonlocal' effects in the model are explained to arise only on the epistemic level of description by the agent whose knowledge is incomplete due to the restricted means of investigating the system. This shows that dismissing local realism only on the basis of arguments concerning a qubit in the interferometric setup is premature.
Simplicity of the interferometric setup for a qubit makes it an attractive framework for discussion of various paradoxes and quantum effects. For example, it is often taken as a prototypical situation illustrating quantum interference [3,4], interaction-free measurements [5][6][7], quantum Zeno effect [6][7][8][9], delayed-choice experiments [10,11] or discussions of Leggett-Garg inequalities [12,13]. In view of the presented model arguments of this type lose on their original allure. All these effects have 'classical' analogues which can be explained by incomplete knowledge and state disturbance, and hence are not reserved exclusively for the quantum realm. The main point of interest here is that single-particle effects in two-path quantum interferometric circuits can be simulated in a classical manner without resort to non-locality.
Indeed, recent research show that many phenomena typically associated with strictly quantum mechanical effects have analogues in classical models with epistemic restrictions [17][18][19][20][21][22][23][24][25]. Most notable in this respect is the Spekkens' toy model [20] which reproduces a surprisingly large array of quantum phenomena in a simple discrete system constrained by the so-called 'knowledge balance principle'. This idea has been taken further to a continuous model which reconstructs Gaussian quantum mechanics from the so-called Liouville mechanics of the classical phase-space subject to an epistemic restriction [21]. The main point of these models is that they are ψ-epistemic in the classification of Ref. [26]. Following these lines a considerable effort has been taken towards understanding to what extent quantum states can be seen as states of knowledge and the related project of reconstructing quantum theory based on this premise. However, there are strong results, see Ref. [27], which suggest that it is not possible within ψ-epistemic theories in the classification of Ref. [26]. We note that our construction falls into the category of ψ-ontic models wherein these objections do not apply. Another important characteristic of ontological models which aim at reconstructing quantum predictions is their preparation, transformation and measurement contextuality [28]. One can easily convince himself that the presented model has all these properties.
Let us remark that various ontological models of a qubit exist, e.g. models of Beltrametti-Bugajski [29], Bell-Mermin [1,30] or Kochen-Specker [31]; see Refs. [26,32] for a review. Their focus is, however, on the mathematical formalism rather than on conceptual issues concerning the interpretation. In particular, it is not clear how to cast them into the framework of particles and interferometric circuits without violation of the locality principle. The problem lies in interpreting ontic states as characteristics of local objects in order to avoid paradoxes associated with the collapse of the wave function. In this paper we take a different approach by settling conceptual questions first. Here, we work from the outset with well-defined local objects (particles and gates) and show that operational description of the model is equivalent to a qubit. In this way we get the correct mathematical formalism of a qubit and unproblematic interpretation in terms of particles and gates which conform to the paradigm of locality. Let us point out that our model of a qubit differs from the existing proposals which can be immediately observed by comparing the respective ontic state spaces.
A general conclusion from the paper is that singleparticle effects in two-path interferometric circuits are not enough to establish non-locality. It seems that quantum non-locality is genuinely multi-particle phenomenon, i.e. requires at least two quantum particles to manifest nontrivial effects as it is considered in the Bell-type arguments. Of course, our model being restricted only to two-paths interferometric circuits is insufficient to support this statement in full generality. It would be interesting from the foundational point of view to complete the picture by considering the possibility of local simulation of a single quantum particle in a general multi-path interferometric setup which corresponds to a qudit [16].
where n = (θ , ϕ ) = R x (ξ)ẑ. In the second equality ( * * ) we have changed the variable˜ n = − n and consequently substituted: cos 2 θ 2 = sin 2 θ 2 and sin 2 θ 2 = cos 2 θ 2 ; cf. the same trick used in justifying equality ( * ) above. In this way we have checked action of phase shifters P j (ω) and beam splitters B(ξ) on each distribution p ∈ E, which concludes the proof of Lemma 1.

Proof of Lemma 2.
First we will consider the case of a single detector D j placed in the j-th path. The detector 'Clicks' only if there is real particle in the j-th path and 'No Click' testifies to presence of the real particle in the otherj-th path, cf. Def. 4 (iii). Therefore, for all distributions p ∈ [ N ] with N ∈ S 2 probability of outcome corresponding to the real particle being in path i = 0, 1 is given by (cf. Eqs. (54) and (55)) P N (i = 0) = cos 2 θ 2 = 1 2 1 + cos θ = 1 2 1 +ẑ · N , P N (i = 1) = sin 2 θ 2 = 1 2 1 − cos θ = 1 2 1 −ẑ · N , where N = (θ, ϕ) and we have cos θ =ẑ · N . In particular, we have P +ẑ (0) = P −ẑ (1) = 1 and P +ẑ (1) = P −ẑ (0) = 0. We observe that according to Def. 4 (iii) state after detection depends on the outcome. By conditioning on whether the detector 'Clicked' or did not 'Click' we get the following description of post-measurement states. In the case of single detector D 0 , we get and for the remaining distributions in [±ẑ] * we have Similarly, for single detector D 1 , we get and [+ẑ] * δ 0 δ n δ ∅ D1 / / δ 0 δ n δ ∅ ∈ [+ẑ] * , detector never 'Clicks' , . This is because there is only one real particle in the system (either in path i = 0 or path i = 1). Clearly, probabilities of outcomes follow the pattern of Eq. (65) and conditioned on the outcomes post-measurement states are given by This exhausts all distributions in the restricted set E and concludes the proof of Lemma 2.

Proof of Proposition 3
Suppose that we have two different decompositions of a mixture p ∈ conv E, i.e. p = k p k p k = l q l q l with p k ∈ [ N k ] and q l ∈ [ N l ]. In a more explicit form it writes as follows (cf. Eqs. with indices ranging over N ∈ S 2 , α, β ∈ [0, 2π), i ∈ {0, 1}, n ∈ S 2 \{−ẑ}, and a finite number of non-zero coefficients p Then, following Eq. (35) we can associate two vectors to the same distribution p, i.e. we have We need to prove uniqueness, which means that the associated vectors are the same N p = N p . Let us first write out supports of the following distributions (cf. Eq. (10) and (12)) supp δ i δ n δ ∅ = (i, n, ∅) , and summing over α gives