Local model of a qudit: Single particle in optical circuits

It has been said about quantum interference, that “in reality, it contains the only mystery”. Together with nonlocality, it is often considered the characteristic feature of quantum theory challenging our classical understanding of the world. In this work, we are concerned with the restricted setting of a single particle propagating in multipath interferometric circuits—that is, the physical realization of a qudit—which is host to many typically quantum mechanical effects including collapse of the wave function and contextuality. In this paper, we show that this framework can be simulated with classical resources without violating the locality principle. We present a local ontological model whose predictions are indistinguishable from the quantum case. In the model, ‘nonlocality' appears merely as an epistemic effect arising from a level of description by agents whose knowledge is incomplete. It is notably different from the multiparticle scenarios where entanglement leads to nonlocal correlations on an ontological level. This result exposes the conceptual difference between single- and multiparticle phenomena, pointing to the latter as a deeper quantum mystery.

Figure 1

Ontology of the model and interferometric circuits. On the right, the circuits describe propagation of a particle through a network of (spatially separated) paths and gates which represent a sequence of transformations. A basic interferometric toolkit consists of free evolution (empty path), phase shifters $S_j$, beam splitters $B_{st}$ (on which two paths cross), and detectors $D_j$, which inform (Click or No Click) about the presence or absence of a particle in a given path. This selection of gates is general enough to provide a physical realization of any unitary and projective measurement described by quantum formalism in $\mathcal{H}={\mathbb{C}}^N$ [44]. On the left, ontology of the model consists of a single particle and local fields propagating in each path of the circuit. Each time the particle has the well-defined position $q=1,\cdots ,N$ and the fields are characterized by amplitude $u_j$ and strength ${\tau }_j$ with $j=1,\cdots ,N$ labeling the paths. In this paper, we show that this ontology, completed with appropriately defined local stochastic gates, fully reconstructs quantum mechanical predictions for a single particle in the interferometric circuits.

Figure 2

Evolution of classes $\left[\vec{z}\right]\subset \mathcal{P}\left(\mathrm{\Lambda }\right)$. On the left is an illustration of the whole space of probability distributions over $\mathrm{\Lambda }$, denoted by $\mathcal{P}\left(\mathrm{\Lambda }\right)$, with disjoint subsets representing classes of interest $\left[\vec{z}\right]$. These classes transform congruently (as a whole) under action of the gates in the model, as explained in Theorem t1. In the picture, the initial class $\left[{\mathit{e}}_j\right]$ undergoes a sequence of transformations On the right, evolution of a single class $\left[\vec{z}\right]$ depends on the composition of gates at a given time step in the experiment: conditioned on a Click in detector $D_k$ we have $\left[\vec{z}\right]⟶\left[{\mathit{e}}_k\right]$; otherwise, in a case when all detectors remain silent No Click (or there are no detectors at all) we get $\left[\vec{z}\right]⟶\left[\vec{z}{}^{\prime }\right]$ with $\vec{z}{}^{\prime }$ determined by the implemented gates.

Figure 3

Operational account of interferometric experiment. A source ascertains the presence of a particle in a given path $j=2$. After initial preparation ${\mathcal{P}}_{{}^j}^{{}^{{}_{in}}}$ the system undergoes a sequence of transformations ${\mathcal{T}}^{\mathcal{G}}$ and measurements ${\mathcal{M}}^{\mathcal{G}}$ which depend on the composition of gates $\mathcal{G}$ at a given time in the circuit. In the box at the bottom, a schematic illustration of quantum vs ontological model description is given. Quantum theory describes preparation procedures by state vectors $|\psi \rangle \in {\mathbb{C}}^N$ (with initial preparations associated with vectors in the computational basis), transformations ${\mathcal{T}}^{\mathcal{G}}$ are represented by unitaries $U$ and measurements ${\mathcal{M}}^{\mathcal{G}}\equiv \left\{{\mathcal{M}}_k^{\mathcal{G}}\right\}$ by PVM's $\left\{{\mathrm{\Pi }}_k\right\}$, with Born's rule describing the statistics of outcomes—see Eqs. (A2, A3, A4, A5). The ontological model describes the experiment by evolution of a distribution $\mathit{p}\in \mathcal{P}\left(\mathrm{\Lambda }\right)$ over the underlying ontic state space $\mathrm{\Lambda }$ which starts from some well-defined initial distribution ${\mathit{p}}_{{}^j}^{{}^{{}_{in}}}$ with transformations ${\mathcal{T}}^{\mathcal{G}}$ represented by stochastic mappings ${\mathrm{\Gamma }}_{{\mathcal{T}}^{\mathcal{G}}}$ and measurements ${\mathcal{M}}^{\mathcal{G}}\equiv \left\{{\mathcal{M}}_k^{\mathcal{G}}\right\}$ represented by collections of stochastic mappings $\left\{{\mathrm{\Gamma }}_{{\mathcal{M}}^{\mathcal{G}},k}\right\}$ with outcomes specified by the response functions ${\xi }_k^{{\mathcal{M}}^{\mathcal{G}}}$—see Eqs. (A7)–(A10).

Figure 4

Construction of classes of interest $\left[\vec{z}\right]\subset \mathcal{P}\left(\mathrm{\Lambda }\right)$. Distributions $\mathit{p}\in \left[\vec{z}\right]$ are defined to have support in ${\bigcup }_{i=1}^N{\mathrm{\Lambda }}_{\vec{z}}^i$ with cumulative probability over the respective subsets ${\mathrm{\Lambda }}_{\vec{z}}^i$ equal to $|z_i{|}^2$, where $\vec{z}\in {\mathbb{C}}^N$ is a normalized vector defined up to the overall phase—see Definition t4 and construction in Steps t11, t12, t13. Since all ${\mathrm{\Lambda }}_{\vec{z}}^i$ are disjoint, distributions in different classes have nonoverlapping supports, e.g., compare $\mathit{p}\in \left[\vec{z}\right]$ and ${\mathit{p}}^{\prime }\in \left[\vec{z}{}^{\prime }\right]$. Among all these classes there are a few special ones $\left[{\mathit{e}}_j\right]$ for $j=1,\cdots ,N$, which describe initial preparations ${\mathcal{P}}_{{}^j}^{{}^{{}_{in}}}$ with the particle in a given path.

Figure 5

Evolution of distributions in auxiliary classes ${\left[\vec{z}\right]}_i\subset \mathcal{P}\left(\mathrm{\Lambda }\right)$. Distributions $\mathit{p}\in {\left[\vec{z}\right]}_i$ are defined to have support in ${\mathrm{\Lambda }}_{\vec{z}}^i$ and their evolution $\mathit{p}⟶{\mathit{p}}^{\prime }$ depends on the composition of gates $\mathcal{G}\equiv \{\mathcal{F},\mathcal{D},\mathcal{S},\mathcal{B}\}$. If there is a detector in the $i\mathrm{th}$ path (i.e., for $i\in \mathcal{D}$), then it Clicks and afterwards ${\mathit{p}}^{\prime }\in {\left[{\mathit{e}}_i\right]}_i$ (at the top left). In the remaining cases (i.e., for $i\notin \mathcal{D}$) all the detectors remain silent and the evolution is as follows. If the $i\mathrm{th}$ path does not go into a beam splitter (i.e., for $i\notin \bigcup \mathcal{B}$), then we get ${\mathit{p}}^{\prime }\in {\left[\vec{z}{}^{\prime }\right]}_i$. Otherwise, if the $i\mathrm{th}$ path goes into the beam splitter $B_{st}$ (i.e., for $i\in \{s,t\}\in \mathcal{B}$), then distribution $\mathit{p}$ evolves into a mixture of distributions ${\mathit{p}}_s^{\prime }\in {\left[\vec{z}{}^{\prime }\right]}_s$ and ${\mathit{p}}_t^{\prime }\in {\left[\vec{z}{}^{\prime }\right]}_t$, which is weighted with probabilities proportional to $|z_s^{\prime }{|}^2$ and $|z_t^{\prime }{|}^2$.