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Simulation of quantum entangled states by classical probabilistic models

Andrey Akhmeteli

Published Sep 15, 2026• 0 citations
OpenAlex

Abstract

We propose a classical probabilistic plasma-like model of bipartite quantum states, including entangled states. In a finite-dimensional approximation, a bipartite density operator can be written as a finite signed sum of product-state projectors,$$ \rho=\sum_k q_k |a_k,b_k\rangle\langle a_k,b_k|, $$where an entangled state necessarily requires negative coefficients in every such decomposition. By linearity and factorization of the Wigner transform on product states, the corresponding bipartite Wigner function has the form$$ W(\mathbf{x}_1,\mathbf{p}_1,\mathbf{x}_2,\mathbf{p}_2)=\sum_{k=1}^{K}q_kW_{1k}(\mathbf{x}_1,\mathbf{p}_1)W_{2k}(\mathbf{x}_2,\mathbf{p}_2). $$Building on an earlier plasma-like model of one-particle Wigner functions in terms of classical particles and antiparticles, we interpret \(|q_k|\) as the expected value of the number of copies of the $k$-th plasma pair in a realization of the ensemble, while the sign of \(q_k\) distinguishes particle- and antiparticle-type contributions. Negative quasiprobabilities therefore enter the construction at two distinct levels: through possible negativity of the one-particle Wigner functions and through negative coefficients in the product-state decomposition associated with entanglement. Averaging over the ensemble of plasma copies reproduces the bipartite Wigner function and therefore the corresponding quantum state.Building on an earlier plasma-like model of one-particle Wigner functions in terms of classical particles and antiparticles, we interpret the coefficients of the sum as the expected values of the number of copies of the bipartite plasmas in a realization of the ensemble, while the sign of the coefficients distinguishes particle- and antiparticle-type contributions. Negative quasiprobabilities therefore enter the construction at two distinct levels: through possible negativity of the one-particle Wigner functions and through negative coefficients in the product-state decomposition associated with entanglement. The ensemble-averaged signed phase-space distribution reproduces the bipartite Wigner function and therefore the corresponding quantum state.

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We propose a classical probabilistic plasma-like model of bipartite quantum states, including entangled states. In a finite-dimensional approximation, a bipartite density operator can be written as a finite signed sum of product-state projectors,ρ=∑kqk∣ak,bk⟩⟨ak,bk∣,\rho=\sum_k q_k |a_k,b_k\rangle\langle a_k,b_k|,ρ=∑k​qk​∣ak​,bk​⟩⟨ak​,bk​∣,where an entangled state necessarily requires negative coefficients in every such decomposition. By linearity and factorization of the Wigner transform on product states, the corresponding bipartite Wigner function has the formW(x1,p1,x2,p2)=∑k=1KqkW1k(x1,p1)W2k(x2,p2).W(\mathbf{x}_1,\mathbf{p}_1,\mathbf{x}_2,\mathbf{p}_2)=\sum_{k=1}^{K}q_kW_{1k}(\mathbf{x}_1,\mathbf{p}_1)W_{2k}(\mathbf{x}_2,\mathbf{p}_2).W(x1​,p1​,x2​,p2​)=∑k=1K​qk​W1k​(x1​,p1​)W2k​(x2​,p2​).Building on an earlier plasma-like model of one-particle Wigner functions in terms of classical particles and antiparticles, we interpret (|q_k|) as the expected value of the number of copies of the kkk-th plasma pair in a realization of the ensemble, while the sign of (q_k) distinguishes particle- and antiparticle-type contributions. Negative quasiprobabilities therefore enter the construction at two distinct levels: through possible negativity of the one-particle Wigner functions and through negative coefficients in the product-state decomposition associated with entanglement. Averaging over the ensemble of plasma copies reproduces the bipartite Wigner function and therefore the corresponding quantum state.Building on an earlier plasma-like model of one-particle Wigner functions in terms of classical particles and antiparticles, we interpret the coefficients of the sum as the expected values of the number of copies of the bipartite plasmas in a realization of the ensemble, while the sign of the coefficients distinguishes particle- and antiparticle-type contributions. Negative quasiprobabilities therefore enter the construction at two distinct levels: through possible negativity of the one-particle Wigner functions and through negative coefficients in the product-state decomposition associated with entanglement. The ensemble-averaged signed phase-space distribution reproduces the bipartite Wigner function and therefore the corresponding quantum state.
One of the crucial differences between mathematical models of classical and quantum mechanics is the use of the tensor product of the state spaces of subsystems as the state space of the corresponding composite system. (To describe an ensemble of classical composite systems one uses random variables taking values in the Cartesian product of the state spaces of subsystems.) We show that, nevertheless, it is possible to establish a natural correspondence between the classical and quantum probabilistic descriptions of composite systems. Quantum averages for composite systems (including entangled) can be represented as averages with respect to classical random fields. It is essentially what Albert Einstein was dreamed of. Quantum mechanics is represented as classical statistical mechanics with infinite-dimensional phase space. While the mathematical construction is completely rigorous, its physical interpretation is a complicated problem (which will not be discussed in this paper).
A recently introduced classical simulation method for universal quantum computation with magic states operates by repeated sampling from probability functions [M. Zurel et al. PRL 260404 (2020)]. This method is closely related to sampling algorithms based on Wigner functions, with the important distinction that Wigner functions can take negative values obstructing the sampling. Indeed, negativity in Wigner functions has been identified as a precondition for a quantum speed-up. However, in the present method of classical simulation, negativity of quasiprobability functions never arises. This model remains probabilistic for all quantum computations. In this paper, we analyze the amount of classical data that the simulation procedure must track. We find that this amount is small. Specifically, for any number n of magic states, the number of bits that describe the quantum system at any given time is 2n2+O(n).
A classical simulation scheme of quantum computation given a restricted set of states and measurements may be---occasionally, but only occasionally---interpreted naturally as a statistical simulation of positive quasi-probability distributions on a phase space. In this dissertation, we explore phase space representations for finite-dimensional quantum systems and their negativities beyond the usual analogues of the Wigner function. The first line of study focuses on a characterization tool for valid quasi-probability distributions of (possibly mixed) quantum states. A quantum generalization of Bochner's theorem from classical probability theory simultaneously characterizes both the set of valid Wigner functions and the subset of positive ones. We extend this theorem to discrete phase spaces based on projective representations of abelian groups, including the discrete Wigner functions of Gross (J. Math. Phys. 47, 122107 (2006)) and Gibbons et al. (Phys. Rev. A 70, 062101 (2004)). The rest of the dissertation is then dedicated to phase space representations covariant with respect to a general Lie group. This means, in particular, that operators that lie in the same orbit of the group have the same amount of negativity in their quasi-probabilities---a natural requirement if one would like to interpret negativity as non-classicality, and the group action as a classical dynamics. We show that the construction due to Brif and Mann (Phys. Rev. A 59, 971 (1999)) of group covariant phase space representations satisfying the so-called Stratonovich-Weyl correspondence is essentially unique, given the group and an input state in the relevant representation. As an application, we construct quasi-probability representations on a compact phase space of fermionic Gaussian states tailored to a classically simulatable problem of fermionic linear optics and find that, in the first non-trivial case of four fermionic modes, the quasi-probabilities of the Gaussian states exhibit a considerable amount of negativity.
A recently introduced classical simulation method for universal quantum computation with magic states operates by repeated sampling from probability functions [M. Zurel PRL 260404 (2020)]. This method is closely related to sampling algorithms based on Wigner functions, with the important distinction that Wigner functions can take negative values obstructing the sampling. Indeed, negativity in Wigner functions has been identified as a precondition for a quantum speed-up. However, in the present method of classical simulation, negativity of quasiprobability functions never arises. This model remains probabilistic for all quantum computations. In this paper, we analyze the amount of classical data that the simulation procedure must track. We find that this amount is small. Specifically, for any number n of magic states, the number of bits that describe the quantum system at any given time is 2 n 2 + O ( n ) . Published by the American Physical Society 2024
We develop a path integral representation for the dynamics of quantum systems with a finite-dimensional Hilbert space, formulated entirely within a discrete phase space. Starting from the discrete Wigner function on Zd×Zd{\mathbb{Z}_d} \times {\mathbb{Z}_d}Zd​×Zd​ (ddd an odd prime) and the associated Weyl transform built from generalized displacement operators, we derive an exact kernel that propagates the discrete Wigner function in time and, by iterating its composition law through a short-time approximation, obtain a sum-over-paths expression weighted by a discrete phase-space action -- the natural finite-dimensional counterpart of Marinov's functional. For Hamiltonians linear in the phase-space coordinates and at times strictly commensurate with the lattice, the fluctuation sum collapses to a deterministic shift, realizing the discrete analog of classical Hamiltonian flow. Applying the formalism to one and to two interacting qutrits (d=3d=3d=3), we show that the full entanglement dynamics -- captured by a closed-form linear entropy valid for all times -- requires the coherent contribution of all fluctuation sectors; the boundary-term (mean-field) sector alone fails to reproduce it. For a non-stabilizer Hamiltonian, where the short-time kernel is only approximate, the time-sliced path integral converges to the exact dynamics, including the dynamical generation of Wigner negativity. We discuss implications for the semiclassical simulation of many-body spin systems and for the characterization of non-classicality through Wigner negativity.
We introduce a general method for the construction of quasiprobability representations for arbitrary notions of quantum coherence. Our technique yields a non-negative probability distribution for the decomposition of any classical state. Conversely, quantum phenomena are certified in terms of signed distributions, i.e., quasiprobabilities, and a residual component unaccessible via classical states. Our unifying method combines well-established concepts, such as phase-space distributions in quantum optics, with resources of quantumness relevant for quantum technologies. We apply our approach to analyze various forms of quantum coherence in different physical systems. Moreover, our framework renders it possible to uncover complex quantum correlations between systems, for example, via quasiprobability representations of multipartite entanglement.