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Simulation of quantum entangled states by classical probabilistic models

A new preprint:

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|,

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(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).

Building on an earlier plasma-like model of one-particle Wigner functions in terms of classical particles and antiparticles, we interpret |qk||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 qkq_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.