Local Hidden Variable Models for Entangled Quantum States Using Finite Shared Randomness

The statistics of local measurements performed on certain entangled states can be reproduced using a local hidden variable (LHV) model. While all known models make use of an infinite amount of shared randomness, we show that essentially all entangled states admitting a LHV model can be simulated with finite shared randomness. Our most economical model simulates noisy two-qubit Werner states using only ${\log }_2(12)\simeq 3.58$ bits of shared randomness. We also discuss the case of positive operator valued measures, and the simulation of nonlocal states with finite shared randomness and finite communication. Our work represents a first step towards quantifying the cost of LHV models for entangled quantum states.

Figure 1

Simulation of two-qubit Werner states ${\rho }_W(\alpha )$ with finite shared randomness. The graph shows the relation between the visibility $\alpha$ (essentially the degree of entanglement) and the amount of shared randomness, quantified in bits. For $\alpha \le 1/3$ (below the solid line) the state is separable; hence 2 bits of shared randomness suffice (triangle). For $1/3<\alpha \lesssim 0.43$, the state can be simulated with ${\log }_2(12)$ bits of shared randomness using protocol 1. For $0.43\lesssim \alpha \lesssim 0.66$, ${\rho }_W(\alpha )$ can be simulated with a larger (but nevertheless finite) amount of shared randomness. For $0.43\lesssim \alpha <0.5$, we have a LHS model (using protocol 2). For $0.5<\alpha \lesssim 0.66$ the state becomes steerable but can nevertheless be simulated by a LHV model using finite shared randomness, by applying Result 1 to the model of Ref. [10] (see main text).