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arXiv:quant-ph/0507120 · Physical Review A

Simulating quantum correlations as a distributed sampling problem

Julien Degorre, Sophie Laplante, Jérémie Roland

Published Dec 9, 2005• 92 citations
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Abstract

It is known that quantum correlations exhibited by a maximally entangled qubit pair can be simulated with the help of shared randomness, supplemented with additional resources, such as communication, postselection or nonlocal boxes. For instance, in the case of projective measurements, it is possible to solve this problem with protocols using one bit of communication or making one use of a nonlocal box. We show that this problem reduces to a distributed sampling problem. We give a new method to obtain samples from a biased distribution, starting with shared random variables following a uniform distribution, and use it to build distributed sampling protocols. This approach allows us to derive, in a simpler and unified way, many existing protocols for projective measurements, and extend them to positive operator value measurements. Moreover, this approach naturally leads to a local hidden variable model for Werner states.

DOI
10.1103/physreva.72.062314

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It is known that quantum correlations exhibited by a maximally entangled qubit pair can be simulated with the help of shared randomness, supplemented with additional resources, such as communication, postselection or nonlocal boxes. For instance, in the case of projective measurements, it is possible to solve this problem with protocols using one bit of communication or making one use of a nonlocal box. We show that this problem reduces to a distributed sampling problem. We give a new method to obtain samples from a biased distribution, starting with shared random variables following a uniform distribution, and use it to build distributed sampling protocols. This approach allows us to derive, in a simpler and unified way, many existing protocols for projective measurements, and extend them to positive operator value measurements. Moreover, this approach naturally leads to a local hidden variable model for Werner states.
A longstanding question in Bell nonlocality is whether finite-alphabet nonsignalling resources can reproduce the continuum of measurement correlations of partially entangled qubits. Here we resolve this problem by showing that every fixed pure two-qubit state admits an exact simulation of all local projective-measurement statistics using shared randomness and finitely many Popescu-Rohrlich (PR) boxes, without communication or postselection. The boxes operate in parallel, with a worst-case bound independent of the measurement settings: at most 128cot⁡4θ128\cot^4θ128cot4θ calls for the Schmidt state cos⁡θ∣00⟩+sin⁡θ∣11⟩\cosθ|00\rangle+\sinθ|11\ranglecosθ∣00⟩+sinθ∣11⟩, 0<θ<π/40<θ<π/40<θ<π/4. The key is an exact finite-selection method that reduces input-dependent continuous sampling to a choice from a fixed-length shared random list, implemented through a finite PR-box network. The construction extends to arbitrary mixed two-qubit states and local binary-outcome POVMs. This reveals a fundamental separation between the continuity of quantum measurement correlations and the alphabet size of the resources needed to simulate them, shedding light on the resource-theoretic structure of Bell nonlocality.
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)≃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.
It is known that all causal correlations between two parties which output each 1 bit, a and b, when receiving each 1 bit, x and y, can be expressed as convex combinations of local correlations (i.e., correlations that can be simulated with local random variables) and nonlocal correlations of the form a+b=xy mod 2. We show that a single instance of the latter elementary nonlocal correlation suffices to simulate exactly all possible projective measurements that can be performed on a maximally entangled state of two qubits, with no communication needed at all. This elementary nonlocal correlation thus defines some unit of nonlocality, which we call a nl bit.
We propose a simple simulation of nonlocal quantum correlations among NNN qubits using a local hidden variable source with a positive probability distribution, given that each of the NNN observers has access to a local negative bit. Notably, unlike the Toner-Bacon protocol, no exchange of classical bits between the observers is required. Moreover, our simulation can be extended to include Popescu-Rohrlich box correlations.