Finite-alphabet nonsignalling simulation of entangled qubits

Problem

For every partially entangled two-qubit pure state, does some fixed finite-alphabet nonsignalling resource give an exact noncommunicating simulation of all local projective measurements? Fix

\begin{equation} \lvert\psi_\theta\rangle =\cos\theta\,\lvert00\rangle+\sin\theta\,\lvert11\rangle, \qquad 0<\theta<\frac{\pi}{4}. \tag{1} \end{equation}

The target correlations for Bloch directions \(\mathbf x,\mathbf y\in S^2\) are

\begin{equation} P_\theta(a,b\mid\mathbf x,\mathbf y) =\operatorname{Tr}\!\left[ \lvert\psi_\theta\rangle\!\langle\psi_\theta\rvert \bigl(M_{a\mid\mathbf x}\otimes M_{b\mid\mathbf y}\bigr) \right], \qquad a,b\in\{0,1\}, \tag{2} \end{equation}

where \(M_{a\mid\mathbf x}=(I+(-1)^a\mathbf x\cdot\boldsymbol\sigma)/2\) and similarly for Bob. For each \(\theta\) in Eq. (1), one may choose a nonsignalling box \(R(u,v\mid s,t)\) with finite input and output alphabets and a finite number of copies of it. The box, the number of copies, and the local wiring may depend on \(\theta\) but not on \(\mathbf x\) or \(\mathbf y\), and the wiring may use unlimited shared randomness but no communication. It must reproduce Eq. (2) exactly.

Source

Gisin explicitly poses finite-alphabet nonsignalling simulation, while Brunner et al. isolate the same finiteness gap after giving a continuous-input construction [Gis09], [BGPS08].

Progress

Gisin posed the existence of a finite-input, finite-output nonsignalling resource for this task [Gis09].
At the excluded maximally entangled endpoint \(\theta=\pi/4\), one finite-alphabet PR box and shared randomness reproduce all local projective measurements exactly [CGMP05].
Brunner, Gisin, Popescu, and Scarani constructed an exact simulation for every state in Eq. (1), but their protocol uses a continuous-input millionaire box that compares real parameters. They explicitly leave replacement by finite-input resources open [BGPS08].
Some finite measurement sets obtained from partially entangled states cannot be simulated with a single PR box. This rules out one particular resource budget, not all finite-alphabet resources or all finite numbers of uses [BGS05].

Comment

Continuous-input nonsignalling simulation is known, whereas exact simulation by any finite-alphabet box used finitely many times is not. The finiteness requirement applies to both input and output alphabets and to the number of resource uses.

References

[Gis09]: N. Gisin, “Bell Inequalities: Many Questions, a Few Answers,” in W. C. Myrvold and J. Christian (eds.), Quantum Reality, Relativistic Causality, and Closing the Epistemic Circle, The Western Ontario Series in Philosophy of Science 73, 125–138 (Springer, 2009).DOI arXiv
[CGMP05]: N. J. Cerf, N. Gisin, S. Massar, and S. Popescu, “Simulating Maximal Quantum Entanglement without Communication,” Physical Review Letters 94, 220403 (2005).DOI arXiv
[BGPS08]: N. Brunner, N. Gisin, S. Popescu, and V. Scarani, “Simulation of Partial Entanglement with Nonsignaling Resources,” Physical Review A 78, 052111 (2008).DOI arXiv
[BGS05]: N. Brunner, N. Gisin, and V. Scarani, “Entanglement and Non-locality Are Different Resources,” New Journal of Physics 7, 88 (2005).DOI arXiv

Finite-alphabet nonsignalling simulation of entangled qubits

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