Universal simulation with two PR boxes

Problem

Can shared randomness and at most two Popescu–Rohrlich boxes exactly simulate every pair of local projective measurements on every two-qubit state, without communication? A Popescu–Rohrlich box is the binary nonsignalling behavior

\begin{equation} P_{\mathrm{PR}}(u,v\mid s,t) =\begin{cases} \tfrac12,&u\oplus v=st,\\ 0,&u\oplus v\neq st, \end{cases} \qquad s,t,u,v\in\{0,1\}. \tag{1} \end{equation}

The parties may use arbitrary local, possibly adaptive wirings of two independent copies of Eq. (1). By convexity and Schmidt decomposition, it suffices to solve the simulation for all pure states

\begin{equation} \lvert\psi_\theta\rangle =\cos\theta\,\lvert00\rangle+\sin\theta\,\lvert11\rangle, \qquad 0<\theta\leq\frac{\pi}{4}, \tag{2} \end{equation}

and all local projective measurements on the state in Eq. (2). A complementary finite-scenario objective is to characterize the convex set generated by two-box wirings, for example through its facet inequalities.

Source

Gisin explicitly asks whether two Popescu–Rohrlich boxes suffice to simulate all projective-measurement correlations of arbitrary two-qubit pure states [Gis09].

Progress

Gisin explicitly posed both the universal two-box simulation problem and the finite-scenario inequality problem [Gis09].
One copy of Eq. (1), together with shared randomness, exactly simulates arbitrary projective measurements on a maximally entangled qubit pair [CGMP05]. This covers only the endpoint \(\theta=\pi/4\) in Eq. (2).
Finite-setting inequalities valid for all one-PR-box strategies are violated by correlations of nonmaximally entangled two-qubit states. Hence one PR box is not a universal resource, but these inequalities do not decide whether two boxes suffice [BGS05], [BSG06].
Photonic experiments have violated one-box bounds and thereby certified a two-box lower bound for selected measurements. This does not construct a two-box simulation valid for every state and measurement [CLBGK15].

Comment

Neither a universal exact two-box protocol nor a counterexample requiring more than two boxes is known. In finite input–output scenarios, a complete description of the convex set generated by two-box wirings is likewise open.

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
[BGS05]: N. Brunner, N. Gisin, and V. Scarani, “Entanglement and Non-locality Are Different Resources,” New Journal of Physics 7, 88 (2005).DOI arXiv
[BSG06]: N. Brunner, V. Scarani, and N. Gisin, “Bell-Type Inequalities for Non-local Resources,” Journal of Mathematical Physics 47, 112101 (2006).DOI arXiv
[CLBGK15]: B. G. Christensen, Y.-C. Liang, N. Brunner, N. Gisin, and P. G. Kwiat, “Exploring the Limits of Quantum Nonlocality with Entangled Photons,” Physical Review X 5, 041052 (2015).DOI arXiv

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