One-bit simulation of partially entangled qubits

Problem

Can shared randomness and one classical bit exactly simulate every pair of local projective measurements on every pure entangled two-qubit state? In Schmidt form the state is

\begin{equation} \lvert\psi_p\rangle =\sqrt p\,\lvert00\rangle+\sqrt{1-p}\,\lvert11\rangle, \qquad \frac12<p<1. \tag{1} \end{equation}

For arbitrary Bloch vectors \(\mathbf x,\mathbf y\in S^2\) and outcomes \(a,b\in\{0,1\}\), the target distribution associated with Eq. (1) is

\begin{equation} \begin{aligned} P_p(a,b\mid\mathbf x,\mathbf y) &=\operatorname{Tr}\!\left[ \lvert\psi_p\rangle\!\langle\psi_p\rvert \bigl(M_{a\mid\mathbf x}\otimes M_{b\mid\mathbf y}\bigr) \right],\\ M_{a\mid\mathbf x} &=\frac{I+(-1)^a\mathbf x\cdot\boldsymbol\sigma}{2}, \qquad M_{b\mid\mathbf y} =\frac{I+(-1)^b\mathbf y\cdot\boldsymbol\sigma}{2}. \end{aligned} \tag{2} \end{equation}

The protocol may use unlimited shared randomness; after receiving \(\mathbf x\), Alice sends Bob one bit, and their local outputs must reproduce Eq. (2) for every pair of measurement directions. If no such universal protocol exists, find a finite-setting linear inequality satisfied by all one-bit protocols and violated by one of these target distributions.

Source

Gisin explicitly asks whether one classical bit simulates every partially entangled two-qubit state; Renner and Quintino restate the surviving question after narrowing the parameter range [Gis09], [RQ23].

Progress

Gisin posed the separating-inequality question for one-bit-assisted correlations [Gis09]. At the maximally entangled endpoint \(p=1/2\), Toner and Bacon gave an exact one-bit protocol for all projective measurements [TB03]; this does not cover the nonmaximally entangled interval in Eq. (1).
Renner and Quintino constructed a one-trit protocol for every \(1/2\leq p\leq1\). For \(1/2<p<1\), their protocol uses only one bit whenever

\begin{equation} \frac{2p(1-p)}{2p-1}\log\!\left(\frac{p}{1-p}\right)+2(1-p)\leq1, \tag{3} \end{equation}

which holds approximately for \(0.835\leq p<1\); the product-state endpoint \(p=1\) requires no communication. Thus Eq. (3) leaves the interval \(1/2<p<0.835\) unresolved [RQ23].
Numerical and semianalytical models support one-bit simulation beyond the proved range, but do not supply an exact protocol for all directions [SLYS23]. Conversely, finite-scenario inequalities for one-bit-assisted correlations can be enumerated, but no known inequality separates the remaining targets in Eq. (2) [BT03].

Comment

The unresolved parameter range is approximately \(1/2<p<0.835\). It is not known whether one bit always suffices there or whether a finite-setting inequality can prove that it does not.

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
[TB03]: B. F. Toner and D. Bacon, “Communication Cost of Simulating Bell Correlations,” Physical Review Letters 91, 187904 (2003).DOI arXiv
[RQ23]: M. J. Renner and M. T. Quintino, “The Minimal Communication Cost for Simulating Entangled Qubits,” Quantum 7, 1149 (2023).DOI arXiv
[SLYS23]: P. Sidajaya, A. D. Lim, B. Yu, and V. Scarani, “Neural Network Approach to the Simulation of Entangled States with One Bit of Communication,” Quantum 7, 1150 (2023).DOI arXiv
[BT03]: D. Bacon and B. F. Toner, “Bell Inequalities with Auxiliary Communication,” Physical Review Letters 90, 157904 (2003).DOI arXiv

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