Simulating partially entangled qubits with one bit of communication

Decide whether shared randomness and one bit of communication reproduce all projective measurements on every partially entangled qubit pair.

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	1	+# Simulating partially entangled qubits with one bit of communication
	2	+
	3	+Shared randomness plus a little classical communication can imitate the correlations of an entangled pair. For a maximally entangled pair of qubits, one bit is enough. Whether one bit also suffices for every less entangled pair, which intuition says should be easier, has been open since Gisin listed it in 2007.
	4	+
	5	+## Problem
	6	+
	7	+For $1/2<p<1$, consider the two-qubit state
	8	+
	9	+$$\lvert\psi_p\rangle=\sqrt{p}\,\lvert00\rangle+\sqrt{1-p}\,\lvert11\rangle .$$
	10	+
	11	+Alice receives a direction $\mathbf{x}$ on the unit sphere and Bob a direction $\mathbf{y}$. Measuring the spin along these directions gives outcomes $a,b\in\{0,1\}$ with probabilities
	12	+
	13	+$$P_p(a,b\mid\mathbf{x},\mathbf{y})=\langle\psi_p\rvert\,\Pi_a^{\mathbf{x}}\otimes\Pi_b^{\mathbf{y}}\,\lvert\psi_p\rangle,\qquad \Pi_a^{\mathbf{x}}=\tfrac12\big(I+(-1)^a\,\mathbf{x}\cdot\boldsymbol{\sigma}\big).$$
	14	+
	15	+In a one-bit protocol, Alice and Bob share a random variable $\lambda$; Alice sees $\mathbf{x}$ and $\lambda$, outputs $a$ and sends one bit $c$ to Bob; Bob outputs $b$ from $\mathbf{y}$, $\lambda$ and $c$.
	16	+
	17	+Question. For every $p\in(1/2,1)$, is there a one-bit protocol that reproduces $P_p(a,b\mid\mathbf{x},\mathbf{y})$ exactly for all $\mathbf{x},\mathbf{y}$? If not, find a Bell-type inequality with finitely many settings that holds for all one-bit protocols and is violated by some $\lvert\psi_p\rangle$.
	18	+
	19	+## Background
	20	+
	21	+How much communication does it take to fake quantum correlations? Toner and Bacon showed in 2003 that one bit suffices for all projective measurements on a maximally entangled pair [2]. Gisin then asked whether a Bell inequality exists that holds for every one-bit model but is violated by some partially entangled two-qubit state [1]; the question is item J of IQOQI Open Quantum Problem 32 [6]. A negative answer would mean that some weakly entangled pairs need more communication than a maximally entangled one; a positive one would give a one-bit simulation of every pure two-qubit state.
	22	+
	23	+## What is known
	24	+
	25	+- At $p=1/2$, one bit suffices [2].
	26	+- Renner and Quintino gave a protocol that uses one trit for every $p$, and one bit whenever
	27	+
	28	+ $$\frac{2p(1-p)}{2p-1}\ln\frac{p}{1-p}+2(1-p)\le1,$$
	29	+
	30	+ which holds for roughly $0.835\le p\le1$. The interval $1/2<p<0.835$ remains open [3].
	31	+- Bacon and Toner found the complete set of one-bit inequalities for two two-outcome settings per party, and for the correlation observables with three settings; quantum correlations satisfy all of them [4].
	32	+- Neural-network models, and a semianalytical protocol inspired by them, reproduce the target statistics closely. A search for any quantum correlation that one bit cannot reproduce, over several Bell scenarios with up to 5 inputs and 4 outputs, found none. This is evidence, not an exact protocol or a proof [5].
	33	+- The smallest known correlation that provably needs more than one bit comes from a maximally entangled pair of five-level systems, not from qubits ([Sidajaya and Scarani, 2024](https://doi.org/10.1103/PhysRevA.109.062408)).
	34	+
	35	+## What would count as progress
	36	+
	37	+- An exact one-bit protocol for some $p$ below $0.835$, with a proof that it reproduces $P_p$ for all directions; or one for the whole interval, which settles the question.
	38	+- A finite-setting inequality satisfied by all one-bit protocols, which can be certified by enumerating deterministic one-bit strategies as a linear program, together with directions $\mathbf{x}_i,\mathbf{y}_j$ and a $p$ for which $\lvert\psi_p\rangle$ violates it.
	39	+- Larger complete lists of one-bit inequalities for more settings, which would sharpen the numerical evidence either way.
	40	+
	41	+## References
	42	+
	43	+1. 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, 125–138 (Springer, 2009). [doi:10.1007/978-1-4020-9107-0_9](https://doi.org/10.1007/978-1-4020-9107-0_9)
	44	+2. B. F. Toner and D. Bacon, "Communication cost of simulating Bell correlations", Physical Review Letters 91, 187904 (2003). [doi:10.1103/PhysRevLett.91.187904](https://doi.org/10.1103/PhysRevLett.91.187904)
	45	+3. M. J. Renner and M. T. Quintino, "The minimal communication cost for simulating entangled qubits", Quantum 7, 1149 (2023). [doi:10.22331/q-2023-10-24-1149](https://doi.org/10.22331/q-2023-10-24-1149)
	46	+4. D. Bacon and B. F. Toner, "Bell inequalities with auxiliary communication", Physical Review Letters 90, 157904 (2003). [doi:10.1103/PhysRevLett.90.157904](https://doi.org/10.1103/PhysRevLett.90.157904)
	47	+5. 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:10.22331/q-2023-10-24-1150](https://doi.org/10.22331/q-2023-10-24-1150)
	48	+6. N. Gisin (contact), "Bell inequalities: many questions, a few answers", Open Quantum Problems, IQOQI Vienna, problem 32 (posted 2007). [Archived page](https://web.archive.org/web/20231029013532/https://oqp.iqoqi.oeaw.ac.at/bell-inequalities-many-questions-a-few-answers)
	49	+
	50	+## Source
	51	+
	52	+Imported from [QIQCOP Zoo op_64db785888803dd3](https://qiqc-op.com/problem/op_64db785888803dd3/), curated by Bikun Li, Qicheng Tang, Changhao Li, Chengkai Zhu, Minbo Gao, Zhong-Xia Shang, Bin Cheng and Naixu Guo. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.