Empirical coordination of separable quantum correlations

Husein Natur; Uzi Pereg; Husein Natur; Uzi Pereg

doi:10.3934/math.2025458

AIMS Mathematics

2025, Volume 10, Issue 4: 10028-10061. doi: 10.3934/math.2025458

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Empirical coordination of separable quantum correlations

Husein Natur ^1,2,
Uzi Pereg ^{1,2
,
,}

1.
Faculty of Electrical and Computer Engineering, Technion, Haifa, Israel
2.
Helen Diller Quantum Center, Technion, Haifa, Israel

Received: 22 December 2024 Revised: 24 March 2025 Accepted: 16 April 2025 Published: 28 April 2025
MSC : 94A05, 94A24, 94A29, 94A34, 81P99

We introduce the notion of empirical coordination for quantum correlations. Quantum mechanics enables the calculation of probabilities for experimental outcomes, emphasizing statistical averages rather than detailed descriptions of individual events. Empirical coordination is thus a natural framework for quantum systems. Focusing on the cascade network, the optimal coordination rates are established, indicating the minimal resources required to simulate, on average, a quantum state. As we consider a network with classical communication links, superposition cannot be maintained, hence the quantum correlations are separable (i.e., a convex combination of product states). This precludes entanglement. Providing the users with shared randomness, before communication begins, does not affect the optimal rates for empirical coordination. We begin with a rate characterization for a basic two-node network, and then generalize to a cascade network. The special case of a network with an isolated node is considered as well. The results can be further generalized to other networks as our analysis includes a generic achievability scheme. The optimal rate formula involves optimization over a collection of state extensions. This is a unique feature of the quantum setting, as the classical parallel does not include optimization. As demonstrated through examples, the performance depends heavily on the choice of decomposition. We further discuss the consequences of our results for quantum cooperative games.
- quantum information theory,
- quantum communication,
- empirical coordination,
- quantum measurements,
- reverse Shannon theorem,
- separable correlations
Citation: Husein Natur, Uzi Pereg. Empirical coordination of separable quantum correlations[J]. AIMS Mathematics, 2025, 10(4): 10028-10061. doi: 10.3934/math.2025458

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Abstract

We introduce the notion of empirical coordination for quantum correlations. Quantum mechanics enables the calculation of probabilities for experimental outcomes, emphasizing statistical averages rather than detailed descriptions of individual events. Empirical coordination is thus a natural framework for quantum systems. Focusing on the cascade network, the optimal coordination rates are established, indicating the minimal resources required to simulate, on average, a quantum state. As we consider a network with classical communication links, superposition cannot be maintained, hence the quantum correlations are separable (i.e., a convex combination of product states). This precludes entanglement. Providing the users with shared randomness, before communication begins, does not affect the optimal rates for empirical coordination. We begin with a rate characterization for a basic two-node network, and then generalize to a cascade network. The special case of a network with an isolated node is considered as well. The results can be further generalized to other networks as our analysis includes a generic achievability scheme. The optimal rate formula involves optimization over a collection of state extensions. This is a unique feature of the quantum setting, as the classical parallel does not include optimization. As demonstrated through examples, the performance depends heavily on the choice of decomposition. We further discuss the consequences of our results for quantum cooperative games.

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