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Quantum entanglement monogamy games: Repetition and dimension implications

Moran, Francis

Published Jan 1, 2025• 0 citations
Publisher

Abstract

We study monogamy of entanglement, a fundamental constraint on how entanglement can be distributed across multipartite systems, through the framework of extended nonlocal quantum games. We analyze and compare three classes of strategies and their success probabilities: classical, quantum, and non-signaling. Our results demonstrate how the values of these strategies change in higher-dimensional games as well as with multiple repetitions. In particular, we demonstrate that the ratio between the non-signaling and quantum value can be made arbitrarily large with sufficient game repetitions. These games are subsequently incorporated into one-sided device-independent quantum key distribution protocols, with specific parameters such as key rate clearly defined. Finally, a simulation of a monogamy of entanglement game using IBM's Quantum Information Software Kit demonstrates monogamy of entanglement on real quantum hardware.

DOI
10.7282/t3-7et2-5876

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We study monogamy of entanglement, a fundamental constraint on how entanglement can be distributed across multipartite systems, through the framework of extended nonlocal quantum games. We analyze and compare three classes of strategies and their success probabilities: classical, quantum, and non-signaling. Our results demonstrate how the values of these strategies change in higher-dimensional games as well as with multiple repetitions. In particular, we demonstrate that the ratio between the non-signaling and quantum value can be made arbitrarily large with sufficient game repetitions. These games are subsequently incorporated into one-sided device-independent quantum key distribution protocols, with specific parameters such as key rate clearly defined. Finally, a simulation of a monogamy of entanglement game using IBM's Quantum Information Software Kit demonstrates monogamy of entanglement on real quantum hardware.
Abstract The distribution of entanglement in a multiparty system can be described by the principle of monogamy. Monogamy is a fundamental characteristic of entanglement that restricts its distribution among more than two parties. In this work, our aim is to explore how quantum entanglement is distributed in accordance with monogamy relations, which utilize both genuine multipartite entanglement measures and bipartite entanglement measures. Specifically, we treat the source entanglement or the accessible entanglement as genuine multipartite entanglement measures and use the entanglement of formation for bipartite cases. For GHZ class states, we analytically demonstrate that the square of the source entanglement serves as an upper bound for the sum of the squares of the entanglement of formation of the reduced subsystems, with some exceptions in specific non-generic GHZ states. We will also present numerical evidence supporting this result for W-class states. In addition, we also explore the monogamy relationship by using accessible entanglement as an upper bound. Our analysis reveals an opposite but consistent pattern in the behavior of monogamy depending on whether source entanglement or accessible entanglement is used as the upper bound. This pattern emphasizes the dual nature of these two measures.
Multipartite entanglement remains a challenging and not fully understood aspect of quantum information. Monogamy-of-Entanglement (MoE) games have been highly effective for studying limitations on the usefulness of entanglement imposed by monogamy constraints. To better reveal the extent to which multipartite entanglement can be useful, we introduce a class of quantum guessing games, termed One-at-a-Time Guessing (OTG) games. In these games, quantum players individually guess the outcomes of random measurements performed by a referee on a pre-shared entangled state. Unlike MoE games, OTG games select players individually at random according to a specified probability distribution, thereby probing each player's correlation with the referee. We show that, despite monogamy constraints, players sharing certain entangled states can moderately outperform those relying only on classical uncertainty. This advantage arises even in simple OTG games involving only Pauli measurements on qubits, where optimal entanglement increases the winning probability by at least 4%. This contrasts with MoE games, where shared entanglement has been shown in several settings to provide only limited (if any) advantage over classical strategies. We further establish a majorization property: the value of an OTG game respects the majorization ordering of the player-selection probability distribution. We also analyze in detail a two-player OTG game in which the referee measures one of the three Pauli observables on a qubit, and show that it is optimally played using a specific parameterized family of three-qubit WWW-like states. These results suggest that OTG games provide a useful framework for investigating the usefulness of multipartite entanglement in multiparty quantum correlations.
Bell monogamy relations characterize the trade-offs in Bell inequality violations among pairs of players in multiplayer settings. In this work, we introduce a method for extending monogamy relations from a distinguished set of configurations to monogamy relations in all possible multiplayer settings. Applying this approach, we show that nonlocality in the Clauser, Horne, Shimony, and Holt game arises in only two cases: the original two-player scenario and the four-player scenario on a line. While the bound for this four-player scenario follows from known quadratic monogamy constraints, we also establish two six-party numerical monogamy relations that cannot be derived from existing results. In particular, we show there are points in the intersection of consecutive quadratic Bell monogamy relations which are not quantum realizable. Finally, we present a nonlocal game in which a single player can simultaneously saturate the quantum value with two other parties. This is an example of a nonlocal game unaffected by the monogamous nature of quantum entanglement.
Coordination is a fundamental primitive in communication and information theory, in which distributed systems must collectively generate correlated behavior rather than merely exchange messages. In quantum networks, the nature of entanglement, quantum measurements, and nonclassical correlations introduces coordination possibilities unavailable classically. This article reviews recent advances in coordination over classical and quantum communication networks, focusing on empirical and strong coordination in multi-user settings. We consider the implications of coordination for nonlocal games, showing how it provides a natural framework for understanding the correlations that enable spatially separated players to improve their probability of winning. We present a unified framework for coordination using classical or quantum communication and pre-shared correlation resources. The review covers simulation of both entanglement and separable correlations across a variety of network architectures, including two-node, cascade, broadcast, and multiple-access networks. We study the operational differences between empirical and strong coordination, and the tradeoffs between communication and correlation resources, such as pre-shared randomness and entanglement. Coordination plays a major role in device-independent quantum key distribution (DI-QKD) schemes, in which parties can generate a secret key even if the devices used in the process have been prepared by an adversary. Furthermore, we examine the role of coordination in quantum repeaters, where distributed entanglement serves as a resource for long-distance quantum communication. The review highlights connections between coordination theory and applications such as distributed quantum systems, quantum internet architectures, and future communication networks.
Extended non-local games are a generalization of monogamy-of-entanglement games, played by two quantum parties and a quantum referee that performs a measurement on their local quantum system. Along the lines of the NPA hierarchy, the optimal winning probability of those games can be upper bounded by a hierarchy of semidefinite programs (SDPs) converging to the optimal value. Here, we show that if one extends such games by considering constraints and loss , motivated by experimental errors and loss through quantum communication, the convergence of the SDPs to the optimal value still holds. We give applications of this result, and we compute SDPs that show tighter security of protocols for relativistic bit commitment, quantum key distribution, and quantum position verification.
The monogamy and polygamy properties of quantum entanglement characterize fundamental constraints on the distribution of entanglement in multipartite quantum systems. In this paper, we investigate tighter monogamy and polygamy relations for multipartite entanglement. By establishing a new mathematical inequality, we derive a family of improved monogamy and polygamy inequalities for tripartite quantum systems and further extend these results to general multipartite systems. Comparisons with existing results show that the obtained bounds are tighter. Illustrative examples are provided to demonstrate the effectiveness of the proposed relations.