Two-Qubit Pure Entanglement as Optimal Social Welfare Resource in Bayesian Game

Manik Banik1, Some Sankar Bhattacharya2, Nirman Ganguly3, Tamal Guha4, Amit Mukherjee5, Ashutosh Rai6,7,8, and Arup Roy1

1S.N. Bose National Centre for Basic Sciences, Block JD, Sector III, Salt Lake, Kolkata 700098, India.
2Department of Computer Science, The University of Hong Kong, Pokfulam Road, Hong Kong.
3Department of Mathematics, Birla Institute of Technology and Science Pilani, Hyderabad campus, Telengana 500078,India.
4Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700108, India.
5Optics and Quantum Information Group, The Institute of Mathematical Sciences, HBNI, CIT Campus, Taramani, Chennai 600113, India.
6School of Electrical Engineering, Korea Advanced Institute of Science and Technology (KAIST), 291 Daehak-ro, Yuseong-gu, Daejeon 34141, Republic of Korea.
7International Institute of Physics, Federal University of Rio Grande do Norte, 59070-405 Natal, Brazil.
8Centre for Quantum Computer Science, University of Latvia, Raina Bulv. 19, Riga, LV-1586, Latvia.

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Entanglement is of paramount importance in quantum information theory. Its supremacy over classical correlations has been demonstrated in a numerous information theoretic protocols. Here we study possible adequacy of quantum entanglement in Bayesian game theory, particularly in social welfare solution (SWS), a strategy which the players follow to maximize sum of their payoffs. Given a multi-partite quantum state as an advice, players can come up with several correlated strategies by performing local measurements on their parts of the quantum state. A quantum strategy is called quantum-SWS if it is advantageous over a classical equilibrium (CE) strategy in the sense that none of the players has to sacrifice their CE-payoff rather some have incentive and at the same time it maximizes sum of all players' payoffs over all possible quantum advantageous strategies. Quantum state yielding such a quantum-SWS is called a quantum social welfare advice (SWA). We show that any two-qubit pure entangled state, even if it is arbitrarily close to a product state, can serve as quantum-SWA in some Bayesian game. Our result, thus, gives cognizance to the fact that every two-qubit pure entanglement is the best resource for some operational task.

Quantum entanglement: a social welfare advice in Bayesian game.

Game theory - the study of human conflict and cooperation within a competitive situation – has found applications in a wide range of studies starting from economics, political sciences, biological phenomena, logic, computer science and psychology. The seminal Nash theorem assures the existence of strategies, the so called Nash equilibrium, adopted by the players from which none of the players has any incentive to shift those. In a Bayesian scenario, where each player has some private information unknown to the other players, Robert Aumann proved that the proper notion of equilibrium is not the ordinary Nash equilibrium but a more general – correlated equilibrium. In Kenneth Arrow’s social choice theory it becomes apparent that instead of pursuing solely their own payoffs, considering additional social goals can benefit a person.

Quantum mechanics – the most precise theory of microscopic world - predicts a vast range of astonishing and often strikingly counterintuitive phenomena, such as quantum entanglement. Entanglement captures the physical phenomenon involving composite quantum systems where the quantum states of the subsystems cannot be described independently even when the subsystems are separated by a large distance. Apart from several underlying foundational puzzles of quantum theory, such as, Einstein–Podolsky–Rosen paradox, Schrödinger Steering phenomenon, and Bell Nonlocality, quantum entanglement has also been established as a useful resource for a number of computational and communication tasks.

In the present work, the authors study possible adequacy of quantum entanglement in Bayesian game theory, particularly in social welfare solution (SWS), a strategy which the players follow to maximize the sum of their payoffs. A quantum strategy is called quantum-SWS if it provides an advantage over a classical equilibrium (CE) strategy in the sense that none of the players has to sacrifice their CE-payoffs. Instead some of them have incentive and at the same time it maximizes the sum of all players' payoffs among all possible quantum advantageous strategies. A quantum state yielding such a quantum-SWS is called a quantum social welfare advice (SWA). It has been shown that any pure entangled state of a two-qubit system (the most elementary composite quantum system) can serve as quantum-SWA in some Bayesian game.

In the time, when quantum technologies are on the verge of evolving into their second commercial generation, and researchers are trying to find applications of quantum resources in different new tasks, the present work gives cognizance to the fact that every two-qubit pure entanglement is the optimal resource – the ‘gold coin’ - for some operational task.

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