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Mean-field type quantum filter for a quantum Ising type system

Kentaro Ohki

Published Jan 1, 2020• 0 citations
Publisher

Abstract

Mean-field games or mean-field type control problems are one of distributed control schemes that reduce the computational burden. In this paper, a quantum version of mean-field game settings is developed and the mean-field type quantum filter is derived for quantum Ising models.
DOI
10.1016/j.ifacol.2020.12.1489

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In this paper we are merging the two new branches of game theory: quantum games and mean-field games (MFG). Building a quantum analog of MFGs requires the full reconstruction of its foundations and methodology, because in N-particle quantum evolution particles are not separated in individual dynamics and the key concept of the classical MFG theory, the empirical measure defined as the sum of Dirac masses of the positions of the players, is not applicable in quantum setting. As a preliminary result we derive the new nonlinear stochastic Schrödinger equation, as the limit of the quantum filtering equation describing continuously observed and controlled system of a large number of interacting particles, the result that may have an independent value. We then show that to a control quantum system of interacting particles there corresponds a special system of classical interacting particles with the identical limiting MFG system, defined on an appropriate Riemanian manifold. Solutions of this system are shown to specify approximate Nash equilibria for N-agent quantum games.
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