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arXiv:2405.07389 · IEEE Conference on Decision and Control (CDC)

Observed quantum particles system with graphon interaction

Sofiane Chalal, Nina H. Amini, Gaoyue Guo, Hamed Amini

Published Dec 16, 2024• 0 citations
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Abstract

In this paper, we consider a system of heterogeneously interacting quantum particles subject to indirect continuous measurement. The interaction is assumed to be of the mean-field type. We derive a new limiting quantum graphon system, prove the well-posedness of this system, and establish a stability result.
DOI
10.1109/cdc56724.2024.10885856

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We consider a system of N identical quantum particles coupled to an environment. The coupling is scaled in a mean-field way, and particles interact indirectly through the environment. There is no direct interaction between particles. The time evolution of the mean-field-scaled particles is called the mean-field dynamics and the limit of the evolution as N → ∞ is called the mean-field limit. The study of mean-field limits is an active topic of research in many-body quantum theory. Previously, almost all research in this area has been carried out for directly interacting particles without interaction with an environment. In this thesis, we prove that under a suitable condition on the initial system states the mean-field limit in the above indirectly interacting particle system does exist. The condition is satisfied in particular for spin ½ (qubit) systems. Assuming this condition, we show that the system dynamics is the free dynamics in the mean-field limit N → ∞, and we find the first and second correction terms. We show that the particles have a collective effect on the dynamics of the environment (reservoir) which we derive analytically. We give examples to illustrate our results.
Stochastic Master equations or quantum filtering equations for mixed states are well known objects in quantum physics. Building a mathematically rigorous theory of these equations in infinite-dimensional spaces has been a long standing open problem. The first objective of this paper is to give a solution to this problem under the assumption of bounded operators providing coupling with environment (or a measurement device). Furthermore, recently the author built the theory of the law of large number limit for continuously observed interacting quantum particle systems leading to quantum mean-field games. These limits are described by certain nontrivial extensions of quantum stochastic master equations that can be looked at as infinite-dimensional operator-valued McKean-Vlasov diffusions. The second objective of this paper is to provide a well-posedness result for these new class of McKean-Vlasov diffusions.
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Quantum games and mean-field games (MFG) represent two important new branches of game theory. In a recent paper the author developed quantum MFGs merging these two branches. These quantum MFGs were based on the theory of continuous quantum observations and filtering of diffusive type. In the present paper we develop the analogous quantum MFG theory based on continuous quantum observations and filtering of counting type. However, proving existence and uniqueness of the solutions for resulting limiting forward-backward system based on jump-type processes on manifolds seems to be more complicated than for diffusions. In this paper we only prove that if a solution exists, then it gives an ϵ-Nash equilibrium for the corresponding N-player quantum game. The existence of solutions is suggested as an interesting open problem.
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.