TY - JOUR

T1 - Dynamic Quantum Games

AU - Kolokoltsov, Vassili N.

N1 - Publisher Copyright: © 2021, The Author(s).

PY - 2021/5/10

Y1 - 2021/5/10

N2 - Quantum games represent the really twenty-first century branch of game theory, tightly linked to the modern development of quantum computing and quantum technologies. The main accent in these developments so far was made on stationary or repeated games. In this paper, we aim at initiating the truly dynamic theory with strategies chosen by players in real time. Since direct continuous observations are known to destroy quantum evolutions (so-called quantum Zeno paradox), the necessary new ingredient for quantum dynamic games must be the theory of non-direct observations and the corresponding quantum filtering. Apart from the technical problems in organizing feedback quantum control in real time, the difficulty in applying this theory for obtaining mathematically amenable control systems is due partially to the fact that it leads usually to rather non-trivial jump-type Markov processes and/or degenerate diffusions on manifolds, for which the corresponding control is very difficult to handle. The starting point for the present research is the remarkable discovery (quite unexpected, at least to the author) that there exists a very natural class of homodyne detections such that the diffusion processes on projective spaces resulting by filtering under such arrangements coincide exactly with the standard Brownian motions (BM) on these spaces. In some cases, one can even reduce the process to the plain BM on Euclidean spaces or tori. The theory of such motions is well studied making it possible to develop a tractable theory of related control and games, which can be at the same time practically implemented on quantum optical devices.

AB - Quantum games represent the really twenty-first century branch of game theory, tightly linked to the modern development of quantum computing and quantum technologies. The main accent in these developments so far was made on stationary or repeated games. In this paper, we aim at initiating the truly dynamic theory with strategies chosen by players in real time. Since direct continuous observations are known to destroy quantum evolutions (so-called quantum Zeno paradox), the necessary new ingredient for quantum dynamic games must be the theory of non-direct observations and the corresponding quantum filtering. Apart from the technical problems in organizing feedback quantum control in real time, the difficulty in applying this theory for obtaining mathematically amenable control systems is due partially to the fact that it leads usually to rather non-trivial jump-type Markov processes and/or degenerate diffusions on manifolds, for which the corresponding control is very difficult to handle. The starting point for the present research is the remarkable discovery (quite unexpected, at least to the author) that there exists a very natural class of homodyne detections such that the diffusion processes on projective spaces resulting by filtering under such arrangements coincide exactly with the standard Brownian motions (BM) on these spaces. In some cases, one can even reduce the process to the plain BM on Euclidean spaces or tori. The theory of such motions is well studied making it possible to develop a tractable theory of related control and games, which can be at the same time practically implemented on quantum optical devices.

KW - Belavkin equation

KW - Brownian motion on sphere and complex projective spaces

KW - Classical and mild solutions

KW - Controlled diffusion on Riemannian manifolds

KW - Hamilton–Jacobi–Bellman–Isaacs equation on manifolds

KW - Ito’s formula

KW - Output process and innovation process

KW - Quantum control

KW - Quantum dynamic games

KW - Quantum filtering

KW - Stochastic Schrödinger equation

KW - STOCHASTIC DIFFERENTIAL-GAMES

KW - LIMIT

KW - TIME

KW - Jacobi&#8211

KW - Hamilton&#8211

KW - Bellman&#8211

KW - dinger equation

KW - Isaacs equation on manifolds

KW - Stochastic Schr&#246

KW - s formula

KW - Ito&#8217

UR - http://www.scopus.com/inward/record.url?scp=85105427309&partnerID=8YFLogxK

UR - https://www.mendeley.com/catalogue/f42e487b-36d8-39ff-8aa1-a5131a717204/

U2 - 10.1007/s13235-021-00389-w

DO - 10.1007/s13235-021-00389-w

M3 - Article

AN - SCOPUS:85105427309

JO - Dynamic Games and Applications

JF - Dynamic Games and Applications

SN - 2153-0785

ER -