Abstract
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.
Highlights
Quantum games represent the really twenty-first century branch of game theory, tightly linked to the modern development of quantum computing and quantum technologies
Since direct continuous observations are known to destroy quantum evolutions, the necessary new ingredient for quantum dynamic games must be the theory of non-direct observations and the corresponding quantum filtering
There is an important work under way on the technical side of organizing feedback quantum control in real time, see, e.g., [2,12] and [36]
Summary
Quantum games represent the really twenty-first century branch of game theory, tightly linked to the modern development of quantum computing and quantum technologies. Since direct continuous observations are known to destroy quantum evolutions (socalled quantum Zeno paradox), the necessary new ingredient for quantum dynamic games must be the theory of non-direct observations and the corresponding quantum filtering This theory was essentially developed by Belavkin in the 1980s of the last century, in [5,6,7], see [11] for a readable modern account. In the following two sections, we introduce our main homodyne detection schemes (first for qubits and for qudits) that allow one to turn the problems of dynamic quantum filtering, control and games into the problems of the drift control of the standard Brownian motions on the complex projective spaces.
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