Mathematical Modeling of Finite Quantum Systems

Mingsheng Ying, Yuan Feng, Shenggang Ying

Abstract Markov decision process (MDP) offers a general framework for modelling sequential decision making where outcomes are random. In particular, it serves as a mathematical framework for reinforcement learning. This paper introduces an extension of MDP, namely quantum MDP (qMDP), that can serve as a mathematical model of decision making about quantum systems. We develop dynamic programming algorithms for policy evaluation and finding optimal policies for qMDPs in the case of finite-horizon. The results obtained in this paper provide some useful mathematical tools for reinforcement learning techniques applied to the quantum world.

Hendra I. Nurdin, Naoki Yamamoto

Ulrich Faigle, Alexander Schönhuth

A mathematical framework for observable processes is introduced via the model of systems whose states may be time dependent and described by possibly "negative probabilities". The model generalizes and includes the linearly dependent models or observable operator models for classical discrete stochastic processes. Within this model a general convergence result for finite-dimensional processes, which generalize finite state (hidden) Markov models, is derived. On the philosophical side, the model furthermore offers an explanation for Bell's inequality in quantum mechanics.

Masahito Hayashi

E.O. Samsonov, R. Goncharov, Andrei Gaidash +3 more

In this paper we report a continuous-variable quantum key distribution protocol using multimode coherent states generated on subcarrier frequencies of the optical spectrum. We propose a coherent detection scheme where power from a carrier wave is used as a local oscillator. We compose a mathematical model of the proposed scheme and perform its security analysis in the finite-size regime using fully quantum asymptotic equipartition property technique. We calculate a lower bound on the secret key rate for the system under the assumption that the quantum channel noise is negligible compared to detector dark counts, and an eavesdropper is restricted to collective attacks. Our calculation shows that the current realistic system implementation would allow distributing secret keys over channels with losses up to 9 dB.