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Foundation of Quantum Optimal Transport and Applications

Kazuki Ikeda

TL;DR

The Monge–Kantorovich problem is extended and the folk theorem of the quantum prisoners’ dilemma is established, which claims mutual cooperation can be an equilibrium of the infinitely repeated quantum game.

Abstract

Quantum optimal transportation seeks an operator which minimizes the total cost of transporting a quantum state to another state, under some constraints that should be satisfied during transportation. We formulate this issue by extending the Monge-Kantorovich problem, which is a classical optimal transportation theory, and present some applications. As examples, we address quantum walk, quantum automata and quantum games from a viewpoint of optimal transportation. Moreover we explicitly show the folk theorem of the prisoners' dilemma, which claims mutual cooperation can be an equilibrium of the repeated game. A series of examples would show generic and practical advantages of the abstract quantum optimal transportation theory.

Foundation of Quantum Optimal Transport and Applications

TL;DR

The Monge–Kantorovich problem is extended and the folk theorem of the quantum prisoners’ dilemma is established, which claims mutual cooperation can be an equilibrium of the infinitely repeated quantum game.

Abstract

Quantum optimal transportation seeks an operator which minimizes the total cost of transporting a quantum state to another state, under some constraints that should be satisfied during transportation. We formulate this issue by extending the Monge-Kantorovich problem, which is a classical optimal transportation theory, and present some applications. As examples, we address quantum walk, quantum automata and quantum games from a viewpoint of optimal transportation. Moreover we explicitly show the folk theorem of the prisoners' dilemma, which claims mutual cooperation can be an equilibrium of the repeated game. A series of examples would show generic and practical advantages of the abstract quantum optimal transportation theory.

Paper Structure

This paper contains 15 sections, 2 theorems, 58 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Quantum Optimal Transport
  3. The Monge-Kantorovich Problem
  4. Quantum Optimal Transport
  5. Dynamical Approach
  6. Variant 1
  7. Variant 2
  8. Variant 3
  9. Variant 4
  10. Variant 5
  11. Applications of Quantum Optimal Transportation
  12. Costly Quantum Walk
  13. Quantum Cellular Automata
  14. Automata and Games
  15. Repeated Quantum Games

Key Result

Theorem 3.1

There is such a quantum strategy for the repeated QPD that is an equilibrium of the repeated game.

Figures (2)

  • Figure 1: Strategy profile/state transition diagram of the grim trigger strategy. The bold arrow stands for an initial strategy/state.
  • Figure :

Theorems & Definitions (2)

  • Theorem 3.1: Strategic Efficiency
  • Proposition 3.2