Optimal quantum channels

Abstract

A method to optimize the cost of a quantum channel is developed. The goal is to determine the cheapest channel that produces prescribed output states for a given set of input states. This is essentially a quantum version of optimal transport. To attach a clear conceptual meaning to the cost, channels are viewed in terms of what we call elementary transitions, which are analogous to point-to-point transitions between classical systems. The role of entanglement in optimization of cost is emphasized. We also show how our approach can be applied to theoretically search for channels performing a prescribed set of tasks on the states of a system, while otherwise disturbing the state as little as possible.

v1: 5 pages. v2: More complete discussion of the role of entanglement. Faulty example extended to a separate section. Other minor additions and corrections. References added. 8 pages. v3: Some references and clarifying remarks added. Minor corrections made. 9 pages. v4: Minor typos (missing transposition signs) fixed at end of Sec. VI

References in corpus (5)

• Quasiprobability distribution functions for periodic phase-spaces: I. Theoretical Aspects
• Quantum Monge-Kantorovich problem and transport distance between density matrices
• Quasiprobability distribution functions for finite-dimensional discrete phase spaces: Spin-tunneling effects in a toy model
• Quadratic Wasserstein metrics for von Neumann algebras via transport plans
• A limit relation for entropy and channel capacity per unit cost

Cited by in corpus (12)

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• Quantum Wasserstein isometries on the qubit state space
• Quantum Channel Learning
• The Wasserstein distance of order 1 for quantum spin systems on infinite lattices
• On the metric property of quantum Wasserstein divergences
• Quantum optimal control in quantum technologies. Strategic report on current status, visions and goals for research in Europe
• Machine learning of quantum channels on NISQ devices
• Classical shadows meet quantum optimal mass transport
• Quantum channels, complex Stiefel manifolds, and optimization
• Isometries of the qubit state space with respect to quantum Wasserstein distances
• Quantum detailed balance via elementary transitions