Simulation of Adjoints and Petz Recovery Maps for Unknown Quantum Channels

TL;DR

This paper investigates the physical realizability of quantum channel transformations, establishing a hierarchy of what can be implemented, and introduces protocols for implementing the transpose and Petz recovery map with improved efficiency.

Contribution

It provides a hierarchy of realizability for quantum channel transformations and introduces a probabilistic protocol for the transpose and an optimal virtual protocol for the complex conjugate.

Findings

Exact transpose implementation with a single query

No-go theorems for complex conjugate and adjoint transformations

Optimal virtual protocol for complex conjugate based on quasi-probability decomposition

Abstract

Transformations of quantum channels, such as the transpose, complex conjugate, and adjoint, are fundamental to quantum information theory. Given access to an unknown channel, a central problem is whether these transformations can be implemented physically with quantum supermaps. While such supermaps are known for unitary operations, the situation for general quantum channels is fundamentally different. In this work, we establish a strict hierarchy of physical realizability for the transposition, complex conjugation, and adjoint transformation of an unknown quantum channel. We present a probabilistic protocol that exactly implements the transpose with a single query. In contrast, we prove no-go theorems showing that neither the complex conjugate nor the adjoint can be implemented by any completely positive supermap, even probabilistically. We then overcome this impossibility by designing a virtual protocol for the complex conjugate based on quasi-probability decomposition, and show its optimality in terms of the diamond norm. As a key application, we propose a protocol to estimate the expectation values resulting from the Petz recovery map of an unknown channel, achieving an improved query complexity compared to existing methods.