Simulation of Adjoints and Petz Recovery Maps for Unknown Quantum Channels

TL;DR

This work addresses whether dual maps of an unknown quantum channel can be physically realized. It proves a strict hierarchy: the channel transpose $\mathcal{N}^{T}$ is implementable probabilistically, while the complex conjugate $\mathcal{N}^*$ and the adjoint $\mathcal{N}^{\dag}$ are not universal CP supermaps. To overcome this, the authors introduce virtual combs built from Werner–Holevo channels, enabling quasi-probabilistic simulation of $\mathcal{N}^*$ with optimal base-norm equal to the diamond norm, and show how to compose with transpose to realize $\mathcal{N}^{\dag}$ virtually. As a key application, they develop a sampling-based method to estimate the Petz recovery map $\mathcal{P}_{\sigma,\mathcal{N}}$ for unknown channels with improved query complexity, enabling practical recovery and error mitigation tasks. The results provide a rigorous operational foundation for second-order transformations of channels and offer concrete tools for probing open-system dynamics and quantum thermodynamics.

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.

Figures (3)

• Figure 1: Probabilistic implementation of the transpose of an unknown channel ${\cal N}_{A\to B}$. A maximally entangled state "$\langle$" and an input state $\rho$ are prepared. The black-box channel ${\cal N}_{A\to B}$ is applied and a Bell measurement $\{\Phi_{B'B}, {\mathds{1}}-\Phi_{B'B}\}$ is then performed. The output state on system $A'$ is ${\cal N}_{B\to A}^{{{\mathsf{T}}}}(\rho)$ when the measurement outcome is $\Phi_{B'B}$.
• Figure 2: Quasi-probabilistic simulation of the complex conjugate ${\cal N}^*$ for an unknown channel ${\cal N}$. The protocol brackets the channel with pre- and post-processing Werner-Holevo channels ${\cal W}^{\pm}$, selected according to a specific quasiprobability distribution. The target expectation value $\operatorname{Tr}[O{\cal N}^*(\rho)]$ is reconstructed through classical post-processing of the measurement outcomes from multiple sampling rounds.
• Figure 3: Protocol for estimating $\operatorname{Tr}[O_A{\cal P}_{\sigma,{\cal N}}(\omega_B)]$ of the Petz recovery map. The scheme integrates the sampling-based simulation of the channel adjoint ${\cal N}^\dagger$ and probabilistic transposition, with block-encodings of $\sigma^{1/2}$ and ${\cal N}(\sigma)^{-1/2}$.

Theorems & Definitions (6)

• Theorem 1: No-go for complex conjugation
• Lemma 2
• Theorem 3: Virtual complex conjugation
• Theorem 4
• Lemma \ref{lemma:basenorm=diamondnorm}
• Theorem \ref{thm:min_basenorm}