Twenty-one characterizations of reversible quantum channels

Abstract

Reversible quantum channels play a fundamental role in quantum dynamics and quantum information processing. A quantum channel is reversible if there exists another quantum channel acting as its left inverse. Due to their intrinsic significance and wide applications, it is desirable to characterize reversible quantum channels from diverse perspectives. In this work, we study reversible quantum channels on finite-dimensional Hilbert spaces, with particular emphasis on the case of different input and output dimensions. We systematically present twenty-one equivalent characterizations of reversible quantum channels from algebraic, geometrical, and information-theoretical perspectives. Among these characterizations, some are well known, while others, implicit in the literature or formulated in other contexts, are clarified here; the Choi-state characterization is derived in this work. Specifically, we prove that the Choi states of reversible quantum channels admit three equivalent forms: the spectral, direct-sum, and tensor-product representations. These twenty-one equivalent characterizations establish a comprehensive framework for reversible quantum channels, provide diverse insights into the structural and information-theoretic properties of quantum channels, and facilitate applications of reversibility in quantum information processing such as quantum error correction, quantum teleportation, and quantum thermodynamics.

The paper studies finite-dimensional quantum channels $\mathcal E:\mathcal S(H_a)\to\mathcal S(H_b)$ that admit a channel-valued left inverse on the entire input state space, allowing unequal input and output dimensions. It organizes twenty-one equivalent characterizations spanning Kraus-operator structure, recovery maps, Choi states, complementary channels, operator geometry, and information preservation. The structural description is $\mathcal E(\rho)=\sum_{k=1}^{m}\lambda_kV_k\rho V_k^\dagger$, where $V_k^\dagger V_{k'}=\delta_{kk'}\mathbf1_a$ and the positive weights sum to one. Consequently, $d_a\le d_b$ and $m=\operatorname{rank}J_{\mathcal E}\le\lfloor d_b/d_a\rfloor$. The Choi-state characterization, identified as the new contribution, gives equivalent spectral, direct-sum, and tensor-product forms: the Choi state consists of maximally entangled blocks with orthogonal output supports, or equivalently a maximally entangled state tensored with a fixed ancillary state, up to an output unitary and zero extension. Other equivalences include Knill–Laflamme conditions, Petz and rotated-Petz recovery, a constant complementary channel, scaled Schatten-norm preservation, preservation of suitable distinguishability measures, and saturation of mutual- and coherent-information bounds. Recovery is uniquely fixed by the Petz map on the output support, with arbitrary channel action on its orthogonal complement. Detailed equivalence proofs are assigned to the appendix; many characterizations are collected or reformulated from existing results. The scope is exact, global reversibility in finite dimensions, not recovery of a selected state family. Infinite-dimensional extensions, approximate reversibility, and quantitative measures of irreversibility are left for future work.