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arXiv:2303.11707 · Letters in Mathematical Physics

Recoverability of quantum channels via hypothesis testing

Anna Jenčová

Published Feb 16, 2024• 12 citations
arXivHTMLPDF

Abstract

Abstract A quantum channel is sufficient with respect to a set of input states if it can be reversed on this set. In the approximate version, the input states can be recovered within an error bounded by the decrease of the relative entropy under the channel. Using a new integral representation of the relative entropy in Frenkel (Integral formula for quantum relative entropy implies data processing inequality, Quantum 7, 1102 (2023)), we present an easy proof of a characterization of sufficient quantum channels and recoverability by preservation of optimal success probabilities in hypothesis testing problems, equivalently, by preservation of $$L_1$$ L 1 -distance.

DOI
10.1007/s11005-024-01775-2

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Abstract A quantum channel is sufficient with respect to a set of input states if it can be reversed on this set. In the approximate version, the input states can be recovered within an error bounded by the decrease of the relative entropy under the channel. Using a new integral representation of the relative entropy in Frenkel (Integral formula for quantum relative entropy implies data processing inequality, Quantum 7, 1102 (2023)), we present an easy proof of a characterization of sufficient quantum channels and recoverability by preservation of optimal success probabilities in hypothesis testing problems, equivalently, by preservation of L1L_1L1​ L 1 -distance.
The data processing inequality states that the quantum relative entropy between two states ρ and σ can never increase by applying the same quantum channel N to both states. This inequality can be strengthened with a remainder term in the form of a distance between ρ and the closest recovered state (R∘N)(ρ), where R is a recovery map with the property that σ=(R∘N)(σ). We show the existence of an explicit recovery map that is universal in the sense that it depends only on σ and the quantum channel N to be reversed. This result gives an alternate, information-theoretic characterization of the conditions for approximate quantum error correction.
The data processing inequality states that the quantum relative entropy between two states ρ\rho ρ and σ\sigma σ can never increase by applying the same quantum channel N\mathcal {N}N to both states. This inequality can be strengthened with a remainder term in the form of a distance between ρ\rho ρ and the closest recovered state (R∘N)(ρ)(\mathcal {R} \circ \mathcal {N})(\rho )(R∘N)(ρ) , where R\mathcal {R}R is a recovery map with the property that σ=(R∘N)(σ)\sigma = (\mathcal {R} \circ \mathcal {N})(\sigma )σ=(R∘N)(σ) . We show the existence of an explicit recovery map that is universal in the sense that it depends only on σ\sigma σ and the quantum channel N\mathcal {N}N to be reversed. This result gives an alternate, information-theoretic characterization of the conditions for approximate quantum error correction.
The data processing inequality states that the quantum relative entropy between two states ρ and σ can never increase by applying the same quantum channel N to both states. This inequality can be strengthened with a remainder term in the form of a distance between ρ and the closest recovered state (R∘N)(ρ), where R is a recovery map with the property that σ=(R∘N)(σ). We show the existence of an explicit recovery map that is universal in the sense that it depends only on σ and the quantum channel N to be reversed. This result gives an alternate, information-theoretic characterization of the conditions for approximate quantum error correction.
This thesis addresses the interplay between asymptotic hypothesis testing and entropy inequalities in quantum information theory. In the first part of the thesis we focus on hypothesis testing. We consider two main settings; one can either fix quantum states while optimizing over measurements or fix a measurement and evaluate its capability to discriminate by optimizing over states. In the former setting, we prove a composite quantum Stein's Lemma. We also discuss how this gives an operational interpretation to several quantities of interest. For the latter, we give the optimal asymptotic error rates in several symmetric and asymmetric settings, and discuss properties and examples of these rates. In the second part, the focus is shifted to entropy inequalities. We start with recoverability inequalities. Using tools developed to prove the composite Stein's Lemma, we give a strengthened lower bound on the conditional quantum mutual information (CQMI). Next, we give an operational interpretation to the relative entropy of recovery via hypothesis testing. Then, we discuss some recent counterexamples, which show that the relative entropy of recovery is not a lower bound on the CQMI; we provide more counterexamples where some systems are classical. We then turn to a seemingly different type of entropy inequalities called bounds on information combining. Using a particular recoverability inequality, we give a lower bound and additionally conjecture optimal lower and upper bounds. Furthermore, we discuss implications of our bounds to the finite blocklength behavior of Polar codes. Finally, we discuss Renyi-222 entropy inequalities for Gaussian states, by exploiting their formulation as log-det inequalities to find recoverability related bounds on several quantities. We apply this to Gaussian steerability and entanglement measures, proving their monogamy and several other features.
Abstract We generalize our results in paper I in this series to quantum channels between general von Neumann algebras, proving the approximate recoverability of states which undergo a small change in relative entropy through the channel. To this end, we derive a strengthened form of the quantum data processing inequality for the change in relative entropy of two states under a channel between two von Neumann algebras. Compared to the usual inequality, there is an explicit lower bound involving the fidelity between the original state and a recovery channel.
Hypothesis testing is an important task in mathematics and physics. Hypothesis testing of two random variables is related to the Kullback-Leibler divergence of the two corresponding distributions. Similarly, quantum hypothesis testing of two quantum states is characterized by the quantum relative entropy. While a quantum state can be abstracted as a device that only has outputs but no inputs, the most general quantum device is a quantum channel that also has inputs. In this work, we extend hypothesis testing to general quantum channels. In both the one-shot and asymptotic scenario, we study several quantifiers for hypothesis testing under different assumptions of how the channels are used. As the quantifiers are analogs to the quantum relative entropy of states, we call them the quantum relative entropy of channels. Then, we define the entropy of channels based on relative entropies from the target channel to the completely depolarizing channel. We investigate the properties that the quantum relative entropy of channels should satisfy, and we study its interplay with entanglement. With the broad applications of the quantum relative entropy of states, our results can be useful for understanding general properties of quantum channels.