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arXiv:2302.02341

Sufficient statistic and recoverability via Quantum Fisher Information metrics

Gao Li, Haojian Li, Iman Marvian, Cambyse Rouzé

Published Feb 5, 2023• 0 citations
arXivHTMLPDF

Abstract

We prove that for a large class of quantum Fisher information, a quantum channel is sufficient for a family of quantum states, i.e., the input states can be recovered from the output, if and only if the quantum Fisher information is preserved under the quantum channel. This class, for instance, includes Winger-Yanase-Dyson skew information. On the other hand, interestingly, the SLD quantum Fisher information, as the most popular example of quantum analog of Fisher information, does not satisfy this property. Our recoverability result is obtained by studying Riemannian monotone metrics on the quantum state space, i.e. metrics monotone decreasing under the action of quantum channels, a property often called data processing inequality. For two quantum states, the monotone metric gives the corresponding quantum $χ^2$ divergence. We obtain an approximate recovery result in the sense that, if the quantum $χ^2$ divergence is approximately preserved by a quantum channel, then two states can be approximately recovered by the Petz recovery map. We also obtain a universal recovery bound for the $χ_{\frac{1}{2}}$ divergence. Finally, we discuss applications in the context of quantum thermodynamics and the resource theory of asymmetry.

DOI
10.48550/arxiv.2302.02341

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Abstract The optimized quantum f -divergences form a family of distinguishability measures that includes the quantum relative entropy and the sandwiched Rényi relative quasi-entropy as special cases. In this paper, we establish physically meaningful refinements of the data-processing inequality for the optimized f -divergence. In particular, the refinements state that the absolute difference between the optimized f -divergence and its channel-processed version is an upper bound on how well one can recover a quantum state acted upon by a quantum channel, whenever the recovery channel is taken to be a rotated Petz recovery channel. Not only do these results lead to physically meaningful refinements of the data-processing inequality for the sandwiched Rényi relative entropy, but they also have implications for perfect reversibility (i.e. quantum sufficiency) of the optimized f -divergences. Along the way, we improve upon previous physically meaningful refinements of the data-processing inequality for the standard f -divergence, as established in recent work of Carlen and Vershynina [arXiv: 1710.02409 , arXiv: 1710.08080 ]. Finally, we extend the definition of the optimized f -divergence, its data-processing inequality, and all of our recoverability results to the general von Neumann algebraic setting, so that all of our results can be employed in physical settings beyond those confined to the most common finite-dimensional setting of interest in quantum information theory.
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Any reasonable measure of distinguishability of quantum states must satisfy a data processing inequality, that is, it must not increase under the action of a quantum channel. We can ask about the proportion of information lost or preserved and this leads us to study contraction and expansion coefficients respectively, which can be combined into a single \emph{relative expansion coefficient}. We focus on two prominent families: (i) standard quantum fff divergences and (ii) their local (second-order) behaviour, which induces a monotone Riemannian semi-norm (that is linked to the χ2χ^2χ2 divergence). Building on prior work, we identify new families of fff for which the global (fff divergence) and local (Riemannian) relative expansion coefficients coincide for every pair of channels, and we clarify how exceptional such exact coincidences are. Beyond equality, we introduce an \emph{equivalence} framework that transfers qualitative properties such as strict positivity uniformly across different relative expansion coefficients. Leveraging the link between equality in the data processing inequality (DPI) and channel reversibility, we apply our framework of relative expansion coefficients to approximate recoverability of quantum information. Using our relative expansion results for primitive channels, we prove a reverse quantum Markov convergence theorem, converting positive expansion coefficients into quantitative lower bounds on the convergence rate.
A set of classical or quantum states is equivalent to another one if there exists a pair of classical or quantum channels mapping either set to the other one. For dichotomies (pairs of states), this is closely connected to (classical or quantum) Rényi divergences (RD) and the data-processing inequality: If a RD remains unchanged when a channel is applied to the dichotomy, then there is a recovery channel mapping the image back to the initial dichotomy. Here, we prove for classical dichotomies that equality of the RDs alone is already sufficient for the existence of a channel in any of the two directions and discuss some applications. In the quantum case, all families of quantum RDs are seen to be insufficient because they cannot detect anti-unitary transformations. Thus, including anti-unitaries, we pose the problem of finding a sufficient family. It is shown that the Petz and maximal quantum RD are still insufficient in this more general sense and we provide evidence for sufficiency of the minimal quantum RD. As a side result of our techniques, we obtain an infinite list of inequalities fulfilled by the classical, the Petz quantum, and the maximal quantum RDs. These inequalities are not true for the minimal quantum RDs. Our results further imply that any sufficient set of conditions for state transitions in the resource theory of athermality must be able to detect time-reversal.
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.