The doubly minimized Petz Renyi mutual information of order $α$ is defined as the minimum of the Petz divergence of order $α$ of a given bipartite quantum state relative to all product states. The doubly minimized sandwiched Renyi mutual information is defined analogously, with the Petz divergence replaced by the sandwiched divergence. In this work, we study certain binary quantum state discrimination problems related to correlation detection. We show that the corresponding direct exponent is determined by the doubly minimized Petz Renyi mutual information of order $α\in (1/2,1)$, and that the strong converse exponent is determined by the doubly minimized sandwiched Renyi mutual information of order $α\in (1,\infty)$. This provides an operational interpretation of these types of Renyi mutual information and generalizes previous results for classical probability distributions to the quantum setting. For completeness, we also study the corresponding moderate deviation regime both below and above the threshold, and determine the Stein exponent and the second-order asymptotics.
Tumula information and doubly minimized Petz Renyi lautum information
Institute for Theoretical Physics, ETH Zurich（苏黎世联邦理工学院理论物理研究所）；IBM Quantum, IBM Research Europe – Zurich（IBM量子，IBM欧洲研究-苏黎世）；Scuola Normale Superiore (SNS)（比萨高等师范学院）
We derive a new bound on the effectiveness of the Petz map as a universal recovery channel in approximate quantum error correction using the second sandwiched Rényi relative entropy $\tilde{D}_{2}$. For large Hilbert spaces, our bound implies that the Petz map performs quantum error correction with order-$ε$ accuracy whenever the data processing inequality for $\tilde{D}_{2}$ is saturated up to terms of order $ε^2$ times the inverse Hilbert space dimension. Conceptually, our result is obtained by extending arXiv:2011.03473, in which we studied exact saturation of the data processing inequality…
The quantum relative entropy is a measure of the distinguishability of two quantum states, and it is a unifying concept in quantum information theory: many information measures such as entropy, conditional entropy, mutual information, and entanglement measures can be realized from it. As such, there has been broad interest in generalizing the notion to further understand its most basic properties, one of which is the data processing inequality. The quantum f-divergence of Petz is one generalization of the quantum relative entropy, and it also leads to other relative entropies, such as the Petz…
We provide a transparent, simple and unified treatment of recent results on the equality conditions for the data processing inequality (DPI) of the sandwiched quantum Rényi divergence, including the statement that equality in the data processing implies recoverability via the Petz recovery map for the full range of $α$ recently proven by Jen\v cová. We also obtain a new set of equality conditions, generalizing a previous result by Leditzky et al.
The quantum relative entropy is a measure of the distinguishability of two quantum states, and it is a unifying concept in quantum information theory: many information measures such as entropy, conditional entropy, mutual information, and entanglement measures can be realized from it. As such, there has been broad interest in generalizing the notion to further understand its most basic properties, one of which is the data processing inequality. The quantum f-divergence of Petz is one generalization of the quantum relative entropy, and it also leads to other relative entropies, such as the Petz…
Distance measures between quantum states like the trace distance and the fidelity can naturally be defined by optimizing a classical distance measure over all measurement statistics that can be obtained from the respective quantum states. In contrast, Petz showed that the measured relative entropy, defined as a maximization of the Kullback-Leibler divergence over projective measurement statistics, is strictly smaller than Umegaki's quantum relative entropy whenever the states do not commute. We extend this result in two ways. First, we show that Petz' conclusion remains true if we allow genera…
Quantum information quantities play a substantial role in characterizing operational quantities in various quantum information-theoretic problems. We consider numerical computation of four quantum information quantities: Petz-Augustin information, sandwiched Augustin information, conditional sandwiched Renyi entropy and sandwiched Renyi information. To compute these quantities requires minimizing some order-$α$ quantum Renyi divergences over the set of quantum states. Whereas the optimization problems are obviously convex, they violate standard bounded gradient/Hessian conditions in literature…
Rényi and Augustin information are generalizations of mutual information defined via the Rényi divergence, playing a significant role in evaluating the performance of information processing tasks by virtue of its connection to the error exponent analysis. In quantum information theory, there are three generalizations of the classical Rényi divergence -- the Petz's, sandwiched, and log-Euclidean versions, that possess meaningful operational interpretation. However, the associated quantum Rényi and Augustin information are much less explored compared with their classical counterpart, and lacking…
Operational interpretation of the reverse sandwiched Renyi divergences in composite quantum hypothesis testing
School of Data Science, The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳）数据科学学院）；International Quantum Academy（国际量子学院）；Graduate School of Mathematics, Nagoya University（名古屋大学大学院数学研究科）
The quantum Renyi relative entropies play a prominent role in quantum information theory, finding applications in characterizing error exponents and strong converse exponents for quantum hypothesis testing and quantum communication theory. On a different thread, quantum Gaussian states have been intensely investigated theoretically, motivated by the fact that they are more readily accessible in the laboratory than are other, more exotic quantum states. In this paper, we derive formulas for the quantum Renyi relative entropies of quantum Gaussian states. We consider both the traditional (Petz) …
Institute for Quantum Information, RWTH Aachen University（亚琛工业大学量子信息研究所）；Department of Electrical Engineering and Graduate Institute of Communication Engineering, National Taiwan University（台湾大学电机工程学系暨通信工程学研究所）；Department of Mathematics, National Taiwan University（台湾大学数学系）；Center for Quantum Science and Engineering, National Taiwan University（台湾大学量子科学与工程中心）；Physics/Mathematics Division, National Center for Theoretical Sciences（国家理论科学中心物理/数学组）；Hon Hai (Foxconn) Quantum Computing Center（鸿海（富士康）量子计算中心）
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…
The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））；University of Cambridge（剑桥大学）；Univ Lyon, Inria, ENS Lyon, UCBL, LIP, Lyon, France（里昂大学、法国国家信息与自动化研究所、巴黎高等师范学院里昂分校、里昂第一大学、法国国家科学研究中心里昂实验室）
The concept of classical $f$-divergences gives a unified framework to construct and study measures of dissimilarity of probability distributions; special cases include the relative entropy and the Rényi divergences. Various quantum versions of this concept, and more narrowly, the concept of Rényi divergences, have been introduced in the literature with applications in quantum information theory; most notably Petz' quasi-entropies (standard $f$-divergences), Matsumoto's maximal $f$-divergences, measured $f$-divergences, and sandwiched and $α$-$z$-Rényi divergences. In this paper we give a syste…
Rényi divergences and binary state discrimination error exponents for fermionic quasi-free states
HUN-REN Wigner Research Centre for Physics（匈牙利科学院维格纳物理研究中心）；Department of Analysis and Operations Research, Institute of Mathematics, Budapest University of Technology and Economics（布达佩斯科技大学数学研究所分析与运筹学系）
Additivity of quantum relative entropies as a single-copy criterion
School of Mathematics, Institute for Research in Fundamental Sciences (IPM)（基础科学研究所）；Centre for Quantum Technologies, National University of Singapore（新加坡国立大学量子技术中心）；Department of Electrical and Computer Engineering, College of Design and Engineering, National University of Singapore（新加坡国立大学设计与工程学院电气与计算机工程系）
Quantum Computing and Information Research Centre (QX), Faculty of Science, King Mongkut’s University of Technology Thonburi（泰国皇家理工大学科学技术学院量子计算与信息研究中心）
