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Masahito Hayashi - One of the best experts on this subject based on the ideXlab platform.
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Quantum Hypothesis Testing and Discrimination of Quantum States
Quantum Information Theory, 2016Co-Authors: Masahito HayashiAbstract:Various types of information processing occur in Quantum systems. The most fundamental processes are state discrimination and Hypothesis testing. These problems often form the basis for an analysis of other types of Quantum information processes. The difficulties associated with the noncommutativity of Quantum mechanics appear in the most evident way among these problems. Therefore, we examine state discrimination and Hypothesis testing before examining other types of information processing in Quantum systems in this text. In two-state discrimination, we discriminate between two unknown candidate states by performing a measurement and examining the measurement data. Note that in this case, the two hypotheses for the unknown state are treated symmetrically. In contrast, if the two hypotheses are treated asymmetrically, the process is called Hypothesis testing rather than state discrimination. Hypothesis testing is not only interesting in itself but is also relevant to other topics in Quantum information theory. In particular, the Quantum version of Stein’s lemma, which is the central topic of this chapter, is closely related to Quantum channel coding discussed in Chap. 4. Moreover, Stein’s lemma is also connected to the distillation of maximally entangled states, as discussed in Sect. 8.5, in addition to other topics discussed in Chap. 9. The importance of Stein’s lemma may not be apparent at first sight since it considers the tensor product states of identical states, which rarely appear in real communications. However, the asymptotic analysis for these tensor product states provides the key to the analysis of asymptotic problems in Quantum communications. For these reasons, this topic is discussed in an earlier chapter in this text.
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Classical-Quantum Channel Coding
Introduction to Quantum Information Science, 2014Co-Authors: Masahito Hayashi, Satoshi Ishizaka, Akinori Kawachi, Kimura, Tomohiro OgawaAbstract:Classical-Quantum channel coding is a method to transmit messages over noisy Quantum channels. This chapter is aimed at a transparent understanding of the classical-Quantum channel coding theorem, which is one of the landmarks in Quantum information theory. Since it is required at the receiver’s side to optimize the Quantum measurement to decode the messages, the theory of discriminating Quantum states (Quantum Hypothesis testing) plays an important role.
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Quantum Hypothesis Testing for Gaussian States: Quantum Analogues of χ 2 , t-, and F-Tests
Communications in Mathematical Physics, 2013Co-Authors: Wataru Kumagai, Masahito HayashiAbstract:We consider Quantum counterparts of testing problems for which the optimal tests are the χ2, t-, and F-tests. These Quantum counterparts are formulated as Quantum Hypothesis testing problems concerning Gaussian state families, and they contain nuisance parameters, which have group symmetry. The Quantum Hunt-Stein theorem removes some of these nuisance parameters, but other difficulties remain. In order to remove them, we combine the Quantum Hunt-Stein theorem and other reduction methods to establish a general reduction theorem that reduces a complicated Quantum Hypothesis testing problem to a fundamental Quantum Hypothesis testing problem. Using these methods, we derive Quantum counterparts of the χ2, t-, and F-tests as optimal tests in the respective settings.
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Quantum Hypothesis testing for Quantum gaussian states Quantum analogues of chi square t and f tests
arXiv: Quantum Physics, 2011Co-Authors: Wataru Kumagai, Masahito HayashiAbstract:We treat Quantum counterparts of testing problems whose optimal tests are given by chi-square, t and F tests. These Quantum counterparts are formulated as Quantum Hypothesis testing problems concerning Quantum Gaussian states families, and contain disturbance parameters, which have group symmetry. Quantum Hunt-Stein Theorem removes a part of these disturbance parameters, but other types of difficulty still remain. In order to remove them, combining Quantum Hunt-Stein theorem and other reduction methods, we establish a general reduction theorem that reduces a complicated Quantum Hypothesis testing problem to a fundamental Quantum Hypothesis testing problem. Using these methods, we derive Quantum counterparts of chi-square, t and F tests as optimal tests in the respective settings.
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Quantum Hypothesis testing with group symmetry
Journal of Mathematical Physics, 2009Co-Authors: Fumio Hiai, Milán Mosonyi, Masahito HayashiAbstract:The asymptotic discrimination problem of two Quantum states is studied in the setting where measurements are required to be invariant under some symmetry group of the system. We consider various asymptotic error exponents in connection with the problems of the Chernoff bound, the Hoeffding bound, and Stein’s lemma, and derive bounds on these quantities in terms of their corresponding statistical distance measures. A special emphasis is put on the comparison of the performances of group-invariant and unrestricted measurements.
H Nagaoka - One of the best experts on this subject based on the ideXlab platform.
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An Information-Spectrum Approach to Classical and Quantum Hypothesis Testing for Simple Hypotheses
IEEE Transactions on Information Theory, 2007Co-Authors: H Nagaoka, Masahito HayashiAbstract:The information-spectrum analysis made by Han for classical Hypothesis testing for simple hypotheses is extended to a unifying framework including both classical and Quantum Hypothesis testing. The results are also applied to fixed-length source coding when loosening the normalizing condition for probability distributions and for Quantum states. We establish general formulas for several quantities relating to the asymptotic optimality of tests/codes in terms of classical and Quantum information spectra
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Making Good Codes for Classical-Quantum Channel Coding via Quantum Hypothesis Testing
IEEE Transactions on Information Theory, 2007Co-Authors: Tomohiro Ogawa, H NagaokaAbstract:In this correspondence, we give an alternative proof of the direct part of the classical-Quantum channel coding theorem (the Holevo-Schumacher-Westmoreland (HSW) theorem), using ideas of Quantum Hypothesis testing. In order to show the existence of good codes, we invoke a limit theorem, relevant to the Quantum Stein's lemma, in Quantum Hypothesis testing as the law of large numbers used in the classical case. We also apply a greedy construction of good codes using a packing procedure of noncommutative operators. Consequently we derive an upper bound on the coding error probability, which is used to give an alternative proof of the HSW theorem. This approach elucidates how the Holevo information applies to the classical-Quantum channel coding problems
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The Converse Part of The Theorem for Quantum Hoeffding Bound
arXiv: Quantum Physics, 2006Co-Authors: H NagaokaAbstract:We prove the converse part of the theorem for Quantum Hoeffding bound on the asymptotics of Quantum Hypothesis testing, essentially based on an argument developed by Nussbaum and Szkola in proving the converse part of the Quantum Chernoff bound. Our result complements Hayashi's proof of the direct (achievability) part of the theorem, so that the Quantum Hoeffding bound has now been established.
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GTIT-C - Some mathematical problems related to Quantum Hypothesis testing
Lecture Notes in Computer Science, 2006Co-Authors: H NagaokaAbstract:We present two open problems related to the asymptotics of Quantum Hypothesis testing, together with some discussions about their mutual relation, the classical counterparts and a possible conjecture.
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A new proof of the channel coding theorem via Hypothesis testing in Quantum information theory
Proceedings IEEE International Symposium on Information Theory, 2002Co-Authors: H NagaokaAbstract:A new proof of the direct part of the Quantum channel coding theorem is shown based on a standpoint of Quantum Hypothesis testing. A packing procedure of mutually noncommutative operators is carried out to derive an upper bound on the error probability, which is similar to Feinstein's lemma in classical channel coding. The upper bound is used to show the proof of the direct part along with a variant of Hiai-Petz's theorem in Hypothesis testing.
Tomohiro Ogawa - One of the best experts on this subject based on the ideXlab platform.
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Two Approaches to Obtain the Strong Converse Exponent of Quantum Hypothesis Testing for General Sequences of Quantum States
IEEE Transactions on Information Theory, 2015Co-Authors: Milán Mosonyi, Tomohiro OgawaAbstract:We present two general approaches to obtain the strong converse exponent of simple Quantum Hypothesis testing for correlated Quantum states. One approach requires that the states satisfy a certain factorization property; typical examples of such states are the temperature states of translation-invariant finite-range interactions on a spin chain. The other approach requires the differentiability of a regularized Renyi $\alpha $ -divergence in the parameter $\alpha $ ; typical examples of such states include temperature states of non-interacting fermionic lattice systems, and classical irreducible Markov chains. In all cases, we get that the strong converse exponent is equal to the Hoeffding anti-divergence, which in turn is obtained from the regularized Renyi divergences of the two states.
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Quantum Hypothesis Testing and the Operational Interpretation of the Quantum Rényi Relative Entropies
Communications in Mathematical Physics, 2014Co-Authors: Milán Mosonyi, Tomohiro OgawaAbstract:We show that the new Quantum extension of Renyi’s α-relative entropies, introduced recently by Muller-Lennert et al. (J Math Phys 54:122203, 2013) and Wilde et al. (Commun Math Phys 331(2):593–622, 2014), have an operational interpretation in the strong converse problem of Quantum Hypothesis testing. Together with related results for the direct part of Quantum Hypothesis testing, known as the Quantum Hoeffding bound, our result suggests that the operationally relevant definition of the Quantum Renyi relative entropies depends on the parameter α: for α 1 the right choice is the newly introduced version \({D_\alpha^{(new)}} (\rho \| \sigma) := \frac{1}{\alpha-1}\,{\rm log\,\,Tr}\,\big(\sigma^{\frac{1-\alpha}{2 \alpha}}\rho \sigma^{\frac{1-\alpha}{2 \alpha}}\big)^{\alpha}\).On the way to proving our main result, we show that the new Renyi α-relative entropies are asymptotically attainable by measurements for α > 1. From this, we obtain a new simple proof for their monotonicity under completely positive trace-preserving maps.
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Classical-Quantum Channel Coding
Introduction to Quantum Information Science, 2014Co-Authors: Masahito Hayashi, Satoshi Ishizaka, Akinori Kawachi, Kimura, Tomohiro OgawaAbstract:Classical-Quantum channel coding is a method to transmit messages over noisy Quantum channels. This chapter is aimed at a transparent understanding of the classical-Quantum channel coding theorem, which is one of the landmarks in Quantum information theory. Since it is required at the receiver’s side to optimize the Quantum measurement to decode the messages, the theory of discriminating Quantum states (Quantum Hypothesis testing) plays an important role.
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The strong converse rate of Quantum Hypothesis testing for correlated Quantum states.
arXiv: Quantum Physics, 2014Co-Authors: Milán Mosonyi, Tomohiro OgawaAbstract:We present two general approaches to obtain the strong converse rate of Quantum Hypothesis testing for correlated Quantum states. One approach requires that the states satisfy a certain factorization property; typical examples of such states are the temperature states of translation-invariant finite-range interactions on a spin chain. The other approach requires the differentiability of a regularized R\'enyi $\alpha$-divergence in the parameter $\alpha$; typical examples of such states include temperature states of non-interacting fermionic lattice systems, and classical irreducible Markov chains. In all cases, we get that the strong converse exponent is equal to the Hoeffding anti-divergence, which in turn is obtained from the regularized R\'enyi divergences of the two states.
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Making Good Codes for Classical-Quantum Channel Coding via Quantum Hypothesis Testing
IEEE Transactions on Information Theory, 2007Co-Authors: Tomohiro Ogawa, H NagaokaAbstract:In this correspondence, we give an alternative proof of the direct part of the classical-Quantum channel coding theorem (the Holevo-Schumacher-Westmoreland (HSW) theorem), using ideas of Quantum Hypothesis testing. In order to show the existence of good codes, we invoke a limit theorem, relevant to the Quantum Stein's lemma, in Quantum Hypothesis testing as the law of large numbers used in the classical case. We also apply a greedy construction of good codes using a packing procedure of noncommutative operators. Consequently we derive an upper bound on the coding error probability, which is used to give an alternative proof of the HSW theorem. This approach elucidates how the Holevo information applies to the classical-Quantum channel coding problems
Gonzalo Vazquezvilar - One of the best experts on this subject based on the ideXlab platform.
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multiple Quantum Hypothesis testing expressions and classical Quantum channel converse bounds
International Symposium on Information Theory, 2016Co-Authors: Gonzalo VazquezvilarAbstract:Alternative exact expressions are derived for the minimum error probability of a Hypothesis test discriminating among M Quantum states. The first expression corresponds to the error probability of a binary Hypothesis test with certain parameters; the second involves the optimization of a given information-spectrum measure. Particularized in the classical-Quantum channel coding setting, this characterization implies the tightness of two existing converse bounds; one derived by Matthews and Wehner using Hypothesis-testing, and one obtained by Hayashi and Nagaoka via an information-spectrum approach.
Min-hsiu Hsieh - One of the best experts on this subject based on the ideXlab platform.
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Moderate Deviation Analysis for Classical-Quantum Channels and Quantum Hypothesis Testing
IEEE Transactions on Information Theory, 2018Co-Authors: Hao-chung Cheng, Min-hsiu HsiehAbstract:In this paper, we study the tradeoffs between the error probabilities of classical-Quantum channels and the block-length n when the transmission rates approach the channel capacity at a rate lower than 1/√n, a research topic known as moderate deviation analysis. We show that the optimal error probability vanishes under this rate convergence. Our main technical contributions are a tight Quantum sphere-packing bound, obtained via Chaganty and Sethuraman's concentration inequality in strong large deviation theory, and asymptotic expansions of error-exponent functions. Moderate deviation analysis for Quantum Hypothesis testing is also established. The converse directly follows from our channel coding result, while the achievability relies on a martingale inequality.
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ISIT - Moderate deviations for Quantum Hypothesis testing and a martingale inequality
2017 IEEE International Symposium on Information Theory (ISIT), 2017Co-Authors: Hao-chung Cheng, Min-hsiu HsiehAbstract:“To be considered for the 2017 IEEE Jack Keil Wolf ISIT Student Paper Award.” We study the asymptotic behavior of the type-I error in Quantum Hypothesis testing when the exponent of the type-II error approaches the Quantum relative entropy sufficiently slowly. Our result shows that the moderate deviation principle holds for the testing problem if the Quantum relative variance is positive. Our proof strategy employs strong large deviation theory and a martingale inequality.
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Moderate Deviation Analysis for Classical-Quantum Channels and Quantum Hypothesis Testing
arXiv: Quantum Physics, 2017Co-Authors: Hao-chung Cheng, Min-hsiu HsiehAbstract:In this work, we study the tradeoffs between the error probabilities of classical-Quantum channels and the blocklength $n$ when the transmission rates approach the channel capacity at a rate slower than $1/\sqrt{n}$, a research topic known as moderate deviation analysis. We show that the optimal error probability vanishes under this rate convergence. Our main technical contributions are a tight Quantum sphere-packing bound, obtained via Chaganty and Sethuraman's concentration inequality in strong large deviation theory, and asymptotic expansions of error-exponent functions. Moderate deviation analysis for Quantum Hypothesis testing is also established. The converse directly follows from our channel coding result, while the achievability relies on a martingale inequality.
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A Smooth Entropy Approach to Quantum Hypothesis Testing and the Classical Capacity of Quantum Channels
IEEE Transactions on Information Theory, 2013Co-Authors: Nilanjana Datta, Milán Mosonyi, Min-hsiu Hsieh, Fernando G S L BrandaoAbstract:We use the smooth entropy approach to treat the problems of binary Quantum Hypothesis testing and the transmission of classical information through a Quantum channel. We provide lower and upper bounds on the optimal type II error of Quantum Hypothesis testing in terms of the smooth max-relative entropy of the two states representing the two hypotheses. Then using a relative entropy version of the Quantum asymptotic equipartition property (QAEP), we can recover the strong converse rate of the i.i.d. Hypothesis testing problem in the asymptotics. On the other hand, combining Stein's lemma with our bounds, we obtain a stronger ( e-independent) version of the relative entropy-QAEP. Similarly, we provide bounds on the one-shot e-error classical capacity of a Quantum channel in terms of a smooth max-relative entropy variant of its Holevo capacity. Using these bounds and the e-independent version of the relative entropy-QAEP, we can recover both the Holevo- Schumacher- Westmoreland theorem about the optimal direct rate of a memoryless Quantum channel with product state encoding, as well as its strong converse counterpart.
