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arXiv:1709.07701

Role of Hypothesis Testing in Quantum Information

Masahito Hayashi

Published Sep 22, 2017• 6 citations
arXivHTMLPDF

Abstract

Recently, it is well recognized that hypothesis testing has deep relations with other topics in quantum information theory as well as in classical information theory. These relations enable us to derive precise evaluation in the finite-length setting. However, such usefulness of hypothesis testing is not limited to information theoretical topics. For example, it can be used for verification of entangled state and quantum computer as well as guaranteeing the security of keys generated via quantum key distribution. In this talk, we overview these kinds of applications of hypothesis testing.

DOI
10.48550/arxiv.1709.07701

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Recently, it is well recognized that hypothesis testing has deep relations with other topics in quantum information theory as well as in classical information theory. These relations enable us to derive precise evaluation in the finite-length setting. However, such usefulness of hypothesis testing is not limited to information theoretical topics. For example, it can be used for verification of entangled state and quantum computer as well as guaranteeing the security of keys generated via quantum key distribution. In this talk, we overview these kinds of applications of hypothesis testing.
Recently, there.have lots of work in quantum hypothesis testing that lies in the core of fascinating applications in quantum information theory.They mainly deal with a simple hypothesis because their main concern is the dis- tinguishability of two quantum states.However, in a practical application, composite hypothesis testing is much important as in classical statistics.In the present paper, we focus on the unbiased test in the quantum composite hypothesis testing.We derive necessary and sufficient conditions on exis- tence of a nontrivial unbiased test in the finite-dimensional Hilbert space.As a practical application, we consider the statistical hypothesis testing of entanglements.It is shown that the uniformly most powerful unbiased test is the random guess for the hypothesis testing.It implies that quan- tum fluctuation conflicts with the concept of hypothesis testing in classical statistics.
This expository article gives an overview of the theory of hypothesis testing of quantum states in finite dimensional Hilbert spaces. Optimal measurement strategy for testing binary quantum hypotheses, which result in minimum error probability, is discussed. Collective and individual adaptive measurement strategies in testing hypotheses in the multiple copy scenario, with various upper and lower bounds on error probability, are outlined. A brief account on quantum channel discrimination and the role of entangled states in achieving enhanced precision in the task of channel discrimination is given.
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
In this thesis we investigate the theory of quantum hypothesis testing and its potential applications for the new area of quantum technologies. We first consider the asymmetric formulation of quantum hypothesis testing where the aim is to minimize the probability of false negatives and the main tool is provided by the quantum Hoeffding bound. In this context we provide a general recipe for computing this bound in the most important scenario for continuous variable quantum information, that of Gaussian states. We then study both asymmetric and symmetric quantum hypothesis testing in the context of quantum channel discrimination. Here we show how the use of quantum-correlated light can enhance the detection of small variations of transmissivity in a sample of photodegrabable material, while a classical source of light either cannot retrieve information or would destroy the sample. This non-invasive quantum technique might be useful to realize in-vivo and real-time probing of very fragile biological samples, such as DNA or RNA. We also show that the same principle can be exploited to build next-generation memories for the confidential storage of confidential data, where information can be read only by well-tailored sources of entangled light.
Quantum key distribution (QKD) achieves information-theoretic security for a shared secret key, using distributed quantum states. To establish secret bits, two honest parties apply key distillation protocols to the resulting measurement outcomes. In this work, we establish a rigorous connection between the key rate achievable by applying two-way key distillation, such as advantage distillation, and quantum hypothesis testing, via an integral representation of the conditional entropy in terms of the error probabilities of binary state discrimination. This connection significantly improves bounds on key rates at finite blocklengths relative to existing fidelity-based bounds. Moreover, in an important special case, it closes the gap between the known sufficient and conjectured necessary conditions for key generation in the large-blocklength limit. More generally, it establishes a matching asymptotic necessary condition, leaving the corresponding finite-blocklength result open. More broadly, our work shows how results in quantum hypothesis testing can directly sharpen the security analysis of QKD.
Quantum entanglement stands as one of the most intriguing phenomena in quantum mechanics, defying classical intuition and paving the way for revolutionary advancements in information processing. the fundamentals of quantum entanglement and explores its applications in quantum information processing. Beginning with a concise overview of entanglement theory, we delve into various protocols and techniques that leverage entangled states to perform tasks such as quantum teleportation, superdense coding, and quantum cryptography. Furthermore, we discuss recent developments in the field, including entanglement-based quantum key distribution schemes and entanglement swapping protocols. Through a comprehensive examination of both theoretical principles and experimental implementations, this paper elucidates the pivotal role of quantum entanglement in shaping the landscape of quantum information processing, with implications for secure communication, computational speedup, and future quantum technologies.