Abstract
One of the key tasks in physics is to perform measurements in order to determine the state of a system. Often, measurements are aimed at determining the values of physical parameters, but one can also ask simpler questions, such as ``is the system in state A or state B?''. In quantum mechanics, the latter type of measurements can be studied and optimized using the framework of quantum hypothesis testing. In many cases one can explicitly find the optimal measurement in the limit where one has simultaneous access to a large number n of identical copies of the system, and estimate the expected error as n becomes large. Interestingly, error estimates turn out to involve various quantum information theoretic quantities such as relative entropy, thereby giving these quantities operational meaning. In this paper we consider the application of quantum hypothesis testing to quantum many-body systems and quantum field theory. We review some of the necessary background material, and study in some detail the situation where the two states one wants to distinguish are parametrically close. The relevant error estimates involve quantities such as the variance of relative entropy, for which we prove a new inequality. We explore the optimal measurement strategy for spin chains and two-dimensional conformal field theory, focusing on the task of distinguishing reduced density matrices of subsystems. The optimal strategy turns out to be somewhat cumbersome to implement in practice, and we discuss a possible alternative strategy and the corresponding errors.
Highlights
We begin with three remarks: i) a rigorous framework for the task is quantum hypothesis testing, ii) many results obtained for relative entropy and generalized divergences can be embedded in this framework, giving them an operational interpretation, and iii) hypothesis testing suggests an optimal measurement protocol to minimize the error in distinguishing two states
In this paper we have reviewed some aspects of quantum hypothesis testing and studied a few applications in quantum many-body systems and two-dimensional conformal field theories
We have seen that the error estimates of different types of hypothesis testing involve different interesting quantum information theoretic quantities
Summary
The purpose of this work is to i) introduce and review quantum hypothesis testing for readers with a background in quantum field theory and many-body theory, ii) develop some new results in a perturbative setup, and iii) apply the tools to distinguish in particular two reduced density matrices in a subsystem of a quantum many-body system. We begin with three remarks: i) a rigorous framework for the task is quantum hypothesis testing, ii) many results obtained for relative entropy and generalized divergences can be embedded in this framework, giving them an operational interpretation, and iii) hypothesis testing suggests an optimal measurement protocol to minimize the error in distinguishing two states. An explicit description is difficult as it leads to a challenging combinatorial problem, involving Krawtchouk polynomials and related to the Terwilliger algebra of the Hamming cube We construct optimal measurement protocols for subregions, using techniques of boundary CFT [41] to compute the necessary ingredients This general framework can be applied to distinguish two thermal states from a subregion, and we study explicitly the case of the free fermion. After the completion of this paper, related work studying various properties and applications of relative entropy variance (there called “variance of relative surprisal”) from an information theoretic point of view appeared in [43]
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