Quantum hypothesis testing in many-body systems

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.

Capacity of entanglement in local operators

“…On a somewhat different note, an information-theoretic quantity, known as the capacity of entanglement [20][21][22][23] or capacity in short, recently gained some attention, for example, in the context of the information paradox [24,25]. It was shown that, like entanglement entropy, capacity could also truly probe the phase transition that happens at the Page time [24,25].…”

“…More generally, one can define capacity in terms of the relative entropy variance between two density matrices. When one of the density matrix becomes maximally mixed, i.e., proportional to the identity, the relative entropy variance becomes the capacity[23].…”

Symmetry resolved relative entropies and distances in conformal field theory

“…the examples and discussions in Refs. [38,[61][62][63][64][65]. Also the Schatten distances have been studied in field theories [37,38,[65][66][67][68][69], but not as much as the relative entropy, most likely because of the more difficult replica approach (see below) necessary for their determination.…”

Eigenstate capacity and Page curve in fermionic Gaussian states

“…In this paper, we address this question and find an affirmative answer. We consider a different entanglement measure, the capacity of entanglement (CoE) [36][37][38], first considered in the Kitaev model to describe topologically ordered states [39] and to understand the thermodynamic properties of the entanglement spectrum [40]. Very recently, it has received considerable attention, for example, in local operator excitations [41], random pure states [42], and in the context of holography and black hole information problem, where it is shown to capture partially entangled states in replica wormhole geometry [43,44].…”