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.

76 pages, v2: references added, v3: minor changes, published version

References in corpus (14)

• Quantum Illumination with Gaussian States
• The Quantum Chernoff Bound
• Entanglement spectrum in one-dimensional systems
• Entanglement hamiltonians in two-dimensional conformal field theory
• Relative Entropies in Conformal Field Theory
• Entanglement spectrum degeneracy and Cardy formula in 1+1 dimensional conformal field theories
• Subsystem Trace Distance in Quantum Field Theory
• Error exponents in hypothesis testing for correlated states on a spin chain
• Relative entropy of excited states in conformal field theories of arbitrary dimensions
• Relative Entanglement Entropies in 1+1-dimensional conformal field theories
• Quantum state discrimination bounds for finite sample size
• Generalized Wick's theorem for multiquasiparticle overlaps as a limit of Gaudin's theorem
• Relative entropy in higher spin holography
• Renyi divergences from Euclidean quenches

Cited by in corpus (8)

• Symmetry resolved relative entropies and distances in conformal field theory
• Eigenstate capacity and Page curve in fermionic Gaussian states
• Quantum information geometry of driven CFTs
• Capacity of Entanglement in Local Operators
• Probing RG flows, symmetry resolution and quench dynamics through the capacity of entanglement
• Subsystem distances between quasiparticle excited states
• Trace distance between fermionic Gaussian states from a truncation method
• Quantum phase classification via partial tomography-based quantum hypothesis testing