Abstract
We report on a systematic replica approach to calculate the subsystem trace distance for a quantum field theory. This method has been recently introduced in [J. Zhang, P. Ruggiero and P. Calabrese, Phys. Rev. Lett.122 (2019) 141602], of which this work is a completion. The trace distance between two reduced density matrices ρA and σA is obtained from the moments tr(ρA− σA)n and taking the limit n → 1 of the traces of the even powers. We focus here on the case of a subsystem consisting of a single interval of length ℓ embedded in the low lying eigenstates of a one-dimensional critical system of length L, a situation that can be studied exploiting the path integral form of the reduced density matrices of two-dimensional conformal field theories. The trace distance turns out to be a scale invariant universal function of ℓ/L. Here we complete our previous work by providing detailed derivations of all results and further new formulas for the distances between several low-lying states in two-dimensional free massless compact boson and fermion theories. Remarkably, for one special case in the bosonic theory and for another in the fermionic one, we obtain the exact trace distance, as well as the Schatten n-distance, for an interval of arbitrary length, while in generic case we have a general form for the first term in the expansion in powers of ℓ/L. The analytical predictions in conformal field theories are tested against exact numerical calculations in XX and Ising spin chains, finding perfect agreement. As a byproduct, new results in two-dimensional CFT are also obtained for other entanglement-related quantities, such as the relative entropy and the fidelity.
Highlights
The characterisation of the entanglement content of extended quantum system has become a crucial theme in modern physics [1,2,3] at the level that a few experimental protocols to measure such entanglement have been already set up [4,5,6,7,8]
We report on a systematic replica approach to calculate the subsystem trace distance for a quantum field theory
The method can be applied to many different situations, but in [72] we focused on one-dimensional (1D) systems described by a 2D Conformal Field Theory (CFT), with the subsystem consisting of an interval of length embedded in a circle of length L
Summary
The characterisation of the entanglement content of extended quantum system has become a crucial theme in modern physics [1,2,3] at the level that a few experimental protocols to measure such entanglement have been already set up [4,5,6,7,8]. It is definitely a useful tool in quantum information theory, but is not a metric: it is not symmetric in its inputs, it may be infinite for some density matrices, and does not satisfy the triangle inequality [58] Another quantity, already studied in literature, that provides an indication of the difference of two states is the fidelity [62], defined as [57, 58]. The method can be applied to many different situations, but in [72] we focused on one-dimensional (1D) systems described by a 2D Conformal Field Theory (CFT), with the subsystem consisting of an interval of length embedded in a circle of length L In such setting, entanglement measures as the Renyi and the von Neumann entropy have been considered. In appendix D we provides some details of the analytic continuation
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