Subsystem trace distance in low-lying states of (1 + 1)-dimensional conformal field theories

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, P. Calabrese, Phys. Rev. Lett. 122, 141602 (2019)], of which this work is a completion. The trace distance between two reduced density matrices ρ A and σ A is obtained from the moments tr(ρ A −σ A ) 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 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. Here we complete our previous work by providing detailed derivations of all results and further new formulas for the distances between several lowlying 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. 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.

Subsystem distance after a local operator quench

“…where, again, ρ A,0 is the RDM of the ground state and ρ A (0) = ρ A,0 . Notice that the true Schatten distance is given by D A (t 1 , t 2 )(trρ n A,0 ) 1/n , but we introduced the normalization factor (trρ n A,0 ) 1/n for convenience in the field theoretical calculations (see, e.g., [9]); with some abuse of notation we will refer to D (n) A (t 1 , t 2 ) as Schatten distances, rarely specifying they are normalized as in (2.15). Following the approach of Refs.…”

“…Following the approach of Refs. [8,9], we first calculate the Schatten distance for general even integer n e D (ne) 16) and then take the analytical continuation arbitrary real n e → n to get the Schatten distance. In particular, for n e → 1 we get trace distance…”

“…We focus on those aspects that will be useful to the present paper, more general details can be found in, e.g., Refs. [9,[45][46][47][48][49][50] and references therein.…”

“…For the trace distance instead only small subsystems (say with up to 7) may be worked out by explicitly constructing the entire 2 × 2 density matrix from the correlation one, see for details Refs. [8,9].…”

Negativity spectrum in the random singlet phase

“…With free fermion techniques is not straightforward to calculate the eigenvalues of the sum of Gaussian operators (see anyhow Ref. 151 for a brute force approach). Instead, the traces of arbitrary integer powers of sums of (even non-commuting) gaussian operators can be calculated with, by now, standard methods 145,147 .…”