# Subsystem Trace Distance in Quantum Field Theory

Zhang, Jiaju (SISSA and INFN, Via Bonomea 265, 34136 Trieste, Italy) ; Ruggiero, Paola (SISSA and INFN, Via Bonomea 265, 34136 Trieste, Italy) ; Calabrese, Pasquale (SISSA and INFN, Via Bonomea 265, 34136 Trieste, Italy) (International Centre for Theoretical Physics (ICTP), Strada Costiera 11, 34151 Trieste, Italy)

10 April 2019

Abstract: We develop a systematic method to calculate the trace distance between two reduced density matrices in $1+1$ dimensional quantum field theories. The approach exploits the path integral representation of the reduced density matrices and an ad hoc replica trick. We then extensively apply this method to the calculation of the distance between reduced density matrices of one interval of length $\ell$ in eigenstates of conformal field theories. When the interval is short, using the operator product expansion of twist operators, we obtain a universal form for the leading order in $\ell$ of the trace distance. We compute the trace distances among the reduced density matrices of several low lying states in two-dimensional free massless boson and fermion theories. We compare our analytic conformal results with numerical calculations in $XX$ and Ising spin chains finding perfect agreement.

Published in: Physical Review Letters 122 (2019)