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.

• Received 13 February 2019

DOI:https://doi.org/10.1103/PhysRevLett.122.141602

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Published by the American Physical Society