Relative Entropies in Conformal Field Theory

Relative entropy is a measure of distinguishability for quantum states, and it plays a central role in quantum information theory. The family of Renyi entropies generalizes to Renyi relative entropies that include, as special cases, most entropy measures used in quantum information theory. We construct a Euclidean path-integral approach to Renyi relative entropies in conformal field theory, then compute the fidelity and the relative entropy of states in one spatial dimension at zero and finite temperature using a replica trick. In contrast to the entanglement entropy, the relative entropy is free of ultraviolet divergences, and is obtained as a limit of certain correlation functions. The relative entropy of two states provides an upper bound on their trace distance.

Figure 1

(a) The path integral for $\rho$ can be written as a product of three matrices: ${\sigma }^{1/3}{\rho }_3(\sigma ){\sigma }^{1/3}$. (b) Performing the path integral on only the slice with operator insertions $\mathcal{O}$ and boundary conditions ${\varphi }^{-}$ and ${\varphi }^{+}$ imposed on each end corresponds to computing ${\rho }_3(\sigma )$. (c) The correlation function found by normalizing $\mathrm{tr}[{\rho }_3(\sigma {)}^3]$.

Figure 2

(a) The path integral corresponding to the reduced density matrix at temperature $T$: ${\sigma }_T$. (b) The density matrix of ${\rho }_{T/3}$ after the conformal transformation that maps ${\sigma }_T$ to the complex plane. (c) The $m$-sheeted cover can be computed using the correlation function of twist operators.

Figure 3

The five-sheeted Riemann surface that one constructs to compute the second Renyi relative entropy with $m=3$.