Left-Right Relative Entropy

TL;DR

This paper introduces the left-right relative entropy as a new measure of distinguishability in boundary states of 2D conformal field theories, revealing connections to modular data, quantum anomalies, and boundary symmetries.

Contribution

It formulates a universal expression for left-right relative entropy in boundary CFTs, extends it to Rènyi entropies and fidelity, and uncovers novel phenomena related to quantum anomalies and symmetry sectors.

Findings

Exact formulas for diagonal CFTs in terms of modular data.

Left-right relative entropy can vanish between orthogonal boundary states.

Identification of relative entanglement sectors linked to quantum anomalies.

Abstract

The concept of distinguishability lies at the heart of quantum information theory. We introduce \textit{left-right relative entropy} as a quantitative measure of distinguishability within the space of boundary states in two-dimensional conformal field theories (CFTs). By tracing over either the left- or right-moving modes, we derive a universal formula for arbitrary regularized boundary states defined on a circle. Remarkably, the resulting quantity reduces to a Kullback--Leibler divergence, where the associated probability distribution is determined entirely by the modular $\mathcal{S}$-matrix and the boundary data. For diagonal CFTs, we obtain exact expressions for the left-right relative entropy in terms of modular data, and extend the framework to define left-right R\'enyi relative entropies and quantum fidelity. Applying this formalism to the Ising model, tricritical Ising model, and $\widehat{su}(2)_k$ WZW model, we uncover a striking phenomenon: the left-right relative entropy between certain reduced boundary states vanishes even though the corresponding global boundary states are orthogonal. This observation motivates the introduction of \textit{relative entanglement sectors}, defined as equivalence classes of boundary states that are indistinguishable with respect to left-right relative entropy. These sectors transform as NIM-representations of global symmetries and exhibit level-dependent structures that mirror $\mathbb{Z}_2$ 't Hooft anomalies. Our findings establish an unexpected bridge between quantum information measures, boundary conformal symmetry, and quantum anomaly constraints, establishing a new information-theoretic framework for characterizing exotic phases of matter constrained by quantum anomalies