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arXiv:1807.09448 · Journal of High Energy Physics

Towards an entanglement measure for mixed states in CFTs based on relative entropy

Tadashi Takayanagi, Tomonori Ugajin, Koji Umemoto

Published Oct 1, 2018• 10 citations

Abstract

Abstract Relative entropy of entanglement (REE) is an entanglement measure of bipartite mixed states, defined by the minimum of the relative entropy S(ρ AB ||σ AB ) between a given mixed state ρ AB and an arbitrary separable state σ AB . The REE is always bounded by the mutual information I AB = S(ρ AB ||ρ A ⊗ ρ B ) because the latter measures not only quantum entanglement but also classical correlations. In this paper we address the question of to what extent REE can be small compared to the mutual information in conformal field theories (CFTs). For this purpose, we perturbatively compute the relative entropy between the vacuum reduced density matrix ρ AB 0 on disjoint subsystems A ∪ B and arbitrarily separable state σ AB in the limit where two subsystems A and B are well separated, then minimize the relative entropy with respect to the separable states. We argue that the result highly depends on the spectrum of CFT on the subsystems. When we have a few low energy spectrum of operators as in the case where the subsystems consist of finite number of spins in spin chain models, the REE is considerably smaller than the mutual information. However in general our perturbative scheme breaks down, and the REE can be as large as the mutual information.

DOI: 10.1007/jhep10(2018)166

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Abstract Relative entropy of entanglement (REE) is an entanglement measure of bipartite mixed states, defined by the minimum of the relative entropy S(ρ AB ||σ AB ) between a given mixed state ρ AB and an arbitrary separable state σ AB . The REE is always bounded by the mutual information I AB = S(ρ AB ||ρ A ⊗ ρ B ) because the latter measures not only quantum entanglement but also classical correlations. In this paper we address the question of to what extent REE can be small compared to the mutual information in conformal field theories (CFTs). For this purpose, we perturbatively compute the relative entropy between the vacuum reduced density matrix ρ AB 0 on disjoint subsystems A ∪ B and arbitrarily separable state σ AB in the limit where two subsystems A and B are well separated, then minimize the relative entropy with respect to the separable states. We argue that the result highly depends on the spectrum of CFT on the subsystems. When we have a few low energy spectrum of operators as in the case where the subsystems consist of finite number of spins in spin chain models, the REE is considerably smaller than the mutual information. However in general our perturbative scheme breaks down, and the REE can be as large as the mutual information.

We study the relative entanglement entropies of one interval between excited states of a 1+1 dimensional conformal field theory (CFT). To compute the relative entropy S(ρ 1∥ρ 0) between two given reduced density matrices ρ 1 and ρ 0 of a quantum field theory, we employ the replica trick which relies on the path integral representation of Tr(ρ 1 ρ 0 − 1 ) and define a set of Rényi relative entropies S n (ρ 1∥ρ 0). We compute these quantities for integer values of the parameter n and derive via the replica limit the relative entropy between excited states generated by primary fields of a free massless bosonic field. In particular, we provide the relative entanglement entropy of the state described by the primary operator i∂ϕ, both with respect to the ground state and to the state generated by chiral vertex operators. These predictions are tested against exact numerical calculations in the XX spin-chain finding perfect agreement.

We determine the relative entropy of entanglement of a bipartite mixed state

ρ_{AB}

obtained by tracing out one subsystem of a tripartite Haar-random pure state

|ψ\rangle_{ABC}

, finding

E_R(ρ_{AB})=\log\frac{d_Ad_B}{\max(d_A,d_B,d_C)}+O(1)

. Equivalently, the relative entropy of entanglement nearly saturates the smaller of the entanglement of formation

E_F(ρ_{AB})

and the mutual information

I(A:B)

. The upper bound is achieved by an explicit separable state obtained through one-sided Schmidt dephasing, which is therefore approximately closest.

Extending our previous work, we study the relative entropy between the reduced density matrices obtained from globally excited states in conformal field theories of arbitrary dimensions. We find a general formula in the small subsystem size limit. When one of the states is the vacuum of the CFT, our result matches with the holographic entanglement entropy computations in the corresponding bulk geometries, including AdS black branes. We also discuss the first asymmetric part of the relative entropy and comment on some implications of the results on the distinguishability of black hole microstates in AdS/CFT.

Quantifying entanglement properties of mixed states in quantum field theory via entanglement of purification and reflected entropy is a new and challenging subject. In this work, we study both quantities for two spherical subregions far away from each other in the vacuum of a conformal field theory in any number of dimensions. Using lattice techniques, we find an elementary proof that the decay of both the entanglement of purification and reflected entropy is enhanced with respect to the mutual information behavior by a logarithm of the distance between the subregions. In the case of the Ising spin chain at criticality and the related free fermion conformal field theory, we compute also the overall coefficients numerically for the both quantities of interest.

We derive dynamics of the entanglement wedge cross section directly from the two-dimensional holographic CFTs with a local operator quench. This derivation is based on the reflected entropy, a correlation measure for mixed states. We further compare these results with the mutual information and ones for integrable systems. This comparison directly suggests the classical correlation plays an important role in chaotic systems, unlike integrable ones. Besides a local operator quench, we study the reflected entropy in heavy primary states and find a breaking of the subsystem ETH. We checked the above results also hold for the odd entanglement entropy, which is another measure for mixed states related to the entanglement wedge cross section.