Abstract
Since the work of Ryu and Takayanagi, deep connections between quantum entanglement and spacetime geometry have been revealed. The negative eigenvalues of the partial transpose of a bipartite density operator is a useful diagnostic of entanglement. In this paper, we discuss the properties of the associated entanglement negativity and its Rényi generalizations in holographic duality. We first review the definition of the Rényi negativities, which contain the familiar logarithmic negativity as a special case. We then study these quantities in the random tensor network model and rigorously derive their large bond dimension asymptotics. Finally, we study entanglement negativity in holographic theories with a gravity dual, where we find that Rényi negativities are often dominated by bulk solutions that break the replica symmetry. From these replica symmetry breaking solutions, we derive general expressions for Rényi negativities and their special limits including the logarithmic negativity. In fixed-area states, these general expressions simplify dramatically and agree precisely with our results in the random tensor network model. This provides a concrete setting for further studying the implications of replica symmetry breaking in holography.
Highlights
In this paper, we study entanglement negativity and its Rényi generalization in the holographic duality and its random tensor network toy model
We study these quantities in the random tensor network model and rigorously derive their large bond dimension asymptotics
We found that these quantities are nontrivial in the phase where the entanglement wedge of AB connects A with B, corresponding to a positive mutual information between A and B according to the RyuTakayanagi formula
Summary
Just like one can generalize the von Neumann entropy to the Rényi entropies, S(k)(ρ) =. Since the eigenvalues of ρTABB can be negative, the analytic continuation of Nk to real k needs to be done separately for even and odd k: N2(neven)(ρAB) = |λi|2n , i. Limit of the logarithm of the analytic continuation for even k, since it only depends on the absolute values of the eigenvalues: EN (ρAB ). It is natural to consider its derivative with n, as an analog of the von Neumann entropy: SATBB. We call this quantity the partially transposed entropy.. The partially transposed entropy (2.5) is the k → 1 limit of the refined odd Rényi negativity (2.7). Ρ⊗ABk is the tensor product of k copies of the original density operator, X is a k-cycle (for definiteness, we take the standard one) and X−1 its inverse, and PA(X) and PB(X−1) are both special cases of the general notation PM (g), by which we denote the representation of a permutation group element g ∈ Sk on the k copies of some subsystem M
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