A bstract In this paper, we analyze the question of replica symmetry in the bulk for multi-partite entanglement measures in the vacuum state of two dimensional holographic CFTs. We first define a class of multi-partite local unitary invariants, multi-invariants , with a given replica symmetry that acts freely and transitively on the replicas. We look for a subclass of measures such that the dual bulk geometry also preserves replica symmetry. We obtain the most general solution to this problem if we require the bulk to preserve replica symmetry for general configurations of the regions. Orbifolding the bulk solution with the replica symmetry gives us a bulk geometry with a network of conical singularities. Our approach makes it clear that there are infinitely many infinitely large families of multi-invariants such that each family evaluates identically on the holographic state. Geometrically, these are equalities involving volumes of handlebodies, possibly of different genus, at particular points in the moduli space. In certain cases, we check our bulk computation with an explicit calculation in CFT. Finally we comment on the generalization to higher dimension.
A bstract The entanglement negativity
E E ( A : B ) is a useful measure of quantum entanglement in bipartite mixed states. In random tensor networks (RTNs), which are related to fixed-area states, it was found in ref. [1] that the dominant saddles computing the even Rényi negativity
E(2k) E 2 k generically break the ℤ 2 k replica symmetry. This calls into question previous calculations of holographic negativity using 2D CFT techniques that assumed ℤ 2 k replica symmetry and proposed that the negativity was related to the entanglement wedge cross section. In this paper, we resolve this issue by showing that in general holographic states, the saddles computing
E(2k) E 2 k indeed break the ℤ 2 k replica symmetry. Our argument involves an identity relating
E(2k) E 2 k to the k -th Rényi entropy on subregion AB ∗ in the doubled state
∣ρAB⟩AA∗BB∗ ρ AB A A ∗ BB ∗ , from which we see that the ℤ 2 k replica symmetry is broken down to ℤ k . For k < 1, which includes the case of
E E ( A : B ) at k = 1/2, we use a modified cosmic brane proposal to derive a new holographic prescription for
E(2k) E 2 k and show that it is given by a new saddle with multiple cosmic branes anchored to subregions A and B in the original state. Using our prescription, we reproduce known results for the PSSY model and show that our saddle dominates over previously proposed CFT calculations near k = 1. Moreover, we argue that the ℤ 2 k symmetric configurations previously proposed are not gravitational saddles, unlike our proposal. Finally, we contrast holographic calculations with those arising from RTNs with non-maximally entangled links, demonstrating that the qualitative form of backreaction in such RTNs is different from that in gravity.
A bstract 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.
We investigate mixed state entanglement measures of entanglement negativity and reflected entropy for bipartite states in two dimensional conformal field theories with an anomaly through appropriate replica techniques. Furthermore we propose holographic constructions for these measures from the corresponding bulk dual geometries involving topologically massive gravity in
AdS3 and find exact agreement with the field theory results. In this connection we extend an earlier holographic proposal for the entanglement negativity to the bulk action with a gravitational Chern-Simons term and compute its contribution to the entanglement wedge cross section dual to the reflected entropy.
We discuss two measures of entanglement in quantum field theory and their holographic realizations. For field theories admitting a global symmetry, we introduce a global symmetry entanglement entropy, associated with the partitioning of the symmetry group. This quantity is proposed to be related to the generalized holographic entanglement entropy defined via the partitioning of the internal space of the bulk geometry. Thesecond measure of quantum field theory entanglement is the field space entanglement entropy, obtained by integrating out a subset of the quantum fields. We argue that field space entanglement entropy cannot be precisely realised geometrically in a holographic dual. However, for holographic geometries with interior decoupling regions, the differential entropy provides a close analogue to the field space entanglement entropy. We derive generic descriptions of such inner throat regions in terms of gravity coupled to massive scalars and show how the differential entropy in the throat captures features of the field space entanglement entropy.
A bstract In this work, we develop a systematic formalism to evaluate the upper bound of a large family of holographic entanglement entropy combinations when fixing n subsystems and fine-tuning one other subsystem. The upper bound configurations and values of these entropy combinations can be derived and classified. The upper bound of these entropy combinations reveals holographic n + 1-partite entanglement that n fixed subsystems participate in. In AdS 3 /CFT 2 , AdS 4 /CFT 3 , and even higher-dimensional holography, one can, in principle, find different formulas of upper bound values, reflecting the fundamental difference in entanglement structure in different dimensions.