Abstract
Negativity is a measure of entanglement that can be used both in pure and mixed states. The negativity spectrum is the spectrum of eigenvalues of the partially transposed density matrix, and characterizes the degree and “phase” of entanglement. For pure states, it is simply determined by the entanglement spectrum. We use a diagrammatic method complemented by a modification of the Ford-Fulkerson algorithm to find the negativity spectrum in general random tensor networks with large bond dimensions. In holography, these describe the entanglement of fixed-area states. It was found that many fixed-area states have a negativity spectrum given by a semi-circle. More generally, we find new negativity spectra that appear in random tensor networks, as well as in phase transitions in holographic states, wormholes, and holographic states with bulk matter. The smallest random tensor network is the same as a micro-canonical version of Jackiw-Teitelboim (JT) gravity decorated with end-of-the-world branes. We consider the semi-classical negativity of Hawking radiation and find that contributions from islands should be included. We verify this in the JT gravity model, showing the Euclidean wormhole origin of these contributions.
Highlights
The negativity spectrum is the spectrum of eigenvalues of the partially transposed density matrix, and characterizes the degree and “phase” of entanglement
While the vast majority of the literature focuses on the entanglement entropy as a measure of quantum entanglement, entanglement entropy has a well-known caveat when applied to mixed quantum states, such as thermal states (Gibbs states) and reduced density matrices obtained by tracing out a part of the total system
Every connected piece in figure 12 is what we would have gotten if we were calculating entanglement entropy instead of negativity. This is because the green lines are not crossed, which is what we had to do before in order to implement the partial transpose. Here, such a diagram calculates the trace of the density matrix traced over A, which is used for the entanglement entropy between A and B in the pure state on AB
Summary
Let us begin by considering the case of a single tensor X. For an empty subsystem C (top horizontal line in the phase diagram), this reduces to the random pure state studied by Page [32]. In this case, region III disappears, and the entanglement takes the form of the Page curve shown in figure 2. Where dotted, solid, and dashed lines correspond to subsystems A, B, and C respectively In these diagrams, we perform matrix operations in the lower part of the diagram, and ensemble averaging in the upper part. We show how to derive such loop equations in the presence of the partial transpose in appendix A, and use it for random Haar states Another method is to write consistency relations for all diagrams representing the summation of all moments.
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