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arXiv:2004.04163 · Journal of High Energy Physics

A field theory study of entanglement wedge cross section: odd entropy

Ali Mollabashi, Kotaro Tamaoka

Published Aug 1, 2020• 20 citations
arXivHTMLPDF

Abstract

Abstract We study odd entanglement entropy (odd entropy in short), a candidate of measure for mixed states holographically dual to the entanglement wedge cross section, in two-dimensional free scalar field theories. Our study is restricted to Gaussian states of scale-invariant theories as well as their finite temperature generalizations, for which we show that odd entropy is a well-defined measure for mixed states. Motivated from holographic results, the difference between odd and von Neumann entropy is also studied. In particular, we show that large amounts of quantum correlations ensure the odd entropy to be larger than von Neumann entropy, which is qualitatively consistent with the holographic CFT. In general cases, we also find that this difference is not even a monotonic function with respect to size of (and distance between) subsystems.

DOI
10.1007/jhep08(2020)078

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Abstract We study odd entanglement entropy (odd entropy in short), a candidate of measure for mixed states holographically dual to the entanglement wedge cross section, in two-dimensional free scalar field theories. Our study is restricted to Gaussian states of scale-invariant theories as well as their finite temperature generalizations, for which we show that odd entropy is a well-defined measure for mixed states. Motivated from holographic results, the difference between odd and von Neumann entropy is also studied. In particular, we show that large amounts of quantum correlations ensure the odd entropy to be larger than von Neumann entropy, which is qualitatively consistent with the holographic CFT. In general cases, we also find that this difference is not even a monotonic function with respect to size of (and distance between) subsystems.
Abstract We obtain the odd entanglement entropy (OEE) for bipartite mixed states in ( 1 + 1 ) -dimensional boundary conformal field theories (BCFT 2 s) utilizing an appropriate replica technique. For the generic configuration of two disjoint intervals, the large- c factorization of the twist field correlator leads to several distinct phases depending on the relative sizes and separations of the intervals. The resulting expressions match consistently with the entanglement wedge cross section (EWCS) for the bulk AdS 3 braneworld geometries, thereby extending the proposed holographic duality between the difference of the OEE and the entanglement entropy with the EWCS to BCFT setups. Furthermore, we obtain the OEE for these mixed states in related holographic moving mirror models which simulate Hawking radiation from black holes and verify the above duality in these time-dependent backgrounds. In this context we analyze the time evolution of the OEE and establish the analogues of Page curves for this mixed-state correlation measure.
Abstract The odd entanglement entropy (OEE) for bipartite states in a class of (1+1)(1+1)(1+1) ( 1 + 1 ) -dimensional Galilean conformal field theories ( GCFT1+1GCFT_{1+1}GCFT1+1​ G C F T 1 + 1 ) is obtained through an appropriate replica technique. In this context our results are compared with the entanglement wedge cross section (EWCS) for (2+1)(2+1)(2+1) ( 2 + 1 ) -dimensional asymptotically flat geometries dual to the GCFT1+1GCFT_{1+1}GCFT1+1​ G C F T 1 + 1 in the framework of flat holography. We find that our results are consistent with the duality of the difference between the odd entanglement entropy and the entanglement entropy of bipartite states, with the bulk EWCS for flat holographic scenarios.
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.
We define a new information theoretic quantity called odd entanglement entropy (OEE) which enables us to compute the entanglement wedge cross section in holographic conformal field theories (CFTs). The entanglement wedge cross section has been introduced as a minimal cross section of the entanglement wedge, a natural generalization of the Ryu-Takayanagi surface. By using the replica trick, we explicitly compute the OEE for two-dimensional holographic CFT (three-dimensional anti-de Sitter space and planar Bañados-Teitelboim-Zanelli black hole) and see agreement with the entanglement wedge cross section. We conjecture this relation will hold in general dimensions.
We construct a replica technique to perturbatively compute the odd entanglement entropy (OEE) for bipartite mixed states in TT‾\mathrm{T}\overline{\mathrm{T}}TT deformed CFT2s{\mathrm{CFT}}_{2}\mathrm{s}CFT2​s. This framework is then utilized to obtain the leading-order correction to the OEE for two disjoint intervals, two adjacent intervals, and a single interval in TT‾\mathrm{T}\overline{\mathrm{T}}TT deformed thermal CFT2s{\mathrm{CFT}}_{2}\mathrm{s}CFT2​s in the large central charge limit. The field theory results are subsequently reproduced in the high-temperature limit from holographic computations for the entanglement wedge cross sections in the dual bulk finite cutoff Ba\~nados-Teitelboim-Zanelli geometries. We further show that for finite size TT‾\mathrm{T}\overline{\mathrm{T}}TT deformed CFT2s{\mathrm{CFT}}_{2}\mathrm{s}CFT2​s at zero temperature the corrections to the OEE are vanishing to the leading order from both field theory and bulk holographic computations.
Abstract We consider the time evolution of mixed state correlation measures in two-dimensional conformal field theories, such as logarithmic negativity, odd entropy, and reflected entropy, after quantum quenches of various kinds. These correlation measures, in the holographic context, are all associated to the entanglement wedge cross section. We contrast various classes of conformal field theories, both rational and irrational (pure) conformal field theories. First, for rational conformal field theories, whose dynamics can be well described by the quasi-particle picture, we find all four quantities for disjoint intervals to be proportional, regardless of the specific quench protocol. Second, using the light cone boot- strap, we generalize our results to irrational conformal field theories where we find sharp distinctions from the quasi-particle results and striking differences between mutual information and the other measures. The large surplus of logarithmic negativity relative to mutual information forces us to reconsider what mutual information and logarithmic negativity really measure. We interpret these results as a signature of information scrambling and chaos in irrational theories. These CFT results perfectly agree with our gravitational (holographic) calculations. Furthermore, using holography, we are able to generalize the results to outside of the light cone limit. Finally, due to the breakdown of the quasi-particle picture for irrational theories, we appeal to the “line-tension picture,” motivated by random unitary circuits, as a phenomenological description. We observe that random unitary circuits, with local Hilbert space dimension determined by the Cardy formula, have precisely the same entanglement dynamics as irrational (including holographic) conformal field theories.