Superbalance of holographic entropy inequalities
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Springer
Received: April 29, 2020 Accepted: July 7, 2020 Published: July 31, 2020
Temple He, Veronika E. Hubeny and Mukund Rangamani Center for Quantum Mathematics and Physics (QMAP), Department of Physics, University of California, 1 Shields Ave, Davis, CA 95616, U.S.A.
E-mail: [email protected], [email protected], [email protected] Abstract: The domain of allowed von Neumann entropies of a holographic field theory carves out a polyhedral cone — the holographic entropy cone — in entropy space. Such polyhedral cones are characterized by their extreme rays. For an arbitrary number of parties, it is known that the so-called perfect tensors are extreme rays. In this work, we constrain the form of the remaining extreme rays by showing that they correspond to geometries with vanishing mutual information between any two parties, ensuring the absence of Bell pair type entanglement between them. This is tantamount to proving that besides subadditivity, all non-redundant holographic entropy inequalities are superbalanced, i.e. not only do UV divergences cancel in the inequality itself (assuming smooth entangling surfaces), but also in the purification thereof. Keywords: AdS-CFT Correspondence, Conformal Field Theory ArXiv ePrint: 2002.04558
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP07(2020)245
JHEP07(2020)245
Superbalance of holographic entropy inequalities
Contents 1 Introduction
1
2 Three representations of entropy space
3 5 6 7 10
4 Discussion
14
A Conversion between I and K bases
16
1
Introduction
Entropy inequalities in quantum information theory are linear constraints on the von Neumann (entanglement) entropies of the various subsystems that give a useful characterization of the full system. For instance, given a factorizable Hilbert space H1 ⊗ H2 , the entropy of the combined system is constrained by those of the individual subsystems H1 and H2 via the subadditivity (SA) inequality S1 + S2 ≥ S12 ,
(1.1)
where Si is the entanglement entropy associated to Hi , and Sij is that associated to Hi ⊗Hj . Equivalently, (1.1) can be expressed as the non-negativity of the mutual information I12 ≡ S1 + S2 − S12 ,
(1.2)
which characterizes the total amount of correlation between H1 and H2 . Understanding the general structure of such entropy inequalities provides intuition about the nature of admissible quantum states and thus remains of broad interest both in quantum mechanics and in continuum quantum field theories (QFTs). Progress in this direction has however proven difficult, since for general quantum systems it remains unclear, even for those involving a small number of parties, what the complete set of the entropy inequalities are, or even whether there are finitely many. Moreover, in QFTs the computation of entanglement entropy is notoriously difficult. In recent years, progress has been made by considering a subclass of QFTs that are holographic conformal field theories (CFTs), which are dual to a gravitational theory in an asymptotically
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