Monogamy Of Mutual Information Research Articles

Quantum Extremal Surfaces and the Holographic Entropy Cone

Quantum states with geometric duals are known to satisfy a stricter set of entropy inequalities than those obeyed by general quantum systems. The set of allowed entropies derived using the Ryu-Takayanagi (RT) formula defines the Holographic Entropy Cone (HEC). These inequalities are no longer satisfied once general quantum corrections are included by employing the Quantum Extremal Surface (QES) prescription. Nevertheless, the structure of the QES formula allows for a controlled study of how quantum contributions from bulk entropies interplay with HEC inequalities. In this paper, we initiate an exploration of this problem by relating bulk entropy constraints to boundary entropy inequalities. In particular, we show that requiring the bulk entropies to satisfy the HEC implies that the boundary entropies also satisfy the HEC. Further, we also show that requiring the bulk entropies to obey monogamy of mutual information (MMI) implies the boundary entropies also obey MMI.

Nonarchimedean holographic entropy from networks of perfect tensors

We consider a class of holographic quantum error-correcting codes, built from perfect tensors in network configurations dual to Bruhat-Tits trees and their quotients by Schottky groups corresponding to BTZ black holes. The resulting holographic states can be constructed in the limit of infinite network size. We obtain a p-adic version of entropy which obeys a Ryu-Takayanagi like formula for bipartite entanglement of connected or disconnected regions, in both genus-zero and genus-one p-adic backgrounds, along with a Bekenstein-Hawking-type formula for black hole entropy. We prove entropy inequalities obeyed by such tensor networks, such as subadditivity, strong subadditivity, and monogamy of mutual information (which is always saturated). In addition, we construct infinite classes of perfect tensors directly from semiclassical states in phase spaces over finite fields, generalizing the CRSS algorithm, and give Hamiltonians exhibiting these as vacua.

Radial cutoffs and holographic entanglement

Tensor networks, Toverline{T} , and broader notions of a holographic principle all motivate the idea that some notion of gravitational holography should persist in the presence of a radial cutoff. But in the absence of time-reflection symmetry, the areas of Hubeny-Rangamani-Takayanagi surfaces anchored to the radial cutoff generally violate strong sub-additivity, even when the associated boundary regions are spacelike separated as defined by both bulk and boundary notions of causality. We thus propose an alternate definition of cutoff-holographic entropy using a restricted maximin prescription anchored to a codimension 2 cutoff surface. For bulk solutions that respect the null energy condition, we show that the resulting areas satisfy SSA, entanglement wedge nesting, and monogamy of mutual information in parallel with cutoff free results in AdS. These results hold even when the cutoff surface fails to be convex.

Bit Threads and Holographic Monogamy

Bit threads provide an alternative description of holographic entanglement, replacing the Ryu-Takayanagi minimal surface with bulk curves connecting pairs of boundary points. We use bit threads to prove the monogamy of mutual information (MMI) property of holographic entanglement entropies. This is accomplished using the concept of a so-called multicommodity flow, adapted from the network setting, and tools from the theory of convex optimization. Based on the bit thread picture, we conjecture a general ansatz for a holographic state, involving only bipartite and perfect-tensor type entanglement, for any decomposition of the boundary into four regions. We also give new proofs of analogous theorems on networks.

Bulk locality and cooperative flows

We use the ‘bit thread’ formulation of holographic entanglement entropy to highlight the distinction between the universally-valid strong subadditivity and the more restrictive relation called monogamy of mutual information (MMI), known to hold for geometrical states (i.e. states of holographic theories with gravitational duals describing a classical bulk geometry). In particular, we provide a novel proof of MMI, using bit threads directly. To this end, we present an explicit geometrical construction of cooperative flows which we build out of disjoint thread bundles. We conjecture that our method applies in a wide class of configurations, including ones with non-trivial topology, causal structure, and time dependence. The explicit nature of the construction reveals that MMI is more deeply rooted in bulk locality than is the case for strong subadditivity.

Entanglement structures in qubit systems

Using measures of entanglement such as negativity and tangles we provide a detailed analysis of entanglement structures in pure states of non-interacting qubits. The motivation for this exercise primarily comes from holographic considerations, where entanglement is inextricably linked with the emergence of geometry. We use the qubit systems as toy models to probe the internal structure, and introduce some useful measures involving entanglement negativity to quantify general features of entanglement. In particular, our analysis focuses on various constraints on the pattern of entanglement which are known to be satisfied by holographic sates, such as the saturation of Araki–Lieb inequality (in certain circumstances), and the monogamy of mutual information. We argue that even systems as simple as few non-interacting qubits can be useful laboratories to explore how the emergence of the bulk geometry may be related to quantum information principles.

Maximin surfaces, and the strong subadditivity of the covariant holographic entanglement entropy

The covariant holographic entropy conjecture of AdS/CFT relates the entropy of a boundary region R to the area of an extremal surface in the bulk spacetime. This extremal surface can be obtained by a maximin construction, allowing many new results to be proven. On manifolds obeying the null curvature condition, these extremal surfaces: (i) always lie outside the causal wedge of R, (ii) have less area than the bifurcation surface of the causal wedge, (iii) move away from the boundary as R grows, and (iv) obey strong subadditivity and monogamy of mutual information. These results suggest that the information in R allows the bulk to be reconstructed all the way up to the extremal area surface. The maximin surfaces are shown to exist on spacetimes without horizons, and on black hole spacetimes with Kasner-like singularities.

Monogamy and backflow of mutual information in non-Markovian thermal baths

We investigate the dynamics of information among the parties of tripartite systems. We start by proving two results concerning the monogamy of mutual information. The first one states that mutual information is monogamous for generic tripartite pure states. The second shows that, in general, mutual information is monogamous only if the amount of genuine tripartite correlations is large enough. Then, we analyze the internal dynamics of tripartite systems whose parties do not exchange energy. In particular, we allow for one of the subsystems to play the role of a finite thermal bath. As a result, we find a typical scenario in which local information tends to be converted into delocalized information. Moreover, we show that (i) the information flow is reversible for finite thermal baths at low temperatures, (ii) monogamy of mutual information is respected throughout the dynamics, and (iii) genuine tripartite correlations are typically present. Finally, we analytically calculate a quantity capable of revealing favorable regimes for non-Markovianity in our model.

General properties of holographic entanglement entropy

The Ryu-Takayanagi formula implies many general properties of entanglement entropies in holographic theories. We review the known properties, such as continuity, strong subadditivity, and monogamy of mutual information, and fill in gaps in some of the previously-published proofs. We also add a few new properties, including: properties of the map from boundary regions to bulk regions implied by the RT formula, such as monotonicity; conditions under which subadditivity-type inequalities are saturated; and an inequality concerning reflection-symmetric states. We attempt to draw lessons from these properties about the structure of the reduced density matrix in holographic theories.