We define a new criterion for selecting a specific minimal entanglement purification of given mixed states in generic quantum states using the entanglement of purification. We then propose that its holographic dual is the state living on the boundary of the entanglement wedge in the surface-state correspondence. Finally, we make some remarks about the relationship between this and the complexity equals action proposals.
We define a new criterion for selecting a specific minimal entanglement purification of given mixed states in generic quantum states using the entanglement of purification. We then propose that its holographic dual is the state living on the boundary of the entanglement wedge in the surface-state correspondence. Finally, we make some remarks about the relationship between this and the complexity equals action proposals.
Abstract As a probe of circuit complexity in holographic field theories, we study sub-system analogues based on the entanglement wedge of the bulk quantities appearing in the “complexity = volume” and “complexity = action” conjectures. We calculate these quantities for one exterior region of an eternal static neutral or charged black hole in general dimensions, dual to a thermal state on one boundary with or without chemical potential respectively, as well as for a shock wave geometry. We then define several analogues of circuit complexity for mixed states, and use tensor networks to gain intuition about them. In the action approach, we find two possible cases depending on an ambiguity in the definition of the action associated with a counterterm. In one case, there is a promising qualitative match between the holographic action and what we call the purification complexity, the minimum number of gates required to prepare an arbitrary purification of the given mixed state. In the other case, the match is to what we call the basis complexity, the minimum number of gates required to prepare the given mixed state starting from a minimal complexity state with the same eigenvalue spectrum. One way to fix this ambiguity is to choose an action definition such that UV divergent part is positive, in which case the best match to the action result is the basis complexity. In contrast, the holographic volume does not appear to match any of our definitions of mixed-state complexity.
Abstract Starting from the entanglement wedge of a multipartite mixed state we describe a purification procedure which involves the gluing of several copies. The resulting geometry has non-trivial topology and a single oriented boundary for each original boundary region. In the purified geometry the original multipartite entanglement wedge cross section is mapped to a minimal surface of a particular non-trivial homology class. In contrast, each original bipartite entanglement wedge cross section is mapped to the minimal wormhole throat around each boundary. Using the bit thread formalism we show how maximal flows for the bipartite and multipartite entanglement wedge cross section can be glued together to form maximal multiflows in the purified geometry. The defining feature differentiating the flows is given by the existence of threads which cross between different copies of the original entanglement wedge. Together these demonstrate a possible connection between multipartite entanglement and the topology of holographic spacetimes.
In this paper, we holographically quantify the entanglement and complexity for mixed states by following the prescription of purification. The bulk theory we consider in this work is a hyperscaling violating solution, characterized by two parameters, hyperscaling violating exponent $\ensuremath{\theta}$ and dynamical exponent z. This geometry is due to a nonrelativistic strongly coupled theory with hidden Fermi surfaces. We first compute the holographic analogy of entanglement of purification, denoted as the minimal area of the entanglement wedge cross section and observe the effects of z and $\ensuremath{\theta}$. Then in order to probe the mixed state complexity we compute the mutual complexity for the Banados Teitelboim Zanelli (BTZ) black hole and the hyperscaling violating geometry by incorporating the holographic subregion complexity conjecture. We carry this out for two disjoint subsystems separated by a distance and also when the subsystems are adjacent with subsystems making up the full system. Furthermore, various aspects of holographic entanglement entropy such as the entanglement Smarr relation, the Fisher information metric, and the butterfly velocity have also been discussed.
Abstract The entanglement of purification (EoP), which measures the classical correlations and entanglement of a given mixed state, has been conjectured to be dual to the area of the minimal cross section of the entanglement wedge in holography. Using the surface-state correspondence, we propose a “bit thread” formulation of the EoP. With this formulation, proofs of some known properties of the EoP are performed. Moreover, we show that the quantum advantage of dense code (QAoDC), which reflects the increase in the rate of classical information transmission through quantum channel due to entanglement, also admits a flow interpretation. In this picture, we can prove the monogamy relation of QAoDC with the EoP for tripartite states. We also derive a new lower bound for S(AB) in terms of QAoDC, which is tighter than the one given by the Araki-Lieb inequality.
Abstract We apply the bit thread formulation of holographic entanglement entropy to reduced states describing only the geometry contained within an entanglement wedge. We argue that a certain optimized bit thread configuration, which we construct, gives a purification of the reduced state to a full holographic state obeying a precise set of conditional mutual information relations. When this purification exists, we establish, under certain assumptions, the conjectured E P = E W relation equating the entanglement of purification with the area of the minimal cross section partitioning the bulk entanglement wedge. Along the way, we comment on minimal purifications of holographic states, geometric purifications, and black hole geometries.
We apply the bit thread formulation of holographic entanglement entropy to reduced states describing only the geometry contained within an entanglement wedge. We argue that a certain optimized bit thread configuration, which we construct, gives a purification of the reduced state to a full holographic state obeying a precise set of conditional mutual information relations. When this purification exists, we establish, under certain assumptions, the conjectured EP = EW relation equating the entanglement of purification with the area of the minimal cross section partitioning the bulk entanglement wedge. Along the way, we comment on minimal purifications of holographic states, geometric purifications, and black hole geometries.