10 research outputs found
Assisted Entanglement Distillation
Motivated by the problem of designing quantum repeaters, we study
entanglement distillation between two parties, Alice and Bob, starting from a
mixed state and with the help of "repeater" stations. To treat the case of a
single repeater, we extend the notion of entanglement of assistance to
arbitrary mixed tripartite states and exhibit a protocol, based on a random
coding strategy, for extracting pure entanglement. The rates achievable by this
protocol formally resemble those achievable if the repeater station could merge
its state to one of Alice and Bob even when such merging is impossible. This
rate is provably better than the hashing bound for sufficiently pure tripartite
states. We also compare our assisted distillation protocol to a hierarchical
strategy consisting of entanglement distillation followed by entanglement
swapping. We demonstrate by the use of a simple example that our random
measurement strategy outperforms hierarchical distillation strategies when the
individual helper stations' states fail to individually factorize into portions
associated specifically with Alice and Bob. Finally, we use these results to
find achievable rates for the more general scenario, where many spatially
separated repeaters help two recipients distill entanglement.Comment: 25 pages, 4 figure
Hypergraph min-cuts from quantum entropies
The von Neumann entropy of pure quantum states and the min-cut function of weighted hypergraphs are both symmetric submodular functions. In this article, we explain this coincidence by proving that the min-cut function of any
weighted hypergraph can be approximated (up to an overall rescaling) by the entropies of quantum states known as
stabilizer states. We do so by constructing a novel ensemble of random quantum states, built from tensor networks,
whose entanglement structure is determined by a given hypergraph. This implies that the min-cuts of hypergraphs are
constrained by quantum entropy inequalities, and it follows that the recently defined hypergraph cones are contained in
the quantum stabilizer entropy cones, which confirms a conjecture made in the recent literature
Holographic duality from random tensor networks
Tensor networks provide a natural framework for exploring holographic duality
because they obey entanglement area laws. They have been used to construct
explicit toy models realizing many interesting structural features of the
AdS/CFT correspondence, including the non-uniqueness of bulk operator
reconstruction in the boundary theory. In this article, we explore the
holographic properties of networks of random tensors. We find that our models
naturally incorporate many features that are analogous to those of the AdS/CFT
correspondence. When the bond dimension of the tensors is large, we show that
the entanglement entropy of boundary regions, whether connected or not, obey
the Ryu-Takayanagi entropy formula, a fact closely related to known properties
of the multipartite entanglement of assistance. Moreover, we find that each
boundary region faithfully encodes the physics of the entire bulk entanglement
wedge. Our method is to interpret the average over random tensors as the
partition function of a classical ferromagnetic Ising model, so that the
minimal surfaces of Ryu-Takayanagi appear as domain walls. Upon including the
analog of a bulk field, we find that our model reproduces the expected
corrections to the Ryu-Takayanagi formula: the minimal surface is displaced and
the entropy is augmented by the entanglement of the bulk field. Increasing the
entanglement of the bulk field ultimately changes the minimal surface
topologically in a way similar to creation of a black hole. Extrapolating bulk
correlation functions to the boundary permits the calculation of the scaling
dimensions of boundary operators, which exhibit a large gap between a small
number of low-dimension operators and the rest. While we are primarily
motivated by AdS/CFT duality, our main results define a more general form of
bulk-boundary correspondence which could be useful for extending holography to
other spacetimes.Comment: 57 pages, 13 figure
One-shot holography
Following the work of [2008.03319], we define a generally covariant
max-entanglement wedge of a boundary region , which we conjecture to be the
bulk region reconstructible from . We similarly define a covariant
min-entanglement wedge, which we conjecture to be the bulk region that can
influence the boundary state on . We prove that the min- and
max-entanglement wedges obey various properties necessary for this conjecture,
such as nesting, inclusion of the causal wedge, and a reduction to the usual
quantum extremal surface prescription in the appropriate special cases. These
proofs rely on one-shot versions of the (restricted) quantum focusing
conjecture (QFC) that we conjecture to hold. We argue that this QFC implies a
one-shot generalized second law (GSL) and quantum Bousso bound. Moreover, in a
particular semiclassical limit we prove this one-shot GSL directly using
algebraic techniques. Finally, in order to derive our results, we extend both
the frameworks of one-shot quantum Shannon theory and state-specific
reconstruction to finite-dimensional von Neumann algebras, allowing nontrivial
centers.Comment: 84 pages, 8 figure