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Quantum bit threads and holographic entanglement

Agón, Cesar A., Pedraza, Juan F.·2022·Journal of High Energy Physics
DOI 10.1007/jhep02(2022)180
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Abstract

Quantum corrections to holographic entanglement entropy require knowledge of the bulk quantum state. In this paper, we derive a novel dual prescription for the generalized entropy that allows us to interpret the leading quantum corrections in a geometric way with minimal input from the bulk state. The equivalence is proven using tools borrowed from convex optimization. The new prescription does not involve bulk surfaces but instead uses a generalized notion of a flow, which allows for possible sources or sinks in the bulk geometry. In its discrete version, our prescription can alternatively be interpreted in terms of a set of Planck-thickness bit threads, which can be either classical or quantum. This interpretation uncovers an aspect of the generalized entropy that admits a neat information-theoretic description, namely, the fact that the quantum corrections can be cast in terms of entanglement distillation of the bulk state. We also prove some general properties of our prescription, including nesting and a quantum version of the max multiflow theorem. These properties are used to verify that our proposal respects known inequalities that a von Neumann entropy must satisfy, including subadditivity and strong subadditivity, as well as to investigate the fate of the holographic monogamy. Finally, using the Iyer-Wald formalism we show that for cases with a local modular Hamiltonian there is always a canonical solution to the program that exploits the property of bulk locality. Combining with previous results by Swingle and Van Raamsdonk, we show that the con- sistency of this special solution requires the semi-classical Einstein’s equations to hold for any consistent perturbative bulk quantum state.

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Figures

Figure 1 Family of bulk surfaces m i homologous to the boundary region A that continuously interpolate between γ A and A , and hence foliate Σ A .
Figure 1 Family of bulk surfaces m i homologous to the boundary region A that continuously interpolate between γ A and A , and hence foliate Σ A .CC-BY · From the PDF
Figure 2 Thread representation of the max flow associated to a simply connected boundary region (left). The crosses represent sinks of threads while the dots represent sources. Quantum threads are thus naturally oriented objects. From the global perspective, they are seen to jump across the minimal surface via quantum tunneling. This process can be intuitively understood in double holographic models (right), where the quantum threads are still continuously connected through the an emergent coordinate u . In this context, the sinks and sources arise by a projection of a divergenceless field into a lower dimensional space.
Figure 2 Thread representation of the max flow associated to a simply connected boundary region (left). The crosses represent sinks of threads while the dots represent sources. Quantum threads are thus naturally oriented objects. From the global perspective, they are seen to jump across the minimal surface via quantum tunneling. This process can be intuitively understood in double holographic models (right), where the quantum threads are still continuously connected through the an emergent coordinate u . In this context, the sinks and sources arise by a projection of a divergenceless field into a lower dimensional space.CC-BY · From the PDF
Figure 3 Thread representation of the max flow associated to a boundary region made out the two simply connected parts A and B . Above, we have defined a , b and ab as the bulk homology regions associated with A , B and AB , respectively, and c = ab \ a ∪ b . The entropy density on a , b and c is picked to be positive and thus acts as sinks for the quantum threads. When restricted to the region a ∪ b , though, some of these threads (the ones ending on c ) are interpreted as classical because they contribute exclusively to the area ∂ ( a ∪ b ) .
Figure 3 Thread representation of the max flow associated to a boundary region made out the two simply connected parts A and B . Above, we have defined a , b and ab as the bulk homology regions associated with A , B and AB , respectively, and c = ab \ a ∪ b . The entropy density on a , b and c is picked to be positive and thus acts as sinks for the quantum threads. When restricted to the region a ∪ b , though, some of these threads (the ones ending on c ) are interpreted as classical because they contribute exclusively to the area ∂ ( a ∪ b ) .CC-BY · From the PDF
Figure 4 Thread representation of the max flow associated to a boundary region made out of two simply connected parts A and B . In this case, the entropy density on c = ab \ a ∪ b is negative and thus acts as sources for the quantum threads. When restricted to the region ab , though, some of these threads (the ones starting on c ) are interpreted as classical because they contribute exclusively to the area ∂ ( a ∪ b ) .
Figure 4 Thread representation of the max flow associated to a boundary region made out of two simply connected parts A and B . In this case, the entropy density on c = ab \ a ∪ b is negative and thus acts as sources for the quantum threads. When restricted to the region ab , though, some of these threads (the ones starting on c ) are interpreted as classical because they contribute exclusively to the area ∂ ( a ∪ b ) .CC-BY · From the PDF

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Citations

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Metadata

Authors
Agón, C. A., & Pedraza, J. F.
ORCID
Pedraza, Juan F.
Venue
Journal of High Energy Physics
Publisher
Springer Science and Business Media LLC
Published
2022-02
Volume
2022
Issue
2
ISSN
1029-8479
DOI
10.1007/jhep02(2022)180
Cited by
33
References
13
Language
en
Licence

About these checks

Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.

Statistics and arithmetic are recomputed, not judged.

Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.

What counts as a claim ›How we measure confidence ›How we check this ›

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