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Quantum minimal surfaces from quantum error correction

Akers, Chris, Penington, Geoff·2022·SciPost Physics
DOI 10.21468/scipostphys.12.5.157
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Abstract

We show that complementary state-specific reconstruction of logical (bulk) operators is equivalent to the existence of a quantum minimal surface prescription for physical (boundary) entropies. This significantly generalizes both sides of an equivalence previously shown by Harlow[1]; in particular, we do not require the entanglement wedge to be the same for all states in the code space. In developing this theorem, we construct an emergent bulk geometry for general quantum codes, defining ``areas'' associated to arbitrary logical subsystems, and argue that this definition is ``functionally unique.'' We also formalize a definition of bulk reconstruction that we call ``state-specific product unitary’’ reconstruction. This definition captures the quantum error correction (QEC) properties present in holographic codes and has potential independent interest as a very broad generalization of QEC; it includes most traditional versions of QEC as special cases. Our results extend to approximate codes, and even to the ``non-isometric codes'' that seem to describe the interior of a black hole at late times.

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Crossref · 10.21468/scipostphys.12.5.157

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Figures

Figure 1 An AdS / CFT setup without complementary state-independent recovery. Between the two candidate extremal surfaces lies a black hole with horizon area much greater than the difference in area between the two surfaces, A BH A 2 A 1 . Neither B nor B can reconstruct operators in b 0 in a state-independent way.
Figure 1 An AdS / CFT setup without complementary state-independent recovery. Between the two candidate extremal surfaces lies a black hole with horizon area much greater than the difference in area between the two surfaces, A BH A 2 A 1 . Neither B nor B can reconstruct operators in b 0 in a state-independent way.CC-BY · From the PDF
Figure 2 Two illustrations of the quantum codes used throughout this paper. (a) A linear map V : i H b i ! H B H B . (b) This linear map encompasses situations in holography. Each input leg corresponds to a bulk point or subregion, while the output legs correspond to boundary subregions. The map V is implicit.
Figure 2 Two illustrations of the quantum codes used throughout this paper. (a) A linear map V : i H b i ! H B H B . (b) This linear map encompasses situations in holography. Each input leg corresponds to a bulk point or subregion, while the output legs correspond to boundary subregions. The map V is implicit.CC-BY · From the PDF
Image 3
CC-BY · From the PDF
Figure 4 An AdS / CFT setup in which a unitary acting within an entanglement wedge changes the location of the quantum minimal surface. On the left, we have two black holes in an entangled pure state. Because of the large amount of entanglement, 1 is quantum minimal and therefore both black holes are inside the entanglement wedge of B . On the right, we have the situation after acting on both black holes with a unitary Ubb 0 that maps the original state to a factorized pure state. Now 2 is quantum minimal. This unitary has changed the entropies S ( B ) and S ( B ) and therefore cannot be represented by a unitary on B .
Figure 4 An AdS / CFT setup in which a unitary acting within an entanglement wedge changes the location of the quantum minimal surface. On the left, we have two black holes in an entangled pure state. Because of the large amount of entanglement, 1 is quantum minimal and therefore both black holes are inside the entanglement wedge of B . On the right, we have the situation after acting on both black holes with a unitary Ubb 0 that maps the original state to a factorized pure state. Now 2 is quantum minimal. This unitary has changed the entropies S ( B ) and S ( B ) and therefore cannot be represented by a unitary on B .CC-BY · From the PDF
Image 5
CC-BY · From the PDF
Figure 5 An example tensor network. The blue circles each represent tensors, fourpartite states on the union of the small red circle and three black legs touching that blue circle. Each tensor can be viewed as a map from a state on the red circle to a state on the black legs. A solid black line between tensors represents postselection of the state on one leg from each tensor onto the state ϕ MAX ι . The entire tensor network forms a map from states on the union of the small red circles to the union of the boundary legs, which are those legs crossing the outer circle.
Figure 5 An example tensor network. The blue circles each represent tensors, fourpartite states on the union of the small red circle and three black legs touching that blue circle. Each tensor can be viewed as a map from a state on the red circle to a state on the black legs. A solid black line between tensors represents postselection of the state on one leg from each tensor onto the state ϕ MAX ι . The entire tensor network forms a map from states on the union of the small red circles to the union of the boundary legs, which are those legs crossing the outer circle.CC-BY · From the PDF
Image 7
CC-BY · From the PDF
Image 8
CC-BY · From the PDF
Image 9
CC-BY · From the PDF

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Citations

Cites

Research works printed but not linked: 34

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Metadata

Authors
Akers, C., & Penington, G.
Venue
SciPost Physics
Publisher
Stichting SciPost
Published
2022-05-12
Volume
12
Issue
5
ISSN
2542-4653
DOI
10.21468/scipostphys.12.5.157
Cited by
65
References
17
Language
en
Licence
Funders
  • Air Force Office of Scientific Research (FA9550-19-1-0360)
  • Simons Foundation
  • United States Department of Energy (DE-SC0019380)

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