Space from Hilbert Space: Recovering Geometry from Bulk Entanglement - INSPIRE

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Space from Hilbert Space: Recovering Geometry from Bulk Entanglement

ChunJun Cao
(
- Caltech
)
,
Sean M. Carroll
(
- Caltech
)
,
Spyridon Michalakis
(
- Caltech and
- Caltech, IQI
)

Jun 27, 2016

20 pages

Published in:

Phys.Rev.D 95 (2017) 2, 024031

Published: Jan 27, 2017

e-Print:

1606.08444 [hep-th]

DOI:

10.1103/PhysRevD.95.024031

Report number:

CALT-2016-15

View in:

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Abstract: (APS)

We examine how to construct a spatial manifold and its geometry from the entanglement structure of an abstract quantum state in Hilbert space. Given a decomposition of Hilbert space H into a tensor product of factors, we consider a class of “redundancy-constrained states” in H that generalize the area-law behavior for entanglement entropy usually found in condensed-matter systems with gapped local Hamiltonians. Using mutual information to define a distance measure on the graph, we employ classical multidimensional scaling to extract the best-fit spatial dimensionality of the emergent geometry. We then show that entanglement perturbations on such emergent geometries naturally give rise to local modifications of spatial curvature which obey a (spatial) analog of Einstein’s equation. The Hilbert space corresponding to a region of flat space is finite-dimensional and scales as the volume, though the entropy (and the maximum change thereof) scales like the area of the boundary. A version of the ER=EPR conjecture is recovered, in that perturbations that entangle distant parts of the emergent geometry generate a configuration that may be considered as a highly quantum wormhole.

Note:

37 pages, 5 figures. Updated notation, references, and acknowledgement

geometry: emergence
entropy: entanglement
Hilbert space
perturbation
Einstein-Podolsky-Rosen paradox
Einstein equation
condensed matter
Hamiltonian
curvature
wormhole

References(68)

Figures(7)

[1]

The Quantum Sine-Gordon Equation as the Massive Thirring Model

Sidney R. Coleman
(
- Harvard U.
)

- Phys.Rev.D 11 (1975) 2088,
•
DOI:
- 10.1103/PhysRevD.11.2088

[2]

The Large $N$ limit of superconformal field theories and supergravity

Juan Martin Maldacena
(
- Harvard U.
)

- Int.J.Theor.Phys. 38 (1999) 1113-1133 (reprint),
- Adv.Theor.Math.Phys. 2 (1998) 231-252
•
e-Print:
- hep-th/9711200
•
DOI:
- 10.4310/ATMP.1998.v2.n2.a1

[3]

Relative Entropy and Holography

- JHEP 08 (2013) 060
•
e-Print:
- 1305.3182
•
DOI:
- 10.1007/JHEP08(2013)060

[4]

Gravitation from Entanglement in Holographic CFTs

- JHEP 03 (2014) 051
•
e-Print:
- 1312.7856
•
DOI:
- 10.1007/JHEP03(2014)051

[5]

Universality of Gravity from Entanglement

e-Print:
- 1405.2933

[6]

Locality of Gravitational Systems from Entanglement of Conformal Field Theories

Jennifer Lin
(
- Chicago U., EFI and
- Chicago U.
)
,
Matilde Marcolli
(
- Caltech
)
,
Hirosi Ooguri
(
- Caltech and
- Tokyo U., IPMU
)
,
Bogdan Stoica
(
- Caltech
)

- Phys.Rev.Lett. 114 (2015) 221601
•
e-Print:
- 1412.1879
•
DOI:
- 10.1103/PhysRevLett.114.221601

[7]

Tensor Network States and Geometry

G. Evenbly
(
- Queensland U.
)
,
G. Vidal
(
- Queensland U. and
- Perimeter Inst. Theor. Phys.
)

- J.Statist.Phys. 145 (2011) 4, 891-918
•
DOI:
- 10.1007/s10955-011-0237-4

[8]

Entanglement Renormalization and Holography

Brian Swingle
(
- MIT, LNS and
- Harvard U., Phys. Dept.
)

- Phys.Rev.D 86 (2012) 065007
•
e-Print:
- 0905.1317
•
DOI:
- 10.1103/PhysRevD.86.065007

[9]

Constructing holographic spacetimes using entanglement renormalization

Brian Swingle
(
- Harvard U., Phys. Dept.
)

e-Print:
- 1209.3304

[10]

Exact holographic mapping and emergent space-time geometry

Xiao-Liang Qi
(
- Stanford U., Phys. Dept.
)

e-Print:
- 1309.6282

[11]

Consistency conditions for an AdS multiscale entanglement renormalization ansatz correspondence

et al.

- Phys.Rev.D 91 (2015) 12, 125036
•
e-Print:
- 1504.06632
•
DOI:
- 10.1103/PhysRevD.91.125036

[12]

Tensor network quotient takes the vacuum to the thermal state

Bartlomiej Czech
(
- Stanford U., Phys. Dept.
)
,
Glen Evenbly
(
- UC, Irvine
)
,
Lampros Lamprou
(
- Stanford U., Phys. Dept.
)
,
Samuel McCandlish
(
- Stanford U., Phys. Dept.
)
,
Xiao-Liang Qi
(
- Stanford U., Phys. Dept.
)

et al.

- Phys.Rev.B 94 (2016) 8, 085101
•
e-Print:
- 1510.07637
•
DOI:
- 10.1103/PhysRevB.94.085101

[13]

Tensor Networks from Kinematic Space

Bartlomiej Czech
(
- Stanford U., ITP and
- Stanford U., Phys. Dept.
)
,
Lampros Lamprou
(
- Stanford U., ITP and
- Stanford U., Phys. Dept.
)
,
Samuel McCandlish
(
- Stanford U., ITP and
- Stanford U., Phys. Dept.
)
,
James Sully
(
- SLAC
)

- JHEP 07 (2016) 100
•
e-Print:
- 1512.01548
•
DOI:
- 10.1007/JHEP07(2016)100

[14]

Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence

Fernando Pastawski
(
- Caltech, IQI
)
,
Beni Yoshida
(
- Caltech, IQI
)
,
Daniel Harlow
(
- Princeton U. (main)
)
,
John Preskill
(
- Caltech, IQI
)

- JHEP 06 (2015) 149
•
e-Print:
- 1503.06237
•
DOI:
- 10.1007/JHEP06(2015)149

[15]

Holographic duality from random tensor networks

Patrick Hayden
(
- Stanford U., ITP and
- Stanford U., Phys. Dept.
)
,
Sepehr Nezami
(
- Stanford U., ITP and
- Stanford U., Phys. Dept.
)
,
Xiao-Liang Qi
(
- Stanford U., ITP and
- Stanford U., Phys. Dept.
)
,
Nathaniel Thomas
(
- Stanford U., ITP and
- Stanford U., Phys. Dept.
)
,
Michael Walter
(
- Stanford U., ITP and
- Stanford U., Phys. Dept.
)

et al.

- JHEP 11 (2016) 009
•
e-Print:
- 1601.01694
•
DOI:
- 10.1007/JHEP11(2016)009

[16]

Cool horizons for entangled black holes

- Fortsch.Phys. 61 (2013) 781-811
•
e-Print:
- 1306.0533
•
DOI:
- 10.1002/prop.201300020

[17]

Building up spacetime with quantum entanglement

Mark Van Raamsdonk
(
- British Columbia U.
)

- Gen.Rel.Grav. 42 (2010) 2323-2329,
- Int.J.Mod.Phys.D 19 (2010) 2429-2435
•
e-Print:
- 1005.3035
•
DOI:
- 10.1142/S0218271810018529

[18]

Holographic Dual of an Einstein-Podolsky-Rosen Pair has a Wormhole

Kristan Jensen
(
- Victoria U.
)
,
Andreas Karch
(
- Washington U., Seattle
)

- Phys.Rev.Lett. 111 (2013) 21, 211602
•
e-Print:
- 1307.1132
•
DOI:
- 10.1103/PhysRevLett.111.211602

[19]

Splitting Spacetime and Cloning Qubits: Linking No-Go Theorems across the ER=EPR Duality

Ning Bao
(
- Caltech, IQI and
- Caltech
)
,
Jason Pollack
(
- Caltech
)
,
Grant N. Remmen
(
- Caltech
)

- Fortsch.Phys. 63 (2015) 705-710
•
e-Print:
- 1506.08203
•
DOI:
- 10.1002/prop.201500053

[20]

Wormhole and Entanglement (Non-)Detection in the ER=EPR Correspondence

Ning Bao
(
- Caltech and
- Caltech, IQI
)
,
Jason Pollack
(
- Caltech
)
,
Grant N. Remmen
(
- Caltech
)

- JHEP 11 (2015) 126
•
e-Print:
- 1509.05426
•
DOI:
- 10.1007/JHEP11(2015)126

[21]

Entanglement Conservation, ER=EPR, and a New Classical Area Theorem for Wormholes

Grant Remmen
(
- Caltech
)
,
Ning Bao
(
- Caltech and
- Caltech, IQI
)
,
Jason Pollack
(
- Caltech
)

- JHEP 07 (2016) 048
•
e-Print:
- 1604.08217
•
DOI:
- 10.1007/JHEP07(2016)048

[22]

On the weight of entanglement

David Edward Bruschi

- Phys.Lett.B 754 (2016) 182-186
•
e-Print:
- 1412.4007
•
DOI:
- 10.1016/j.physletb.2016.01.034

[23]

Entanglement Equilibrium and the Einstein Equation

Ted Jacobson
(
- Santa Barbara, KITP and
- Maryland U.
)

- Phys.Rev.Lett. 116 (2016) 20, 201101
•
e-Print:
- 1505.04753
•
DOI:
- 10.1103/PhysRevLett.116.201101

[24]

Comments on Jacobson’s “entanglement equilibrium and the Einstein equation”

- JHEP 03 (2016) 194
•
e-Print:
- 1601.00528
•
DOI:
- 10.1007/JHEP03(2016)194

[25]

What is the Entropy in Entropic Gravity?

Sean M. Carroll
(
- Caltech
)
,
Grant N. Remmen
(
- Caltech
)

- Phys.Rev.D 93 (2016) 12, 124052
•
e-Print:
- 1601.07558
•
DOI:
- 10.1103/PhysRevD.93.124052

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