Entanglement entropy and the colored Jones polynomial

Balasubramanian, Vijay; DeCross, Matthew; Fliss, Jackson; Kar, Arjun; Leigh, Robert G.; Parrikar, Onkar

doi:10.1007/jhep05(2018)038

Title: Entanglement entropy and the colored Jones polynomial

Journal Article · Mon May 07 00:00:00 EDT 2018 · Journal of High Energy Physics (Online)

DOI:https://doi.org/10.1007/jhep05(2018)038· OSTI ID:1507583

Balasubramanian, Vijay ^[1]; DeCross, Matthew ^[2]; Fliss, Jackson ^[3];

^[2]; Leigh, Robert G. ^[3]; Parrikar, Onkar ^[2]

Univ. of Pennsylvania, Philadelphia, PA (United States). David Rittenhouse Lab.; Vrije Univ. Brussel (VUB) and International Solvay Institutes, Brussels (Belgium). Theoretische Natuurkunde
Univ. of Pennsylvania, Philadelphia, PA (United States). David Rittenhouse Lab.
Univ. of Illinois, Urbana, IL (United States). Dept. of Physics

We study the multi-party entanglement structure of states in Chern-Simons theory created by performing the path integral on 3-manifolds with linked torus boundaries, called link complements. For gauge group SU(2), the wavefunctions of these states (in a particular basis) are the colored Jones polynomials of the corresponding links. We first review the case of U(1) Chern-Simons theory where these are stabilizer states, a fact we use to re-derive an explicit formula for the entanglement entropy across a general link bipartition. We then present the following results for SU(2) Chern-Simons theory: (i) The entanglement entropy for a bipartition of a link gives a lower bound on the genus of surfaces in the ambient S3 separating the two sublinks. (ii) All torus links (namely, links which can be drawn on the surface of a torus) have a GHZ-like entanglement structure — i.e., partial traces leave a separable state. By contrast, through explicit computation, we test in many examples that hyperbolic links (namely, links whose complements admit hyperbolic structures) have W-like entanglement — i.e., partial traces leave a non-separable state. (iii) Finally, we consider hyperbolic links in the complexified SL(2,C) Chern-Simons theory, which is closely related to 3d Einstein gravity with a negative cosmological constant. In the limit of small Newton constant, we discuss how the entanglement structure is controlled by the Neumann-Zagier potential on the moduli space of hyperbolic structures on the link complement.

View Accepted Manuscript (DOE)

Cite

Export

Save

Research Organization:: Duke Univ., Durham, NC (United States); Univ. of Illinois at Urbana-Champaign, IL (United States)

Sponsoring Organization:: USDOE Office of Science (SC)

Grant/Contract Number:: FG02-05ER41367; SC0015655

OSTI ID:: 1507583

Journal Information:: Journal of High Energy Physics (Online), Vol. 2018, Issue 5; ISSN 1029-8479

Publisher:: Springer BerlinCopyright Statement

Country of Publication:: United States

Language:: English

Citation Metrics:

Cited by: 23 works

Citation information provided by
Web of Science

References (44)

Multiboundary wormholes and holographic entanglement Balasubramanian, Vijay; Hayden, Patrick; Maloney, Alexander Classical and Quantum Gravity, Vol. 31, Issue 18 https://doi.org/10.1088/0264-9381/31/18/185015	journal	September 2014
Topological quantum field theories Atiyah, Michael Publications mathématiques de l'IHÉS, Vol. 68, Issue 1 https://doi.org/10.1007/BF02698547	journal	January 1988
Topological quantum field theory Witten, Edward Communications in Mathematical Physics, Vol. 117, Issue 3 https://doi.org/10.1007/BF01223371	journal	September 1988
Torus Knots and Mirror Symmetry Brini, Andrea; Mariño, Marcos; Eynard, Bertrand Annales Henri Poincaré, Vol. 13, Issue 8 https://doi.org/10.1007/s00023-012-0171-2	journal	March 2012
Volumes of hyperbolic three-manifolds Neumann, Walter D.; Zagier, Don Topology, Vol. 24, Issue 3 https://doi.org/10.1016/0040-9383(85)90004-7	journal	January 1985
Multi-boundary entanglement in Chern-Simons theory and link invariants Balasubramanian, Vijay; Fliss, Jackson R.; Leigh, Robert G. Journal of High Energy Physics, Vol. 2017, Issue 4 https://doi.org/10.1007/JHEP04(2017)061	journal	April 2017
:{unav) Kashaev, R. M. Letters in Mathematical Physics, Vol. 39, Issue 3, p. 269-275 https://doi.org/10.1023/A:1007364912784	journal	January 1997
Computable measure of entanglement Vidal, G.; Werner, R. F. Physical Review A, Vol. 65, Issue 3 https://doi.org/10.1103/PhysRevA.65.032314	journal	February 2002
Tensor network models of multiboundary wormholes Peach, Alex; Ross, Simon F. Classical and Quantum Gravity, Vol. 34, Issue 10 https://doi.org/10.1088/1361-6382/aa6b0f	journal	April 2017
Hudson’s theorem for finite-dimensional quantum systems Gross, D. Journal of Mathematical Physics, Vol. 47, Issue 12 https://doi.org/10.1063/1.2393152	journal	December 2006
Exact results for perturbative Chern–Simons theory with complex gauge group Dimofte, Tudor; Gukov, Sergei; Lenells, Jonatan Communications in Number Theory and Physics, Vol. 3, Issue 2 https://doi.org/10.4310/CNTP.2009.v3.n2.a4	journal	January 2009
Holographic Derivation of Entanglement Entropy from the anti–de Sitter Space/Conformal Field Theory Correspondence Ryu, Shinsei; Takayanagi, Tadashi Physical Review Letters, Vol. 96, Issue 18 https://doi.org/10.1103/PhysRevLett.96.181602	journal	May 2006
Separability Criterion for Density Matrices Peres, Asher Physical Review Letters, Vol. 77, Issue 8 https://doi.org/10.1103/PhysRevLett.77.1413	journal	August 1996
Analytic continuation of Chern-Simons theory Witten, Edward Chern-Simons Gauge Theory: 20 Years After https://doi.org/10.1090/amsip/050/19	book	January 2010
Generalized volume conjecture and the $A$ -polynomials: The Neumann–Zagier potential function as a classical limit of the partition function Hikami, Kazuhiro Journal of Geometry and Physics, Vol. 57, Issue 9 https://doi.org/10.1016/j.geomphys.2007.03.008	journal	August 2007
Closed surface bundles of least volume Aaber, John W.; Dunfield, Nathan Algebraic & Geometric Topology, Vol. 10, Issue 4 https://doi.org/10.2140/agt.2010.10.2315	journal	January 2010
Knots, links and branes at large N Labastida, José M. F.; Mariño, Marcos; Vafa, Cumrun Journal of High Energy Physics, Vol. 2000, Issue 11 https://doi.org/10.1088/1126-6708/2000/11/007	journal	November 2000
Four qubits can be entangled in nine different ways Verstraete, F.; Dehaene, J.; De Moor, B. Physical Review A, Vol. 65, Issue 5 https://doi.org/10.1103/PhysRevA.65.052112	journal	April 2002
Holographic quantum error-correcting codes: toy models for the bulk/boundary correspondence Pastawski, Fernando; Yoshida, Beni; Harlow, Daniel Journal of High Energy Physics, Vol. 2015, Issue 6 https://doi.org/10.1007/JHEP06(2015)149	journal	June 2015
Three-Dimensional Quantum Gravity, Chern-Simons Theory, and the A-Polynomial Gukov, Sergei Communications in Mathematical Physics, Vol. 255, Issue 3 https://doi.org/10.1007/s00220-005-1312-y	journal	March 2005
Holographic geometry of entanglement renormalization in quantum field theories Nozaki, Masahiro; Ryu, Shinsei; Takayanagi, Tadashi Journal of High Energy Physics, Vol. 2012, Issue 10 https://doi.org/10.1007/JHEP10(2012)193	journal	October 2012
Cyclic surgery, degrees of maps of character curves, and volume rigidity for hyperbolic manifolds Dunfield, Nathan M. Inventiones Mathematicae, Vol. 136, Issue 3 https://doi.org/10.1007/s002220050321	journal	May 1999
Three qubits can be entangled in two inequivalent ways Dür, W.; Vidal, G.; Cirac, J. I. Physical Review A, Vol. 62, Issue 6 https://doi.org/10.1103/PhysRevA.62.062314	journal	November 2000
Closed incompressible surfaces in alternating knot and link complements Menasco, W. Topology, Vol. 23, Issue 1 https://doi.org/10.1016/0040-9383(84)90023-5	journal	January 1984
Hot multiboundary wormholes from bipartite entanglement Marolf, Donald; Maxfield, Henry; Peach, Alex Classical and Quantum Gravity, Vol. 32, Issue 21 https://doi.org/10.1088/0264-9381/32/21/215006	journal	October 2015
Quantization of Chern-Simons gauge theory with complex gauge group Witten, Edward Communications in Mathematical Physics, Vol. 137, Issue 1 https://doi.org/10.1007/BF02099116	journal	March 1991
Quantum field theory and the Jones polynomial Witten, Edward Communications in Mathematical Physics, Vol. 121, Issue 3 https://doi.org/10.1007/BF01217730	journal	September 1989
Multiparty entanglement in graph states Hein, M.; Eisert, J.; Briegel, H. J. Physical Review A, Vol. 69, Issue 6 https://doi.org/10.1103/PhysRevA.69.062311	journal	June 2004
Graph states of prime-power dimension from generalized CNOT quantum circuit Chen, Lin; Zhou, D. L. Scientific Reports, Vol. 6, Issue 1 https://doi.org/10.1038/srep27135	journal	June 2016
Fusion rules and modular transformations in 2D conformal field theory Verlinde, Erik Nuclear Physics B, Vol. 300 https://doi.org/10.1016/0550-3213(88)90603-7	journal	January 1988
Entanglement of formation and monogamy of multi-party quantum entanglement Kim, Jeong San Scientific Reports, Vol. 11, Issue 1 https://doi.org/10.1038/s41598-021-82052-3	journal	January 2021
Exact Results for Perturbative Chern-Simons Theory with Complex Gauge Group Dimofte, Tudor; Gukov, Sergei; Lenells, Jonatan arXiv https://doi.org/10.48550/arxiv.0903.2472	text	January 2009
Closed surface bundles of least volume Aaber, John William; Dunfield, Nathan M. arXiv https://doi.org/10.48550/arxiv.1002.3423	text	January 2010
Quantum Field Theory and the Volume Conjecture Dimofte, Tudor; Gukov, Sergei arXiv https://doi.org/10.48550/arxiv.1003.4808	text	January 2010
Holographic Geometry of Entanglement Renormalization in Quantum Field Theories Nozaki, Masahiro; Ryu, Shinsei; Takayanagi, Tadashi arXiv https://doi.org/10.48550/arxiv.1208.3469	text	January 2012
Multiboundary Wormholes and Holographic Entanglement Balasubramanian, Vijay; Hayden, Patrick; Maloney, Alexander arXiv https://doi.org/10.48550/arxiv.1406.2663	text	January 2014
Hot multiboundary wormholes from bipartite entanglement Marolf, Donald; Maxfield, Henry; Peach, Alex arXiv https://doi.org/10.48550/arxiv.1506.04128	text	January 2015
Multi-Boundary Entanglement in Chern-Simons Theory and Link Invariants Balasubramanian, Vijay; Fliss, Jackson R.; Leigh, Robert G. arXiv https://doi.org/10.48550/arxiv.1611.05460	text	January 2016
Entanglement on linked boundaries in Chern-Simons theory with generic gauge groups Dwivedi, Siddharth; Singh, Vivek Kumar; Dhara, Saswati arXiv https://doi.org/10.48550/arxiv.1711.06474	text	January 2017
Knots, links and branes at large N Labastida, J. M. F.; Marino, Marcos; Vafa, Cumrun arXiv https://doi.org/10.48550/arxiv.hep-th/0010102	text	January 2000
On the Gauge Theory/Geometry Correspondence Gopakumar, Rajesh; Vafa, Cumrun arXiv https://doi.org/10.48550/arxiv.hep-th/9811131	text	January 1998
Cyclic surgery, degrees of maps of character curves, and volume rigidity for hyperbolic manifolds Dunfield, Nathan M. arXiv https://doi.org/10.48550/arxiv.math/9802022	text	January 1998
Hudson's Theorem for finite-dimensional quantum systems Gross, D. arXiv https://doi.org/10.48550/arxiv.quant-ph/0602001	text	January 2006
Polynomials for torus links from Chern-Simons gauge theories Isidro, J. M.; Labastida, J. M. F.; Ramallo, A. V. Nuclear Physics B, Vol. 398, Issue 1 https://doi.org/10.1016/0550-3213(93)90632-y	journal	June 1993

Cited By (6)

From topological to quantum entanglement Melnikov, D.; Mironov, A.; Mironov, S. Journal of High Energy Physics, Vol. 2019, Issue 5 https://doi.org/10.1007/jhep05(2019)116	journal	May 2019
Circuit complexity of knot states in Chern-Simons theory Camilo, Giancarlo; Melnikov, Dmitry; Novaes, Fábio Journal of High Energy Physics, Vol. 2019, Issue 7 https://doi.org/10.1007/jhep07(2019)163	journal	July 2019
Entanglement on multiple S2 boundaries in Chern-Simons theory Dwivedi, Siddharth; Singh, Vivek Kumar; Ramadevi, P. Journal of High Energy Physics, Vol. 2019, Issue 8 https://doi.org/10.1007/jhep08(2019)034	journal	August 2019
From Topological to Quantum Entanglement Melnikov, D.; Mironov, A.; Mironov, S. arXiv https://doi.org/10.48550/arxiv.1809.04574	text	January 2018
Circuit Complexity of Knot States in Chern-Simons theory Camilo, Giancarlo; Melnikov, Dmitry; Novaes, Fábio arXiv https://doi.org/10.48550/arxiv.1903.10609	text	January 2019
Topological Entanglement and Knots Mironov, Sergey Universe, Vol. 5, Issue 2 https://doi.org/10.3390/universe5020060	journal	February 2019

Similar Records

Complexity for link complement states in Chern-Simons theory

Journal Article · Fri Sep 03 00:00:00 EDT 2021 · Physical Review D · OSTI ID:1507583

Leigh, Robert G.; Pai, Pin-Chun

K-decompositions and 3d gauge theories

Journal Article · Thu Nov 24 00:00:00 EST 2016 · Journal of High Energy Physics (Online) · OSTI ID:1507583

Dimofte, Tudor; Gabella, Maxime; Goncharov, Alexander B.

Quantum modularity and complex Chern-Simons theory

Journal Article · Fri Apr 27 00:00:00 EDT 2018 · Communications in Number Theory and Physics · OSTI ID:1507583

Dimofte, Tudor; Garoufalidis, Stavros

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
Chern-Simons Theories
Topological Field Theories
Wilson
't Hooft and Polyakov loops
Conformal Field Theory

Title: Entanglement entropy and the colored Jones polynomial

Citation Formats

References (44)

Cited By (6)

Similar Records

Related Subjects