Abstract
We consider the (Rényi) mutual information, \( {I}^{(n)}\left(A,B\right)={S}_A^{(n)}+{S}_B^{(n)}-{S}_{A\cup B}^{(n)} \), of distant compact spatial regions A and B in the vacuum state of a free scalar field. The distance r between A and B is much greater than their sizes RA,B. It is known that \( {I}^{(n)}\left(A,B\right)\sim {C}_{AB}^{(n)}{\left\langle 0\left|\phi (r)\phi (0)0\right|\right\rangle}^2 \). We obtain the direct expression of \( {C}_{AB}^{(n)} \) for arbitrary regions A and B. We perform the analytical continuation of n and obtain the mutual information. The direct expression is useful for the numerical computation. By using the direct expression, we can compute directly I(A, B) without computing SA, SB and SA∪B respectively, so it reduces significantly the amount of computation.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
S. Ryu and T. Takayanagi, Aspects of holographic entanglement entropy, JHEP 08 (2006) 045 [hep-th/0605073] [INSPIRE].
T. Faulkner, A. Lewkowycz and J. Maldacena, Quantum corrections to holographic entanglement entropy, JHEP 11 (2013) 074 [arXiv:1307.2892] [INSPIRE].
B. Swingle, Entanglement renormalization and holography, Phys. Rev. D 86 (2012) 065007 [arXiv:0905.1317] [INSPIRE].
M. Nozaki, S. Ryu and T. Takayanagi, Holographic geometry of entanglement renormalization in quantum field theories, JHEP 10 (2012) 193 [arXiv:1208.3469] [INSPIRE].
M. Miyaji and T. Takayanagi, Surface/state correspondence as a generalized holography, PTEP 2015 (2015) 073B03 [arXiv:1503.03542] [INSPIRE].
P. Caputa, M. Miyaji, T. Takayanagi and K. Umemoto, Holographic entanglement of purification from conformal field theories, Phys. Rev. Lett. 122 (2019) 111601 [arXiv:1812.05268] [INSPIRE].
N. Shiba and T. Takayanagi, Volume law for the entanglement entropy in non-local QFTs, JHEP 02 (2014) 033 [arXiv:1311.1643] [INSPIRE].
A. Mollabashi, N. Shiba and T. Takayanagi, Entanglement between two interacting CFTs and generalized holographic entanglement entropy, JHEP 04 (2014) 185 [arXiv:1403.1393] [INSPIRE].
M. Miyaji, T. Numasawa, N. Shiba, T. Takayanagi and K. Watanabe, Continuous multiscale entanglement renormalization ansatz as holographic surface-state correspondence, Phys. Rev. Lett. 115 (2015) 171602 [arXiv:1506.01353] [INSPIRE].
M. Miyaji, T. Numasawa, N. Shiba, T. Takayanagi and K. Watanabe, Distance between quantum states and gauge-gravity duality, Phys. Rev. Lett. 115 (2015) 261602 [arXiv:1507.07555] [INSPIRE].
T. Miyagawa, N. Shiba and T. Takayanagi, Double-trace deformations and entanglement entropy in AdS, Fortsch. Phys. 64 (2016) 92 [arXiv:1511.07194] [INSPIRE].
T. Numasawa, N. Shiba, T. Takayanagi and K. Watanabe, EPR pairs, local projections and quantum teleportation in holography, JHEP 08 (2016) 077 [arXiv:1604.01772] [INSPIRE].
M. Levin and X.-G. Wen, Detecting topological order in a ground state wave function, Phys. Rev. Lett. 96 (2006) 110405 [cond-mat/0510613] [INSPIRE].
A. Kitaev and J. Preskill, Topological entanglement entropy, Phys. Rev. Lett. 96 (2006) 110404 [hep-th/0510092] [INSPIRE].
P. Calabrese and J.L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. 0406 (2004) P06002 [hep-th/0405152] [INSPIRE].
N. Shiba, Aharonov-Bohm effect on entanglement entropy in conformal field theory, Phys. Rev. D 96 (2017) 065016 [arXiv:1701.00688] [INSPIRE].
S. Ghosh, R.M. Soni and S.P. Trivedi, On the entanglement entropy for gauge theories, JHEP 09 (2015) 069 [arXiv:1501.02593] [INSPIRE].
S. Aoki, T. Iritani, M. Nozaki, T. Numasawa, N. Shiba and H. Tasaki, On the definition of entanglement entropy in lattice gauge theories, JHEP 06 (2015) 187 [arXiv:1502.04267] [INSPIRE].
Y. Nakai, N. Shiba and M. Yamada, Entanglement entropy and decoupling in the universe, Phys. Rev. D 96 (2017) 123518 [arXiv:1709.02390] [INSPIRE].
L. Bombelli, R.K. Koul, J. Lee and R.D. Sorkin, A quantum source of entropy for black holes, Phys. Rev. D 34 (1986) 373 [INSPIRE].
M. Srednicki, Entropy and area, Phys. Rev. Lett. 71 (1993) 666 [hep-th/9303048] [INSPIRE].
L. Susskind and J. Uglum, Black hole entropy in canonical quantum gravity and superstring theory, Phys. Rev. D 50 (1994) 2700 [hep-th/9401070] [INSPIRE].
D.N. Kabat, Black hole entropy and entropy of entanglement, Nucl. Phys. B 453 (1995) 281 [hep-th/9503016] [INSPIRE].
N. Shiba, Entanglement entropy of two black holes and entanglement entropic force, Phys. Rev. D 83 (2011) 065002 [arXiv:1011.3760] [INSPIRE].
N. Shiba, Entanglement entropy of two spheres, JHEP 07 (2012) 100 [arXiv:1201.4865] [INSPIRE].
M. Creutz, Quarks, gluons and lattices, Cambridge University Press, Cambridge, U.K. (1985).
J. Cardy, Some results on the mutual information of disjoint regions in higher dimensions, J. Phys. A 46 (2013) 285402 [arXiv:1304.7985] [INSPIRE].
N. Shiba, Entanglement entropy of disjoint regions in excited states: an operator method, JHEP 12 (2014) 152 [arXiv:1408.0637] [INSPIRE].
P. Calabrese, J. Cardy and E. Tonni, Entanglement entropy of two disjoint intervals in conformal field theory II, J. Stat. Mech. 1101 (2011) P01021 [arXiv:1011.5482] [INSPIRE].
M. Headrick, Entanglement Rényi entropies in holographic theories, Phys. Rev. D 82 (2010) 126010 [arXiv:1006.0047] [INSPIRE].
H. Casini and M. Huerta, Universal terms for the entanglement entropy in 2 + 1 dimensions, Nucl. Phys. B 764 (2007) 183 [hep-th/0606256] [INSPIRE].
H. Casini, M. Huerta and L. Leitao, Entanglement entropy for a Dirac fermion in three dimensions: vertex contribution, Nucl. Phys. B 814 (2009) 594 [arXiv:0811.1968] [INSPIRE].
T. Hirata and T. Takayanagi, AdS/CFT and strong subadditivity of entanglement entropy, JHEP 02 (2007) 042 [hep-th/0608213] [INSPIRE].
E. Fradkin and J.E. Moore, Entanglement entropy of 2D conformal quantum critical points: hearing the shape of a quantum drum, Phys. Rev. Lett. 97 (2006) 050404 [cond-mat/0605683] [INSPIRE].
G. Vidal and R.F. Werner, Computable measure of entanglement, Phys. Rev. A 65 (2002) 032314 [quant-ph/0102117] [INSPIRE].
P. Calabrese, J. Cardy and E. Tonni, Entanglement negativity in quantum field theory, Phys. Rev. Lett. 109 (2012) 130502 [arXiv:1206.3092] [INSPIRE].
C. Agón and T. Faulkner, Quantum corrections to holographic mutual information, JHEP 08 (2016) 118 [arXiv:1511.07462] [INSPIRE].
C.A. Agón, I. Cohen-Abbo and H.J. Schnitzer, Large distance expansion of mutual information for disjoint disks in a free scalar theory, JHEP 11 (2016) 073 [arXiv:1505.03757] [INSPIRE].
H.J. Schnitzer, Mutual Rényi information for two disjoint compound systems, arXiv:1406.1161 [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 1907.07155
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Shiba, N. Direct calculation of mutual information of distant regions. J. High Energ. Phys. 2020, 182 (2020). https://doi.org/10.1007/JHEP09(2020)182
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP09(2020)182