Abstract
We explore entanglement negativity, a measure of the distillable entanglement contained in a quantum state, in relativistic field theories in various dimensions. We first give a general overview of negativity and its properties and then explain a well known result relating (logarithmic) negativity of pure quantum states to the Rényi entropy (at index 1/2), by exploiting the simple features of entanglement in thermal states. In particular, we show that the negativity of the thermofield double state is given by the free energy difference of the system at temperature T and 2 T respectively. We then use this result to compute the negativity in the vacuum state of conformal field theories in various dimensions, utilizing results that have been derived for free and holographic CFTs in the literature. We also comment upon general lessons to be learnt about negativity in holographic field theories.
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References
J. Bell, On the Einstein-Podolsky-Rosen paradox, Physics 1 (1964) 195 [INSPIRE].
R.F. Werner, Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model, Phys. Rev. A 40 (1989) 4277 [INSPIRE].
S. Popescu, Bell’s inequalities versus teleportation: what is nonlocality?, Phys. Rev. Lett. 72 (1994) 797.
S. Popescu, Bell’s inequalities and density matrices: revealing ‘hidden’ nonlocality, Phys. Rev. Lett. 74 (1995) 2619 [quant-ph/9502005] [INSPIRE].
C.H. Bennett et al., Purification of noisy entanglement and faithful teleportation via noisy channels, Phys. Rev. Lett. 76 (1996) 722 [quant-ph/9511027] [INSPIRE].
M.B. Plenio and S. Virmani, An introduction to entanglement measures, Quant. Inf. Comput. 7 (2007) 1 [quant-ph/0504163] [INSPIRE].
G. Vidal and R.F. Werner, Computable measure of entanglement, Phys. Rev. A 65 (2002) 032314 [INSPIRE].
R. Verch and R.F. Werner, Distillability and positivity of partial transposes in general quantum field systems, Rev. Math. Phys. 17 (2005) 545 [quant-ph/0403089] [INSPIRE].
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].
V.E. Hubeny, M. Rangamani and T. Takayanagi, A covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].
B. Swingle, Entanglement renormalization and holography, Phys. Rev. D 86 (2012) 065007 [arXiv:0905.1317] [INSPIRE].
M. Van Raamsdonk, Comments on quantum gravity and entanglement, arXiv:0907.2939 [INSPIRE].
M. Van Raamsdonk, Building up spacetime with quantum entanglement, Gen. Rel. Grav. 42 (2010) 2323 [arXiv:1005.3035] [INSPIRE].
J. Maldacena and L. Susskind, Cool horizons for entangled black holes, Fortsch. Phys. 61 (2013) 781 [arXiv:1306.0533] [INSPIRE].
T. Hartman and J. Maldacena, Time evolution of entanglement entropy from black hole interiors, JHEP 05 (2013) 014 [arXiv:1303.1080] [INSPIRE].
M. Nozaki, T. Numasawa, A. Prudenziati and T. Takayanagi, Dynamics of entanglement entropy from Einstein equation, Phys. Rev. D 88 (2013) 026012 [arXiv:1304.7100] [INSPIRE].
N. Lashkari, M.B. McDermott and M. Van Raamsdonk, Gravitational dynamics from entanglement ‘thermodynamics’, JHEP 04 (2014) 195 [arXiv:1308.3716] [INSPIRE].
J. Bhattacharya and T. Takayanagi, Entropic counterpart of perturbative Einstein equation, JHEP 10 (2013) 219 [arXiv:1308.3792] [INSPIRE].
T. Faulkner, M. Guica, T. Hartman, R.C. Myers and M. Van Raamsdonk, Gravitation from entanglement in holographic CFTs, JHEP 03 (2014) 051 [arXiv:1312.7856] [INSPIRE].
B. Swingle and M. Van Raamsdonk, Universality of gravity from entanglement, arXiv:1405.2933 [INSPIRE].
H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement entropy, JHEP 05 (2011) 036 [arXiv:1102.0440] [INSPIRE].
P. Calabrese, J. Cardy and E. Tonni, Entanglement negativity in quantum field theory, Phys. Rev. Lett. 109 (2012) 130502 [arXiv:1206.3092] [INSPIRE].
P. Calabrese, J. Cardy and E. Tonni, Entanglement negativity in extended systems: a field theoretical approach, J. Stat. Mech. 02 (2013) P02008 [arXiv:1210.5359] [INSPIRE].
A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, JHEP 08 (2013) 090 [arXiv:1304.4926] [INSPIRE].
V. Balasubramanian, P. Hayden, A. Maloney, D. Marolf and S.F. Ross, Multiboundary wormholes and holographic entanglement, Class. Quant. Grav. 31 (2014) 185015 [arXiv:1406.2663] [INSPIRE].
H. Gharibyan and R.F. Penna, Are entangled particles connected by wormholes? Support for the ER=EPR conjecture from entropy inequalities, Phys. Rev. D 89 (2014) 066001 [arXiv:1308.0289] [INSPIRE].
A. Peres, Separability criterion for density matrices, Phys. Rev. Lett. 77 (1996) 1413 [quant-ph/9604005] [INSPIRE].
M. Horodecki, P. Horodecki and R. Horodecki, On the necessary and sufficient conditions for separability of mixed quantum states, Phys. Lett. A 223 (1996) 1 [quant-ph/9605038] [INSPIRE].
M. Horodecki, P. Horodecki and R. Horodecki, Mixed state entanglement and distillation: is there a ‘bound’ entanglement in nature?, Phys. Rev. Lett. 80 (1998) 5239 [quant-ph/9801069] [INSPIRE].
A. Peres, All the Bell inequalities, Found. Phys. 29 (1999) 589 [quant-ph/9807017] [INSPIRE].
R. Werner and M. Wolf, Bell’s inequalities for states with positive partial transpose, Phys. Rev. A 61 (2000) 062102 [quant-ph/9910063].
B. Terhal, A. Doherty and D. Schwab, Symmetric extensions of quantum states and local hidden variable theories, Phys. Rev. Lett. 90 (2003) 157903.
T. Vértesi and N. Brunner, Disproving the Peres conjecture: Bell nonlocality from bipartite bound entanglement, arXiv:1405.4502.
K. Audenaert, M. Plenio and J. Eisert, The entanglement cost under operations preserving the positivity of partial transpose, Phys. Rev. Lett. 90 (2003) 027901 [quant-ph/0207146].
M. Plenio, The logarithmic negativity: a full entanglement monotone that is not convex, Phys. Rev. Lett. 95 (2005) 090503 [quant-ph/0505071].
H. He and G. Vidal, Disentangling theorem and monogamy for entanglement negativity, arXiv:1401.5843.
Y.-C. Ou and H. Fan, Monogamy inequality in terms of negativity for three-qubit states, Phys. Rev. A 75 (2007) 062308 [quant-ph/0702127].
G. Vidal and R. Tarrach, Robustness of entanglement, Phys. Rev. A 59 (1999) 141 [quant-ph/9806094] [INSPIRE].
M. Headrick, V.E. Hubeny, A. Lawrence and M. Rangamani, Causality and holographic entanglement entropy, arXiv:1408.6300 [INSPIRE].
S.A. Gentle and M. Rangamani, Holographic entanglement and causal information in coherent states, JHEP 01 (2014) 120 [arXiv:1311.0015] [INSPIRE].
H. Casini and M. Huerta, Entanglement entropy for the n-sphere, Phys. Lett. B 694 (2010) 167 [arXiv:1007.1813] [INSPIRE].
I.R. Klebanov, S.S. Pufu, S. Sachdev and B.R. Safdi, Rényi entropies for free field theories, JHEP 04 (2012) 074 [arXiv:1111.6290] [INSPIRE].
D.V. Fursaev, Entanglement Rényi entropies in conformal field theories and holography, JHEP 05 (2012) 080 [arXiv:1201.1702] [INSPIRE].
R. Emparan, AdS/CFT duals of topological black holes and the entropy of zero energy states, JHEP 06 (1999) 036 [hep-th/9906040] [INSPIRE].
L.-Y. Hung, R.C. Myers, M. Smolkin and A. Yale, Holographic calculations of Rényi entropy, JHEP 12 (2011) 047 [arXiv:1110.1084] [INSPIRE].
D.A. Galante and R.C. Myers, Holographic Rényi entropies at finite coupling, JHEP 08 (2013) 063 [arXiv:1305.7191] [INSPIRE].
M. Headrick, Entanglement Rényi entropies in holographic theories, Phys. Rev. D 82 (2010) 126010 [arXiv:1006.0047] [INSPIRE].
T. Hartman, Entanglement entropy at large central charge, arXiv:1303.6955 [INSPIRE].
V.E. Hubeny, H. Maxfield, M. Rangamani and E. Tonni, Holographic entanglement plateaux, JHEP 08 (2013) 092 [arXiv:1306.4004] [INSPIRE].
H. Araki and E.H. Lieb, Entropy inequalities, Commun. Math. Phys. 18 (1970) 160 [INSPIRE].
M. Headrick, General properties of holographic entanglement entropy, JHEP 03 (2014) 085 [arXiv:1312.6717] [INSPIRE].
L. Zhang and J. Wu, On conjectures of classical and quantum correlations in bipartite states, J. Phys. A 45 (2012) 025301 [arXiv:1105.2993].
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Rangamani, M., Rota, M. Comments on entanglement negativity in holographic field theories. J. High Energ. Phys. 2014, 60 (2014). https://doi.org/10.1007/JHEP10(2014)060
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DOI: https://doi.org/10.1007/JHEP10(2014)060