Abstract
Entanglement is a physical phenomenon that each state cannot be described individually. Entanglement entropy gives quantitative understanding to the entanglement. We use decomposition of the Hilbert space to discuss properties of the entanglement. Therefore, partial trace operator becomes important to define the reduced density matrix from different centers, which commutes with all elements in the Hilbert space, corresponding to different entanglement choices or different observations on entangling surface. Entanglement entropy is expected to satisfy the strong subadditivity. We discuss decomposition of the Hilbert space for the strong subadditivity and other related inequalities. The entanglement entropy with centers can be computed from the Hamitonian formulations systematically, provided that we know wavefunctional. In the Hamitonian formulation, it is easier to obtain symmetry structure. We consider massless p-form theory as an example. The massless p-form theory in (2p + 2)-dimensions has global symmetry, similar to the electric-magnetic duality, connecting centers in ground state. This defines a duality structure in centers. Because it is hard to exactly compute the entanglement entropy from partial trace operator, we propose the Lagrangian formulation from the Hamitonian formulation to compute the entanglement entropy with centers. From the Lagrangian method and saddle point approximation, the codimension two surface term (leading order) in the Einstein gravity theory or holographic entanglement entropy should correspond to non-tensor product decomposition (center is not identity). Finally, we compute the entanglement entropy of the SU(N) Yang-Mills lattice gauge theory in the fundamental representation using the strong coupling expansion in the extended lattice model to obtain spatial area term in total dimensions larger than two for N > 1.
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References
P.-M. Ho, C.-T. Ma and C.-H. Yeh, BPS States on M5-brane in Large C-field Background, JHEP 08 (2012) 076 [arXiv:1206.1467] [INSPIRE].
P.-M. Ho and C.-T. Ma, Effective Action for Dp-brane in Large RR (p − 1)-Form Background, JHEP 05 (2013) 056 [arXiv:1302.6919] [INSPIRE].
C.-T. Ma and C.-H. Yeh, Supersymmetry and BPS States on D4-brane in Large C-field Background, JHEP 03 (2013) 131 [arXiv:1210.4191] [INSPIRE].
J.-K. Ho and C.-T. Ma, Dimensional Reduction of the Generalized DBI, Nucl. Phys. B 897 (2015) 479 [arXiv:1410.0972] [INSPIRE].
P.-M. Ho and Y. Matsuo, M5 from M2, JHEP 06 (2008) 105 [arXiv:0804.3629] [INSPIRE].
B. Zwiebach, Curvature Squared Terms and String Theories, Phys. Lett. B 156 (1985) 315 [INSPIRE].
B. Zwiebach, Closed string field theory: Quantum action and the B-V master equation, Nucl. Phys. B 390 (1993) 33 [hep-th/9206084] [INSPIRE].
M. Saadi and B. Zwiebach, Closed String Field Theory from Polyhedra, Annals Phys. 192 (1989) 213.
C. Hull and B. Zwiebach, Double Field Theory, JHEP 09 (2009) 099 [arXiv:0904.4664] [INSPIRE].
O. Hohm, C. Hull and B. Zwiebach, Background independent action for double field theory, JHEP 07 (2010) 016 [arXiv:1003.5027] [INSPIRE].
W. Siegel, Two vierbein formalism for string inspired axionic gravity, Phys. Rev. D 47 (1993) 5453 [hep-th/9302036] [INSPIRE].
W. Siegel, Superspace duality in low-energy superstrings, Phys. Rev. D 48 (1993) 2826 [hep-th/9305073] [INSPIRE].
C. Hull and B. Zwiebach, The Gauge algebra of double field theory and Courant brackets, JHEP 09 (2009) 090 [arXiv:0908.1792] [INSPIRE].
O. Hohm, C. Hull and B. Zwiebach, Generalized metric formulation of double field theory, JHEP 08 (2010) 008 [arXiv:1006.4823] [INSPIRE].
C.-T. Ma and C.-M. Shen, Cosmological Implications from O(D, D), Fortsch. Phys. 62 (2014) 921 [arXiv:1405.4073] [INSPIRE].
C.-T. Ma, One-Loop β Function of the Double σ-model with Constant Background, JHEP 04 (2015) 026 [arXiv:1412.1919] [INSPIRE].
C.-T. Ma, Gauge Transformation of Double Field Theory for Open String, Phys. Rev. D 92 (2015) 066004 [arXiv:1411.0287] [INSPIRE].
M.J. Duff, Duality Rotations in String Theory, Nucl. Phys. B 335 (1990) 610 [INSPIRE].
A.A. Tseytlin, Duality symmetric closed string theory and interacting chiral scalars, Nucl. Phys. B 350 (1991) 395 [INSPIRE].
A.A. Tseytlin, Duality Symmetric Formulation of String World Sheet Dynamics, Phys. Lett. B 242 (1990) 163 [INSPIRE].
W. Siegel, Manifest Lorentz Invariance Sometimes Requires Nonlinearity, Nucl. Phys. B 238 (1984) 307 [INSPIRE].
C.-T. Ma, Boundary Conditions and the Generalized Metric Formulation of the Double σ-model, Nucl. Phys. B 898 (2015) 30 [arXiv:1502.02378] [INSPIRE].
M. Gualtieri, Generalized complex geometry, math/0401221.
N. Hitchin, Generalized Calabi-Yau manifolds, Quart. J. Math. 54 (2003) 281 [math/0209099] [INSPIRE].
P.-M. Ho and C.-T. Ma, S-duality for D3-Brane in NS-NS and R-R Backgrounds, JHEP 11 (2014) 142 [arXiv:1311.3393] [INSPIRE].
J.-K. Ho and C.-T. Ma, Electric-Magnetic Dualities in Gauge Theories, arXiv:1507.05378 [INSPIRE].
S. Deser and C. Teitelboim, Duality Transformations of Abelian and Nonabelian Gauge Fields, Phys. Rev. D 13 (1976) 1592 [INSPIRE].
D.S. Berman and M.J. Perry, Generalized Geometry and M-theory, JHEP 06 (2011) 074 [arXiv:1008.1763] [INSPIRE].
H. Casini, M. Huerta and J.A. Rosabal, Remarks on entanglement entropy for gauge fields, Phys. Rev. D 89 (2014) 085012 [arXiv:1312.1183] [INSPIRE].
H. Casini and M. Huerta, Entanglement entropy for a Maxwell field: Numerical calculation on a two dimensional lattice, Phys. Rev. D 90 (2014) 105013 [arXiv:1406.2991] [INSPIRE].
J.M. Maldacena, The Large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].
H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement entropy, JHEP 05 (2011) 036 [arXiv:1102.0440] [INSPIRE].
J. Maldacena and L. Susskind, Cool horizons for entangled black holes, Fortsch. Phys. 61 (2013) 781 [arXiv:1306.0533] [INSPIRE].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
W. Donnelly, Decomposition of entanglement entropy in lattice gauge theory, Phys. Rev. D 85 (2012) 085004 [arXiv:1109.0036] [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].
R.M. Soni and S.P. Trivedi, Aspects of Entanglement Entropy for Gauge Theories, arXiv:1510.07455 [INSPIRE].
J. Dixmier, Von Neumann algebras, North Holland Publishing Company (1981).
H. Araki and E.H. Lieb, Entropy inequalities, Commun. Math. Phys. 18 (1970) 160 [INSPIRE].
E.H. Lieb and M.B. Ruskai, Proof of the strong subadditivity of quantum-mechanical entropy, J. Math. Phys. 14 (1973) 1938 [INSPIRE].
H. Casini and M. Huerta, Entanglement entropy in free quantum field theory, J. Phys. A 42 (2009) 504007 [arXiv:0905.2562] [INSPIRE].
D. Radicevic, Notes on Entanglement in Abelian Gauge Theories, arXiv:1404.1391 [INSPIRE].
W. Donnelly and A.C. Wall, Geometric entropy and edge modes of the electromagnetic field, arXiv:1506.05792 [INSPIRE].
W. Donnelly and A.C. Wall, Entanglement entropy of electromagnetic edge modes, Phys. Rev. Lett. 114 (2015) 111603 [arXiv:1412.1895] [INSPIRE].
A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, JHEP 08 (2013) 090 [arXiv:1304.4926] [INSPIRE].
D.V. Fursaev and S.N. Solodukhin, On the description of the Riemannian geometry in the presence of conical defects, Phys. Rev. D 52 (1995) 2133 [hep-th/9501127] [INSPIRE].
P.V. Buividovich and M.I. Polikarpov, Entanglement entropy in gauge theories and the holographic principle for electric strings, Phys. Lett. B 670 (2008) 141 [arXiv:0806.3376] [INSPIRE].
D. Radicevic, Entanglement in Weakly Coupled Lattice Gauge Theories, arXiv:1509.08478 [INSPIRE].
J.-W. Chen, S.-H. Dai and J.-Y. Pang, Strong Coupling Expansion of the Entanglement Entropy of Yang-Mills Gauge Theories, arXiv:1503.01766 [INSPIRE].
P.V. Buividovich and M.I. Polikarpov, Numerical study of entanglement entropy in SU(2) lattice gauge theory, Nucl. Phys. B 802 (2008) 458 [arXiv:0802.4247] [INSPIRE].
Y. Nakagawa, A. Nakamura, S. Motoki and V.I. Zakharov, Entanglement entropy of SU(3) and SU(2) Yang-Mills theories at finite temperature, PoS(LAT2009)188 [arXiv:0911.2596] [INSPIRE].
Y. Nakagawa, A. Nakamura, S. Motoki and V.I. Zakharov, Quantum entanglement in SU(3) lattice Yang-Mills theory at zero and finite temperatures, PoS(Lattice 2010)281 [arXiv:1104.1011] [INSPIRE].
A. Kitaev and J. Preskill, Topological entanglement entropy, Phys. Rev. Lett. 96 (2006) 110404 [hep-th/0510092] [INSPIRE].
M. Levin and X.-G. Wen, Detecting Topological Order in a Ground State Wave Function, Phys. Rev. Lett. 96 (2006) 110405 [INSPIRE].
M. Pretko and T. Senthil, Entanglement Entropy of U(1) Quantum Spin Liquids, arXiv:1510.03863 [INSPIRE].
W. Donnelly, Entanglement entropy in loop quantum gravity, Phys. Rev. D 77 (2008) 104006 [arXiv:0802.0880] [INSPIRE].
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Ma, CT. Entanglement with centers. J. High Energ. Phys. 2016, 70 (2016). https://doi.org/10.1007/JHEP01(2016)070
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DOI: https://doi.org/10.1007/JHEP01(2016)070