Entanglement entropy of disjoint spacetime intervals in causal set theory. (English) Zbl 1487.83012
Summary: A more complete understanding of entanglement entropy in a covariant manner could inform the search for quantum gravity. We build on work in this direction by extending previous results to disjoint regions in 1 + 1D. We investigate the entanglement entropy of a scalar field in disjoint intervals within the causal set framework, using the spacetime commutator and correlator, \(i\boldsymbol{\Delta}\) and \(\mathbf{W}\) (or the Pauli-Jordan and Wightman functions). A new truncation scheme for disjoint causal diamonds is presented, which follows from the single diamond truncation scheme. We investigate setups including two and three disjoint causal diamonds, as well as a single causal diamond that shares a boundary with a larger global causal diamond. In all the cases that we study, our results agree with the expected area laws. In addition, we study the mutual information in the two disjoint diamond setup. The ease of our calculations indicate our methods to be a useful tool for numerically studying such systems. We end with a discussion of some of the strengths and future applications of the spacetime formulation we use in our entanglement entropy computations, both in causal set theory and in the continuum.
MSC:
| 83C27 | Lattice gravity, Regge calculus and other discrete methods in general relativity and gravitational theory |
| 62D20 | Causal inference from observational studies |
| 81P42 | Entanglement measures, concurrencies, separability criteria |
| 62H20 | Measures of association (correlation, canonical correlation, etc.) |
| 16S15 | Finite generation, finite presentability, normal forms (diamond lemma, term-rewriting) |
| 81P17 | Quantum entropies |
References:
| [1] | Sorkin, R. D.; Bertotti, B.; de Felice, F.; Pascolini, A., On the entropy of the vacuum outside a horizon, General Relativity and Gravitation, vol 1, p 734 (1983), Rome: Consiglio Nazionale delle Ricerche, Rome |
| [2] | Bekenstein, J. D., Black holes and entropy, Phys. Rev. D, 7, 2333-2346 (1973) · Zbl 1369.83037 · doi:10.1103/physrevd.7.2333 |
| [3] | Mathur, S. D., The fuzzball proposal for black holes: an elementary review, Fortschr. Phys., 53, 793-827 (2005) · Zbl 1116.83300 · doi:10.1002/prop.200410203 |
| [4] | Horowitz, G. T., The origin of black hole entropy in string theory, Astrophys. Space Sci. Libr., 211, 46 (1997) · doi:10.1007/bf00644178 |
| [5] | Rovelli, C., Black hole entropy from loop quantum gravity, Phys. Rev. Lett., 77, 3288-3291 (1996) · Zbl 0955.83506 · doi:10.1103/physrevlett.77.3288 |
| [6] | Carlip, S., A note on black hole entropy in loop quantum gravity, Class. Quantum Grav., 32 (2015) · Zbl 1327.83142 · doi:10.1088/0264-9381/32/15/155009 |
| [7] | Dou, D.; Sorkin, R. D., Black hole entropy as causal links, Found. Phys., 33, 279-296 (2003) · doi:10.1023/a:1023781022519 |
| [8] | Bombelli, L.; Koul, R. K.; Lee, J.; Sorkin, R. D., Quantum source of entropy for black holes, Phys. Rev. D, 34, 373-383 (1986) · Zbl 1222.83077 · doi:10.1103/physrevd.34.373 |
| [9] | Jacobson, T.; Satz, A., Black hole entanglement entropy and the renormalization group, Phys. Rev. D, 87 (2013) · doi:10.1103/physrevd.87.084047 |
| [10] | Solodukhin, S. N., Entanglement entropy of black holes, Living Rev. Relativ., 14, 8 (2011) · Zbl 1320.83015 · doi:10.12942/lrr-2011-8 |
| [11] | Emparan, R., Black hole entropy as entanglement entropy: a holographic derivation, J. High Energy Phys. (2006) · doi:10.1088/1126-6708/2006/06/012 |
| [12] | Casini, H.; Huerta, M., A finite entanglement entropy and the c-theorem, Phys. Lett. B, 600, 142-150 (2004) · Zbl 1247.81021 · doi:10.1016/j.physletb.2004.08.072 |
| [13] | Calabrese, P.; Cardy, J., Entanglement entropy and quantum field theory, J. Stat. Mech., 2004 (2004) · Zbl 1082.82002 · doi:10.1088/1742-5468/2004/06/p06002 |
| [14] | Myers, R. C.; Sinha, A., Seeing a c-theorem with holography, Phys. Rev. D, 82 (2010) · doi:10.1103/physrevd.82.046006 |
| [15] | Ryu, S.; Takayanagi, T., Aspects of holographic entanglement entropy, J. High Energy Phys. (2006) · doi:10.1088/1126-6708/2006/08/045 |
| [16] | Ryu, S.; Takayanagi, T., Holographic derivation of entanglement entropy from the anti-de sitter space/conformal field theory correspondence, Phys. Rev. Lett., 96 (2006) · Zbl 1228.83110 · doi:10.1103/physrevlett.96.181602 |
| [17] | Kitaev, A.; Preskill, J., Topological entanglement entropy, Phys. Rev. Lett., 96 (2006) · doi:10.1103/physrevlett.96.110404 |
| [18] | Levin, M.; Wen, X-G, Detecting topological order in a ground state wave function, Phys. Rev. Lett., 96 (2006) · doi:10.1103/physrevlett.96.110405 |
| [19] | Swingle, B., Entanglement entropy and the Fermi surface, Phys. Rev. Lett., 105 (2010) · doi:10.1103/physrevlett.105.050502 |
| [20] | Vidal, G.; Werner, R. F., Computable measure of entanglement, Phys. Rev. A, 65 (2002) · doi:10.1103/physreva.65.032314 |
| [21] | Callan, C.; Wilczek, F., On geometric entropy, Phys. Lett. B, 333, 55-61 (1994) · doi:10.1016/0370-2693(94)91007-3 |
| [22] | Hertzberg, M. P., Entanglement entropy in scalar field theory, J. Phys. A: Math. Theor., 46 (2012) · Zbl 1260.81155 · doi:10.1088/1751-8113/46/1/015402 |
| [23] | Rosenhaus, V.; Smolkin, M., Entanglement entropy: a perturbative calculation, J. High Energy Phys. (2014) · doi:10.1007/jhep12(2014)179 |
| [24] | Rosenhaus, V.; Smolkin, M., Entanglement entropy for relevant and geometric perturbations, J. High Energy Phys. (2015) · Zbl 1388.81424 · doi:10.1007/jhep02(2015)015 |
| [25] | Peschel, I., Calculation of reduced density matrices from correlation functions, J. Phys. A: Math. Gen., 36, L205-L208 (2003) · Zbl 1049.82011 · doi:10.1088/0305-4470/36/14/101 |
| [26] | Sorkin, R. D., Expressing entropy globally in terms of (4D) field-correlations, J. Phys.: Conf. Ser., 484 (2014) · doi:10.1088/1742-6596/484/1/012004 |
| [27] | Chen, Y.; Hackl, L.; Kunjwal, R.; Moradi, H.; Yazdi, Y. K.; Zilhão, M., Towards spacetime entanglement entropy for interacting theories, J. High Energy Phys. (2020) · Zbl 1456.83017 · doi:10.1007/jhep11(2020)114 |
| [28] | Afshordi, N.; Buck, M.; Dowker, F.; Rideout, D.; Sorkin, R. D.; Yazdi, Y. K., A ground state for the causal diamond in two dimensions, J. High Energy Phys. (2012) · doi:10.1007/jhep10(2012)088 |
| [29] | Saravani, M.; Sorkin, R. D.; Yazdi, Y. K., Spacetime entanglement entropy in 1 + 1 dimensions, Class. Quantum Grav., 31 (2014) · Zbl 1304.81128 · doi:10.1088/0264-9381/31/21/214006 |
| [30] | Sorkin, R. D.; Yazdi, Y. K., Entanglement entropy in causal set theory, Class. Quantum Grav., 35 (2018) · Zbl 1390.83072 · doi:10.1088/1361-6382/aab06f |
| [31] | Belenchia, A.; Benincasa, D. M T.; Letizia, M.; Liberati, S., On the entanglement entropy of quantum fields in causal sets, Class. Quantum Grav., 35 (2018) · Zbl 1388.83062 · doi:10.1088/1361-6382/aaae27 |
| [32] | Surya, S.; Yazdi, Y. K., Entanglement entropy of causal set de sitter horizons, Class. Quantum Grav., 38 (2021) · Zbl 1480.83098 · doi:10.1088/1361-6382/abf279 |
| [33] | Mathur, A.; Surya, S., A spacetime calculation of the Calabrese-Cardy entanglement entropy, Phys. Lett. B, 820 (2021) · Zbl 07414559 · doi:10.1016/j.physletb.2021.136567 |
| [34] | Johnston, S., Particle propagators on discrete spacetime, Class. Quantum Grav., 25 (2008) · Zbl 1152.83364 · doi:10.1088/0264-9381/25/20/202001 |
| [35] | Johnston, S. P., Quantum fields on causal sets, PhD Thesis (2010) |
| [36] | Afshordi, N.; Aslanbeigi, S.; Sorkin, R. D., A distinguished vacuum state for a quantum field in a curved spacetime: formalism, features, and cosmology, J. High Energy Phys. (2012) · Zbl 1397.81188 · doi:10.1007/jhep08(2012)137 |
| [37] | Sorkin, R. D., From green function to quantum field, Int. J. Geom. Methods Mod. Phys., 14, 1740007 (2017) · Zbl 1373.81269 · doi:10.1142/s0219887817400072 |
| [38] | Bombelli, L.; Lee, J.; Meyer, D.; Sorkin, R. D., Space-time as a causal set, Phys. Rev. Lett., 59, 521-524 (1987) · doi:10.1103/physrevlett.59.521 |
| [39] | Surya, S., The causal set approach to quantum gravity, Living Rev. Relativ., 22, 5 (2019) · Zbl 1442.83029 · doi:10.1007/s41114-019-0023-1 |
| [40] | Yazdi, Y. K., Entanglement entropy of scalar fields in causal set theory, PhD Thesis (2017) |
| [41] | Chandran, A.; Laumann, C.; Sorkin, R., When is an area law not an area law?, Entropy, 18, 240 (2016) · doi:10.3390/e18070240 |
| [42] | Arias, R. E.; Casini, H.; Huerta, M.; Pontello, D., Entropy and modular Hamiltonian for a free chiral scalar in two intervals, Phys. Rev. D, 98 (2018) · doi:10.1103/physrevd.98.125008 |
| [43] | Alba, V.; Tagliacozzo, L.; Calabrese, P., Entanglement entropy of two disjoint blocks in critical Ising models, Phys. Rev. B, 81 (2010) · doi:10.1103/physrevb.81.060411 |
| [44] | Calabrese, P.; Cardy, J.; Tonni, E., Entanglement entropy of two disjoint intervals in conformal field theory II, J. Stat. Mech. (2011) · Zbl 1456.81361 · doi:10.1088/1742-5468/2011/01/p01021 |
| [45] | Ruggiero, P.; Tonni, E.; Calabrese, P., Entanglement entropy of two disjoint intervals and the recursion formula for conformal blocks, J. Stat. Mech. (2018) · Zbl 1456.81090 · doi:10.1088/1742-5468/aae5a8 |
| [46] | Jeong, H. S.; Kim, K. Y.; Nishida, M., Entanglement and Rényi entropy of multiple intervals in \(####\)-deformed CFT and holography, Phys. Rev. D, 100 (2019) · doi:10.1103/physrevd.100.106015 |
| [47] | Hartman, T., Entanglement entropy at large central charge (2013) |
| [48] | Hollands, S.; Sanders, K., Entanglement Measures and Their Properties in Quantum Field Theory, vol 34 (2018), Berlin: Springer, Berlin · Zbl 1408.81005 |
| [49] | Witten, E., APS medal for exceptional achievement in research: invited article on entanglement properties of quantum field theory, Rev. Mod. Phys., 90 (2018) · doi:10.1103/revmodphys.90.045003 |
| [50] | Swingle, B., Mutual information and the structure of entanglement in quantum field theory (2010) |
| [51] | Witten, E., A mini-introduction to information theory, La Rivista del Nuovo Cimento, 43, 187-227 (2020) · doi:10.1007/s40766-020-00004-5 |
| [52] | Furukawa, S.; Pasquier, V.; Shiraishi, J., Mutual information and boson radius in a c = 1 critical system in one dimension, Phys. Rev. Lett., 102 (2009) · doi:10.1103/physrevlett.102.170602 |
| [53] | Hamma, A.; Ionicioiu, R.; Zanardi, P., Bipartite entanglement and entropic boundary law in lattice spin systems, Phys. Rev. A, 71 (2005) · doi:10.1103/physreva.71.022315 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
