Abstract
In this article we define a new information theoretical quantity for any bipartite mixed state ρAB. We call it the balanced partial entanglement (BPE). The BPE is the partial entanglement entropy, which is an integral of the entanglement contour in a subregion, that satisfies certain balance requirements. The BPE depends on the purification hence is not intrinsic. However, the BPE could be a useful way to classify the purifications. We discuss the entropy relations satisfied by BPE and find they are quite similar to those satisfied by the entanglement of purification. We show that in holographic CFT2 the BPE equals to the area of the entanglement wedge cross section (EWCS) divided by 4G. More interestingly, when we consider the canonical purification the BPE is just half of the reflected entropy, which also directly relate to the EWCS. The BPE can be considered as an generalization of the reflected entropy for a generic purification of the mixed state ρAB. We interpret the correspondence between the BPE and EWCS using the holographic picture of the entanglement contour.
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].
J. M. Maldacena, The Large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [Int. J. Theor. Phys. 38 (1999) 1113] [hep-th/9711200] [INSPIRE].
S. S. Gubser, I. R. Klebanov and A. M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
J. Eisert and M. B. Plenio, A Comparison of entanglement measures, J. Mod. Opt. 46 (1999) 145 [quant-ph/9807034] [INSPIRE].
G. Vidal and R. F. Werner, Computable measure of entanglement, Phys. Rev. A 65 (2002) 032314 [quant-ph/0102117] [INSPIRE].
M. B. Plenio, Logarithmic Negativity: A Full Entanglement Monotone That is not Convex, Phys. Rev. Lett. 95 (2005) 090503 [quant-ph/0505071] [INSPIRE].
B. M. Terhal, M. Horodecki, D. W. Leung and D. P. DiVincenzo, The entanglement of purification, J. Math. Phys. 43 (2002) 4286.
Y. Chen and G. Vidal, Entanglement contour, J. Stat. Mech. 2014 (2014) P10011, [arXiv:1406.1471].
Q. Wen, Fine structure in holographic entanglement and entanglement contour, Phys. Rev. D 98 (2018) 106004 [arXiv:1803.05552] [INSPIRE].
J. Kudler-Flam, I. MacCormack and S. Ryu, Holographic entanglement contour, bit threads, and the entanglement tsunami, J. Phys. A 52 (2019) 325401 [arXiv:1902.04654] [INSPIRE].
Q. Wen, Entanglement contour and modular flow from subset entanglement entropies, JHEP 05 (2020) 018 [arXiv:1902.06905] [INSPIRE].
Q. Wen, Formulas for Partial Entanglement Entropy, Phys. Rev. Res. 2 (2020) 023170 [arXiv:1910.10978] [INSPIRE].
B. Czech, J. L. Karczmarek, F. Nogueira and M. Van Raamsdonk, The Gravity Dual of a Density Matrix, Class. Quant. Grav. 29 (2012) 155009 [arXiv:1204.1330] [INSPIRE].
A. C. Wall, Maximin Surfaces, and the Strong Subadditivity of the Covariant Holographic Entanglement Entropy, Class. Quant. Grav. 31 (2014) 225007 [arXiv:1211.3494] [INSPIRE].
M. Headrick, V. E. Hubeny, A. Lawrence and M. Rangamani, Causality & holographic entanglement entropy, JHEP 12 (2014) 162 [arXiv:1408.6300] [INSPIRE].
T. Takayanagi and K. Umemoto, Entanglement of purification through holographic duality, Nature Phys. 14 (2018) 573 [arXiv:1708.09393] [INSPIRE].
P. Nguyen, T. Devakul, M. G. Halbasch, M. P. Zaletel and B. Swingle, Entanglement of purification: from spin chains to holography, JHEP 01 (2018) 098 [arXiv:1709.07424] [INSPIRE].
M. Miyaji and T. Takayanagi, Surface/State Correspondence as a Generalized Holography, PTEP 2015 (2015) 073B03 [arXiv:1503.03542] [INSPIRE].
S. Dutta and T. Faulkner, A canonical purification for the entanglement wedge cross-section, JHEP 03 (2021) 178 [arXiv:1905.00577] [INSPIRE].
J. Kudler-Flam and S. Ryu, Entanglement negativity and minimal entanglement wedge cross sections in holographic theories, Phys. Rev. D 99 (2019) 106014 [arXiv:1808.00446] [INSPIRE].
Y. Kusuki, J. Kudler-Flam and S. Ryu, Derivation of Holographic Negativity in AdS3/CFT2, Phys. Rev. Lett. 123 (2019) 131603 [arXiv:1907.07824] [INSPIRE].
K. Tamaoka, Entanglement Wedge Cross Section from the Dual Density Matrix, Phys. Rev. Lett. 122 (2019) 141601 [arXiv:1809.09109] [INSPIRE].
R. Espíndola, A. Guijosa and J. F. Pedraza, Entanglement Wedge Reconstruction and Entanglement of Purification, Eur. Phys. J. C 78 (2018) 646 [arXiv:1804.05855] [INSPIRE].
C. A. Agón, J. De Boer and J. F. Pedraza, Geometric Aspects of Holographic Bit Threads, JHEP 05 (2019) 075 [arXiv:1811.08879] [INSPIRE].
J. Harper and M. Headrick, Bit threads and holographic entanglement of purification, JHEP 08 (2019) 101 [arXiv:1906.05970] [INSPIRE].
N. Bao, A. Chatwin-Davies, J. Pollack and G. N. Remmen, Towards a Bit Threads Derivation of Holographic Entanglement of Purification, JHEP 07 (2019) 152 [arXiv:1905.04317] [INSPIRE].
Y.-Y. Lin, J.-R. Sun and Y. Sun, Bit thread, entanglement distillation, and entanglement of purification, arXiv:2012.05737 [INSPIRE].
D.-H. Du, C.-B. Chen and F.-W. Shu, Bit threads and holographic entanglement of purification, JHEP 08 (2019) 140 [arXiv:1904.06871] [INSPIRE].
K. Umemoto and Y. Zhou, Entanglement of Purification for Multipartite States and its Holographic Dual, JHEP 10 (2018) 152 [arXiv:1805.02625] [INSPIRE].
K. Babaei Velni, M. R. Mohammadi Mozaffar and M. H. Vahidinia, Some Aspects of Entanglement Wedge Cross-Section, JHEP 05 (2019) 200 [arXiv:1903.08490] [INSPIRE].
M. Ghodrati, X.-M. Kuang, B. Wang, C.-Y. Zhang and Y.-T. Zhou, The connection between holographic entanglement and complexity of purification, JHEP 09 (2019) 009 [arXiv:1902.02475] [INSPIRE].
M. Ghodrati, Entanglement Wedge Reconstruction and Correlation Measures in Mixed States, Modular Flows versus Quantum Recovery Channels, arXiv:2012.04386 [INSPIRE].
J. Kumar Basak, H. Parihar, B. Paul and G. Sengupta, Covariant holographic negativity from the entanglement wedge in AdS3/CFT2, arXiv:2102.05676 [INSPIRE].
J. Kumar Basak, V. Malvimat, H. Parihar, B. Paul and G. Sengupta, On minimal entanglement wedge cross section for holographic entanglement negativity, arXiv:2002.10272 [INSPIRE].
J. Chu, R. Qi and Y. Zhou, Generalizations of Reflected Entropy and the Holographic Dual, JHEP 03 (2020) 151 [arXiv:1909.10456] [INSPIRE].
S. Khoeini-Moghaddam, F. Omidi and C. Paul, Aspects of Hyperscaling Violating Geometries at Finite Cutoff, JHEP 02 (2021) 121 [arXiv:2011.00305] [INSPIRE].
C. Akers and P. Rath, Entanglement Wedge Cross Sections Require Tripartite Entanglement, JHEP 04 (2020) 208 [arXiv:1911.07852] [INSPIRE].
G. Vidal, J. I. Latorre, E. Rico and A. Kitaev, Entanglement in quantum critical phenomena, Phys. Rev. Lett. 90 (2003) 227902 [quant-ph/0211074] [INSPIRE].
A. Botero and B. Reznik, Spatial structures and localization of vacuum entanglement in the linear harmonic chain, Phys. Rev. A 70 (2004) 052329.
I. Frérot and T. Roscilde, Area law and its violation: A microscopic inspection into the structure of entanglement and fluctuations, Phys. Rev. B 92 (2015) 115129.
A. Coser, C. De Nobili and E. Tonni, A contour for the entanglement entropies in harmonic lattices, J. Phys. A 50 (2017) 314001 [arXiv:1701.08427] [INSPIRE].
E. Tonni, J. Rodríguez-Laguna and G. Sierra, Entanglement hamiltonian and entanglement contour in inhomogeneous 1D critical systems, J. Stat. Mech. 1804 (2018) 043105 [arXiv:1712.03557] [INSPIRE].
V. Alba, S. N. Santalla, P. Ruggiero, J. Rodriguez-Laguna, P. Calabrese and G. Sierra, Unusual area-law violation in random inhomogeneous systems, J. Stat. Mech. 1902 (2019) 023105 [arXiv:1807.04179] [INSPIRE].
Q. Wen, Towards the generalized gravitational entropy for spacetimes with non-Lorentz invariant duals, JHEP 01 (2019) 220 [arXiv:1810.11756] [INSPIRE].
G. Di Giulio, R. Arias and E. Tonni, Entanglement hamiltonians in 1D free lattice models after a global quantum quench, J. Stat. Mech. 1912 (2019) 123103 [arXiv:1905.01144] [INSPIRE].
M. Han and Q. Wen, Entanglement entropies from entanglement contour: annuli and spherical shells, arXiv:1905.05522 [INSPIRE].
D. S. Ageev, On the entanglement and complexity contours of excited states in the holographic CFT, arXiv:1905.06920 [INSPIRE].
J. Kudler-Flam, H. Shapourian and S. Ryu, The negativity contour: a quasi-local measure of entanglement for mixed states, SciPost Phys. 8 (2020) 063 [arXiv:1908.07540] [INSPIRE].
S. Singha Roy, S. N. Santalla, J. Rodríguez-Laguna and G. Sierra, Entanglement as geometry and flow, Phys. Rev. B 101 (2020) 195134 [arXiv:1906.05146] [INSPIRE].
N. Samos Sáenz de Buruaga, S. N. Santalla, J. Rodríguez-Laguna and G. Sierra, Piercing the rainbow state: Entanglement on an inhomogeneous spin chain with a defect, Phys. Rev. B 101 (2020) 205121 [arXiv:1912.10788] [INSPIRE].
I. MacCormack, M. T. Tan, J. Kudler-Flam and S. Ryu, Operator and entanglement growth in non-thermalizing systems: many-body localization and the random singlet phase, arXiv:2001.08222 [INSPIRE].
H. Casini and M. Huerta, Remarks on the entanglement entropy for disconnected regions, JHEP 03 (2009) 048 [arXiv:0812.1773] [INSPIRE].
R. Abt, J. Erdmenger, M. Gerbershagen, C. M. Melby-Thompson and C. Northe, Holographic Subregion Complexity from Kinematic Space, JHEP 01 (2019) 012 [arXiv:1805.10298] [INSPIRE].
C. Berthiere and W. Witczak-Krempa, Relating bulk to boundary entanglement, Phys. Rev. B 100 (2019) 235112 [arXiv:1907.11249] [INSPIRE].
H. Casini, C. D. Fosco and M. Huerta, Entanglement and alpha entropies for a massive Dirac field in two dimensions, J. Stat. Mech. 0507 (2005) P07007 [cond-mat/0505563] [INSPIRE].
H. Casini and M. Huerta, A Finite entanglement entropy and the c-theorem, Phys. Lett. B 600 (2004) 142 [hep-th/0405111] [INSPIRE].
P. Calabrese and J. L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. 0406 (2004) P06002 [hep-th/0405152] [INSPIRE].
V. E. Hubeny and M. Rangamani, Holographic entanglement entropy for disconnected regions, JHEP 03 (2008) 006 [arXiv:0711.4118] [INSPIRE].
W.-Z. Guo, Entanglement of purification and projection operator in conformal field theories, Phys. Lett. B 797 (2019) 134934 [arXiv:1901.00330] [INSPIRE].
P. Hayden, M. Headrick and A. Maloney, Holographic Mutual Information is Monogamous, Phys. Rev. D 87 (2013) 046003 [arXiv:1107.2940] [INSPIRE].
J. M. Maldacena, Eternal black holes in anti-de Sitter, JHEP 04 (2003) 021 [hep-th/0106112] [INSPIRE].
N. Engelhardt and A. C. Wall, Coarse Graining Holographic Black Holes, JHEP 05 (2019) 160 [arXiv:1806.01281] [INSPIRE].
N. Engelhardt and A. C. Wall, Decoding the Apparent Horizon: Coarse-Grained Holographic Entropy, Phys. Rev. Lett. 121 (2018) 211301 [arXiv:1706.02038] [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: 2103.00415
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
Wen, Q. Balanced partial entanglement and the entanglement wedge cross section. J. High Energ. Phys. 2021, 301 (2021). https://doi.org/10.1007/JHEP04(2021)301
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP04(2021)301