Abstract
As a generalization of entanglement entropy, pseudo entropy is not always real. The real-valued pseudo entropy has promising applications in holography and quantum phase transition. We apply the notion of pseudo-Hermiticity to formulate the reality condition of pseudo entropy. We find the general form of the transition matrix for which the eigenvalues of the reduced transition matrix possess real or complex pairs of eigenvalues. Further, we find a class of transition matrices for which the pseudo (Rényi) entropies are non-negative. Some known examples which give real pseudo entropy in quantum field theories can be explained in our framework. Our results offer a novel method to generate the transition matrix with real pseudo entropy. Finally, we show the reality condition for pseudo entropy is related to the Tomita-Takesaki modular theory for quantum field theory.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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. 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 [cond-mat/0510613] [INSPIRE].
M. Srednicki, Entropy and area, Phys. Rev. Lett. 71 (1993) 666 [hep-th/9303048] [INSPIRE].
J. Eisert, M. Cramer and M.B. Plenio, Area laws for the entanglement entropy — a review, Rev. Mod. Phys. 82 (2010) 277 [arXiv:0808.3773] [INSPIRE].
P. Calabrese and J.L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. 0406 (2004) P06002 [hep-th/0405152] [INSPIRE].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
V.E. Hubeny, M. Rangamani and T. Takayanagi, A covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].
J. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200].
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].
M. Van Raamsdonk, Building up spacetime with quantum entanglement, Gen. Rel. Grav. 42 (2010) 2323 [arXiv:1005.3035] [INSPIRE].
M. Rangamani and T. Takayanagi, Holographic entanglement entropy, arXiv:1609.01287 [https://doi.org/10.1007/978-3-319-52573-0] [INSPIRE].
A. Almheiri, X. Dong and D. Harlow, Bulk locality and quantum error correction in AdS/CFT, JHEP 04 (2015) 163 [arXiv:1411.7041] [INSPIRE].
X. Dong, D. Harlow and A.C. Wall, Reconstruction of bulk operators within the entanglement wedge in gauge-gravity duality, Phys. Rev. Lett. 117 (2016) 021601 [arXiv:1601.05416] [INSPIRE].
G. Penington, Entanglement wedge reconstruction and the information paradox, JHEP 09 (2020) 002 [arXiv:1905.08255] [INSPIRE].
A. Almheiri, N. Engelhardt, D. Marolf and H. Maxfield, The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole, JHEP 12 (2019) 063 [arXiv:1905.08762] [INSPIRE].
Y. Nakata et al., New holographic generalization of entanglement entropy, Phys. Rev. D 103 (2021) 026005 [arXiv:2005.13801] [INSPIRE].
A. Mollabashi et al., Pseudo entropy in free quantum field theories, Phys. Rev. Lett. 126 (2021) 081601 [arXiv:2011.09648] [INSPIRE].
G. Camilo and A. Prudenziati, Twist operators and pseudo entropies in two-dimensional momentum space, arXiv:2101.02093 [https://doi.org/10.48550/arXiv.2101.02093].
A. Mollabashi et al., Aspects of pseudoentropy in field theories, Phys. Rev. Res. 3 (2021) 033254 [arXiv:2106.03118] [INSPIRE].
T. Nishioka, T. Takayanagi and Y. Taki, Topological pseudo entropy, JHEP 09 (2021) 015 [arXiv:2107.01797] [INSPIRE].
K. Goto, M. Nozaki and K. Tamaoka, Subregion spectrum form factor via pseudoentropy, Phys. Rev. D 104 (2021) L121902 [arXiv:2109.00372] [INSPIRE].
M. Miyaji, Island for gravitationally prepared state and pseudo entanglement wedge, JHEP 12 (2021) 013 [arXiv:2109.03830] [INSPIRE].
I. Akal et al., Page curve under final state projection, Phys. Rev. D 105 (2022) 126026 [arXiv:2112.08433] [INSPIRE].
J. Mukherjee, Pseudo entropy in U(1) gauge theory, JHEP 10 (2022) 016 [arXiv:2205.08179] [INSPIRE].
W.-Z. Guo, S. He and Y.-X. Zhang, On the real-time evolution of pseudo-entropy in 2d CFTs, JHEP 09 (2022) 094 [arXiv:2206.11818] [INSPIRE].
Y. Ishiyama, R. Kojima, S. Matsui and K. Tamaoka, Notes on pseudo entropy amplification, PTEP 2022 (2022) 093B10 [arXiv:2206.14551] [INSPIRE].
A. Bhattacharya, A. Bhattacharyya and S. Maulik, Pseudocomplexity of purification for free scalar field theories, Phys. Rev. D 106 (2022) 086010 [arXiv:2209.00049] [INSPIRE].
K. Doi et al., Pseudoentropy in dS/CFT and timelike entanglement entropy, Phys. Rev. Lett. 130 (2023) 031601 [arXiv:2210.09457].
Y. Aharonov, P.G. Bergmann and J.L. Lebowitz, Time symmetry in the quantum process of measurement, Phys. Rev. 134 (1964) B1410.
Y. Aharonov, D.Z. Albert and L. Vaidman, How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100, Phys. Rev. Lett. 60 (1988) 1351 [INSPIRE].
J. Dressel et al., Colloquium. Understanding quantum weak values: basics and applications, Rev. Mod. Phys. 86 (2014) 307 [arXiv:1305.7154].
Y. Aharonov and L. Vaidman, Complete description of a quantum system at a given time, J. Phys. A 24 (1991) 2315.
Y. Aharonov and L. Vaidman, The two-state vector formalism: an updated review, Lect. Notes Phys. 734 (2007) 399.
Y. Ashida, Z. Gong and M. Ueda, Non-Hermitian physics, Adv. Phys. 69 (2021) 249 [arXiv:2006.01837] [INSPIRE].
M. Bertolotti, PT symmetry in quantum and classical physics, Contemp. Phys. 60 (2019) 196.
C.M. Bender and S. Boettcher, Real spectra in non-Hermitian Hamiltonians having PT symmetry, Phys. Rev. Lett. 80 (1998) 5243 [physics/9712001] [INSPIRE].
C.M. Bender, Making sense of non-Hermitian Hamiltonians, Rept. Prog. Phys. 70 (2007) 947 [hep-th/0703096] [INSPIRE].
A. Mostafazadeh, Pseudo-Hermiticity versus PT symmetry. The necessary condition for the reality of the spectrum, J. Math. Phys. 43 (2002) 205 [math-ph/0107001] [INSPIRE].
A. Mostafazadeh, Pseudo-Hermiticity versus PT symmetry 2. A complete characterization of non-Hermitian Hamiltonians with a real spectrum, J. Math. Phys. 43 (2002) 2814 [math-ph/0110016] [INSPIRE].
A. Mostafazadeh, Pseudo-Hermitian representation of quantum mechanics, Int. J. Geom. Meth. Mod. Phys. 7 (2010) 1191 [arXiv:0810.5643] [INSPIRE].
C.M. Bender, S.F. Brandt, J.-H. Chen and Q.-H. Wang, Ghost busting: PT-symmetric interpretation of the Lee model, Phys. Rev. D 71 (2005) 025014 [hep-th/0411064] [INSPIRE].
C.M. Bender, N. Hassanpour, S.P. Klevansky and S. Sarkar, PT-symmetric quantum field theory in D dimensions, Phys. Rev. D 98 (2018) 125003 [arXiv:1810.12479] [INSPIRE].
Y. Ashida, S. Furukawa and M. Ueda, Parity-time-symmetric quantum critical phenomena, Nature Commun. 8 (2017) 15791.
C.M. Bender, A. Felski, S.P. Klevansky and S. Sarkar, PT symmetry and renormalisation in quantum field theory, J. Phys. Conf. Ser. 2038 (2021) 012004 [arXiv:2103.14864] [INSPIRE].
H. Reeh and S. Schlieder, Bemerkungen zur Unitaäräquivalenz von Lorentzinvarienten Feldern (in German), Nuovo Cim. 20 (1961) 1051.
E. Witten, APS medal for exceptional achievement in research: invited article on entanglement properties of quantum field theory, Rev. Mod. Phys. 90 (2018) 045003 [arXiv:1803.04993] [INSPIRE].
R. Haag, Local quantum physics: fields, particles, algebras, second edition, Springer, Berlin, Heidelberg, Germany (1996).
J.J. Bisognano and E.H. Wichmann, On the duality condition for quantum fields, J. Math. Phys. 17 (1976) 303 [INSPIRE].
R. Blumenhagen and E. Plauschinn, Introduction to conformal field theory: with applications to string theory, Springer (2009) [INSPIRE].
K. Osterwalder and R. Schrader, Axioms for Euclidean Green’s functions, Commun. Math. Phys. 31 (1973) 83 [INSPIRE].
K. Osterwalder and R. Schrader, Axioms for Euclidean Green’s functions. 2, Commun. Math. Phys. 42 (1975) 281 [INSPIRE].
A. Mostafazadeh, Pseudo-Hermiticity for a class of nondiagonalizable Hamiltonians, J. Math. Phys. 43 (2002) 6343 [Erratum ibid. 44 (2003) 943] [math-ph/0207009] [INSPIRE].
P. Calabrese and J.L. Cardy, Evolution of entanglement entropy in one-dimensional systems, J. Stat. Mech. 0504 (2005) P04010 [cond-mat/0503393] [INSPIRE].
Author information
Authors and Affiliations
Corresponding authors
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2209.07308
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
Guo, Wz., He, S. & Zhang, YX. Constructible reality condition of pseudo entropy via pseudo-Hermiticity. J. High Energ. Phys. 2023, 21 (2023). https://doi.org/10.1007/JHEP05(2023)021
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2023)021