Abstract
The entanglement entropy of a free scalar field in its ground state is dominated by an area law term. It is noteworthy, however, that the study of entanglement in scalar field theory has not advanced far beyond the ground state. In this paper, we extend the study of entanglement of harmonic systems, which include free scalar field theory as a continuum limit, to the case of the most general Gaussian states, namely the squeezed states. We find the eigenstates and the spectrum of the reduced density matrix and we calculate the entanglement entropy. We show that our method is equivalent to the correlation matrix method. Finally, we apply our method to free scalar field theory in 1+1 dimensions and show that, for very squeezed states, the entanglement entropy is dominated by a volume term, unlike the ground-state case. Even though the state of the system is time-dependent in a non-trivial manner, this volume term is time-independent. We expect this behaviour to hold in higher dimensions as well, as it emerges in a large-squeezing expansion of the entanglement entropy for a general harmonic system.
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References
T. Jacobson, Thermodynamics of space-time: The Einstein equation of state, Phys. Rev. Lett. 75 (1995) 1260 [gr-qc/9504004] [INSPIRE].
M. Van Raamsdonk, Building up spacetime with quantum entanglement, Gen. Rel. Grav. 42 (2010) 2323 [arXiv:1005.3035] [INSPIRE].
T. Jacobson, Entanglement Equilibrium and the Einstein Equation, Phys. Rev. Lett. 116 (2016) 201101 [arXiv:1505.04753] [INSPIRE].
N. Lashkari, M.B. McDermott and M. Van Raamsdonk, Gravitational dynamics from entanglement ‘thermodynamics’, JHEP 04 (2014) 195 [arXiv:1308.3716] [INSPIRE].
T. Faulkner et al., Gravitation from Entanglement in Holographic CFTs, JHEP 03 (2014) 051 [arXiv:1312.7856] [INSPIRE].
L. Bombelli, R.K. Koul, J. Lee and R.D. Sorkin, A Quantum Source of Entropy for Black Holes, Phys. Rev. D 34 (1986) 373 [INSPIRE].
M. Srednicki, Entropy and area, Phys. Rev. Lett. 71 (1993) 666 [hep-th/9303048] [INSPIRE].
P. Calabrese and J.L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. 0406 (2004) P06002 [hep-th/0405152] [INSPIRE].
H. Casini and M. Huerta, Entanglement entropy in free quantum field theory, J. Phys. A 42 (2009) 504007 [arXiv:0905.2562] [INSPIRE].
P. Calabrese and J. Cardy, Entanglement entropy and conformal field theory, J. Phys. A 42 (2009) 504005 [arXiv:0905.4013] [INSPIRE].
D. Katsinis and G. Pastras, An Inverse Mass Expansion for Entanglement Entropy in Free Massive Scalar Field Theory, Eur. Phys. J. C 78 (2018) 282 [arXiv:1711.02618] [INSPIRE].
D. Katsinis and G. Pastras, Area Law Behaviour of Mutual Information at Finite Temperature, arXiv:1907.04817 [INSPIRE].
D. Katsinis and G. Pastras, An Inverse Mass Expansion for the Mutual Information in Free Scalar QFT at Finite Temperature, JHEP 02 (2020) 091 [arXiv:1907.08508] [INSPIRE].
E. Benedict and S.-Y. Pi, Entanglement entropy of nontrivial states, Annals Phys. 245 (1996) 209 [hep-th/9505121] [INSPIRE].
D. Katsinis and G. Pastras, Entanglement in harmonic systems at coherent states, arXiv:2206.05781 [INSPIRE].
D.N. Page, Average entropy of a subsystem, Phys. Rev. Lett. 71 (1993) 1291 [gr-qc/9305007] [INSPIRE].
D.N. Page, Information in black hole radiation, Phys. Rev. Lett. 71 (1993) 3743 [hep-th/9306083] [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].
E. Bianchi, L. Hackl and M. Kieburg, Page curve for fermionic Gaussian states, Phys. Rev. B 103 (2021) L241118 [arXiv:2103.05416] [INSPIRE].
E. Bianchi, L. Hackl and N. Yokomizo, Entanglement entropy of squeezed vacua on a lattice, Phys. Rev. D 92 (2015) 085045 [arXiv:1507.01567] [INSPIRE].
G. Adesso, S. Ragy and A.R. Lee, Continuous variable quantum information: Gaussian states and beyond, arXiv:1401.4679 [https://doi.org/10.1142/S1230161214400010].
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].
V.F. Mukhanov, H.A. Feldman and R.H. Brandenberger, Theory of cosmological perturbations. Part 1. Classical perturbations. Part 2. Quantum theory of perturbations. Part 3. Extensions, Phys. Rept. 215 (1992) 203 [INSPIRE].
A. Albrecht, P. Ferreira, M. Joyce and T. Prokopec, Inflation and squeezed quantum states, Phys. Rev. D 50 (1994) 4807 [astro-ph/9303001] [INSPIRE].
D. Polarski and A.A. Starobinsky, Semiclassicality and decoherence of cosmological perturbations, Class. Quant. Grav. 13 (1996) 377 [gr-qc/9504030] [INSPIRE].
L.P. Grishchuk and Y.V. Sidorov, Squeezed quantum states of relic gravitons and primordial density fluctuations, Phys. Rev. D 42 (1990) 3413 [INSPIRE].
R.H. Brandenberger, T. Prokopec and V.F. Mukhanov, The Entropy of the gravitational field, Phys. Rev. D 48 (1993) 2443 [gr-qc/9208009] [INSPIRE].
R.H. Brandenberger, V.F. Mukhanov and T. Prokopec, Entropy of a classical stochastic field and cosmological perturbations, Phys. Rev. Lett. 69 (1992) 3606 [astro-ph/9206005] [INSPIRE].
T. Prokopec, Entropy of the squeezed vacuum, Class. Quant. Grav. 10 (1993) 2295 [INSPIRE].
A.L. Matacz, The Coherent state representation of quantum fluctuations in the early universe, Phys. Rev. D 49 (1994) 788 [gr-qc/9212008] [INSPIRE].
M. Gasperini and M. Giovannini, Entropy production in the cosmological amplification of the vacuum fluctuations, Phys. Lett. B 301 (1993) 334 [gr-qc/9301010] [INSPIRE].
M. Gasperini and M. Giovannini, Quantum squeezing and cosmological entropy production, Class. Quant. Grav. 10 (1993) L133 [gr-qc/9307024] [INSPIRE].
C. Kiefer, D. Polarski and A.A. Starobinsky, Entropy of gravitons produced in the early universe, Phys. Rev. D 62 (2000) 043518 [gr-qc/9910065] [INSPIRE].
D. Campo and R. Parentani, Decoherence and entropy of primordial fluctuations. I: Formalism and interpretation, Phys. Rev. D 78 (2008) 065044 [arXiv:0805.0548] [INSPIRE].
K. Boutivas, G. Pastras and N. Tetradis, Entanglement and expansion, JHEP 05 (2023) 199 [arXiv:2302.14666] [INSPIRE].
G. Lindblad, On the generators of quantum dynamical semigroups, Comm. Math. Phys 48 (1976) 119.
I. Peschel, Calculation of reduced density matrices from correlation functions, cond-mat/0212631 [https://doi.org/10.1088/0305-4470/36/14/101].
A. Coser, E. Tonni and P. Calabrese, Entanglement negativity after a global quantum quench, J. Stat. Mech. 1412 (2014) P12017 [arXiv:1410.0900] [INSPIRE].
J.S. Cotler, M.P. Hertzberg, M. Mezei and M.T. Mueller, Entanglement Growth after a Global Quench in Free Scalar Field Theory, JHEP 11 (2016) 166 [arXiv:1609.00872] [INSPIRE].
R.D. Sorkin, Expressing entropy globally in terms of (4D) field-correlations, J. Phys. Conf. Ser. 484 (2014) 012004 [arXiv:1205.2953] [INSPIRE].
Acknowledgments
The research of D.Katsinis was supported by FAPESP Grant No. 2021/01819-0. The research of N. Tetradis was supported by the Hellenic Foundation for Research and Innovation (H.F.R.I.) under the “First Call for H.F.R.I. Research Projects to support Faculty members and Researchers and the procurement of high-cost research equipment grant” (Project Number: 824).
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Katsinis, D., Pastras, G. & Tetradis, N. Entanglement of harmonic systems in squeezed states. J. High Energ. Phys. 2023, 39 (2023). https://doi.org/10.1007/JHEP10(2023)039
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DOI: https://doi.org/10.1007/JHEP10(2023)039