Abstract
We point out an important difference between continuum relativistic quantum field theory (QFT) and lattice models with dramatic consequences for the theory of multi-partite entanglement. On a lattice given a collection of density matrices ρ(1), ρ(2), ⋯, ρ(n) there is no guarantee that there exists an n-partite pure state |Ω〉12⋯n that reduces to these marginals. The state |Ω〉12⋯n exists only if the eigenvalues of the density matrices ρ(i) satisfy certain polygon inequalities. We show that in QFT, as opposed to lattice systems, splitting the space into n non-overlapping regions any collection of local states ω(1), ω(2), ⋯ ω(n) come from the restriction of a global pure state. The reason is that rotating any local state ω(i) by unitary Ui localized in the ith region we come arbitrarily close to any other local state ψ(i). We construct explicit examples of such local unitaries using the cocycle.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
S. Bravyi, Requirements for compatibility between local and multipartite quantum states, quant-ph/0301014.
A. Higuchi, A. Sudbery and J. Szulc, One-qubit reduced states of a pure many-qubit state: polygon inequalities, Phys. Rev. Lett. 90 (2003) 107902 [quant-ph/0209085].
F. Ceyhan and T. Faulkner, Recovering the QNEC from the ANEC, Commun. Math. Phys. 377 (2020) 999 [arXiv:1812.04683] [INSPIRE].
L. Zhang and S.M. Fei, Quantum fidelity and relative entropy between unitary orbits, J. Phys. A 47 (2014) 055301 [arXiv:1305.1472].
R. Haag, Local quantum physics: fields, particles, algebras, Springer, Germany (2012).
A. Connes, Une classification des facteurs de type III, Ann. Sci. E.N.S. 6 (1973) 133.
E. Witten, APS medal for exceptional achievement in research: invited article on entanglement properties of quantum field theory, Rev. Mod. Phys. 90 (2018) 045003.
A. Connes and E. Størmer, Homogeneity of the state space of factors of type III1, J. Funct. Anal. 28 (1978) 187.
N. Lashkari, H. Liu and S. Rajagopal, Modular flow of excited states, arXiv:1811.05052 [INSPIRE].
O. Bratelli and D.W. Robinson, Operator algebras and quantum statistical mechanics, Bull. Amer. Math. Soc 7 (1982) 425
J.J. Bisognano and E.H. Wichmann, On the duality condition for quantum fields, J. Math. Phys. 17 (1976) 303.
S. Balakrishnan, T. Faulkner, Z.U. Khandker and H. Wang, A general proof of the quantum null energy condition, JHEP 09 (2019) 020 [arXiv:1706.09432]
E.P. Wigner, Die Messung quantenmechanischer Operatoren, Z. Physik 131 (1952) 101.
E.P. Wigner and M.M. Yanase, Information contents of distributions, Proc. Nat. Acad. Sci. USA 19 (1963) 910
G.K. Pedersen et al., The Radon-Nikodym theorem for von Neumann algebras, Acta Math. 130 (1973) 53.
N. Lashkari, Constraining quantum fields using modular theory, JHEP 01 (2019) 059 [arXiv:1810.09306].
R. Longo, Algebraic and modular structure of von Neumann algebras of physics, Commun. Math. Phys. 38 (1982) 551 [INSPIRE].
T. Faulkner and A. Lewkowycz, Bulk locality from modular flow, JHEP 07 (2017) 151 [arXiv:1704.05464].
M. Walter, Multipartite quantum states and their marginals, arXiv:1410.6820 [INSPIRE].
H. Araki, Relative entropy of states of von Neumann algebras, Publ. Res. Inst. Math. Sci. 11 (1976) 809.
M. Van Raamsdonk, Building up spacetime with quantum entanglement II: It from BC-bit, arXiv:1809.01197 [INSPIRE].
D. Marolf, CFT sewing as the dual of AdS cut-and-paste, JHEP 02 (2020) 152 [arXiv:1909.09330] [INSPIRE].
N. Lashkari, Entanglement at a scale and renormalization monotones, JHEP 01 (2019) 219 [arXiv:1704.05077].
K. Furuya, N. Lashkari and S. Ouseph, Generalized entanglement entropy, charges, and intertwiners, JHEP 08 (2020) 046 [arXiv:2005.11389] [INSPIRE].
M.A. Nielsen and I. Chuang, Quantum computation and quantum information, Cambridge University Press, Cambridge U.K. (2002).
E.A. Carlen and E.H. Lieb, Remainder terms for some quantum entropy inequalities, J. Math. Phys. 55 (2014) 042201 [arXiv:1402.3840].
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: 1911.11153
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
Lashkari, N. Modular zero modes and sewing the states of QFT. J. High Energ. Phys. 2021, 189 (2021). https://doi.org/10.1007/JHEP04(2021)189
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP04(2021)189