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2019

Entanglement of Purification and Projective Measurement in CFT

Guo, Wu-zhong

Understand

We investigate entanglement of purification in conformal field theory.

By using Reeh-Schlieder theorem, we construct a set of the purification states for $\rho_{AB}$, where $\rho_{AB}$ is reduced density matrix for subregion $AB$ of a global state $\rho$.
The set can be approximated by acting all the unitary observables,located in the complement of subregion $AB$, on the global state $\rho$, as long as the global state $\rho$ is \text{cyclic} for every local algebra, e.g., the vacuum state.
Combining with the gravity explanation of unitary operations in the context of the so-called surface/state correspondence, we prove the holographic EoP formula.

Built on

J. L. Cardy, ¡°Boundary Conditions, Fusion Rules and the Verlinde Formula,¡± Nucl. Phys. B 324 (1989) 581
1989
Earlier work this paper cites.
B. S. Kay and R. M. Wald, Physics Reports, 207(2), 49-136 (1991)
1991
Earlier work this paper cites.
J. M. Maldacena, Int. J. Theor. Phys. 38
1998
Earlier work this paper cites.
B. M. Terhal, M. Horodecki, D. W. Leung and D. P. DiVincenzo, Journal of Mathematical Physics, 43(9), 4286-4298 (2002). [arXiv:0202044[quant-ph]]
2002
Earlier work this paper cites.
S. Ryu and T. Takayanagi, Phys. Rev. Lett. 96
2006
Earlier work this paper cites.
V. E. Hubeny, M. Rangamani and T. Takayanagi, JHEP 0707
Original
2007
Earlier work this paper cites.
R. Haag, Local quantum physics: Fields, particles, algebras. Springer Science Business Media, 2012
2012
Earlier work this paper cites.
B. Czech, J. L. Karczmarek, F. Nogueira and M. Van Raamsdonk, Class. Quant. Grav. 29
Original
2012
Earlier work this paper cites.

Similar

A. C. Wall, Class. Quant. Grav. 31
Original
2014
Cited alongside, same era.
M. Headrick, V. E. Hubeny, A. Lawrence and M. Rangamani, JHEP 1412
Original
2014
Cited alongside, same era.
M. Miyaji and T. Takayanagi, PTEP 2015
Original
2015
Cited alongside, same era.
M. Miyaji, T. Numasawa, N. Shiba, T. Takayanagi and K. Watanabe, Phys. Rev. Lett. 115
Original
2015
Cited alongside, same era.
M. A. Rajabpour, Phys. Rev. B 92
Original
2015
Cited alongside, same era.

Then

M. A. Rajabpour, J. Stat. Mech. 1606
Original
2016
Later among the works it cites.
K. Najafi and M. A. Rajabpour, JHEP 1612
Original
2016
Later among the works it cites.
T. Numasawa, N. Shiba, T. Takayanagi and K. Watanabe, JHEP 1608
Original
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Later among the works it cites.
T. Takayanagi and K. Umemoto, Nature Phys. 14
Original
2018
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P. Nguyen, T. Devakul, M. G. Halbasch, M. P. Zaletel and B. Swingle, JHEP 1801
Original
2018
Later among the works it cites.

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M. Miyaji, S. Ryu, T. Takayanagi and X. Wen, JHEP 1505

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2015

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R. F. Streater and A. S. Wightman, PCT, spin and statistics, and all that. Princeton University Press,2016

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J. Hauschild, Johannes, E. Leviatan; J. Bardarson, E. Altman, M. Zaletel, F. Pollmann, ¡°Finding purifications with minimal entanglement ,¡± eprint arXiv:1711.01288

Original

Cited in the paper.

More precisely, for any state | ψ ⟩ |\psi\rangle one always could find an operator 𝒪 A B ¯ ( ψ ) \mathcal{O}_{\overline{AB}}(\psi) such that the | | ψ ⟩ − 𝒪 A B ¯ ( ψ ) | 0 ⟩ | < ϵ ||\psi\rangle-\mathcal{O}_{\overline{AB}}(\psi)|0\rangle|<\epsilon , where ϵ \epsilon is an aribrary positive constant

Cited in the paper.

In fact this follows by another important property of vacuum state, that is the separating property. A state | Ω ⟩ |\Omega\rangle is said to be separating for the local algebra ℛ ⁡ ( O ) \mathscr{R}(O) if 𝒪 | Ω ⟩ = 0 ⇒ 𝒪 = 0 \mathcal{O}|\Omega\rangle=0\Rightarrow\mathcal{O}=0

Cited in the paper.

A. Bhattacharyya, T. Takayanagi and K. Umemoto, JHEP 1804

Original

2018

Later among the works it cites.

E. Witten, Rev. Mod. Phys. 90

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K. Umemoto and Y. Zhou, JHEP 1810

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2018

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