Built on

G. Aldazabal, D. Marques, and C. Nunez, “Double Field Theory: A Pedagogical Review,” Class. Quant. Grav
Original
1907
Earlier work this paper cites.
P. A. Dirac, “Wave equations in conformal space,” Annals of Mathematics
1936
Earlier work this paper cites.
S. Ferrara, A. F. Grillo, G. Parisi, and R. Gatto, “The shadow operator formalism for conformal algebra. vacuum expectation values and operator products,” Lett. Nuovo Cim
1972
Earlier work this paper cites.
A. Wehrl, “General properties of entropy,” Reviews of Modern Physics
1978
Earlier work this paper cites.
M. M. Roberts, “Time evolution of entanglement entropy from a pulse,” JHEP
Original
1982
Earlier work this paper cites.
J. de Boer and J. Goeree, “W gravity from Chern-Simons theory,” Nucl. Phys
1992
Earlier work this paper cites.
C. Holzhey, F. Larsen, and F. Wilczek, “Geometric and renormalized entropy in conformal field theory,” Nucl. Phys
1994
Earlier work this paper cites.
V. Balasubramanian, P. Kraus, and A. E. Lawrence, “Bulk versus boundary dynamics in anti-de Sitter space-time,” Phys. Rev
1999
Earlier work this paper cites.
S. de Haro, S. N. Solodukhin, and K. Skenderis, “Holographic reconstruction of space-time and renormalization in the AdS / CFT correspondence,” Commun. Math. Phys
2001
Earlier work this paper cites.
V. Vedral, “The role of relative entropy in quantum information theory,” Reviews of Modern Physics
2002
Earlier work this paper cites.
P. Calabrese and J. L. Cardy, “Entanglement entropy and quantum field theory,” J. Stat. Mech
2004
Earlier work this paper cites.
S. Ryu and T. Takayanagi, “Holographic derivation of entanglement entropy from AdS/CFT,” Phys. Rev. Lett
2006
Earlier work this paper cites.
S. Ryu and T. Takayanagi, “Aspects of Holographic Entanglement Entropy,” JHEP
2006
Earlier work this paper cites.
A. Hamilton, D. N. Kabat, G. Lifschytz, and D. A. Lowe, “Holographic representation of local bulk operators,” Phys. Rev
2006
Earlier work this paper cites.
V. A. Fateev and A. V. Litvinov, “Correlation functions in conformal Toda field theory. I.,” JHEP
Original
2007
Earlier work this paper cites.
Elsevier/Academic Press, Amsterdam, seventh ed., 2007
I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products · 2007
Earlier work this paper cites.
I. R. Klebanov, D. Kutasov, and A. Murugan, “Entanglement as a probe of confinement,” Nucl. Phys
Original
2008
Earlier work this paper cites.
G. Vidal, “Class of quantum many-body states that can be efficiently simulated,” Phys. Rev. Lett
2008
Earlier work this paper cites.
J. L. Cardy, O. A. Castro-Alvaredo, and B. Doyon, “Form factors of branch-point twist fields in quantum integrable models and entanglement entropy,” J. Statist. Phys
Original
2008
Earlier work this paper cites.
J. Gomis and T. Okuda, “S-duality, ’t Hooft operators and the operator product expansion,” JHEP
Original
2009
Earlier work this paper cites.
J. Abajo-Arrastia, J. Aparicio, and E. Lopez, “Holographic Evolution of Entanglement Entropy,” JHEP
Original
2010
Earlier work this paper cites.
R. C. Myers and A. Sinha, “Seeing a c-theorem with holography,” Phys. Rev
Original
2010
Earlier work this paper cites.
M. Van Raamsdonk, “Building up spacetime with quantum entanglement,” Gen. Rel. Grav
Original
2010
Earlier work this paper cites.
S. Weinberg, “Six-dimensional Methods for Four-dimensional Conformal Field Theories,” Phys. Rev
Original
2010
Earlier work this paper cites.
T. Albash and C. V. Johnson, “Evolution of Holographic Entanglement Entropy after Thermal and Electromagnetic Quenches,” New J. Phys
Original
2011
Earlier work this paper cites.
V. Balasubramanian, A. Bernamonti, J. de Boer, N. Copland, B. Craps, E. Keski-Vakkuri, B. Muller, A. Schafer, M. Shigemori, and W. Staessens, “Thermalization of Strongly Coupled Field Theories,” Phys. Rev. Lett
Original
2011
Earlier work this paper cites.

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V. Balasubramanian, A. Bernamonti, J. de Boer, N. Copland, B. Craps, E. Keski-Vakkuri, B. Muller, A. Schafer, M. Shigemori, and W. Staessens, “Holographic Thermalization,” Phys. Rev
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2011
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R. C. Myers and A. Sinha, “Holographic c-theorems in arbitrary dimensions,” JHEP
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Cited alongside, same era.
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Then

T. Faulkner, M. Guica, T. Hartman, R. C. Myers, and M. Van Raamsdonk, “Gravitation from Entanglement in Holographic CFTs,” JHEP
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2014
Later among the works it cites.
D. Simmons-Duffin, “Projectors, Shadows, and Conformal Blocks,” JHEP
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2014
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I. A. Morrison, “Boundary-to-bulk maps for AdS causal wedges and the Reeh-Schlieder property in holography,” JHEP
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2014
Later among the works it cites.
L.-Y. Hung, R. C. Myers, and M. Smolkin, “Twist operators in higher dimensions,” JHEP
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2014
Later among the works it cites.
E. Hijano and P. Kraus, “A new spin on entanglement entropy,” JHEP
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2016

Entanglement, Holography and Causal Diamonds

de Boer, Jan, Haehl, Felix M., Heller, Michal P. et al.

Understand

We argue that the degrees of freedom in a d-dimensional CFT can be re-organized in an insightful way by studying observables on the moduli space of causal diamonds (or equivalently, the space of pairs of timelike separated points).

This 2d-dimensional space naturally captures some of the fundamental nonlocality and causal structure inherent in the entanglement of CFT states.
For any primary CFT operator, we construct an observable on this space, which is defined by smearing the associated one-point function over causal diamonds.
Known examples of such quantities are the entanglement entropy of vacuum excitations and its higher spin generalizations.

Built on

G. Aldazabal, D. Marques, and C. Nunez, “Double Field Theory: A Pedagogical Review,” Class. Quant. Grav
Original
1907
Earlier work this paper cites.
P. A. Dirac, “Wave equations in conformal space,” Annals of Mathematics
1936
Earlier work this paper cites.
S. Ferrara, A. F. Grillo, G. Parisi, and R. Gatto, “The shadow operator formalism for conformal algebra. vacuum expectation values and operator products,” Lett. Nuovo Cim
1972
Earlier work this paper cites.
A. Wehrl, “General properties of entropy,” Reviews of Modern Physics
1978
Earlier work this paper cites.
M. M. Roberts, “Time evolution of entanglement entropy from a pulse,” JHEP
Original
1982
Earlier work this paper cites.
J. de Boer and J. Goeree, “W gravity from Chern-Simons theory,” Nucl. Phys
1992
Earlier work this paper cites.
C. Holzhey, F. Larsen, and F. Wilczek, “Geometric and renormalized entropy in conformal field theory,” Nucl. Phys
1994
Earlier work this paper cites.
V. Balasubramanian, P. Kraus, and A. E. Lawrence, “Bulk versus boundary dynamics in anti-de Sitter space-time,” Phys. Rev
1999
Earlier work this paper cites.
S. de Haro, S. N. Solodukhin, and K. Skenderis, “Holographic reconstruction of space-time and renormalization in the AdS / CFT correspondence,” Commun. Math. Phys
2001
Earlier work this paper cites.
V. Vedral, “The role of relative entropy in quantum information theory,” Reviews of Modern Physics
2002
Earlier work this paper cites.
P. Calabrese and J. L. Cardy, “Entanglement entropy and quantum field theory,” J. Stat. Mech
2004
Earlier work this paper cites.
S. Ryu and T. Takayanagi, “Holographic derivation of entanglement entropy from AdS/CFT,” Phys. Rev. Lett
2006
Earlier work this paper cites.
S. Ryu and T. Takayanagi, “Aspects of Holographic Entanglement Entropy,” JHEP
2006
Earlier work this paper cites.
A. Hamilton, D. N. Kabat, G. Lifschytz, and D. A. Lowe, “Holographic representation of local bulk operators,” Phys. Rev
2006
Earlier work this paper cites.
V. A. Fateev and A. V. Litvinov, “Correlation functions in conformal Toda field theory. I.,” JHEP
Original
2007
Earlier work this paper cites.
Elsevier/Academic Press, Amsterdam, seventh ed., 2007
I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products · 2007
Earlier work this paper cites.
I. R. Klebanov, D. Kutasov, and A. Murugan, “Entanglement as a probe of confinement,” Nucl. Phys
Original
2008
Earlier work this paper cites.
G. Vidal, “Class of quantum many-body states that can be efficiently simulated,” Phys. Rev. Lett
2008
Earlier work this paper cites.
J. L. Cardy, O. A. Castro-Alvaredo, and B. Doyon, “Form factors of branch-point twist fields in quantum integrable models and entanglement entropy,” J. Statist. Phys
Original
2008
Earlier work this paper cites.
J. Gomis and T. Okuda, “S-duality, ’t Hooft operators and the operator product expansion,” JHEP
Original
2009
Earlier work this paper cites.
J. Abajo-Arrastia, J. Aparicio, and E. Lopez, “Holographic Evolution of Entanglement Entropy,” JHEP
Original
2010
Earlier work this paper cites.
R. C. Myers and A. Sinha, “Seeing a c-theorem with holography,” Phys. Rev
Original
2010
Earlier work this paper cites.
M. Van Raamsdonk, “Building up spacetime with quantum entanglement,” Gen. Rel. Grav
Original
2010
Earlier work this paper cites.
S. Weinberg, “Six-dimensional Methods for Four-dimensional Conformal Field Theories,” Phys. Rev
Original
2010
Earlier work this paper cites.
T. Albash and C. V. Johnson, “Evolution of Holographic Entanglement Entropy after Thermal and Electromagnetic Quenches,” New J. Phys
Original
2011
Earlier work this paper cites.
V. Balasubramanian, A. Bernamonti, J. de Boer, N. Copland, B. Craps, E. Keski-Vakkuri, B. Muller, A. Schafer, M. Shigemori, and W. Staessens, “Thermalization of Strongly Coupled Field Theories,” Phys. Rev. Lett
Original
2011
Earlier work this paper cites.