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Entanglement of Initial State and Pseudo Entanglement Wedge

Masamichi Miyaji

Published Sep 8, 2021• 1 citation
OpenAlex

Abstract

We consider spacetime initiated by a finite-sized boundary on which a pure initial matter state is set as a natural generalization of the Hartle-Hawking no-boundary state. We study entanglement entropy of the gravitationally prepared matter state at the final time slice. We find that the entropy of the initial state or the entanglement island gives the entropy for large subregions on the final time slice. Consequently, we find the entanglement entropy is bounded from above by the boundary area of the island, leading to an entropy bound in terms of the island formula. The island $I$ appears in the analytically continued spacetime, either at the bra or the ket part of the spacetime in Schwinger-Keldysh formalism, and the entropy is given by an average of pseudo entropy of each entanglement island. We find a necessary condition of the initial state to be consistent with the strong sub-additivity. The condition requires that any probe degrees of freedom are thermally entangled with the rest of the system. We then study which initial condition leads to our finite-sized initial boundary or the Hartle-Hawking no-boundary state. Due to the absence of a moment of time reflection symmetry, the island in our setup requires a generalization of the entanglement wedge, which we call $pseudo~entanglement~wedge$. In pseudo entanglement wedge reconstruction, we consider reconstructing the bulk matter transition matrix on $A\cup I$, from a fine-grained state on $A$. The bulk transition matrix is given by a thermofield double state with a projection by the initial state. We provide an AdS/BCFT model, which provides a double holography model of our setup by considering EOW branes with corners. We also find the exponential hardness of such reconstruction task using a generalization of Python's lunch conjecture to pseudo generalized entropy.

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Island for gravitationally prepared state and pseudo entanglement wedge

Masamichi Miyaji

Abstract We consider spacetime initiated by a finite-sized initial boundary as a generalization of the Hartle-Hawking no-boundary state. We study entanglement entropy of matter state prepared by such spacetime. We find that the entanglement entropy for large subregion is given either by the initial state entanglement or the entanglement island, preventing the entropy to grow arbitrarily large. Consequently, the entanglement entropy is always bounded from above by the boundary area of the island, leading to an entropy bound in terms of the island. The island I is located in the analytically continued spacetime, either at the bra or the ket part of the spacetime in Schwinger-Keldysh formalism. The entanglement entropy is given by an average of complex pseudo generalized entropy for each entanglement island. We find a necessary condition of the initial state to be consistent with the strong sub-additivity, which requires that any probe degrees of freedom are thermally entangled with the rest of the system. We then find a large parameter region where the spacetime with finite-sized initial boundary, which does not have the factorization puzzle at leading order, dominates over the Hartle-Hawking no-boundary state or the bra-ket wormhole. Due to the absence of a moment of time reflection symmetry, the island in our setup is a generalization of the entanglement wedge, called pseudo entanglement wedge. In pseudo entanglement wedge reconstruction, we consider reconstructing the bulk matter transition matrix on A ∪ I, from a fine-grained state on A. The bulk transition matrix is given by a thermofield double state with a projection by the initial state. We also provide an AdS/BCFT model by considering EOW branes with corners. We also find the exponential hardness of such reconstruction task using a generalization of Python’s lunch conjecture to pseudo generalized entropy.

arXiv:2109.03830•Journal of High Energy Physics•Dec 3, 2021•44 citations

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A bstract The concept of the generalized entanglement wedge was recently proposed by Bousso and Penington, which states that any bulk gravitational region a possesses an associated generalized entanglement wedge E ( a ) ⊃ a on a static Cauchy surface M in general gravitational spacetimes, where E ( a ) may contain an entanglement island I ( a ). It suggests that the fine-grained entropy for bulk region a is given by the generalized entropy S gen ( E ( a )). Motivated by this proposal, we extend the quantum bit thread description to general gravitational spacetimes, no longer limited to the AdS spacetime. By utilizing the convex optimization techniques, a dual flow description for the generalized entropy S gen ( E ( a )) of a bulk gravitational region a is established on the static Cauchy surface M , such that S gen ( E ( a )) is equal to the maximum flux of any flow that starts from the boundary ∂M and ends at bulk region a , or equivalently, the maximum number of bit threads that connect the boundary ∂M to the bulk region a . In addition, the nesting property of flows is also proved. Thus the basic properties of the entropy for bulk regions, i.e. the monotonicity, subadditivity, Araki-Lieb inequality and strong subadditivity, can be verified from flow perspectives by using properties of flows, such as the nesting property. Moreover, in max thread configurations, we find that there exists some lower bounds on the bulk entanglement entropy of matter fields in the region E ( a ) \ a , particularly on an entanglement island region I ( a ) ⊂ ( E ( a ) \ a ), as required by the existence of a nontrivial generalized entanglement wedge. Our quantum bit thread formulation may provide a way to investigate more fine-grained entanglement structures in general spacetimes.

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Entanglement Wedge Reconstruction and the Information Paradox

Geoff Penington

When absorbing boundary conditions are used to evaporate a black hole in AdS/CFT, we show that there is a phase transition in the location of the quantum Ryu-Takayanagi surface, at precisely the Page time. The new RT surface lies slightly inside the event horizon, at an infalling time approximately the scrambling time $β/2π\log S_{BH}$ into the past. We can immediately derive the Page curve, using the Ryu-Takayanagi formula, and the Hayden-Preskill decoding criterion, using entanglement wedge reconstruction. Because part of the interior is now encoded in the early Hawking radiation, the decreasing entanglement entropy of the black hole is exactly consistent with the semiclassical bulk entanglement of the late-time Hawking modes, despite the absence of a firewall. By studying the entanglement wedge of highly mixed states, we can understand the state dependence of the interior reconstructions. A crucial role is played by the existence of tiny, non-perturbative errors in entanglement wedge reconstruction. Directly after the Page time, interior operators can only be reconstructed from the Hawking radiation if the initial state of the black hole is known. As the black hole continues to evaporate, reconstructions become possible that simultaneously work for a large class of initial states. Using similar techniques, we generalise Hayden-Preskill to show how the amount of Hawking radiation required to reconstruct a large diary, thrown into the black hole, depends on both the energy and the entropy of the diary. Finally we argue that, before the evaporation begins, a single, state-independent interior reconstruction exists for any code space of microstates with entropy strictly less than the Bekenstein-Hawking entropy, and show that this is sufficient state dependence to avoid the AMPSS typical-state firewall paradox.

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Bernard S. Kay

In 2006, Ryu and Takayanagi (RT) pointed out that (with a suitable cutoff) the entanglement entropy between two complementary regions of an equal-time surface of a d+1-dimensional conformal field theory on the conformal boundary of AdS_{d+2} is, when the AdS radius is appropriately related to the parameters of the CFT, equal to 1/4G times the area of the d-dimensional minimal surface in the AdS bulk which has the junction of those complementary regions as its boundary, where G is the bulk Newton constant. We point out here that the RT-equality implies that, in the quantum theory on the bulk AdS background which is related to the boundary CFT according to Rehren's 1999 algebraic holography theorem, the entanglement entropy between two complementary bulk Rehren wedges is equal to 1/4G times the (suitably cut off) area of their shared ridge. (This follows because of the geometrical fact that, for complementary ball-shaped regions, the RT minimal surface is precisely the shared ridge of the complementary bulk Rehren wedges which correspond, under Rehren's bulk-wedge to boundary double-cone bijection, to the complementary boundary double-cones whose bases are the RT complementary balls.) This is consistent with the Bianchi-Meyers conjecture -- that, in a theory of quantum gravity, the entanglement entropy, S, between the degrees of freedom of a given region with those of its complement is S = A/4G (+ lower order terms) -- but only if the phrase 'degrees of freedom' is replaced by 'matter degrees of freedom'. It also supports related previous arguments of the author -- consistent with the author's 'matter-gravity entanglement hypothesis' -- that the AdS/CFT correspondence is actually only a bijection between just the matter (i.e. non-gravity) sector operators of the bulk and the boundary CFT operators.

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Effective entropy of quantum fields coupled with gravity

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Entanglement entropy, or von Neumann entropy, quantifies the amount of uncertainty of a quantum state. For quantum fields in curved space, entanglement entropy of the quantum field theory degrees of freedom is well-defined for a fixed background geometry. In this paper, we propose a generalization of the quantum field theory entanglement entropy by including dynamical gravity. The generalized quantity named effective entropy, and its Renyi entropy generalizations, are defined by analytic continuation of a replica calculation. The replicated theory is defined as a gravitational path integral with multiple copies of the original boundary conditions, with a co-dimension-2 brane at the boundary of region we are studying. We discuss different approaches to define the region in a gauge invariant way, and show that the effective entropy satisfies the quantum extremal surface formula. When the quantum fields carry a significant amount of entanglement, the quantum extremal surface can have a topology transition, after which an entanglement island region appears. Our result generalizes the Hubeny-Rangamani-Takayanagi formula of holographic entropy (with quantum corrections) to general geometries without asymptotic AdS boundary, and provides a more solid framework for addressing problems such as the Page curve of evaporating black holes in asymptotic flat spacetime. We apply the formula to two example systems, a closed two-dimensional universe and a four-dimensional maximally extended Schwarzchild black hole. We discuss the analog of the effective entropy in random tensor network models, which provides more concrete understanding of quantum information properties in general dynamical geometries. We show that, in absence of a large boundary like in AdS space case, it is essential to introduce ancilla that couples to the original system, in order for correctly characterizing quantum states and correlation functions in the random tensor network. Using the superdensity operator formalism, we study the system with ancilla and show how quantum information in the entanglement island can be reconstructed in a state-dependent and observer-dependent map. We study the closed universe (without spatial boundary) case and discuss how it is related to open universe.

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Entanglement Wedge Reconstruction via Universal Recovery Channels

Jordan S. Cotler, Patrick M. Hayden, Geoff Penington +3 more

In the context of quantum theories of spacetime, one overarching question is how quantum information in the bulk spacetime is encoded holographically in boundary degrees of freedom.It is particularly interesting to understand the correspondence between bulk subregions and boundary subregions in order to address the emergence of locality in the bulk quantum spacetime.For the AdS/CFT correspondence, it is known that this bulk information is encoded redundantly on the boundary in the form of an error-correcting code.Having access only to a subregion of the boundary is as if part of the holographic code has been damaged by noise and rendered inaccessible.In quantum-information science, the problem of recovering information from a damaged code is addressed by the theory of universal recovery channels.We apply and extend this theory to address the problem of relating bulk and boundary subregions in AdS/CFT, focusing on a conjecture known as entanglement wedge reconstruction.Existing work relies on the exact equivalence between bulk and boundary relative entropies, but these are only approximately equal in bulk effective field theory, and in similar situations it is known that predictions from exact entropic equalities can be qualitatively incorrect.We show that the framework of universal recovery channels provides a robust demonstration of the entanglement wedge reconstruction conjecture as well as new physical insights.Most notably, we find that a bulk operator acting in a given boundary region's entanglement wedge can be expressed as the response of the boundary region's modular Hamiltonian to a perturbation of the bulk state in the direction of the bulk operator.This formula can be interpreted as a noncommutative version of Bayes's rule that attempts to undo the noise induced by restricting to only a portion of the boundary.To reach these conclusions, we extend the theory of universal recovery channels to finite-dimensional operator algebras and demonstrate that recovery channels approximately preserve the multiplicative structure of the operator algebra.

arXiv:1704.05839•Physical Review X•Jul 24, 2019•221 citations
We consider spacetime initiated by a finite-sized boundary on which a pure initial matter state is set as a natural generalization of the Hartle-Hawking no-boundary state. We study entanglement entropy of the gravitationally prepared matter state at the final time slice. We find that the entropy of the initial state or the entanglement island gives the entropy for large subregions on the final time slice. Consequently, we find the entanglement entropy is bounded from above by the boundary area of the island, leading to an entropy bound in terms of the island formula. The island III appears in the analytically continued spacetime, either at the bra or the ket part of the spacetime in Schwinger-Keldysh formalism, and the entropy is given by an average of pseudo entropy of each entanglement island. We find a necessary condition of the initial state to be consistent with the strong sub-additivity. The condition requires that any probe degrees of freedom are thermally entangled with the rest of the system. We then study which initial condition leads to our finite-sized initial boundary or the Hartle-Hawking no-boundary state. Due to the absence of a moment of time reflection symmetry, the island in our setup requires a generalization of the entanglement wedge, which we call pseudo entanglement wedgepseudo~entanglement~wedgepseudo entanglement wedge. In pseudo entanglement wedge reconstruction, we consider reconstructing the bulk matter transition matrix on A∪IA\cup IA∪I, from a fine-grained state on AAA. The bulk transition matrix is given by a thermofield double state with a projection by the initial state. We provide an AdS/BCFT model, which provides a double holography model of our setup by considering EOW branes with corners. We also find the exponential hardness of such reconstruction task using a generalization of Python's lunch conjecture to pseudo generalized entropy.
Abstract We consider spacetime initiated by a finite-sized initial boundary as a generalization of the Hartle-Hawking no-boundary state. We study entanglement entropy of matter state prepared by such spacetime. We find that the entanglement entropy for large subregion is given either by the initial state entanglement or the entanglement island, preventing the entropy to grow arbitrarily large. Consequently, the entanglement entropy is always bounded from above by the boundary area of the island, leading to an entropy bound in terms of the island. The island I is located in the analytically continued spacetime, either at the bra or the ket part of the spacetime in Schwinger-Keldysh formalism. The entanglement entropy is given by an average of complex pseudo generalized entropy for each entanglement island. We find a necessary condition of the initial state to be consistent with the strong sub-additivity, which requires that any probe degrees of freedom are thermally entangled with the rest of the system. We then find a large parameter region where the spacetime with finite-sized initial boundary, which does not have the factorization puzzle at leading order, dominates over the Hartle-Hawking no-boundary state or the bra-ket wormhole. Due to the absence of a moment of time reflection symmetry, the island in our setup is a generalization of the entanglement wedge, called pseudo entanglement wedge. In pseudo entanglement wedge reconstruction, we consider reconstructing the bulk matter transition matrix on A ∪ I, from a fine-grained state on A. The bulk transition matrix is given by a thermofield double state with a projection by the initial state. We also provide an AdS/BCFT model by considering EOW branes with corners. We also find the exponential hardness of such reconstruction task using a generalization of Python’s lunch conjecture to pseudo generalized entropy.
A bstract The concept of the generalized entanglement wedge was recently proposed by Bousso and Penington, which states that any bulk gravitational region a possesses an associated generalized entanglement wedge E ( a ) ⊃ a on a static Cauchy surface M in general gravitational spacetimes, where E ( a ) may contain an entanglement island I ( a ). It suggests that the fine-grained entropy for bulk region a is given by the generalized entropy S gen ( E ( a )). Motivated by this proposal, we extend the quantum bit thread description to general gravitational spacetimes, no longer limited to the AdS spacetime. By utilizing the convex optimization techniques, a dual flow description for the generalized entropy S gen ( E ( a )) of a bulk gravitational region a is established on the static Cauchy surface M , such that S gen ( E ( a )) is equal to the maximum flux of any flow that starts from the boundary ∂M and ends at bulk region a , or equivalently, the maximum number of bit threads that connect the boundary ∂M to the bulk region a . In addition, the nesting property of flows is also proved. Thus the basic properties of the entropy for bulk regions, i.e. the monotonicity, subadditivity, Araki-Lieb inequality and strong subadditivity, can be verified from flow perspectives by using properties of flows, such as the nesting property. Moreover, in max thread configurations, we find that there exists some lower bounds on the bulk entanglement entropy of matter fields in the region E ( a ) \ a , particularly on an entanglement island region I ( a ) ⊂ ( E ( a ) \ a ), as required by the existence of a nontrivial generalized entanglement wedge. Our quantum bit thread formulation may provide a way to investigate more fine-grained entanglement structures in general spacetimes.
When absorbing boundary conditions are used to evaporate a black hole in AdS/CFT, we show that there is a phase transition in the location of the quantum Ryu-Takayanagi surface, at precisely the Page time. The new RT surface lies slightly inside the event horizon, at an infalling time approximately the scrambling time β/2πlog⁡SBHβ/2π\log S_{BH}β/2πlogSBH​ into the past. We can immediately derive the Page curve, using the Ryu-Takayanagi formula, and the Hayden-Preskill decoding criterion, using entanglement wedge reconstruction. Because part of the interior is now encoded in the early Hawking radiation, the decreasing entanglement entropy of the black hole is exactly consistent with the semiclassical bulk entanglement of the late-time Hawking modes, despite the absence of a firewall. By studying the entanglement wedge of highly mixed states, we can understand the state dependence of the interior reconstructions. A crucial role is played by the existence of tiny, non-perturbative errors in entanglement wedge reconstruction. Directly after the Page time, interior operators can only be reconstructed from the Hawking radiation if the initial state of the black hole is known. As the black hole continues to evaporate, reconstructions become possible that simultaneously work for a large class of initial states. Using similar techniques, we generalise Hayden-Preskill to show how the amount of Hawking radiation required to reconstruct a large diary, thrown into the black hole, depends on both the energy and the entropy of the diary. Finally we argue that, before the evaporation begins, a single, state-independent interior reconstruction exists for any code space of microstates with entropy strictly less than the Bekenstein-Hawking entropy, and show that this is sufficient state dependence to avoid the AMPSS typical-state firewall paradox.
In 2006, Ryu and Takayanagi (RT) pointed out that (with a suitable cutoff) the entanglement entropy between two complementary regions of an equal-time surface of a d+1-dimensional conformal field theory on the conformal boundary of AdS_{d+2} is, when the AdS radius is appropriately related to the parameters of the CFT, equal to 1/4G times the area of the d-dimensional minimal surface in the AdS bulk which has the junction of those complementary regions as its boundary, where G is the bulk Newton constant. We point out here that the RT-equality implies that, in the quantum theory on the bulk AdS background which is related to the boundary CFT according to Rehren's 1999 algebraic holography theorem, the entanglement entropy between two complementary bulk Rehren wedges is equal to 1/4G times the (suitably cut off) area of their shared ridge. (This follows because of the geometrical fact that, for complementary ball-shaped regions, the RT minimal surface is precisely the shared ridge of the complementary bulk Rehren wedges which correspond, under Rehren's bulk-wedge to boundary double-cone bijection, to the complementary boundary double-cones whose bases are the RT complementary balls.) This is consistent with the Bianchi-Meyers conjecture -- that, in a theory of quantum gravity, the entanglement entropy, S, between the degrees of freedom of a given region with those of its complement is S = A/4G (+ lower order terms) -- but only if the phrase 'degrees of freedom' is replaced by 'matter degrees of freedom'. It also supports related previous arguments of the author -- consistent with the author's 'matter-gravity entanglement hypothesis' -- that the AdS/CFT correspondence is actually only a bijection between just the matter (i.e. non-gravity) sector operators of the bulk and the boundary CFT operators.
Entanglement entropy, or von Neumann entropy, quantifies the amount of uncertainty of a quantum state. For quantum fields in curved space, entanglement entropy of the quantum field theory degrees of freedom is well-defined for a fixed background geometry. In this paper, we propose a generalization of the quantum field theory entanglement entropy by including dynamical gravity. The generalized quantity named effective entropy, and its Renyi entropy generalizations, are defined by analytic continuation of a replica calculation. The replicated theory is defined as a gravitational path integral with multiple copies of the original boundary conditions, with a co-dimension-2 brane at the boundary of region we are studying. We discuss different approaches to define the region in a gauge invariant way, and show that the effective entropy satisfies the quantum extremal surface formula. When the quantum fields carry a significant amount of entanglement, the quantum extremal surface can have a topology transition, after which an entanglement island region appears. Our result generalizes the Hubeny-Rangamani-Takayanagi formula of holographic entropy (with quantum corrections) to general geometries without asymptotic AdS boundary, and provides a more solid framework for addressing problems such as the Page curve of evaporating black holes in asymptotic flat spacetime. We apply the formula to two example systems, a closed two-dimensional universe and a four-dimensional maximally extended Schwarzchild black hole. We discuss the analog of the effective entropy in random tensor network models, which provides more concrete understanding of quantum information properties in general dynamical geometries. We show that, in absence of a large boundary like in AdS space case, it is essential to introduce ancilla that couples to the original system, in order for correctly characterizing quantum states and correlation functions in the random tensor network. Using the superdensity operator formalism, we study the system with ancilla and show how quantum information in the entanglement island can be reconstructed in a state-dependent and observer-dependent map. We study the closed universe (without spatial boundary) case and discuss how it is related to open universe.
In the context of quantum theories of spacetime, one overarching question is how quantum information in the bulk spacetime is encoded holographically in boundary degrees of freedom.It is particularly interesting to understand the correspondence between bulk subregions and boundary subregions in order to address the emergence of locality in the bulk quantum spacetime.For the AdS/CFT correspondence, it is known that this bulk information is encoded redundantly on the boundary in the form of an error-correcting code.Having access only to a subregion of the boundary is as if part of the holographic code has been damaged by noise and rendered inaccessible.In quantum-information science, the problem of recovering information from a damaged code is addressed by the theory of universal recovery channels.We apply and extend this theory to address the problem of relating bulk and boundary subregions in AdS/CFT, focusing on a conjecture known as entanglement wedge reconstruction.Existing work relies on the exact equivalence between bulk and boundary relative entropies, but these are only approximately equal in bulk effective field theory, and in similar situations it is known that predictions from exact entropic equalities can be qualitatively incorrect.We show that the framework of universal recovery channels provides a robust demonstration of the entanglement wedge reconstruction conjecture as well as new physical insights.Most notably, we find that a bulk operator acting in a given boundary region's entanglement wedge can be expressed as the response of the boundary region's modular Hamiltonian to a perturbation of the bulk state in the direction of the bulk operator.This formula can be interpreted as a noncommutative version of Bayes's rule that attempts to undo the noise induced by restricting to only a portion of the boundary.To reach these conclusions, we extend the theory of universal recovery channels to finite-dimensional operator algebras and demonstrate that recovery channels approximately preserve the multiplicative structure of the operator algebra.