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The black hole interior from non-isometric codes and complexity

Akers, Chris, Engelhardt, Netta, Harlow, Daniel, Penington, Geoff, Vardhan, Shreya·2024·Journal of High Energy Physics
DOI 10.1007/jhep06(2024)155
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Abstract

A bstract: Quantum error correction has given us a natural language for the emergence of spacetime, but the black hole interior poses a challenge for this framework: at late times the apparent number of interior degrees of freedom in effective field theory can vastly exceed the true number of fundamental degrees of freedom, so there can be no isometric (i.e. inner-product preserving) encoding of the former into the latter. In this paper we explain how quantum error correction nonetheless can be used to explain the emergence of the black hole interior, via the idea of “non-isometric codes protected by computational complexity”. We show that many previous ideas, such as the existence of a large number of “null states”, a breakdown of effective field theory for operations of exponential complexity, the quantum extremal surface calculation of the Page curve, post-selection, “state-dependent/state-specific” operator reconstruction, and the “simple entropy” approach to complexity coarse-graining, all fit naturally into this framework, and we illustrate all of these phenomena simultaneously in a soluble model.

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Figures

Figure 1 The holographic encoding map in AdS/CFT: an approximate isometry V : H bulk → H Boundary maps the effective bulk degrees of freedom to the fundamental boundary degrees of freedom.
Figure 1 The holographic encoding map in AdS/CFT: an approximate isometry V : H bulk → H Boundary maps the effective bulk degrees of freedom to the fundamental boundary degrees of freedom.CC-BY · From the PDF
Figure 2 Subregion duality in AdS/CFT: given a boundary subregion B , its minimal quantum extremal surface X min B divides the bulk space into the entanglement wedge b of B and the entanglement wedge b of B , and all bulk observables in b (or b ) on this low-energy subspace can be represented as operators in B (or B ).
Figure 2 Subregion duality in AdS/CFT: given a boundary subregion B , its minimal quantum extremal surface X min B divides the bulk space into the entanglement wedge b of B and the entanglement wedge b of B , and all bulk observables in b (or b ) on this low-energy subspace can be represented as operators in B (or B ).CC-BY · From the PDF
Figure 3 Holography for a black hole B radiating into a reservoir R . In the effective picture there are left- and right-moving modes ℓ and r on a 'nice slice' Cauchy surface, and the holographic map V : H ℓ ⊗ H r → H B tells us how these interior modes are encoded in the fundamental degrees of freedom. We will always take V to be a linear map, but in general it cannot be an isometry.
Figure 3 Holography for a black hole B radiating into a reservoir R . In the effective picture there are left- and right-moving modes ℓ and r on a 'nice slice' Cauchy surface, and the holographic map V : H ℓ ⊗ H r → H B tells us how these interior modes are encoded in the fundamental degrees of freedom. We will always take V to be a linear map, but in general it cannot be an isometry.CC-BY · From the PDF
Figure 4 Our model holographic map V : a Haar-typical unitary U acts on the effective field theory degrees of freedom ℓ and r in the black hole interior, together with some fixed additional state | ψ 0 〉 f , and then post-selects on some auxilliary degrees of freedom P , resulting in a state of the fundamental degrees of freedom B .
Figure 4 Our model holographic map V : a Haar-typical unitary U acts on the effective field theory degrees of freedom ℓ and r in the black hole interior, together with some fixed additional state | ψ 0 〉 f , and then post-selects on some auxilliary degrees of freedom P , resulting in a state of the fundamental degrees of freedom B .CC-BY · From the PDF
Figure 5 Computing the typical overlap between V | ψ 1 〉 and V | ψ 2 〉 . By equation (A.4) the unitary integral connects the indices going into the top/bottom of U to those going into the bottom/top of U † , and then divides by the total dimensionality | P || B | . This factor of the dimensionality is cancelled by the factor of | P | in V † V and a factor of | B | coming from a B index loop.
Figure 5 Computing the typical overlap between V | ψ 1 〉 and V | ψ 2 〉 . By equation (A.4) the unitary integral connects the indices going into the top/bottom of U to those going into the bottom/top of U † , and then divides by the total dimensionality | P || B | . This factor of the dimensionality is cancelled by the factor of | P | in V † V and a factor of | B | coming from a B index loop.CC-BY · From the PDF
Figure 6 Computing the nontrivial term in the fluctuation of the overlap. The integral connects the indices going into the top/bottom of each U to the indices going into the bottom/top of a U † , but now there are four ways to do these connections: two for the ℓfr indices and two for the BP indices, with the different choices are weighted as in (A.4).
Figure 6 Computing the nontrivial term in the fluctuation of the overlap. The integral connects the indices going into the top/bottom of each U to the indices going into the bottom/top of a U † , but now there are four ways to do these connections: two for the ℓfr indices and two for the BP indices, with the different choices are weighted as in (A.4).CC-BY · From the PDF
Figure 7 Given a cut γ separating the sphere into two equal-volume halves, the volume of the small neighbourhood γ ε = { x : ∃ y ∈ γ. || x -y || ≤ ε } is minimized when γ lies on an equator of the sphere (left) rather than oscillating in a complex way (right). Somewhat counterintuitively, in high dimensions the fractional volume of γ ε is exponentially close to one, even in the former case.
Figure 7 Given a cut γ separating the sphere into two equal-volume halves, the volume of the small neighbourhood γ ε = { x : ∃ y ∈ γ. || x -y || ≤ ε } is minimized when γ lies on an equator of the sphere (left) rather than oscillating in a complex way (right). Somewhat counterintuitively, in high dimensions the fractional volume of γ ε is exponentially close to one, even in the former case.CC-BY · From the PDF
Figure 8 Computing the second Renyi entropy of the radiation. B indices are shaded red, L, ℓ indices are shaded blue, and r, R indices are shaded green. The four different ways of contracting the indices through the unitaries as in (A.4) lead to the four terms in (4.4).
Figure 8 Computing the second Renyi entropy of the radiation. B indices are shaded red, L, ℓ indices are shaded blue, and r, R indices are shaded green. The four different ways of contracting the indices through the unitaries as in (A.4) lead to the four terms in (4.4).CC-BY · From the PDF
Figure 9 The quantum extremal surface calculation of the radiation entropy: at early times the entanglement wedge of the radiation includes only the domain of dependence of R , shaded green in the left diagram, while at late times it also includes an 'island' in the black hole interior. The 'rim' of the island, shown as the second black dot in the right diagram, is the new quantum extremal surface found in [33, 34]. It has an area which is approximately that of the horizon on this time slice, and thus has Area 4 ≈ log | B | .
Figure 9 The quantum extremal surface calculation of the radiation entropy: at early times the entanglement wedge of the radiation includes only the domain of dependence of R , shaded green in the left diagram, while at late times it also includes an 'island' in the black hole interior. The 'rim' of the island, shown as the second black dot in the right diagram, is the new quantum extremal surface found in [33, 34]. It has an area which is approximately that of the horizon on this time slice, and thus has Area 4 ≈ log | B | .CC-BY · From the PDF
Figure 10 Computing the n th Renyi entropy. There are two terms which can dominate: the one which contracts the ingoing/outgoing indices for each U with the outoing/ingoing indices of the U † just below it, which maximizes the number of red loops, and the one which contracts the ingoing/outgoing indices of each U with the outoing/ingoing indices of the U † which is in the previous column (wrapping around to the last U † for the first U ), which maximizes the number of green loops.
Figure 10 Computing the n th Renyi entropy. There are two terms which can dominate: the one which contracts the ingoing/outgoing indices for each U with the outoing/ingoing indices of the U † just below it, which maximizes the number of red loops, and the one which contracts the ingoing/outgoing indices of each U with the outoing/ingoing indices of the U † which is in the previous column (wrapping around to the last U † for the first U ), which maximizes the number of green loops.CC-BY · From the PDF
Figure 11 Converting a bulk measurement to a boundary measurement in AdS/CFT. On the left a bulk observer Diana is holding an apparatus which she has just looked at to learn the result of a measurement, but a boundary observer Wendy thinks that Diana and the apparatus are together in a coherent superposition and no definite outcome has been realized. On the right Wendy uses local operators in the fundamental description to sends a 'space mission' into the bulk which retrieves Diana's apparatus and swaps it out of the system, after which Wendy can look at it to learn the measurement outcome.
Figure 11 Converting a bulk measurement to a boundary measurement in AdS/CFT. On the left a bulk observer Diana is holding an apparatus which she has just looked at to learn the result of a measurement, but a boundary observer Wendy thinks that Diana and the apparatus are together in a coherent superposition and no definite outcome has been realized. On the right Wendy uses local operators in the fundamental description to sends a 'space mission' into the bulk which retrieves Diana's apparatus and swaps it out of the system, after which Wendy can look at it to learn the measurement outcome.CC-BY · From the PDF
Figure 12 The full time-evolution of an evaporating black hole in the fundamental picture. The red lines represent qudits that fall into the black hole from the bath, while the blue lines represent qudits that come out of the black hole as radiation. After n 0 time steps only radiation remains. Note that the outgoing qudits appear in the order they would on the nice slice in figure 3, while the ingoing qudits are reversed (qudits to the right fell in earlier).
Figure 12 The full time-evolution of an evaporating black hole in the fundamental picture. The red lines represent qudits that fall into the black hole from the bath, while the blue lines represent qudits that come out of the black hole as radiation. After n 0 time steps only radiation remains. Note that the outgoing qudits appear in the order they would on the nice slice in figure 3, while the ingoing qudits are reversed (qudits to the right fell in earlier).CC-BY · From the PDF
Figure 13 Re-interpreting the evolution in the fundamental description as a non-isometric map from the effective description. Here n 0 = 7, m 0 = 3, and t = 2. The factor of √ | P (2) | = | r (2) | is necessary to ensure that the maximally-mixed states contract to give the identity map on R (2) . The diagram on the right is precisely the action of a non-isometric holographic map V 2 ⊗ I R (2) on an effective description state which is arbitrary on ℓ (2) and maximally mixed on r (2) R (2) , with V 2 : H ℓ (2) ⊗H r (2) → H B (2) given by V 2 = √ | P (2) |〈 MAX | R (2) ,r (2) U 2 U 1 U 0 | ψ 0 〉 f . Note the clear resemblance to figure 4.
Figure 13 Re-interpreting the evolution in the fundamental description as a non-isometric map from the effective description. Here n 0 = 7, m 0 = 3, and t = 2. The factor of √ | P (2) | = | r (2) | is necessary to ensure that the maximally-mixed states contract to give the identity map on R (2) . The diagram on the right is precisely the action of a non-isometric holographic map V 2 ⊗ I R (2) on an effective description state which is arbitrary on ℓ (2) and maximally mixed on r (2) R (2) , with V 2 : H ℓ (2) ⊗H r (2) → H B (2) given by V 2 = √ | P (2) |〈 MAX | R (2) ,r (2) U 2 U 1 U 0 | ψ 0 〉 f . Note the clear resemblance to figure 4.CC-BY · From the PDF
Figure 14 The recursive step in defining the holographic map V t : H ℓ ( t ) ⊗H r ( t ) → H B ( t ) . To obtain V t +1 we act on V t with U t +1 and then post-select the two new outgoing qudits in R ( t +1) down to become the new qudits in r ( t +1) in the effective description.
Figure 14 The recursive step in defining the holographic map V t : H ℓ ( t ) ⊗H r ( t ) → H B ( t ) . To obtain V t +1 we act on V t with U t +1 and then post-select the two new outgoing qudits in R ( t +1) down to become the new qudits in r ( t +1) in the effective description.CC-BY · From the PDF
Figure 15 Equivariance of the holographic map under time evolution: acting with the effective-description dynamics at time t and then encoding with V t +1 is the same as encoding with V t at time t and then acting with the fundamental-description dynamics.
Figure 15 Equivariance of the holographic map under time evolution: acting with the effective-description dynamics at time t and then encoding with V t +1 is the same as encoding with V t at time t and then acting with the fundamental-description dynamics.CC-BY · From the PDF
Figure 16 Outermost wedges of various subsets of the fundamental degrees of freedom for an evaporating black hole.
Figure 16 Outermost wedges of various subsets of the fundamental degrees of freedom for an evaporating black hole.CC-BY · From the PDF
Figure 17 Back-reaction on the global geometry of an evaporating black hole caused by disturbing a right-moving interior mode. The orange line is the collapsing shell which formed the black hole, the thin vertical black line is the join between the radiation system and the black hole region, the red line is the shockwave which is created by disturbing the mode, the horizontal thin black line is the Cauchy slice on which we disturb the mode, and the purple lines are singularities. The dashed lines are future/past horizons. Note in particular the appearance of a past singularity in the back-reacted geometry, which didn't exist for the black hole formed from collapse. The green arrow is an infalling mode which in the unperturbed geometry can be evolved back out to the radiation, but which evolves back to the past singularity in the back-reacted geometry.
Figure 17 Back-reaction on the global geometry of an evaporating black hole caused by disturbing a right-moving interior mode. The orange line is the collapsing shell which formed the black hole, the thin vertical black line is the join between the radiation system and the black hole region, the red line is the shockwave which is created by disturbing the mode, the horizontal thin black line is the Cauchy slice on which we disturb the mode, and the purple lines are singularities. The dashed lines are future/past horizons. Note in particular the appearance of a past singularity in the back-reacted geometry, which didn't exist for the black hole formed from collapse. The green arrow is an infalling mode which in the unperturbed geometry can be evolved back out to the radiation, but which evolves back to the past singularity in the back-reacted geometry.CC-BY · From the PDF
Figure 18 Outermost wedges for B , R , and BR in the back-reacted geometry, shaded blue. Note that the outermost QES for B is not the same as the outermost QES for BR ; the former is the QES used in [33, 34] to compute the Page curve, while the latter is the new QES constructed in [53].
Figure 18 Outermost wedges for B , R , and BR in the back-reacted geometry, shaded blue. Note that the outermost QES for B is not the same as the outermost QES for BR ; the former is the QES used in [33, 34] to compute the Page curve, while the latter is the new QES constructed in [53].CC-BY · From the PDF
Figure 19 Acting with an interior unitary W ℓr in the dynamical model. The unitary acts on the two orange lines.
Figure 19 Acting with an interior unitary W ℓr in the dynamical model. The unitary acts on the two orange lines.CC-BY · From the PDF
Figure 20 Two possible bulges for B and R for the evaporating black hole perturbed by a generic W ℓr , indicated by the black dots. The shaded regions are their outer wedges assuming we are looking to construct X bulge R , the causal complements of these are their outer wedges as candidates for X bulge B . We have also shown the analogous cuts through the quantum circuit for our dynamical model.
Figure 20 Two possible bulges for B and R for the evaporating black hole perturbed by a generic W ℓr , indicated by the black dots. The shaded regions are their outer wedges assuming we are looking to construct X bulge R , the causal complements of these are their outer wedges as candidates for X bulge B . We have also shown the analogous cuts through the quantum circuit for our dynamical model.CC-BY · From the PDF
Figure 21 Interior modes in a one-sided AdS collapse. The left boundary of the diagram is the origin of polar coordinates, while the right boundary is the AdS boundary. The collapsing shell is shown in orange. Neighboring left and right movers are entangled on the nice slice, since the left movers arise from Hawking quanta which are reflected back into the black hole as shown with the dashed lines.
Figure 21 Interior modes in a one-sided AdS collapse. The left boundary of the diagram is the origin of polar coordinates, while the right boundary is the AdS boundary. The collapsing shell is shown in orange. Neighboring left and right movers are entangled on the nice slice, since the left movers arise from Hawking quanta which are reflected back into the black hole as shown with the dashed lines.CC-BY · From the PDF
Figure 22 A few time steps in a dynamical model of a one-sided AdS black hole formed from a fast collapse. On the left we have the fundamental CFT dynamics. The initial state of the infalling shell is encoded into a few adjustable qudits ℓ 0 and then a fixed state | ψ 0 〉 f . It evolves unitarily, with the radiated qudits, shown in red, being reabsorbed by the black hole after a time of order ℓ ads (each time step is of order ℓ ads ). On the right we re-interpret this evolution as defining a non-isometric map from the interior modes ℓ and r to the CFT degrees of freedom B , with the non-isometry arising from the post-selection on these modes just as in figure 13. We have allowed the unitary at each time step to be different, but to model the usual situation with no sources in the dual CFT it is natural to take all the U i to be the same.
Figure 22 A few time steps in a dynamical model of a one-sided AdS black hole formed from a fast collapse. On the left we have the fundamental CFT dynamics. The initial state of the infalling shell is encoded into a few adjustable qudits ℓ 0 and then a fixed state | ψ 0 〉 f . It evolves unitarily, with the radiated qudits, shown in red, being reabsorbed by the black hole after a time of order ℓ ads (each time step is of order ℓ ads ). On the right we re-interpret this evolution as defining a non-isometric map from the interior modes ℓ and r to the CFT degrees of freedom B , with the non-isometry arising from the post-selection on these modes just as in figure 13. We have allowed the unitary at each time step to be different, but to model the usual situation with no sources in the dual CFT it is natural to take all the U i to be the same.CC-BY · From the PDF
Figure 23 The black hole final state proposal: left-moving and right-moving interior modes are jointly post-selected on at the singularity, leading to a circuit which unitarily maps the state of the collapsing matter to the Hawking radiation. Note the similarity to figure 4 and especially figure 13.
Figure 23 The black hole final state proposal: left-moving and right-moving interior modes are jointly post-selected on at the singularity, leading to a circuit which unitarily maps the state of the collapsing matter to the Hawking radiation. Note the similarity to figure 4 and especially figure 13.CC-BY · From the PDF
Figure 24 The sums over the s m and s ′ m associated with each U m appearing the average of (I.5) can be seen as a partition function on the above lattice (we have shown the case where n 0 = 7, m 0 = 3, and t = 3). Each red vertex on the graph, other those fixed at the boundaries, can have either an ↑ or ↓ spin, and the total weight of any spin configuration can be obtained by multiplying the weights associated with each of the interactions, which are as shown on the right. The contraction of | ψ 0 , ↑〉 with either 〈↑ | or 〈↓ | contributes a factor of 1.
Figure 24 The sums over the s m and s ′ m associated with each U m appearing the average of (I.5) can be seen as a partition function on the above lattice (we have shown the case where n 0 = 7, m 0 = 3, and t = 3). Each red vertex on the graph, other those fixed at the boundaries, can have either an ↑ or ↓ spin, and the total weight of any spin configuration can be obtained by multiplying the weights associated with each of the interactions, which are as shown on the right. The contraction of | ψ 0 , ↑〉 with either 〈↑ | or 〈↓ | contributes a factor of 1.CC-BY · From the PDF
Figure 25 The two configurations, (a) and (b), which can dominate in the partition function (I.5), as well as a third (c) which is suppressed compared to (b) by a factor of 1 /q .
Figure 25 The two configurations, (a) and (b), which can dominate in the partition function (I.5), as well as a third (c) which is suppressed compared to (b) by a factor of 1 /q .CC-BY · From the PDF
Figure 26 Partition function for exp( -S 2 (Ψ R )), with two possible dominant configurations. Note that | χ { in } ℓ , ↑〉 on the lower boundary gives a factor of 1 when connected to ↑ , and exp( -S 2 ( χ { in } ℓ )) when connected to ↓ .
Figure 26 Partition function for exp( -S 2 (Ψ R )), with two possible dominant configurations. Note that | χ { in } ℓ , ↑〉 on the lower boundary gives a factor of 1 when connected to ↑ , and exp( -S 2 ( χ { in } ℓ )) when connected to ↓ .CC-BY · From the PDF

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Metadata

Authors
Akers, C., Engelhardt, N., Harlow, D., Penington, G., & Vardhan, S.
Venue
Journal of High Energy Physics
Publisher
Springer Science and Business Media LLC
Published
2024-06-24
Volume
2024
Issue
6
ISSN
1029-8479
DOI
10.1007/jhep06(2024)155
Cited by
75
References
32
Language
en
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CoChat. (2026). Citation check: The black hole interior from non-isometric codes and complexity. https://app.cochat.ai/papers/10.1007/jhep06(2024)155
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Akers, C., Engelhardt, N., Harlow, D., Penington, G., & Vardhan, S. (2024). The black hole interior from non-isometric codes and complexity. Journal of High Energy Physics. https://doi.org/10.1007/jhep06(2024)155