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Alpha-bits, Teleportation and Black Holes
Geoffrey Penington, Stanford University
ArXiv:1706.09434, ArXiv:1807.06041
Alpha-bits: Teleportation and Black Holes
Geoffrey Penington, Stanford University
ArXiv:1706.09434, ArXiv:1807.06041
Alpha-bits, Teleportation and Black Holes ArXiv:1706.09434, - - PowerPoint PPT Presentation
Alpha-bits, Teleportation and Black Holes ArXiv:1706.09434, - - PowerPoint PPT Presentation
Alpha-bits, Teleportation and Black Holes ArXiv:1706.09434, ArXiv:1807.06041 Geoffrey Penington, Stanford University Alpha-bits: Teleportation and Black Holes ArXiv:1706.09434, ArXiv:1807.06041 Geoffrey Penington, Stanford University Why
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Why should I care about this talk?
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Why should I care about this talk?
❑ Qubits are composite resources.
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Why should I care about this talk?
❑ Qubits are composite resources. ❑ Another resource (that you have never heard of) is more
fundamental than a qubit.
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Why should I care about this talk?
❑ Qubits are composite resources. ❑ Another resource (that you have never heard of) is more
fundamental than a qubit.
❑ Sending qubits at the quantum capacity does not exhaust the ability
- f a channel to send information.
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Why should I care about this talk?
❑ Qubits are composite resources. ❑ Another resource (that you have never heard of) is more fundamental
than a qubit.
❑ Sending qubits at the quantum capacity does not exhaust the ability of a
channel to send information.
❑ There is no need to use classical bits to do entanglement-distillation, state-
merging, remote state preparation, channel simulation or teleportation.
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Why should I care about this talk?
❑ Qubits are composite resources. ❑ Another resource (that you have never heard of) is more fundamental
than a qubit.
❑ Sending qubits at the quantum capacity does not exhaust the ability of a
channel to send information.
❑ There is no need to use classical bits to do entanglement-distillation, state-
merging, remote state preparation, channel simulation or teleportation.
❑ Quantum error correction in AdS/CFT is only approximate and bulk
- perators are state-dependent.
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Why should I care about this talk?
❑ Qubits are composite resources. ❑ Another resource (that you have never heard of) is more fundamental
than a qubit.
❑ Sending qubits at the quantum capacity does not exhaust the ability of a
channel to send information.
❑ There is no need to use classical bits to do entanglement-distillation, state-
merging, remote state preparation, channel simulation or teleportation.
❑ Quantum error correction in AdS/CFT is only approximate and bulk
- perators are state-dependent.
❑ It solves the black hole information paradox?
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Part I: Alpha-bits and Teleportation
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Quantum Communication Resource Inequalities
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Quantum Communication Resource Inequalities
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Quantum Communication Resource Inequalities
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Quantum Communication Resource Inequalities
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Quantum Communication Resource Inequalities
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Quantum Communication Resource Inequalities
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Quantum Communication Resource Inequalities
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Quantum Communication Resource Inequalities
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Quantum Communication Resource Inequalities
weakened version
- f qubits
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Quantum Communication Resource Inequalities
weakened version
- f qubits
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Quantum Communication Resource Inequalities
weakened version
- f qubits
asymptotic
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Quantum Communication Resource Inequalities
weakened version
- f qubits
asymptotic
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Quantum Communication Resource Inequalities
weakened version
- f qubits
asymptotic
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Quantum Communication Resource Inequalities
coherence communication
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What are zero-bits?
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What are zero-bits?
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What are zero-bits?
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What are zero-bits?
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What are zero-bits?
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What are zero-bits?
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What are zero-bits?
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What are zero-bits?
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What are zero-bits?
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What are zero-bits?
isometric
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What are zero-bits?
isometric
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What can you do with zero-bits?
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What can you do with zero-bits?
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What can you do with zero-bits?
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What can you do with zero-bits?
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What can you do with zero-bits?
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Definition of zero-bits
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Definition of qubits
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Definition of qubits
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Definition of qubits
What do we need to be true about the channel?
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Definition of qubits
What do we need to be true about the channel?
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Definition of qubits
What do we need to be true about the channel? Bob can always error correct so long as error correction is possible
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Definition of zero-bits
OK now what about zero-bits? Now Bob only has to be able to error correct any two-dimensional subspace
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Definition of zero-bits
Huh?
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Definition of zero-bits
Need to make definition approximate if zero-bits are to be different from qubits
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Definition of zero-bits
[Hayden, Winter 2012]
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Definition of zero-bits
[Hayden, Winter 2012]
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Definition of zero-bits
Why do I never get told anything interesting [Hayden, Winter 2012]
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Definition of zero-bits
Why do I never get told anything interesting [Hayden, Winter 2012]
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Definition of alpha-bits
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Definition of alpha-bits
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Definition of alpha-bits
“Subspace decoupling duality”
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Transmitting alpha-bits
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Transmitting alpha-bits
bigger smaller
Necessary condition to send alpha-bits. Also sufficient (with some subtleties about needing to use shared randomness and block coding).
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Transmitting alpha-bits
bigger smaller
Necessary condition to send alpha-bits. Also sufficient (with some subtleties about needing to use shared randomness and block coding).
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Transmitting alpha-bits
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Transmitting alpha-bits
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Transmitting alpha-bits
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Transmitting alpha-bits
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Transmitting alpha-bits
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Transmitting alpha-bits
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Alpha-bit resource equalities
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Alpha-bit resource equalities
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Alpha-bit resource equalities
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Alpha-bit resource equalities
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Alpha-bit resource equalities
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What is a cobit?
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What is a cobit?
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What is a cobit?
Alice keeps purification
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(Coherent) super-dense coding
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(Coherent) super-dense coding
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(Coherent) super-dense coding
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(Coherent) super-dense coding
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(Coherent) alpha-bit super-dense coding
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(Coherent) alpha-bit super-dense coding
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(Coherent) alpha-bit super-dense coding
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Zero-bits and ebits as fundamental resources
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Zero-bits and ebits as fundamental resources
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Alpha-bit Capacities
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Alpha-bit Capacities
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Amortised and entanglement-assisted capacities
Single letter!
Unconstrained by ebits and so only zero-bits matter. This explains why all entanglement-assisted capacities are proportional to mutual information.
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Amortised and entanglement-assisted capacities
Single letter!
Unconstrained by ebits and so only zero-bits matter. This explains why all entanglement-assisted capacities are proportional to mutual information.
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Amortised and entanglement-assisted capacities
Single letter!
Unconstrained by ebits and so only zero-bits matter. This explains why all entanglement-assisted capacities are proportional to mutual information.
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Further Applications
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Further Applications
Non-additivity of quantum capacity?
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Further Applications
Non-additivity of quantum capacity?
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Part II: Alpha-bits and Black Holes
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AdS/CFT
Duality between an ordinary quantum field theory, specifically a CFT, known as the ‘boundary’ theory, and quantum gravity in asymptotically anti-de Sitter space in one higher dimension, the ‘bulk’.
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AdS/CFT
Duality between an ordinary quantum field theory, specifically a CFT, known as the ‘boundary’ theory, and quantum gravity in asymptotically anti-de Sitter space in one higher dimension, the ‘bulk’. What does this have to do with quantum information? Also what does it have to do with our universe which is not anti-de Sitter space?
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The Ryu-Takayanagi formula
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The Ryu-Takayanagi formula
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The Ryu-Takayanagi formula
“Information = Geometry”
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Error correction and AdS/CFT
Bulk operators in the central region can be represented by a boundary
- perator acting only on any two of
the three boundary regions A, B and C
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Error correction and AdS/CFT
Bulk operators in the central region can be represented by a boundary
- perator acting only on any two of
the three boundary regions A, B and C (Operator algebra) quantum error correction
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Error correction and AdS/CFT
Bulk operators in the central region can be represented by a boundary
- perator acting only on any two of
the three boundary regions A, B and C (Operator algebra) quantum error correction Bulk states with some particular geometry = code subspace of larger boundary Hilbert space
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Entanglement Wedge Reconstruction
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Entanglement Wedge Reconstruction
Conventional techniques only allow
- perators in top and bottom region to
be reconstructed in region A.
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Entanglement Wedge Reconstruction
Conventional techniques only allow
- perators in top and bottom region to
be reconstructed in region A. BUT Ryu-Takayanagi formula suggests region A ‘knows’ about the area of the solid line
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Entanglement Wedge Reconstruction
Conventional techniques only allow
- perators in top and bottom region to
be reconstructed in region A. BUT Ryu-Takayanagi formula suggests region A ‘knows’ about the area of the solid line Entanglement wedge reconstruction conjecture: Actually the entire region between A and the RT surface can be reconstructed
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The DHW Proof of Entanglement Wedge Reconstruction
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The DHW Proof of Entanglement Wedge Reconstruction
RT formula implies: (JLMS 2016)
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The DHW Proof of Entanglement Wedge Reconstruction
RT formula implies: (JLMS 2016) Hence:
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The DHW Proof of Entanglement Wedge Reconstruction
RT formula implies: (JLMS 2016) Hence:
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The DHW Proof of Entanglement Wedge Reconstruction
RT formula implies: (JLMS 2016) Hence: Hence:
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The DHW Proof of Entanglement Wedge Reconstruction
RT formula implies: (JLMS 2016) Hence: Hence:
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The DHW Proof of Infinite Quantum Compression?
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The DHW Proof of Infinite Quantum Compression?
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The DHW Proof of Infinite Quantum Compression?
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The DHW Proof of Infinite Quantum Compression?
Hence:
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The DHW Proof of Infinite Quantum Compression?
Hence: Successfully compressed number of qubits by a factor of two
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The DHW Proof of Infinite Quantum Compression?
Hence: Successfully compressed number of qubits by a factor of two
ITERATE!
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The DHW Proof of Infinite Quantum Compression?
Hence: Successfully compressed number of qubits by a factor of two
ITERATE!
Exact zero-bits = exact qubits Approximate zero-bits ≠ approximate qubits
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The (slightly corrected) DHW Proof of Entanglement Wedge Reconstruction
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The (slightly corrected) DHW Proof of Entanglement Wedge Reconstruction
Theorem by Cedric Beny:
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The (slightly corrected) DHW Proof of Entanglement Wedge Reconstruction
Theorem by Cedric Beny: Bulk operators that we want to reconstruct must lie within the entanglement wedge of A, even for states that are entangled with a reference system
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Black holes and Alpha-bit Codes
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Black holes and Alpha-bit Codes
Large code space dimensions means that including a reference system can make a big difference
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Black holes and Alpha-bit Codes
Large code space dimensions means that including a reference system can make a big difference
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Black holes and Alpha-bit Codes
For pure black hole states, region a’ is always contained in the entanglement wedge of region A Large code space dimensions means that including a reference system can make a big difference
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Black holes and Alpha-bit Codes
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Black holes and Alpha-bit Codes
REMINDER:
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Black holes and Alpha-bit Codes
For states entangled with a reference system, region a’ is not always contained in the entanglement wedge if the reference system dimension REMINDER:
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Black holes and Alpha-bit Codes
Region A encodes the alpha-bits of region a’
For states entangled with a reference system, region a’ is not always contained in the entanglement wedge if the reference system dimension REMINDER:
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Black holes and Alpha-bit Codes
Region A encodes the alpha-bits of region a’
For states entangled with a reference system, region a’ is not always contained in the entanglement wedge if the reference system dimension We can only reconstruct operators if we know that the state lies in a sufficiently small subspace: the reconstruction is ‘state-dependent’ REMINDER:
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Tensor Network Toy Models: The HaPPY Code with a black hole
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Tensor Network Toy Models: The HaPPY Code with a black hole
Perfect tensor = unitary map from any three legs to the other three legs.
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Tensor Network Toy Models: The HaPPY Code with a black hole
Perfect tensor = unitary map from any three legs to the other three legs. Can ‘push’ operators past the tensor
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Tensor Network Toy Models: The HaPPY Code with a black hole
Perfect tensor = unitary map from any three legs to the other three legs. Can ‘push’ operators past the tensor
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Tensor Network Toy Models: The HaPPY Code with a black hole
Perfect tensor = unitary map from any three legs to the other three legs. Can ‘push’ operators past the tensor
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Tensor Network Toy Models: The HaPPY Code with a black hole
Perfect tensor = unitary map from any three legs to the other three legs. Can ‘push’ operators past the tensor
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Tensor Network Toy Models: The HaPPY Code with a black hole
Tensor network formed by tiling perfect tensors to create AdS space
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Tensor Network Toy Models: The HaPPY Code with a black hole
Tensor network formed by tiling perfect tensors to create AdS space Each tensor has a single ‘bulk’ leg
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Tensor Network Toy Models: The HaPPY Code with a black hole
Tensor network formed by tiling perfect tensors to create AdS space Each tensor has a single ‘bulk’ leg Black hole described by a random unitary with one large ‘bulk’ leg and many
- utward-flowing legs
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Alpha-bits in the HaPPY Code
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Alpha-bits in the HaPPY Code
Operators in the equivalent of region a’ can only be pushed to region A by pushing them through the black hole
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Alpha-bits in the HaPPY Code
Operators in the equivalent of region a’ can only be pushed to region A by pushing them through the black hole Only possible if the black hole bulk leg is sufficiently small
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Consequences of Alpha-bit Codes in AdS/CFT
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Consequences of Alpha-bit Codes in AdS/CFT
Qualitative features of AdS/CFT are only possible because the error correction is only approximate
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Consequences of Alpha-bit Codes in AdS/CFT
Qualitative features of AdS/CFT are only possible because the error correction is only approximate However the errors are tiny: they are non-perturbatively small
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Consequences of Alpha-bit Codes in AdS/CFT
Qualitative features of AdS/CFT are only possible because the error correction is only approximate However the errors are tiny: they are non-perturbatively small RT surface can be made state-dependent at scales much larger than the Planck scale
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Consequences of Alpha-bit Codes in AdS/CFT
Qualitative features of AdS/CFT are only possible because the error correction is only approximate However the errors are tiny: they are non-perturbatively small RT surface can be made state-dependent at scales much larger than the Planck scale Operator reconstruction is state-dependent. State-dependence is also believed (by some) to be necessary to describe operators behind a black hole horizon
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The Information Paradox
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The Information Paradox
Suppose we extract the Hawking radiation from a black hole into an auxiliary Hilbert space
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The Information Paradox
Suppose we extract the Hawking radiation from a black hole into an auxiliary Hilbert space The interior is still contained in the entanglement wedge of the black hole system , so long as we are before the Page time
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The Information Paradox
Suppose we extract the Hawking radiation from a black hole into an auxiliary Hilbert space The interior is still contained in the entanglement wedge of the black hole system , so long as we are before the Page time Hawking radiation is thermally entangled with and so the entanglement entropy increases (agrees with RT formula)
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The Information Paradox
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The Information Paradox
However we cannot reconstruct the interior for codespaces with
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The Information Paradox
However we cannot reconstruct the interior for codespaces with The interior becomes increasingly state-dependent as the black hole evaporates
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The Information Paradox
However we cannot reconstruct the interior for codespaces with The interior becomes increasingly state-dependent as the black hole evaporates Also implies reconstruction is only approximate
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After the Page Time
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After the Page Time
After the Page time, the interior is encoded in the Hawking radiation
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After the Page Time
After the Page time, the interior is encoded in the Hawking radiation The (new) Hawking radiation is now entangled with state-dependent degrees of freedom in . Hence the entanglement entropy will decrease. This agrees with the Ryu-Takayanagi formula and the Page curve.
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After the Page Time
After the Page time, the interior is encoded in the Hawking radiation The (new) Hawking radiation is now entangled with state-dependent degrees of freedom in . Hence the entanglement entropy will decrease. This agrees with the Ryu-Takayanagi formula and the Page curve. By carefully analysing the location in spacetime of the covariant Ryu-Takayanagi surface, we find that information falling into the black hole appears in the Hawking radiation after exactly the scrambling time (Hayden-Preskill).
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