ExU collaboration

Linking quantum information to physics
Yoshifumi Nakata
The University of Tokyo & JST PRESTO

March 7th @ Annual meeting of ExU collaboration

Outline of this talk
A journey of a thousand miles begins with a single step

Information, Chaos, and Black Holes in the Quantum regime.

Outline of this talk
A journey of a thousand miles begins with a single step

Information, Chaos, and Black Holes in the Quantum regime.
1. Information and Chaos in the classical regime.

Outline of this talk
A journey of a thousand miles begins with a single step

Information, Chaos, and Black Holes in the Quantum regime.
1. Information and Chaos in the classical regime.

2. The Hayden-Preskill thought experiment.

Outline of this talk
A journey of a thousand miles begins with a single step

Information, Chaos, and Black Holes in the Quantum regime.
1. Information and Chaos in the classical regime.

2. The Hayden-Preskill thought experiment.
3. Beyond the Hayden-Preskill.

Outline of this talk
A journey of a thousand miles begins with a single step

Information, Chaos, and Black Holes in the Quantum regime.
1. Information and Chaos in the classical regime.

2. The Hayden-Preskill thought experiment.
3. Beyond the Hayden-Preskill.
4. Conclusion and Outlooks.

1. Information and Chaos

in the classical regime.

1. Quantum Error Correction (QEC)
2. Chaotic dynamics and Error Correction in classical3. Quantum chaos and QEC?

1. Information and Chaos

in the classical regime.

1. Quantum Error Correction (QEC)
2. Chaotic dynamics and Error Correction in classical3. Quantum chaos and QEC?

Quantum Error Correction
An introduction in a nutshell

What is Quantum Error Correction (QEC)?
 QEC is a method to effectively cancel quantum noise by ENCODING and DECODING.
Today, we consider to use
chaotic dynamics for encoding!

Encoding

Alice

Decoding

Noise

CPTP map

CPTP map

Bob

Time
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1. Information and Chaos

in the classical regime.

1. Quantum Error Correction (QEC)
2. Chaotic dynamics and Error Correction in classical3. Quantum chaos and QEC?

Chaos and Error Correction in classicalHow can we use chaotic dynamics for correcting errors?

What is chaos in the classical regime?
 Chaos is a dynamical system highly sensitive to initial conditions.
𝑡=𝑇
𝑡=𝑇

𝑡=𝑇

𝑡=0

Simple dynamics

𝑡=0

Chaotic dynamics

QEC in a nutshell

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Chaos and Error Correction in classicalHow can we use chaotic dynamics for correcting errors?

Chaos and Error Correction: intuition 1
 Chaos is a dynamical system highly sensitive to initial conditions.
𝑡=𝑇

0

1

𝑡=0

1. Write 0 or 1 into the system.
2. “Time evolution” of the information by the simple dynamics.
3. Due to the uncertainty induced by the noise, the info. gets unclear.
4. The huge overlap makes it difficult to decode the info. (0 or 1).
→ Failure of decoding!

Simple dynamics

0

or

1

QEC in a nutshell

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Chaos and Error Correction in classicalHow can we use chaotic dynamics for correcting errors?

Chaos and Error Correction: intuition 2
 Chaos is a dynamical system highly sensitive to initial conditions.
𝑡=𝑇
𝑡=𝑇

0
1

𝑡=0

1. Write 0 or 1 into the system.
2. “Time evolution” of the information by the chaotic dynamics.
3. Due to the uncertainty induced by the noise, the info. gets unclear.
4. No overlap makes it easy to decode the info. (0 or 1).
→ Success of decoding!

Chaotic dynamics

0

or

1

QEC in a nutshell

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Chaos and Error Correction in classicalHow can we use chaotic dynamics for correcting errors?

Important remark on Chaos and Error Correction
 The sensitivity of chaos to the initial condition makes it easy to read out information!
One needs to know all the details of the dynamics!
◼ In physics, it is common to assume that the details of the system are our of control.
Hence, chaos is unpredictable.
◼ In quantum information, everything is under control.
➢ Chaotic dynamics is a powerful tool to encode information!

0

or

1

QEC in a nutshell

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1. Information and Chaos

in the classical regime.

1. Quantum Error Correction (QEC)
2. Chaotic dynamics and Error Correction in classical3. Quantum chaos and QEC?

Chaos and Error Correction in classicalHow can we use chaotic dynamics for correcting errors?

Chaos in the Quantum regime 1
 “Quantum chaos” is non-trivial.
➢(At least) two inequivalent defs of “quantum chaos”.

By Prof. Hashimoto
3rd Colloquium of ExU(available on the ExU website)

QEC in a nutshell

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Chaos and Error Correction in classicalHow can we use chaotic dynamics for correcting errors?

Chaos in the Quantum regime 2
 “Quantum chaos” is non-trivial.
➢ We model the “chaotic dynamics” by a Haar random unitary (a.k.a. circular unitary ensemble).

A Haar random unitary = Circular Unitary Ensemble (CUE) A random unitary uniformly distributed over a unitary group

Unitary group➢ A typical dynamics of quantum chaos.

A Haar random unitary= uniform distribution of unitaries

QEC in a nutshell

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Chaos and Error Correction in classicalHow can we use chaotic dynamics for correcting errors?

Chaos and Quantum Error Correctionin quantum

 “Quantum chaos” is non-trivial.

➢ We model the “chaotic dynamics” by a Haar random unitary (a.k.a. circular unitary ensemble).

 “QEC by chaotic dynamics” is non-trivial.
1. Unitarity preserves the distance between two states.
2. We need to recover an unknown quantum state (e.g. |𝛹 ۧ = 𝛼 0 + 𝛽 1 ), not 0 or 1.

 It turns out that quantum chaotic dynamics is useful for QEC.

[Dupuis et al, CMP ‘14]

➢ Finally, established by the “one-shot decoupling theorem”.

A Haar random unitary

As an example, we see the Hayden-Preskill thought experiment.

QEC in a nutshell

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Outline of this talk
A journey of a thousand miles begins with a single step

Information, Chaos, and Black Holes in the Quantum regime.
1. Information and Chaos in the classical regime.

2. The Hayden-Preskill thought experiment.
3. Beyond the Hayden-Preskill.

4. Conclusion and Outlooks.

19

How?

The Hayden-Preskill thought experiment
Information-theoretic toy model of quantum black holes

Information paradox of quantum black holes (BHs) Information paradox is a puzzle about the Hawking radiation.
Whole universe

Unitary dynamics

Whole universe

Hawking radiation

𝜚

BH
|Ψۧ

after a long timeBH|Ψۧ

(thermal)should containthe information of |Ψۧ“Information” of the BH

If you look at the beginning and the end, you will notice that

Apparent contradiction
a pure state |Ψۧ becomes a mixed state 𝜚 by unitary dynamics.

The state |Ψۧ should be recoverable from the radiation (if we are in the quantum side)

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The Hayden-Preskill thought experiment
Information-theoretic toy model of quantum black holes

How can we recover quantum information from the radiation?
 In 2007, Hayden and Preskill addressed this question based on a qubit-toy model of a “BH”.

P. Hayden

J. Preskill

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The Hayden-Preskill thought experiment
Information-theoretic toy model of quantum black holes

Information paradox and the Hyden-Preskill thought experimentradiationℓ qubits|𝜉 ۧ

Old radiation

Q info
Time

|Ψۧ

BH
𝑁 qubits

Unitary𝑈

BH

DecoderNew

𝑁 + 𝑘 − ℓ qubits

Remaining BH

 A qubit-toy model of a quantum “BH”.
|Ψۧ?
A quantum BH of 𝑵 qubitsመ𝐴

New Hawking radiation ofℓ qubits

Unitary
dynamics
Try to recover |𝚿ۧQuantum information of𝑘 qubits

Old Hawking
radiation

To decode the 𝒌-qubit quantum info., how large should ℓ be?

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The Hayden-Preskill thought experiment
Information-theoretic toy model of quantum black holes

Hayden-Preskill thought experiment 1
To decode the 𝒌-qubit quantum info., how large should ℓ be?
1. What is quantum information?
➢ See the newsletter of ExU (available at the end of March).

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The Hayden-Preskill thought experiment
Information-theoretic toy model of quantum black holes

Hayden-Preskill thought experiment 1
To decode the 𝒌-qubit quantum info., how large should ℓ be?
1. What is quantum information?
➢ See the newsletter of ExU (available at the end of March).

Recovery error:
How does this scale with ℓ?
2. How do we model the unitary dynamics 𝑈 of the BH?
➢ The unitary 𝑈 is assumed to be fully chaotic (a Haar random unitary).

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The Hayden-Preskill thought experiment
Information-theoretic toy model of quantum black holes

Hayden-Preskill thought experiment 2
If the dynamic of the BH is chaotic (Haar random),

Here,

, and

[Hayden & Preskill, JHEP, ‘07]is the collision entropyof the initial black hole 𝐵𝑖𝑛 .
𝑁−𝐻2 (𝜉) If ℓ ≫ ℓth = 𝑘 +,𝑘-qubitquantum
informationisrecoverable.
2– Entropy of the initial BH determines the threshold ℓth .

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The Hayden-Preskill thought experiment
Information-theoretic toy model of quantum black holes

Hayden-Preskill thought experiment 3
If the dynamic of the BH is chaotic (Haar random),[Hayden & Preskill, JHEP, ‘07]

𝑁−𝐻2 (𝜉) If ℓ ≫ ℓth = 𝑘 +,𝑘-qubitquantum
informationisrecoverable.
2– Entropy of the initial BH determines the threshold ℓth .

Independent of 𝑁

– If the BH is initially pure (𝑇 = 0), 𝐻2 𝜉 = 0 and ℓth = 𝑘 + 𝑁/2.
– If the BH is initially completely mixed (𝑇 = ∞), 𝐻2 𝜉 = 𝑁 and ℓth = 𝑘.

The 𝑘-qubit info is recoverable when O(𝑘) qubits are radiated.

The Hayden-Preskill recovery

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The Hayden-Preskill thought experiment
Information-theoretic toy model of quantum black holes

Hayden-Preskill & Quantum Error Correction
 The Hayden-Preskill recovery is a special case of QEC by chaotic dynamics.
➢ Noise = Partial trace over the remaining BH.
➢ Encoding = Adding a state |𝜉ۧ and applying a chaotic (Haar random) unitary 𝑈.

Decoding

We “lost” the remaining BH.
Noise = Partial trace

Noise
Encoding

Time
Hayden-Preskill

QEC in a nutshell

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The Hayden-Preskill thought experiment
Information-theoretic toy model of quantum black holes

Hayden-Preskill & Quantum Error Correction
 The Hayden-Preskill recovery is a special case of QEC by chaotic dynamics.
➢ Noise = Partial trace over the remaining BH.
➢ Encoding = Adding a state |𝜉ۧ and applying a chaotic (Haar random) unitary 𝑈.

The Hayden-Preskill thought experiment is a good playground overquantum error correction, quantum chaos, and a quantum black hole!!

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Outline of this talk
A journey of a thousand miles begins with a single step

Information, Chaos, and Black Holes in the Quantum regime.
1. Information and Chaos in the classical regime.

2. The Hayden-Preskill thought experiment.

3. Beyond the Hayden-Preskill.
4. Conclusion and Outlooks.

29

3. Beyond the Hayden-Preskill.

Beyond the Hayden-Preskill
Beyond the Hayden-Preskill 1

Open questions in the Hayden-Preskill thought experiment
The Hayden-Preskill thought experiment is a good playground overquantum error correction, quantum chaos, and a quantum black hole!!

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Beyond the Hayden-Preskill
Beyond the Hayden-Preskill 1

Open questions in the Hayden-Preskill thought experiment
The Hayden-Preskill thought experiment is a good playground overquantum error correction, quantum chaos, and a quantum black hole!!
1. No decoder is given.
➢ How can we actually decode information?

[YN & Koashi, in prep]

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Decoding the Hayden-Preskill
Beyond the Hayden-Preskill 1

How do we decode information?
 In the original work by Hayden&Preskill, no decoder was provided.
➢ How can we explicitly construct a decoder?

 Two decoders in the literature, and their relations to physics.
➢ Teleportation-type decoder for special cases, related to OTOC [Yoshida & Kitaev ’17].
➢ Petz decoders, related to spacetime geometry [Penington, et al’19].
A decoder is a key link to bridge quantum information to physics.

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Decoding the Hayden-Preskill
Beyond the Hayden-Preskill 1

New results by the Classical-to-Quantum decoder We propose an intuitive decoder, called a Classical-to-Quantum decoder.
➢ It is universal since it works for general noise.
➢ It is related to the Renyi-2 entropy.
 A chaotic BH simultaneously emits classical and quantum information.
➢ There is NO difference b/t C and Q info” in the QEC by chaotic dynamics.
Recovery error for
quantum information

Recovery error forclassical information

N.A.

Hayden & Preskill ‘07
Our results

Here,

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Beyond the Hayden-Preskill
Beyond the Hayden-Preskill 1

Open questions in the Hayden-Preskill thought experiment
The Hayden-Preskill thought experiment is a good playground overquantum error correction, quantum chaos, and a quantum black hole!!
1. No decoder is given.
➢ How can we actually decode information?

[YN & Koashi, in prep]

A decoder gives a “link” b/t quantum information and physics.

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Beyond the Hayden-Preskill
Beyond the Hayden-Preskill 1

Open questions in the Hayden-Preskill thought experiment
The Hayden-Preskill thought experiment is a good playground overquantum error correction, quantum chaos, and a quantum black hole!!
1. No decoder is given.
➢ How can we actually decode information?

[YN & Koashi, in prep]

A decoder gives a “link” b/t quantum information and physics.

2. No energy conservation due to the Haar assumption.
➢ What if something is conserved?

[YN, Wakakuwa & Koashi, ‘19&’20]

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The HP with energy conservation
Beyond the Hayden-Preskill 2

Haar random dynamics “scrambles” everything Hayden & Preskill assume a fully chaotic dynamics = a Haar random unitary.

➢ Such a unitary 𝑈Haar scrambles everything = nothing (e.g. energy) is conserved.

 What if a conservation law is taken into account?
➢ Energy conservation (in some sense) was heuristically discussed in [Yoshida,
PRD100:086001 ’19] and [Liu, PRR 2:043164, ’20].
➢ Comprehensive analysis was done in [YN, Wakakuwa, and Koashi,’19 &’20].
Hayden & Preskill

Our model of dynamics

𝑗: conserved quantity

Haar random

Haar random in each sector

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The HP with energy conservation
Beyond the Hayden-Preskill 2

What changes would occur in the HP recovery? 1
 Two big changes by symmetry.
1. Delay of the threshold: the threshold ℓ𝑡ℎ changes to ℓ∗𝑡ℎ .
2. Information remnant: some information cannot be recovered from the radiation.

Delay[Hayden & Preskill, ‘07]

Recovery errorΔ
1Info. remnant

No conservation

0

ℓ𝑡ℎ

ℓ∗𝑡ℎ

Size of new radiationℓ

A formal expression is given by entropic quantities

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The HP with energy conservation
Beyond the Hayden-Preskill 2

What changes would occur in the HP recovery? 2
 Two big changes by symmetry.
1. Delay of the threshold: the threshold ℓ𝑡ℎ changes to ℓ∗𝑡ℎ .
➢ In the case of energy conservation, it is characterized by the head capacity of the BH.
• 𝐸: Energy of the BH• 𝑆(𝐸): Entropy of the initial BH• 𝐶𝑉 : Heat capacity of the BH

Delay

Recovery errorΔ
1Info. remnant

No conservation

0

ℓ𝑡ℎ

ℓ∗𝑡ℎ

Size of new radiationℓ

A formal expression is given by entropic quantities

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The HP with energy conservation
Beyond the Hayden-Preskill 2

What changes would occur in the HP recovery? 3
 Two big changes by symmetry.
1. Delay of the threshold: the threshold ℓ𝑡ℎ changes to ℓ∗𝑡ℎ .
2. Information remnant: a certain amount of information cannot be recovered from the radiation.
➢ Amount of the remnant ∝ (symmetry breaking of the initial BH)−1 ≈ Variance of energy

Interesting relations between information and physics
Delay

Recovery errorΔ
1Info. remnant

No conservation

0

ℓ𝑡ℎ

ℓ∗𝑡ℎ

Size of new radiationℓ

A formal expression is given by entropic quantities

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Beyond the Hayden-Preskill
Beyond the Hayden-Preskill 1

Open questions in the Hayden-Preskill thought experiment
The Hayden-Preskill thought experiment is a good playground overquantum error correction, quantum chaos, and a quantum black hole!!
1. No decoder is given.
➢ How can we actually decode information?

[YN & Koashi, in prep]

A decoder gives a “link” b/t quantum information and physics.

2. No energy conservation due to the Haar assumption.
➢ What if something is conserved?

[YN, Wakakuwa & Koashi, ‘19&’20]

Non-trivial “links” b/t quantum information and physics are revealed.

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Beyond the Hayden-Preskill
Beyond the Hayden-Preskill 1

Open questions in the Hayden-Preskill thought experiment
The Hayden-Preskill thought experiment is a good playground overquantum error correction, quantum chaos, and a quantum black hole!!
1. No decoder is given.
➢ How can we actually decode information?

[YN & Koashi, in prep]

A decoder gives a “link” b/t quantum information and physics.

2. No energy conservation due to the Haar assumption.
➢ What if something is conserved?

[YN, Wakakuwa & Koashi, ‘19&’20]

Non-trivial “links” b/t quantum information and physics are revealed.

3. No Hamiltonian (or Lagrangian).
➢ What if the dynamics is given by a Hamiltonian dynamics?
[YN & Tezuka, in prep]

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The Hayden-Preskill recovery in SYK model
Beyond the Hayden-Preskill 3

Hayden-Preskill recovery by a Hamiltonian dynamics
 No Hamiltonian (or Lagrangian) so far.
𝑒 −𝑖𝐻𝑡➢ The analysis is basically based on a Haar random unitary 𝑈.
➢ What if the dynamics is given by a Hamiltonian, 𝑈 = 𝑒 −𝑖𝐻𝑡 ?

This is a non-trivial question.
 A Haar random unitary cannot be generated by a time-indep. Hamiltonian.
➢ No formal proof (to my knowledge), but highly believed.

……however, who knows the Hamiltonian of a black hole?

In collaboration with M. Tezuka

Sachedev-Ye-Kitaev (SYK) model
as a quantum dual!

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Outline of this talk
A journey of a thousand miles begins with a single step

Information, Chaos, and Black Holes in the Quantum regime.
1. Information and Chaos in the classical regime.

2. The Hayden-Preskill thought experiment.

3. Beyond the Hayden-Preskill.

4. Conclusion and Outlooks.
44

Conclusion and outlooks
Towards the collaboration over QI, condensed-matter, and high energy physics

Conclusion
Error Correction, Chaos, and Black Holes in the Quantum regime.
 Chaotic dynamics (a Haar random unitary) is useful for correcting quantum noise.
 The Hayden-Preskill thought experiment is a canonical toy model about quantum error
correction, quantum chaos, and a quantum black hole.
1. Decoding the Hayden-Preskill [YN & Koashi, in prep]
2. Energy conservation and the Hayden-Preskill recovery
3. The Hayden-Preskill recovery and SYK models
1.

aaa[YN, Wakakuwa & Koashi, ‘19&’20][YN & Tezuka, in prep]
2.

aaaa

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Conclusion and outlooks
Towards the collaboration over QI, condensed-matter, and high energy physics

Open questions
Error Correction, Chaos, and Black Holes in the Quantum regime.
1. The Hayden-Preskill thought experiment has a little flavor of black holes….
➢ It’s about quantum error correction, quantum chaos, and a quantum black hole.
➢ More flavor of quantum black holes?

2. Quantum phase vs the Hayden-Preskill recovery?
➢ The HP recovery may succeed/fail in chaotic, MBL, and integrable phases.
➢ Characterize quantum phases by the recovery?

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for your attention

48

Classical-to-Quantum decoder
Decoding Hayden-Preskill

Classical-to-Quantum decoder１
Classical-to-Quantum decoder𝐴መ
Decoder𝒟𝐴መ

{𝑀𝑍,𝑗 }

𝑋𝑗=

Correcting errors by 𝑀Z

𝐴መ𝑉𝑋

A coherent meas. constructed from 𝑀𝑋

NewOld
radiation radiation

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