One-shot partial decoupling & its application to complex quantum many-body systems

第38回量子情報技術研究会(QIT38)の発表資料

Transcript

1. One-shot partial decoupling & its application to complex quantum many-body systems Yoshifumi Nakata The University of Tokyo In collaboration with Eyuri Wakakuwa [1] and Masato Koashi [2] [1] The University of Electro-Communications [2] The University of Tokyo This work was supported by CREST, JST, Grant No. JPMJCR1671.
2. Outline 2 1. Introduction 2. Main results 3. Summary and Future direction – A decoupling approach to the black hole information paradox – Information paradox when BH has a symmetry – One-shot partial decoupling theorem
3. Black hole Information paradox Time | ۧ Ψ Alice Hawking radiation Life of a black hole Φ Maximally entangled state Quantum information approach by Hayden and Preskill [2007]. Bob How can Bob recover from and ??
4. Hayden-Preskill’s toy model (qubit-BH) Question: How large should be for ෡ Ψ to be approximately Ψ? = the quantum capacity with random encoding. Alice Reference Bob Black hole ( qubits) qubits qubits Time = HP’s scenario: If , ෡ Ψ ≈ Ψ.
5. HP’s solution: Assuming that the dynamics of the BH is sufficiently random, Hayden-Preskill’s toy model (qubit-BH) (evaporated qubits) Θ( ) 0 BH is completely evaporated. Bob can recover Ψ. (i.e. ෡ Ψ ≈ Ψ) Alice Reference Bob Black hole ( qubits) qubits qubits Time “A black hole is hardly black at all” The information dumped into the BH leaks out almost as quickly as possible. ⟹ Scrambling, OTOC, quantum duality….
6. Hayden-Preskill’s toy model (qubit-BH) Alice Reference Bob Black hole ( qubits) qubits qubits Time However, HP scenario is too naïve b/c is assumed to be fully random. ▪ What if a BH has a symmetry? ➢ A conservation law!! e.g.) charges, angular momentum, spins, etc… ➢ Irreducible representation: ℋ =⨁ (ℋ ⨂ ℋ ). ➢ Unitary respecting the symmetry: = ⨁ ( ⨂ ). ➢ In this talk, we consider the simplest Abelian case: = ⨁ . = 1 = 1 = 1 Multiplicity = HP’s scenario: = Symmetric scenario: ( : random) Irreps.
7. One-shot partial decoupling HP approach in brief 1. Assume that is fully random. 2. Use the one-shot decoupling theorem [Dupuis et.al. 2014]. Our approach: 1. Assume that = ⨁ ( : random). 2. Prove one-shot partial decoupling theorem ➢ This generalization is highly non-trivial, and is of independent interest. Information paradox of the BH with symmetry: When the black hole dynamics is = ⨁ ( : random), what state can Bob recover?
8. Alice Reference Bob qubits qubits qubits Black hole “Symmetric” HP’s toy model The original state: , A “block-dephased” state: , where and . Our solution (BH with symmetry): Assuming that the dynamics of the BH is = ⨁ , (evaporated qubits) Θ( ) 0 BH is completely evaporated. Bob can recover Ψ.. (≠ Ψ) = 1
9. Our solution (BH with symmetry): Assuming that the dynamics of the BH is = ⨁ , (evaporated qubits) Θ( ) 0 BH is completely evaporated. Bob can recover Ψ.. (≠ Ψ) = 1 “Symmetric” HP’s toy model A “block-dephased” state: , where and . Regardless of what symmetry the BH has, 1. the BH retains “quantum” info. (coherence) about the conserved quantity. 2. All other info. quickly leaks out. Alice Reference Bob qubits qubits qubits Black hole When does this leaks out? (so that Bob can fully recover ) Depends on the symmetry.
10. Example: rotational symmetry ▪ The # of up-spins is conserved. – Note that it’s not the (3)-symmetry (non-Abelian). Alice Reference Bob qubits qubits qubits Black hole The information paradox when the BH has rotational symmetry: Assuming that the dynamics of the BH is = 0 ⨁ 1 ⨁ 2 ⨁ ⋯ (evaporated qubits) ( ) 0 Bob can fully recover Ψ. ( ∈ (0,1]) “A black hole is hardly black at all” The information dumped into the BH leaks out almost as quickly as possible. ⟹ Scrambling, OTOC, quantum duality…. Bob can recover Ψ.. (≠ Ψ)
11. Summary When a black hole has a symmetry, how does the information leak out from the BH? ▪ Based on the one-shot partial decoupling. ▪ For any symmetry, all except “quantum” info. of the conserved quantity leak out quickly from the BH. ▪ To fully recover Ψ, qubits should leak out for the rotational symmetry. Alice Reference Bob qubits qubits qubits Black hole “A symmetric black hole can be black”
12. Future directions ▪ Non-abelian symmetries: = ⨁ ( ⨂ )? ▪ Applying the one-shot partial decoupling to Q. information theory? ➢ It’s already done (but time was limited today….) ➢ in progress…. (maybe, in next QIT) Alice Reference Bob qubits qubits qubits Black hole When a black hole has a symmetry, how does the information leak out from the BH?
13. Thank you Paper in preparation by Yoshifumi Nakata, Eyuri Wakakuwa, and Masato Koashi.