Symmetric random unitary -its application to the black hole information paraeox-

Slides for the 2018 Mini-Workshop on Design Theory and Quantum Information Shanghai University, Shanghai, China, June 13-15, 2018.

Transcript

1. Symmetric random unitary - its application to the black hole information paradox - Yoshifumi Nakata The University of Tokyo
2. Outline 2/32 Part I. Random unitary in quantum information science 1. Haar random unitary in QI 2. Unitary designs in QI Part II. Symmetric random unitary 1. Symmetric random unitary from the representation theory 2. The information paradox when a black hole has a symmetry Part III. Conclusion As an organizer, to promote discussions b/t math. and Q.I. As a researcher working in the intersection b/t math. and Q.I.
3. Part 1. Random unitary in quantum information science Randomness meets Quantum!!
4. A Haar random unitary Unitary group The uniform distribution of Haar random unitary (Unique unitarily invariant measure) A Haar random untiary is the unique unitarily invariant probability measure H on the unitary group (). Namely, (A) H( ) = 1, (B) for any ∈ () and any (Borel) set ⊆ , H = H = H . 4/32
5. Applications of a Haar random unitary Haar random unitary is very useful in QIP and in fundamental physics. In QIP 1. Q. communication [Hayden et.al. ‘07] 2. Q. computation [Boixo et.al. (aka google) ‘16] 3. Benchmarking Q. gates [Knill et.al. ‘08] 4. Q. sensing [Oszmaniec et.al. ‘16] 5. Q. randomized algorithm Quantum communication Quantum computation In fundamental physics 1. Disordered systems 2. Pre-thermalization [Reimann ‘16] 3. Q. black holes [Hayden&Preskill ‘07] 4. Q. chaos -OTOC- [Roberts&Yoshida ‘16] 5. Quantum duality? [Kitaev ‘15] 5/32
6. ▪ Quantum communication – Two people in different places want to communicate in a quantum manner. Application 1. Haar random unitary in Q. communication Haar random unitary Coherent State merging (aka Fully Quantum Slepian-Wolf protocol) Mother protocol Entanglement distillation Noisy Quantum Teleportation Noisy Superdense coding Father protocol Quantum capacity Entanglement-assisted classical capacity Quantum reverse Shannon theory Quantum multiple access capacities Quantum broadcast channels Distributed compression Family tree of information protocols See Hayden’s tutorial talk in QIP2011 6/32
7. ▪ Quantum computer HOPEFULLY outperforms classical computer! Application 2. Haar random unitary in Q. computation Quantum computational supremacy [Preskill ‘12]: Can a tiny Q. computer beat the best-available supercomputer? Random unitary on ~100 qubits will do the job [Boixo et.al. ’16: Bouland et.al.‘18]. − Factorization algorithm, Hidden subgroup algorithm etc…. − How can we show Quantum ≫ Classical? Faster than known classical algorithms. v.s. By google ~ 50 qubits ~ 1 peta byte 7/32
8. Haar random unitary –a pie in the sky- ▪ Haar is everywhere in QIP and physics. − However, a Haar random unitary is “a pie in the sky”. ▪ The basic unit of QIP is a qubit = a unit vector in ℂ2. – qubits are described by a Hilbert space ℋ = (ℂ2)⨂. – But, Haar random unitaries on (2) are not easy to generate by Q. computers, requiring at least (2) q. gates. ▪ Approximate versions are needed. − Unitary designs come into the play! One qubit by google 8/32
9. Physics meets Mathematics Random unitary Fundamental physics Quantum commun. Quantum comp. Rand. benchmarking Quantum sensing Quantum randomized algorithm Disordered systems Pre-thermalization Quantum black holes Quantum chaos Quantum duality Quantum information science Unitary designs Mathematics comes into the play. Physics is interested in. Applications Approximation • Necessary and sufficient randomness? • Unified framework? • How to generate by Q. computers? − “Efficiency” • Constructing exact ones? 9/32
10. Part 2. Symmetric random unitary - its application to the black hole science - Randomness meets Black holes!! In collaboration with Eyuri Wakakuwa [1] and Masato Koashi [2] [1] The University of Electro-Communications [2] The University of Tokyo
11. ▪ So far, Haar random unitaries on ℋ = (ℂ2)⨂. ▪ Physical systems often have a symmetry. – Rotational symmetry, U(1) symmetry, etc… – Tensor product representation of a group G. – Irreducible decomposition: Random unitary with a symmetry multiplicity multiplicity ℋ = ⨁ (ℋ ⨂ ℋ ) = 1 Hilbert space “commutable” with the symmetry. ℋ = ⨁ (ℋ )⨁ = 1 e.g.) Spin-spin coupling (spin-1/2 × 3): ℋ = 4 ⨁ 2 ⨁ 2 4-dimensional irrep. (dim ℋ1 = 4, dim ℋ1 = 1) 2-dim. irreps with multiplicity 2. (dim ℋ2 = 2, dim ℋ2 = 2) dim(ℋ ) = 11/32
12. ▪ So far, Haar random unitaries on ℋ = (ℂ2)⨂. ▪ Physical systems often have a symmetry. – Rotational symmetry, U(1) symmetry, etc… – Tensor product representation of a group G. – Irreducible decomposition: ▪ “Symmetric” random unitaries. – = ⨁ ( ⨂ ), where is the Haar random unitary on ℋ . Random unitary with a symmetry multiplicity multiplicity ℋ = ⨁ (ℋ ⨂ ℋ ) = 1 Hilbert space “commutable” with the symmetry. ℋ = ⨁ (ℋ )⨁ = 1 = 1 e.g.) Spin-spin coupling (spin-1/2 × 3): ℋ = 4 ⨁ 2 ⨁ 2 { = 1/2, = 1/2 , = 1/2, = −1/2 } { = 1/2, = 1/2 , = 1/2, = −1/2 } 2 dim(ℋ ) = 12/32
13. ▪ Symmetry is the essence of physics!! – Black holes, pre-thermalization, quantum chaos, etc…. ▪ Interesting in Q. information theory. – To “hybrid” quantum and classical information theory. Why symmetric random unitaries? We’ll talk about this, today ;-) = ⨁ ( ⨂ ), where is the Haar random unitary on ℋ . Symmetric random unitary (a group G is given) = 1 To be explored…. 13/32
14. ▪ Can be generalized to “symmetric” unitary designs – How can we implement by Q. circuit? Q. Circuits for “symmetric unitary designs”? Quantum circuits for symmetric unitary designs? – Applying symmetric gates? – Generators of Lie algebra? – Symmetric Hamiltonian + Suzuki-Trotter decomposition? = ⨁ ( ⨂ ), where is the Haar random unitary on ℋ . Symmetric random unitary (a group G is given) = 1 14/32
15. Let’s get into the Black hole!!
16. Black hole Information paradox Stephen Hawking “A black hole eventually evaporates” = Hawking radiation 16/32
17. Black hole Information paradox Time | ۧ Ψ Alice Hawking radiation Life of a black hole Φ Maximally entangled state What happens to “| ۧ Ψ ” after the black hole is completely evaporated? 1. Completely lost? (natural but not likely…) 2. In the Hawking radiation? 17/32
18. Black hole Information paradox Time | ۧ Ψ Alice Hawking radiation Life of a black hole Φ Maximally entangled state Bob Can Bob recover from and ?? Quantum information approach by Hayden and Preskill [2007]. 18/32
19. Hayden-Preskill’s toy model (qubit-BH) Alice Reference Bob Black hole ( ≫ qubits) qubits qubits Time Assumption: is a Haar random unitary. Question: How large should be for ෡ Ψ to be approximately Ψ? is a Haar random unitary. Same situation as “quantum capacity” with random encoding. 19/32
20. HP’s solution: Assuming that the dynamics is Haar random, there exists a CPTP map 1→, with high probability, such that ෡ Ψ − Ψ 1 ≈ (2−). Hayden-Preskill’s toy model (qubit-BH) Alice Reference Bob Black hole ( ≫ qubits) qubits qubits Time If ≫ , ෡ Ψ ≈ Ψ. 20/32
21. HP’s solution: If the dynamics of the BH is Haar random, with high probability, Hayden-Preskill’s toy model (qubit-BH) (evaporated qubits) ( ) 0 BH is completely evaporated. Bob can recover Ψ. (i.e. ෡ Ψ ≈ Ψ) Alice Reference Bob Black hole ( ≫ qubits) qubits qubits Time A BH is an “information mirror”… “A black hole is hardly black at all”. 21/32
22. Hayden-Preskill’s toy model (qubit-BH) Alice Reference Bob Black hole ( ≫ qubits) qubits qubits Time A HP approach was pioneering, however….. ▪ HP scenario is too naïve b/c is assumed to be Haar random. ▪ What if a BH has a symmetry? ➢ Conserved quantities, e.g., charges, angular momentum, spins, etc… ➢ Symmetric random unitary: = ⨁ ( ⨂ ). ➢ For simplicity, we consider Abelian symmetries: = ⨁ . = 1 = 1 is a Haar random unitary. HP’s scenario: = ⨁ ( are Haar random.) Symmetric case: 22/32 Question: How large should be for ෡ Ψ to be approximately Ψ?
23. How to approach the problem? Due to the Haar random unitary . HP approach in detail: 1. Assume that is Haar random. 2. Use the one-shot decoupling theorem. Information paradox of the BH with symmetry: When = ⨁ ( : Haar random), what happens? = 1 Our approach: 1. Assume that = ⨁ ( : Haar random). 2. Use the one-shot partial decoupling theorem. ➢ Non-trivial generalization of the decoupling theorem. 23/32
24. Alice Reference Bob qubits qubits qubits Black hole “Symmetric” HP’s toy model Our solution (BH with symmetry): Assuming that the dynamics of the BH is = ⨁ , there exists a CPTP map 1→, with high probability, such that ෡ Ψ − Ψ.. 1 = 2− , where Ψ.. is a “block-dephased” state. The original state: , = 1 A “block-dephased” state: , where and . 24/32
25. 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 When can Bob get Ψ recovered??? Alice Reference Bob qubits qubits qubits Black hole 25/32
26. 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 Θ() Bob can fully recover Ψ. Bob can recover Ψ.. (≠ Ψ) A BH is an “information mirror”… “A black hole is hardly black at all”. 26/32
27. Summary of the symmetric BHs When a black hole has a symmetry, how does the information leak out from the BH? ▪ Based on the one-shot partial decoupling. ▪ To fully recover Ψ, Θ() qubits should leak out when the BH has a rotational symmetry. Alice Reference Bob qubits qubits qubits Black hole “A symmetric black hole can be black” 27/32
28. Part 3. Future of randomness in quantum Randomness will make the future!! 28
29. Physics meets Mathematics Random unitary Fundamental physics Quantum commun. Quantum comp. Rand. benchmarking Quantum sensing Quantum randomized algorithm Disordered systems Pre-thermalization Quantum black holes Quantum chaos Quantum duality Quantum information science Unitary designs Mathematics comes into the play. Physics is responsible. Applications Approximation 29/32 Let’s develop quantum information with a mathematical approach!!
30. Possible future direction: symmetry Symmetric Random unitary Fundamental physics Quantum commun. Quantum comp. Rand. benchmarking Quantum sensing Quantum randomized algorithm Disordered systems Pre-thermalization Quantum chaos Quantum duality Quantum information science Symmetric Unitary designs? Applications Approximation Symmetric Quantum black holes 30/32
31. Thank you Paper of Part II, in preparation by Yoshifumi Nakata, Eyuri Wakakuwa, and Masato Koashi.
33. How to approach the problem? Due to the Haar random unitary . HP approach in detail: 1. Assume that is fully random. 2. Use the one-shot decoupling. 3. A decoder from the Uhlmann’s trick. Information paradox of the BH with symmetry: When = ⨁ ( : Haar random), what happens? = 1 33/32
34. HP approach in brief How to approach the problem? with high probability, where : state representation of → and min ( |)⨂ is the conditional min-entropy. CPTP map S R E Haar random One-shot decoupling theorem [Dupuis et.al. 2014] ▪ If min ( |)⨂ ≫ 1, →(†) ≈ ⨂ with high probability. ▪ When and are fully decoupled, the existence of a perfect decoder follows from the uniqueness of purification (Uhlmann’s theorem). and are “decoupled”. 34/32
35. Our approach in brief How to approach the problem? One-shot partial decoupling theorem with high probability, where : state representation of → and min ( |)∗ is the conditional min-entropy. CPTP map S R E Symmetric random unitary Symmetric random unitary Sum of product states (i.e. (special) separable state) for decoupling Katrhi-Rao product ∗ (“block-wise” tensor product) Tensor product for decoupling 35/32
36. Partial decoupling theorem Information paradox of the BH with symmetry: When = ⨁ ( : Haar random), what happens? = 1 ▪ If min ( |)෤ ∗ ≫ 1, w.h.p. and are “partially” decoupled. Our approach in brief 1. When and are partially decoupled, what can Bob decode? ➢ By Uhlmann’s theorem in a clever way. 2. How to calculate min ( |)෤ ∗ ? min ( |)෤ ∗ ≥ min ( |)⨂ One-shot partial decoupling theorem with high probability, where : state representation of → and min ( |)∗ is the conditional min-entropy. 36/32
37. Example 1. ℤ2 -symmetry ▪ The parity of the state (= the # of up spins is even/odd) is conserved. e.g.) Ising model with transversal magnetic field etc. ▪ We can show that no dephasing happen!! Alice Reference Bob qubits qubits qubits Black hole The information paradox when the BH has ℤ -symmetry: Assuming that the dynamics of the BH is = even ⨁ odd , (evaporated qubits) ( ) 0 BH is completely evaporated. Bob can fully recover Ψ!! A ℤ2 -symmetric black hole is hardly black at all. 37/32