The computational complexity of entanglement detection

1. The computational complexity of entanglement detection Patrick Hayden Stanford University Based on 1211.6120, 1301.4504 and 1308.5788 With Gus Gutoski, Daniel Harlow, Kevin Milner and Mark Wilde

2. How hard is entanglement detection? • Given a matrix describing a bipartite state, is the state separable or entangled? • NP-hard for d x d, promise gap 1/poly(d) [Gurvits’04 + Gharibian ‘10] • Quasipolynomial time for constant gap [Brandao et al. ’10] • Probably not the right question for large systems. • Given a description of a physical process for preparing a quantum state (i.e. quantum circuit), is the state separable or entangled? • Variants: • Pure versus mixed • State versus channel • Product versus separable • Choice of distance measure (equivalently, nature of promise)

3. Why ask? • Provides a natural set of complete problems for many widely studied classes in quantum complexity • Personal motivation: • Quantum gravity! • Personal frustration at inability to find a “fast scrambler” • Possible implications for the black hole firewall problem

4. Entanglement detection: The platonic ideal NO α α YES β

5. Some complexity classes… P / BPP / BQP P / BPP / BQP = QIP(0) NP / MA / QMA NP / MA / QMA = QIP(1) AM / QIP(2) Cryptographic variant: Zero-knowledge Verifier, in YES instances, can “simulate” prover ZK / SZK / QSZK = QSZK(2) QMA(2) QIP = QIP(3) QIP = QIP(3) = PSPACE [Jain et al. ‘09]

6. Pure state circuit Product output? Trace distance BQP-complete Mixed state circuit Product output? Trace distance QSZK-complete Results: States Mixed state circuit Separable output? 1-LOCC distance (1/poly) NP-hard QSZK-hard In QIP(2)

7. Isometric channel Separable output? 1-LOCC distance QMA-complete Isometric channel Separable output? Trace distance QMA(2)-complete Results: Channels Noisy channel Separable output? 1-LOCC distance QIP-complete

8. The computational universe through the entanglement lens

9. Pure state circuit Product output? Trace distance BQP-complete Mixed state circuit Product output? Trace distance QSZK-complete Results: States Mixed state circuit Separable output? 1-LOCC distance NP-hard QSZK-hard In QIP(2)

10. Baby steps: Detecting pure product states

11. Baby steps:Detecting pure product states

12. 1. QPROD-PURE-STATE is in BQP

13. 2. QPROD-PURE-STATE is BQP-hard

14. 2. QPROD-PURE-STATE is BQP-hard

15. Pure state circuit Product output? Trace distance BQP-complete Mixed state circuit Product output? Trace distance QSZK-complete Results: States Mixed state circuit Separable output? 1-LOCC distance NP-hard QSZK-hard In QIP(2)

16. Jaunty stroll:Detecting mixed product states

17. Jaunty stroll:Detecting mixed product states

18. Jaunty stroll:Detecting mixed product states

19. Completeness: YES instances

20. Soundness: NO instances

21. Zero-knowledge (YES instances):Verifier can simulate prover output

22. QPROD-STATE is QSZK-hard

23. Reduction from co-QSD to QPROD-STATE

24. QPROD-STATE and Quantum Error Correction QPROD-STATE: QEC: R: “System” A: “Reference” B: “Environment” These are the SAME problem!

25. Cloning, Black Holes and Firewalls Quantum information appears to be cloned U V Spacetime structure prevents comparison of the clones (?) Singularity Msg Hawking Radiation Is unitarity safe? Horizon 2007: H & Preskill study old black holes. Radial light rays: (Only just) safe In Out 2012: Almheiri et al. considerφto be entanglement with late time Hawking photon Firewalls! [Page, Preskill, Susskind 93][Susskind, Thorlacius, Uglum 93]

26. Cloning, Black Holes and Firewalls If infalling Bob is to experience the vacuum as he crosses the horizon, φ must be in infalling Hawking partner. U V Singularity If black hole entropy is to decrease, φ must be present in early Hawking radiation. φ φ Early Hawking Radiation But has cloning really occurred? Do two copies of φ exist? To test, Bob would need to decode (QEC) the early Hawking radiation: QSZK-hard but BH lifetime is poly(# qubits). Horizon Radial light rays: In Out 2012: Almheiri et al. considerφto be entanglement with late time Hawking photon Firewalls! [Page, Preskill, Susskind 93][Susskind, Thorlacius, Uglum 93]

27. Pure state circuit Product output? Trace distance BQP-complete Mixed state circuit Product output? Trace distance QSZK-complete Results: States Mixed state circuit Separable output? 1-LOCC distance NP-hard QSZK-hard In QIP(2)

28. Jogging:Detecting mixed separable states ρAB close to separable iff it has a suitable k-extension for sufficiently large k. [BCY ‘10] Send R to the prover, who will try to produce the k-extension. Use phase estimation to verify that the resulting state is a k-extension.

29. Summary • Entanglement detection provides a unifying paradigm for parametrizing quantum complexity classes • Tunable knobs: • State versus channel • Pure versus mixed • Trace norm versus 1-LOCC norm • Product versus separable • Implications for the (worst case) complexity of decoding quantum error correcting codes • Provides challenge to the black hole firewall argument

30. Entanglement detection: The platonic ideal NO α α YES β

31. Complexity of QSEP-STATE? Who knows?

33. Soundness: NO instances