1. T he Separability Problem and its Variants in Quantum Entanglement Theory Nathaniel Johnston Institute for Quantum Computing University of Waterloo

2. Overview What is Quantum Entanglement? The Separability Problem The Bound Entanglement Problem The Separability from Spectrum Problem

3. Overview What is Quantum Entanglement? Weird physical phenomenon Linear algebra works! The Separability Problem The Bound Entanglement Problem The Separability from Spectrum Problem

4. Particles can be “linked” Always get correlated measurement results That’s weird! Tensor product of finite- dimensional vector spaces Tensors of rank > 1 exist That’s obvious! Physicist Mathematician What is Quantum Entanglement?

5. Pure quantum state: with i.e., with Dual (row) vector: Inner product:

6. What is Quantum Entanglement? Tensor product: = = =

7. What is Quantum Entanglement? Outer product tensor product: Obtained via “stacking columns”:

8. What is Quantum Entanglement? Definition A pure state is separable if it can be written as Otherwise, it is entangled. rank 1 rank 2 (thus entangled)

9. What is Quantum Entanglement? Pure state (again): Rank 1 Trace 1 Positive semidefinite equivalent Mixed quantum state: Trace 1 Positive semidefinite equivalent

10. What is Quantum Entanglement? Definition A mixed state is separable if it can be written as with each separable. Otherwise, it is entangled. Equivalent: for some positive semidefinite is the “maximally mixed” state. convex combination separable

11. Overview What is Quantum Entanglement? The Separability Problem How to determine separability? Positive matrix-valued maps Funky matrix norms The Bound Entanglement Problem The Separability from Spectrum Problem

12. The Separability Problem Recall: is separable if we can write for some Definition Given the separability problem is the problem of determining whether ρ is separable or entangled. This is a hard problem! This is an NP-hard problem! (Gurvits, 2003)

13. The Separability Problem All states Separable states ρ

14. The Separability Problem Define a linear map Γ on by Method 1: “Partial” transpose In matrices:

15. The Separability Problem Apply Γ to a separable state: is positive semidefinite Not true for some entangled states: which has eigenvalues 1, 1, 1, and -1. We say that ρ has positive partial transpose (PPT).

16. The Separability Problem All states Separable states ρ PPT states

17. The Separability Problem Theorem (Størmer, 1963; Woronowicz, 1976; Peres, 1996) Let be a quantum state. If ρ is separable then Furthermore, the converse holds if and only if mn ≤ 6. Separability problem is completely solved when mn ≤ 6 Higher dimensions?

18. The Separability Problem Given, define a linear map on by Method 1.1: Positive maps In matrices:

19. The Separability Problem Transpose map: Theorem (Horodecki 3, 1996) A quantum state is separable if and only if for all positive maps Definition is positive if whenever positive semidefinite

20. The Separability Problem All states Separable states ρ Transpose map

21. The Separability Problem The problem: Coming up with positive maps is hard! Proving that a map is positive is NP-hard Current status: Dozens of papers Only a handful of known positive maps

22. The Separability Problem Method 2: Norms Definition The operator norm and trace norm of a matrix are defined by: where are the singular values of X.

23. The Separability Problem All states ≈ unit ball of Separable states ≈ unit ball of

24. The Separability Problem Definition Given define the S(1)-norm via dual Separable version of

25. The Separability Problem Theorem (Rudolph, 2000) Let be the dual of the S(1)-norm, defined by A quantum state ρ is separable if and only if

26. The Separability Problem All states ≈ unit ball of Separable states ≈ unit ball of

27. The Separability Problem The goal: derive bounds for “Swap” operator: “Realignment” map: = 1 if ρ separable because

28. ρ σ The Separability Problem Theorem (Chen–Wu, 2003) If then ρ is entangled. The goal: Come up with more bounds on Lower bounds entanglement Upper bounds separability

29. Overview What is Quantum Entanglement? The Separability Problem The Bound Entanglement Problem Not all entanglement is “useful” Partial transpose is awesome The Separability from Spectrum Problem

30. Bound Entanglement Can we turn mixed entanglement into pure entanglement? ρ ρ ρ

31. Bound Entanglement Not always! Theorem (Horodecki 3, 1998) If the quantum state has positive partial transpose then it is bound entangled (i.e., many copies of ρ can not be locally converted into an entangled pure state). Question: Are there more? Or is this “iff”?

32. Bound Entanglement All states Separable states PPT states = Bound entangled states Bound entangled states

33. Bound Entanglement Let’s phrase the problem mathematically! Recall: for we have Similarly, “Rank 1” and “full rank” versions of same norm

34. Bound Entanglement We now want the “rank 2” version of this norm: Also need the “maximally entangled state”: standard basis of

35. Bound Entanglement Theorem Define a family of projections P 1, P 2, … recursively as follows: Then there exists non-positive partial transpose bound entanglement (more or less) if and only if up to minor technical details (e.g., n ≥ 4 only)

36. Bound Entanglement What do we know so far? Big gap! n = 4, k = 2: equality when k = 1

37. Overview What is Quantum Entanglement? The Separability Problem The Bound Entanglement Problem The Separability from Spectrum Problem We only know eigenvalues Want to determine separable/entangled

38. Separability from Spectrum Only given eigenvalues of ρ Can we prove ρ is entangled/separable? No: diagonal separable Prove entangled? arbitrary eigenvalues, but always separable

39. Separability from Spectrum Sometimes: If all eigenvalues are equal then Prove separable? a separable decomposition Only given eigenvalues of ρ Can we prove ρ is entangled/separable?

40. Separability from Spectrum Can also prove separability if ρ is close to Theorem (Gurvits–Barnum, 2002) Let be a mixed state. If then ρ is separable, where is the Frobenius norm. Frobenius norm: eigenvalues of ρ

41. Separability from Spectrum All states Separable states Gurvits–Barnum ball

42. Separability from Spectrum Definition A quantum state is called separable from spectrum if all quantum states with the same eigenvalues as ρ are separable. only depends on eigenvalues of ρ States in the Gurvits–Barnum ball are separable from spectrum: But there are more!

43. Separability from Spectrum All states Separable states Gurvits–Barnum ball Separable from spectrum

44. Separability from Spectrum The case of two qubits (i.e., m = n = 2) was solved long ago: Theorem (Verstraete–Audenaert–Moor, 2001) A state is separable from spectrum if and only if What about higher-dimensional systems? eigenvalues, sorted so that λ 1 ≥ λ 2 ≥ λ 3 ≥ λ 4 ≥ 0

45. Separability from Spectrum Replace “separable” by “positive partial transpose”. Definition A quantum state is called positive partial transpose (PPT) from spectrum if all quantum states with the same eigenvalues as ρ are PPT.

46. Separability from Spectrum All states Separable states Gurvits–Barnum ball Separable from spectrum PPT from spectrum

47. Separability from Spectrum PPT from spectrum is completely solved (but complicated) Theorem (Hildebrand, 2007) A state is PPT from spectrum if and only if Recall: separability = PPT when m = 2, n ≤ 3 Thus is separable from spectrum if and only if

48. Separability from Spectrum Can PPT from spectrum tell us more about separability from spectrum? Theorem (J., 2013) A state is separable from spectrum if and only if it is PPT from spectrum. Yes! obvious when n ≤ 3 weird when n ≥ 4

49. Separability from Spectrum All states Separable states Gurvits–Barnum ball Separable from spectrum PPT from spectrum =

50. Separability from Spectrum Sketch of proof. Lemma If then ρ is separable. Write as a block matrix: ρ becomes “more positive” as B becomes small compared to A and C

51. Separability from Spectrum Want: every PPT from spectrum to satisfy hypotheses of Lemma. Lemma If then ρ is separable. Not true!

52. Separability from Spectrum Instead: for every PPT from spectrum there exists a 2×2 unitary matrix U such that satisfies hypotheses of Lemma. Lemma If then ρ is separable. works for

53. Separability from Spectrum Lemma If then ρ is separable. Define Then some intermediate value of t works

54. Separability from Spectrum What about separability from spectrum for when m, n ≥ 3? Don’t know! All states Separable states Gurvits–Barnum ball Separable from spectrum PPT from spectrum = All states Separable states Gurvits–Barnum ball Separable from spectrum PPT from spectrum