1. Absolute Separability and the Possible Spectra of Entanglement Witnesses Nathaniel Johnston — joint work with S. Arunachalam and V. Russo Mount Allison University Sackville, New Brunswick, Canada

2. The Separability Problem Recall: is separable if we can write for some Definition Given the separability problem is the problem of determining whether or not ρ is separable. This is a hard problem!

3. Separability Criteria Define a linear map Γ on by Separable states satisfy several one-sided tests, called separability criteria: Method 1: The partial transpose 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.

4. The Separability Problem Method 2: Everything else “Realignment criterion”: based on computing the trace norm of a certain matrix. “Choi map”: a positive map on 3-by-3 matrices that can be used to prove entanglement of certain 3 ⊗ 3 states. “Breuer–Hall map”: a positive map on 2n-by-2n matrices that can be used to prove entanglement of certain 2n ⊗ 2n states.

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

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

7. Absolute Separability Definition A quantum state is called absolutely separable if all quantum states with the same eigenvalues as ρ are separable. Theorem (Gurvits–Barnum, 2002) Let be a mixed state. If then ρ is separable, where is the Frobenius norm. But there are more!

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

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

10. Absolute Separability Absolutely PPT is completely solved (but complicated) Theorem (Hildebrand, 2007) A state is absolutely PPT if and only if Recall: separability = PPT when m = 2 and n ≤ 3 Thus is absolutely separable if and only if

11. Absolute Separability Can absolutely PPT states tell us more about absolute separability? Theorem (J., 2013) A state is absolutely separable if and only if it is absolutely PPT. Yes! obvious when n ≤ 3 weird when n ≥ 4

12. Absolute Separability What about absolute separability for when m, n ≥ 3? Question Is a state that is absolutely PPT necessarily “absolutely separable” as well? If YES: nice characterization of absolute separability If NO: there exist states that are “globally” bound entangled (can apply any global quantum gate to the state, always remains bound entangled) (weird!)

13. Absolute Separability Theorem (Arunachalam–J.–Russo, 2014) A state that is absolutely PPT is also necessarily: “absolutely realignable” “absolutely Choi map” “absolutely Breuer – Hall” “absolutely ” (you get the idea) Replace “separable” by “realignable”. (and other separability criteria too)

14. Absolute Separability These separability criteria are weaker than the PPT test in the “absolute” regime. Regular separability: sepPPT Absolute separability: abs. sepabs. PPT realignable Breuer–Hall abs. realignable abs. Breuer–Hall

15. Entanglement Witnesses An entanglement witness is a Hermitian operator such that for all separable σ, but for some entangled ρ. (W is a hyperplane that separates ρ from the convex set of separable states) Entanglement witnesses are “not as positive” as positive semidefinite matrices. How not as positive? How negative can their eigenvalues be? Some known results:, (Życzkowski et. al.)

16. Spectra of Entanglement Witnesses Theorem The eigenvalues of an entanglement witness satisfy the following inequalities: If n = 2 (qubits), these are the only inequalities! If n ≥ 3, there are other inequalities. (given any eigenvalues satisfying those inequalities, we can find an e.w.) (but I don’t know what they are)

17. Questions What are the remaining eigenvalue inequalities for entanglement witnesses when n ≥ 2? Do these inequalities hold for entanglement witnesses ? What about absolutely separability for when m, n ≥ 3? Don’t know!