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

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!

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.

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.

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

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?

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!

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

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.

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

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

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!)

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)

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

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.)

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)

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!