Absolute Separability and the Possible Spectra of Entanglement

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 Separable states satisfy several one-sided tests, called separability criteria: Method 1: The partial transpose Define a linear map Γ on by 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 2 n-by-2 n matrices that can be used to prove entanglement of certain 2 n ⊗ 2 n states.

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

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

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 But there are more! then ρ is separable, where is the Frobenius norm.

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? Yes! weird when n ≥ 4 obvious when n ≤ 3 Theorem (J. , 2013) A state absolutely PPT. is absolutely separable if and only if it is

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

Absolute Separability These separability criteria are weaker than the PPT test in the “absolute” regime. Regular separability: Absolute separability: abs. realignable sep PPT abs. sep abs. PPT Breuer–Hall ll Bre. s b a Ha – r e u

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 • Some known results: be? , (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! (given any eigenvalues satisfying those inequalities, we can find an e. w. ) • If n ≥ 3, there are other inequalities. (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 m, n ≥ 3? Don’t know! when