Absolute Separability andthe 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 λ1 ≥ λ2 ≥ λ3 ≥ λ4 ≥ 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 <some other junk>” (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:

sep PPT

Absolute separability:

abs. sep abs. 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!