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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!