Download The Separability Problem and its Variants in

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 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?
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 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?
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
realignable
sep
Breuer–Hall
PPT
abs. sep
abs. PPT
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