(Pairwise) Completely Positive MatricesNecessary and Su�cient Conditions
(Absolute) Separability and Entanglement
Pairwise Completely Positive Matrices
and Quantum Entanglement
Nathaniel Johnston and Olivia MacLean
2019 Meeting of the International Linear Algebra Society
Rio de Janeiro, Brazil
July 10, 2019
N. Johnston PCP Matrices
(Pairwise) Completely Positive MatricesNecessary and Su�cient Conditions
(Absolute) Separability and Entanglement
Completely Positive MatricesPairwise Completely Positive Matrices
Completely Positive Matrices
De�nition
A matrix X ∈ Mn(R) is called completely positive (CP) if there
exists some entrywise non-negative B ∈ Mn,m(R) (with marbitrary) such that
X = BBT .
Every CP matrix is positive semide�nite and entrywise
non-negative.
Converse holds if (and only if) n ≤ 4.
Determining complete positivity is NP-hard.
Studied for decades, important in convex optimization.
N. Johnston PCP Matrices
(Pairwise) Completely Positive MatricesNecessary and Su�cient Conditions
(Absolute) Separability and Entanglement
Completely Positive MatricesPairwise Completely Positive Matrices
Completely Positive Matrices
De�nition
A matrix X ∈ Mn(R) is called completely positive (CP) if there
exists some entrywise non-negative B ∈ Mn,m(R) (with marbitrary) such that
X = BBT .
Every CP matrix is positive semide�nite and entrywise
non-negative.
Converse holds if (and only if) n ≤ 4.
Determining complete positivity is NP-hard.
Studied for decades, important in convex optimization.
N. Johnston PCP Matrices
(Pairwise) Completely Positive MatricesNecessary and Su�cient Conditions
(Absolute) Separability and Entanglement
Completely Positive MatricesPairwise Completely Positive Matrices
Completely Positive Matrices
De�nition
A matrix X ∈ Mn(R) is called completely positive (CP) if there
exists some entrywise non-negative B ∈ Mn,m(R) (with marbitrary) such that
X = BBT .
Every CP matrix is positive semide�nite and entrywise
non-negative.
Converse holds if (and only if) n ≤ 4.
Determining complete positivity is NP-hard.
Studied for decades, important in convex optimization.
N. Johnston PCP Matrices
(Pairwise) Completely Positive MatricesNecessary and Su�cient Conditions
(Absolute) Separability and Entanglement
Completely Positive MatricesPairwise Completely Positive Matrices
Completely Positive Matrices
De�nition
A matrix X ∈ Mn(R) is called completely positive (CP) if there
exists some entrywise non-negative B ∈ Mn,m(R) (with marbitrary) such that
X = BBT .
Every CP matrix is positive semide�nite and entrywise
non-negative.
Converse holds if (and only if) n ≤ 4.
Determining complete positivity is NP-hard.
Studied for decades, important in convex optimization.
N. Johnston PCP Matrices
(Pairwise) Completely Positive MatricesNecessary and Su�cient Conditions
(Absolute) Separability and Entanglement
Completely Positive MatricesPairwise Completely Positive Matrices
Completely Positive Matrices
De�nition
A matrix X ∈ Mn(R) is called completely positive (CP) if there
exists some entrywise non-negative B ∈ Mn,m(R) (with marbitrary) such that
X = BBT .
Every CP matrix is positive semide�nite and entrywise
non-negative.
Converse holds if (and only if) n ≤ 4.
Determining complete positivity is NP-hard.
Studied for decades, important in convex optimization.
N. Johnston PCP Matrices
(Pairwise) Completely Positive MatricesNecessary and Su�cient Conditions
(Absolute) Separability and Entanglement
Completely Positive MatricesPairwise Completely Positive Matrices
Pairwise Completely Positive Matrices
De�nition
An ordered pair of matrices (X ,Y ) ∈ Mn(C)×Mn(C) is called
pairwise completely positive (PCP) if there exist matrices
A,B ∈ Mn,m(C) (with m arbitrary) such that
X = (A� B)(A� B)∗ and Y = (A� A)(B � B)∗.
Above, ��� is the Hadamard (entrywise) product.
If (X ,Y ) is PCP then X is positive semide�nite and Y is
entrywise non-negative.
X is CP if and only if (X ,X ) is PCP (not quite trivial).
N. Johnston PCP Matrices
(Pairwise) Completely Positive MatricesNecessary and Su�cient Conditions
(Absolute) Separability and Entanglement
Completely Positive MatricesPairwise Completely Positive Matrices
Pairwise Completely Positive Matrices
De�nition
An ordered pair of matrices (X ,Y ) ∈ Mn(C)×Mn(C) is called
pairwise completely positive (PCP) if there exist matrices
A,B ∈ Mn,m(C) (with m arbitrary) such that
X = (A� B)(A� B)∗ and Y = (A� A)(B � B)∗.
Above, ��� is the Hadamard (entrywise) product.
If (X ,Y ) is PCP then X is positive semide�nite and Y is
entrywise non-negative.
X is CP if and only if (X ,X ) is PCP (not quite trivial).
N. Johnston PCP Matrices
(Pairwise) Completely Positive MatricesNecessary and Su�cient Conditions
(Absolute) Separability and Entanglement
Completely Positive MatricesPairwise Completely Positive Matrices
Pairwise Completely Positive Matrices
De�nition
An ordered pair of matrices (X ,Y ) ∈ Mn(C)×Mn(C) is called
pairwise completely positive (PCP) if there exist matrices
A,B ∈ Mn,m(C) (with m arbitrary) such that
X = (A� B)(A� B)∗ and Y = (A� A)(B � B)∗.
Above, ��� is the Hadamard (entrywise) product.
If (X ,Y ) is PCP then X is positive semide�nite and Y is
entrywise non-negative.
X is CP if and only if (X ,X ) is PCP (not quite trivial).
N. Johnston PCP Matrices
(Pairwise) Completely Positive MatricesNecessary and Su�cient Conditions
(Absolute) Separability and Entanglement
Completely Positive MatricesPairwise Completely Positive Matrices
Pairwise Completely Positive Matrices
De�nition
An ordered pair of matrices (X ,Y ) ∈ Mn(C)×Mn(C) is called
pairwise completely positive (PCP) if there exist matrices
A,B ∈ Mn,m(C) (with m arbitrary) such that
X = (A� B)(A� B)∗ and Y = (A� A)(B � B)∗.
Above, ��� is the Hadamard (entrywise) product.
If (X ,Y ) is PCP then X is positive semide�nite and Y is
entrywise non-negative.
X is CP if and only if (X ,X ) is PCP (not quite trivial).
N. Johnston PCP Matrices
(Pairwise) Completely Positive MatricesNecessary and Su�cient Conditions
(Absolute) Separability and Entanglement
Necessary ConditionsAnswer in Small DimensionsSu�cient Conditions
Necessary Conditions
Showing (X ,Y ) is (not) PCP is hard. Let's develop one-sided tests!
Theorem (†)If (X ,Y ) ∈ Mn(C)×Mn(C) is PCP then:
a) X is positive semide�nite.
b) Y is real and entrywise non-negative.
c) xi ,i = yi ,i for all 1 ≤ i ≤ n.
d) |xi ,j |2 ≤ yi ,jyj ,i for all 1 ≤ i , j ≤ n.
e) X is �more diagonal� than Y : ‖X‖1 − ‖X‖tr ≤ ‖Y ‖1 − ‖Y ‖tr.
‖ · ‖tr is the trace norm, ‖ · ‖1 is the entrywise 1-norm
N. Johnston PCP Matrices
(Pairwise) Completely Positive MatricesNecessary and Su�cient Conditions
(Absolute) Separability and Entanglement
Necessary ConditionsAnswer in Small DimensionsSu�cient Conditions
Necessary Conditions
Showing (X ,Y ) is (not) PCP is hard. Let's develop one-sided tests!
Theorem (†)If (X ,Y ) ∈ Mn(C)×Mn(C) is PCP then:
a) X is positive semide�nite.
b) Y is real and entrywise non-negative.
c) xi ,i = yi ,i for all 1 ≤ i ≤ n.
d) |xi ,j |2 ≤ yi ,jyj ,i for all 1 ≤ i , j ≤ n.
e) X is �more diagonal� than Y : ‖X‖1 − ‖X‖tr ≤ ‖Y ‖1 − ‖Y ‖tr.
‖ · ‖tr is the trace norm, ‖ · ‖1 is the entrywise 1-norm
N. Johnston PCP Matrices
(Pairwise) Completely Positive MatricesNecessary and Su�cient Conditions
(Absolute) Separability and Entanglement
Necessary ConditionsAnswer in Small DimensionsSu�cient Conditions
Necessary Conditions
Showing (X ,Y ) is (not) PCP is hard. Let's develop one-sided tests!
Theorem (†)If (X ,Y ) ∈ Mn(C)×Mn(C) is PCP then:
a) X is positive semide�nite.
b) Y is real and entrywise non-negative.
c) xi ,i = yi ,i for all 1 ≤ i ≤ n.
d) |xi ,j |2 ≤ yi ,jyj ,i for all 1 ≤ i , j ≤ n.
e) X is �more diagonal� than Y : ‖X‖1 − ‖X‖tr ≤ ‖Y ‖1 − ‖Y ‖tr.
‖ · ‖tr is the trace norm, ‖ · ‖1 is the entrywise 1-norm
N. Johnston PCP Matrices
(Pairwise) Completely Positive MatricesNecessary and Su�cient Conditions
(Absolute) Separability and Entanglement
Necessary ConditionsAnswer in Small DimensionsSu�cient Conditions
Necessary Conditions
Showing (X ,Y ) is (not) PCP is hard. Let's develop one-sided tests!
Theorem (†)If (X ,Y ) ∈ Mn(C)×Mn(C) is PCP then:
a) X is positive semide�nite.
b) Y is real and entrywise non-negative.
c) xi ,i = yi ,i for all 1 ≤ i ≤ n.
d) |xi ,j |2 ≤ yi ,jyj ,i for all 1 ≤ i , j ≤ n.
e) X is �more diagonal� than Y : ‖X‖1 − ‖X‖tr ≤ ‖Y ‖1 − ‖Y ‖tr.
‖ · ‖tr is the trace norm, ‖ · ‖1 is the entrywise 1-norm
N. Johnston PCP Matrices
(Pairwise) Completely Positive MatricesNecessary and Su�cient Conditions
(Absolute) Separability and Entanglement
Necessary ConditionsAnswer in Small DimensionsSu�cient Conditions
Necessary Conditions
Showing (X ,Y ) is (not) PCP is hard. Let's develop one-sided tests!
Theorem (†)If (X ,Y ) ∈ Mn(C)×Mn(C) is PCP then:
a) X is positive semide�nite.
b) Y is real and entrywise non-negative.
c) xi ,i = yi ,i for all 1 ≤ i ≤ n.
d) |xi ,j |2 ≤ yi ,jyj ,i for all 1 ≤ i , j ≤ n.
e) X is �more diagonal� than Y : ‖X‖1 − ‖X‖tr ≤ ‖Y ‖1 − ‖Y ‖tr.
‖ · ‖tr is the trace norm, ‖ · ‖1 is the entrywise 1-norm
N. Johnston PCP Matrices
(Pairwise) Completely Positive MatricesNecessary and Su�cient Conditions
(Absolute) Separability and Entanglement
Necessary ConditionsAnswer in Small DimensionsSu�cient Conditions
Answer in Small Dimensions
When n = 2, the �rst four necessary conditions of the previous
theorem are actually su�cient as well:
Theorem
A pair (X ,Y ) ∈ M2(C)×M2(C) is PCP if and only if conditions(a)�(d) of Theorem (†) hold.
Analogous to X ∈ M4(R) being CP if and only if it is positive
semide�nite and entrywise non-negative.
The �if� direction fails for PCP matrices whenever n ≥ 3:
X =
1 1 1
1 1 1
1 1 1
, Y =
1 2 1/21/2 1 2
2 1/2 1
.
N. Johnston PCP Matrices
(Pairwise) Completely Positive MatricesNecessary and Su�cient Conditions
(Absolute) Separability and Entanglement
Necessary ConditionsAnswer in Small DimensionsSu�cient Conditions
Answer in Small Dimensions
When n = 2, the �rst four necessary conditions of the previous
theorem are actually su�cient as well:
Theorem
A pair (X ,Y ) ∈ M2(C)×M2(C) is PCP if and only if conditions(a)�(d) of Theorem (†) hold.
Analogous to X ∈ M4(R) being CP if and only if it is positive
semide�nite and entrywise non-negative.
The �if� direction fails for PCP matrices whenever n ≥ 3:
X =
1 1 1
1 1 1
1 1 1
, Y =
1 2 1/21/2 1 2
2 1/2 1
.
N. Johnston PCP Matrices
(Pairwise) Completely Positive MatricesNecessary and Su�cient Conditions
(Absolute) Separability and Entanglement
Necessary ConditionsAnswer in Small DimensionsSu�cient Conditions
Answer in Small Dimensions
When n = 2, the �rst four necessary conditions of the previous
theorem are actually su�cient as well:
Theorem
A pair (X ,Y ) ∈ M2(C)×M2(C) is PCP if and only if conditions(a)�(d) of Theorem (†) hold.
Analogous to X ∈ M4(R) being CP if and only if it is positive
semide�nite and entrywise non-negative.
The �if� direction fails for PCP matrices whenever n ≥ 3:
X =
1 1 1
1 1 1
1 1 1
, Y =
1 2 1/21/2 1 2
2 1/2 1
.
N. Johnston PCP Matrices
(Pairwise) Completely Positive MatricesNecessary and Su�cient Conditions
(Absolute) Separability and Entanglement
Necessary ConditionsAnswer in Small DimensionsSu�cient Conditions
Answer in Small Dimensions
When n = 2, the �rst four necessary conditions of the previous
theorem are actually su�cient as well:
Theorem
A pair (X ,Y ) ∈ M2(C)×M2(C) is PCP if and only if conditions(a)�(d) of Theorem (†) hold.
Analogous to X ∈ M4(R) being CP if and only if it is positive
semide�nite and entrywise non-negative.
The �if� direction fails for PCP matrices whenever n ≥ 3:
X =
1 1 1
1 1 1
1 1 1
, Y =
1 2 1/21/2 1 2
2 1/2 1
.
N. Johnston PCP Matrices
(Pairwise) Completely Positive MatricesNecessary and Su�cient Conditions
(Absolute) Separability and Entanglement
Necessary ConditionsAnswer in Small DimensionsSu�cient Conditions
Su�cient Conditions
Let's develop a one-sided test in the other direction too. To do so,
we need...
De�nition
The comparison matrix of X ∈ Mn(C) is the matrix
M(X ) =
|x1,1| −|x1,2| · · · −|x1,n|−|x2,1| |x2,2| · · · −|x2,n|
......
. . ....
−|xn,1| −|xn,2| · · · |xn,n|
.
N. Johnston PCP Matrices
(Pairwise) Completely Positive MatricesNecessary and Su�cient Conditions
(Absolute) Separability and Entanglement
Necessary ConditionsAnswer in Small DimensionsSu�cient Conditions
Su�cient Conditions
Let's develop a one-sided test in the other direction too. To do so,
we need...
De�nition
The comparison matrix of X ∈ Mn(C) is the matrix
M(X ) =
|x1,1| −|x1,2| · · · −|x1,n|−|x2,1| |x2,2| · · · −|x2,n|
......
. . ....
−|xn,1| −|xn,2| · · · |xn,n|
.
N. Johnston PCP Matrices
(Pairwise) Completely Positive MatricesNecessary and Su�cient Conditions
(Absolute) Separability and Entanglement
Necessary ConditionsAnswer in Small DimensionsSu�cient Conditions
Su�cient Conditions
We now recall a su�cient condition for complete positivity:
Theorem (Drew�Johnson�Loewy (1994))
If X ∈ Mn(R) is positive semide�nite, entrywise non-negative, and
such that M(X ) is positive semide�nite, then X is CP.
We have a natural generalization for PCP matrices:
Theorem
If (X ,Y ) ∈ Mn(C)×Mn(C) satis�es conditions (a)�(d) of
Theorem (†), and M(X ) is positive semide�nite, then (X ,Y ) is
PCP.
N. Johnston PCP Matrices
(Pairwise) Completely Positive MatricesNecessary and Su�cient Conditions
(Absolute) Separability and Entanglement
Necessary ConditionsAnswer in Small DimensionsSu�cient Conditions
Su�cient Conditions
We now recall a su�cient condition for complete positivity:
Theorem (Drew�Johnson�Loewy (1994))
If X ∈ Mn(R) is positive semide�nite, entrywise non-negative, and
such that M(X ) is positive semide�nite, then X is CP.
We have a natural generalization for PCP matrices:
Theorem
If (X ,Y ) ∈ Mn(C)×Mn(C) satis�es conditions (a)�(d) of
Theorem (†), and M(X ) is positive semide�nite, then (X ,Y ) is
PCP.
N. Johnston PCP Matrices
(Pairwise) Completely Positive MatricesNecessary and Su�cient Conditions
(Absolute) Separability and Entanglement
Necessary ConditionsAnswer in Small DimensionsSu�cient Conditions
Su�cient Conditions
We now recall a su�cient condition for complete positivity:
Theorem (Drew�Johnson�Loewy (1994))
If X ∈ Mn(R) is positive semide�nite, entrywise non-negative, and
such that M(X ) is positive semide�nite, then X is CP.
We have a natural generalization for PCP matrices:
Theorem
If (X ,Y ) ∈ Mn(C)×Mn(C) satis�es conditions (a)�(d) of
Theorem (†), and M(X ) is positive semide�nite, then (X ,Y ) is
PCP.
N. Johnston PCP Matrices
(Pairwise) Completely Positive MatricesNecessary and Su�cient Conditions
(Absolute) Separability and Entanglement
Necessary ConditionsAnswer in Small DimensionsSu�cient Conditions
Su�cient Conditions
We now recall a su�cient condition for complete positivity:
Theorem (Drew�Johnson�Loewy (1994))
If X ∈ Mn(R) is positive semide�nite, entrywise non-negative, and
such that M(X ) is positive semide�nite, then X is CP.
We have a natural generalization for PCP matrices:
Theorem
If (X ,Y ) ∈ Mn(C)×Mn(C) satis�es conditions (a)�(d) of
Theorem (†), and M(X ) is positive semide�nite, then (X ,Y ) is
PCP.
N. Johnston PCP Matrices
(Pairwise) Completely Positive MatricesNecessary and Su�cient Conditions
(Absolute) Separability and Entanglement
Separability and EntanglementConnection with PCP MatricesAbsolute Separability
Separability and Entanglement
De�nition
A matrix Z ∈ Mn(C)⊗Mn(C) is called separable if there exist
positive semide�nite {Xi}, {Yi} ∈ Mn(C) such that
Z =∑i
Xi ⊗ Yi .
Otherwise, Z is called entangled.
Separable matrices are positive semide�nite.
Characterizing these matrices is one of the central problems in
quantum information theory.
Determining separability is NP-hard.
N. Johnston PCP Matrices
(Pairwise) Completely Positive MatricesNecessary and Su�cient Conditions
(Absolute) Separability and Entanglement
Separability and EntanglementConnection with PCP MatricesAbsolute Separability
Separability and Entanglement
De�nition
A matrix Z ∈ Mn(C)⊗Mn(C) is called separable if there exist
positive semide�nite {Xi}, {Yi} ∈ Mn(C) such that
Z =∑i
Xi ⊗ Yi .
Otherwise, Z is called entangled.
Separable matrices are positive semide�nite.
Characterizing these matrices is one of the central problems in
quantum information theory.
Determining separability is NP-hard.
N. Johnston PCP Matrices
(Pairwise) Completely Positive MatricesNecessary and Su�cient Conditions
(Absolute) Separability and Entanglement
Separability and EntanglementConnection with PCP MatricesAbsolute Separability
Separability and Entanglement
De�nition
A matrix Z ∈ Mn(C)⊗Mn(C) is called separable if there exist
positive semide�nite {Xi}, {Yi} ∈ Mn(C) such that
Z =∑i
Xi ⊗ Yi .
Otherwise, Z is called entangled.
Separable matrices are positive semide�nite.
Characterizing these matrices is one of the central problems in
quantum information theory.
Determining separability is NP-hard.
N. Johnston PCP Matrices
(Pairwise) Completely Positive MatricesNecessary and Su�cient Conditions
(Absolute) Separability and Entanglement
Separability and EntanglementConnection with PCP MatricesAbsolute Separability
Separability and Entanglement
De�nition
A matrix Z ∈ Mn(C)⊗Mn(C) is called separable if there exist
positive semide�nite {Xi}, {Yi} ∈ Mn(C) such that
Z =∑i
Xi ⊗ Yi .
Otherwise, Z is called entangled.
Separable matrices are positive semide�nite.
Characterizing these matrices is one of the central problems in
quantum information theory.
Determining separability is NP-hard.
N. Johnston PCP Matrices
(Pairwise) Completely Positive MatricesNecessary and Su�cient Conditions
(Absolute) Separability and Entanglement
Separability and EntanglementConnection with PCP MatricesAbsolute Separability
Separability and Entanglement
De�nition
A matrix Z ∈ Mn(C)⊗Mn(C) is called separable if there exist
positive semide�nite {Xi}, {Yi} ∈ Mn(C) such that
Z =∑i
Xi ⊗ Yi .
Otherwise, Z is called entangled.
Separable matrices are positive semide�nite.
Characterizing these matrices is one of the central problems in
quantum information theory.
Determining separability is NP-hard.
N. Johnston PCP Matrices
(Pairwise) Completely Positive MatricesNecessary and Su�cient Conditions
(Absolute) Separability and Entanglement
Separability and EntanglementConnection with PCP MatricesAbsolute Separability
Necessary Conditions for Separability
Again, we use one-sided tests. The most popular such test is based
on the partial transpose, which is the linear map Γ on
Mn(C)⊗Mn(C) de�ned by
Γ(X ⊗ Y ) = X ⊗ Y T .
Theorem (Horodecki, Peres, Størmer, Woronowicz?)
If Z ∈ Mn(C)⊗Mn(C) is separable then ZΓ is positive semide�nite.
If ZΓ is positive semide�nite, we say that Z has positivepartial transpose (PPT).
N. Johnston PCP Matrices
(Pairwise) Completely Positive MatricesNecessary and Su�cient Conditions
(Absolute) Separability and Entanglement
Separability and EntanglementConnection with PCP MatricesAbsolute Separability
Necessary Conditions for Separability
Again, we use one-sided tests. The most popular such test is based
on the partial transpose, which is the linear map Γ on
Mn(C)⊗Mn(C) de�ned by
Γ(X ⊗ Y ) = X ⊗ Y T .
Theorem (Horodecki, Peres, Størmer, Woronowicz?)
If Z ∈ Mn(C)⊗Mn(C) is separable then ZΓ is positive semide�nite.
If ZΓ is positive semide�nite, we say that Z has positivepartial transpose (PPT).
N. Johnston PCP Matrices
(Pairwise) Completely Positive MatricesNecessary and Su�cient Conditions
(Absolute) Separability and Entanglement
Separability and EntanglementConnection with PCP MatricesAbsolute Separability
Necessary Conditions for Separability
Again, we use one-sided tests. The most popular such test is based
on the partial transpose, which is the linear map Γ on
Mn(C)⊗Mn(C) de�ned by
Γ(X ⊗ Y ) = X ⊗ Y T .
Theorem (Horodecki, Peres, Størmer, Woronowicz?)
If Z ∈ Mn(C)⊗Mn(C) is separable then ZΓ is positive semide�nite.
If ZΓ is positive semide�nite, we say that Z has positivepartial transpose (PPT).
N. Johnston PCP Matrices
(Pairwise) Completely Positive MatricesNecessary and Su�cient Conditions
(Absolute) Separability and Entanglement
Separability and EntanglementConnection with PCP MatricesAbsolute Separability
Connection with PCP Matrices
Let's establish the connection that makes this talk about PCP
matrices make sense in a quantum information theory session.
Given a pair (X ,Y ) ∈ Mn(C)×Mn(C) with xi ,i = yi ,i for all1 ≤ i ≤ n, we de�ne ZX ,Y ∈ Mn(C)⊗Mn(C) by
ZX ,Y =n∑
i ,j=1
xi ,j |i〉〈j | ⊗ |i〉〈j |+n∑
i 6=j=1
yi ,j |i〉〈i | ⊗ |j〉〈j |.
N. Johnston PCP Matrices
(Pairwise) Completely Positive MatricesNecessary and Su�cient Conditions
(Absolute) Separability and Entanglement
Separability and EntanglementConnection with PCP MatricesAbsolute Separability
Connection with PCP Matrices
Let's establish the connection that makes this talk about PCP
matrices make sense in a quantum information theory session.
Given a pair (X ,Y ) ∈ Mn(C)×Mn(C) with xi ,i = yi ,i for all1 ≤ i ≤ n, we de�ne ZX ,Y ∈ Mn(C)⊗Mn(C) by
ZX ,Y =n∑
i ,j=1
xi ,j |i〉〈j | ⊗ |i〉〈j |+n∑
i 6=j=1
yi ,j |i〉〈i | ⊗ |j〉〈j |.
N. Johnston PCP Matrices
(Pairwise) Completely Positive MatricesNecessary and Su�cient Conditions
(Absolute) Separability and Entanglement
Separability and EntanglementConnection with PCP MatricesAbsolute Separability
Connection with PCP Matrices
For example, if n = 3 then ZX ,Y has the form (where · means 0)...
ZX ,Y =
x1,1 · · · x1,2 · · · x1,3· y1,2 · · · · · · ·· · y1,3 · · · · · ·· · · y2,1 · · · · ·
x2,1 · · · x2,2 · · · x2,3· · · · · y2,3 · · ·· · · · · · y3,1 · ·· · · · · · · y3,2 ·
x3,1 · · · x3,2 · · · x3,3
.
N. Johnston PCP Matrices
(Pairwise) Completely Positive MatricesNecessary and Su�cient Conditions
(Absolute) Separability and Entanglement
Separability and EntanglementConnection with PCP MatricesAbsolute Separability
Connection with PCP Matrices
For example, if n = 3 then ZX ,Y has the form (where · means 0)...
ZX ,Y =
x1,1 · · · x1,2 · · · x1,3· y1,2 · · · · · · ·· · y1,3 · · · · · ·· · · y2,1 · · · · ·
x2,1 · · · x2,2 · · · x2,3· · · · · y2,3 · · ·· · · · · · y3,1 · ·· · · · · · · y3,2 ·
x3,1 · · · x3,2 · · · x3,3
.
N. Johnston PCP Matrices
(Pairwise) Completely Positive MatricesNecessary and Su�cient Conditions
(Absolute) Separability and Entanglement
Separability and EntanglementConnection with PCP MatricesAbsolute Separability
Connection with PCP Matrices
Many properties of the pair (X ,Y ) correspond naturally with
properties of ZX ,Y :
ZX ,Y is separable if and only if (X ,Y ) is pairwise completely
positive.
ZX ,Y is positive semide�nite if and only if X is positive
semide�nite and Y is real and entrywise non-negative.
(Properties (a) and (b) in Theorem (†).)
ZΓX ,Y is positive semide�nite if and only if |xi ,j |2 ≤ yi ,jyj ,i for
all 1 ≤ i , j ≤ n.(Property (d) in Theorem (†).)
N. Johnston PCP Matrices
(Pairwise) Completely Positive MatricesNecessary and Su�cient Conditions
(Absolute) Separability and Entanglement
Separability and EntanglementConnection with PCP MatricesAbsolute Separability
Connection with PCP Matrices
Many properties of the pair (X ,Y ) correspond naturally with
properties of ZX ,Y :
ZX ,Y is separable if and only if (X ,Y ) is pairwise completely
positive.
ZX ,Y is positive semide�nite if and only if X is positive
semide�nite and Y is real and entrywise non-negative.
(Properties (a) and (b) in Theorem (†).)
ZΓX ,Y is positive semide�nite if and only if |xi ,j |2 ≤ yi ,jyj ,i for
all 1 ≤ i , j ≤ n.(Property (d) in Theorem (†).)
N. Johnston PCP Matrices
(Pairwise) Completely Positive MatricesNecessary and Su�cient Conditions
(Absolute) Separability and Entanglement
Separability and EntanglementConnection with PCP MatricesAbsolute Separability
Connection with PCP Matrices
Many properties of the pair (X ,Y ) correspond naturally with
properties of ZX ,Y :
ZX ,Y is separable if and only if (X ,Y ) is pairwise completely
positive.
ZX ,Y is positive semide�nite if and only if X is positive
semide�nite and Y is real and entrywise non-negative.
(Properties (a) and (b) in Theorem (†).)
ZΓX ,Y is positive semide�nite if and only if |xi ,j |2 ≤ yi ,jyj ,i for
all 1 ≤ i , j ≤ n.(Property (d) in Theorem (†).)
N. Johnston PCP Matrices
(Pairwise) Completely Positive MatricesNecessary and Su�cient Conditions
(Absolute) Separability and Entanglement
Separability and EntanglementConnection with PCP MatricesAbsolute Separability
Connection with PCP Matrices
Many properties of the pair (X ,Y ) correspond naturally with
properties of ZX ,Y :
ZX ,Y is separable if and only if (X ,Y ) is pairwise completely
positive.
ZX ,Y is positive semide�nite if and only if X is positive
semide�nite and Y is real and entrywise non-negative.
(Properties (a) and (b) in Theorem (†).)
ZΓX ,Y is positive semide�nite if and only if |xi ,j |2 ≤ yi ,jyj ,i for
all 1 ≤ i , j ≤ n.(Property (d) in Theorem (†).)
N. Johnston PCP Matrices
(Pairwise) Completely Positive MatricesNecessary and Su�cient Conditions
(Absolute) Separability and Entanglement
Separability and EntanglementConnection with PCP MatricesAbsolute Separability
Connection with PCP Matrices
However, the su�cient condition for PCP-ness that we presented
gives us a completely new way of showing that matrices of this
special form are separable:
Theorem
If each of ZX ,Y , ZΓX ,Y , and M(ZX ,Y ) are positive semide�nite, then
ZX ,Y is separable.
N. Johnston PCP Matrices
(Pairwise) Completely Positive MatricesNecessary and Su�cient Conditions
(Absolute) Separability and Entanglement
Separability and EntanglementConnection with PCP MatricesAbsolute Separability
Connection with PCP Matrices
However, the su�cient condition for PCP-ness that we presented
gives us a completely new way of showing that matrices of this
special form are separable:
Theorem
If each of ZX ,Y , ZΓX ,Y , and M(ZX ,Y ) are positive semide�nite, then
ZX ,Y is separable.
N. Johnston PCP Matrices
(Pairwise) Completely Positive MatricesNecessary and Su�cient Conditions
(Absolute) Separability and Entanglement
Separability and EntanglementConnection with PCP MatricesAbsolute Separability
Absolute Separability
OK, but why do we care about matrices of the form ZX ,Y in the
�rst place? This separability test only applies to very specially
cooked up states. Well...
De�nition
A matrix Z ∈ Mn(C)⊗Mn(C) is called absolutely separable if
UZU∗ is separable for all unitary U ∈ Mn(C)⊗Mn(C).
For example, identity matrix is absolutely separable, but
(somewhat surprisingly?), so are many other matrices.
N. Johnston PCP Matrices
(Pairwise) Completely Positive MatricesNecessary and Su�cient Conditions
(Absolute) Separability and Entanglement
Separability and EntanglementConnection with PCP MatricesAbsolute Separability
Absolute Separability
OK, but why do we care about matrices of the form ZX ,Y in the
�rst place? This separability test only applies to very specially
cooked up states. Well...
De�nition
A matrix Z ∈ Mn(C)⊗Mn(C) is called absolutely separable if
UZU∗ is separable for all unitary U ∈ Mn(C)⊗Mn(C).
For example, identity matrix is absolutely separable, but
(somewhat surprisingly?), so are many other matrices.
N. Johnston PCP Matrices
(Pairwise) Completely Positive MatricesNecessary and Su�cient Conditions
(Absolute) Separability and Entanglement
Separability and EntanglementConnection with PCP MatricesAbsolute Separability
Absolute Separability
OK, but why do we care about matrices of the form ZX ,Y in the
�rst place? This separability test only applies to very specially
cooked up states. Well...
De�nition
A matrix Z ∈ Mn(C)⊗Mn(C) is called absolutely separable if
UZU∗ is separable for all unitary U ∈ Mn(C)⊗Mn(C).
For example, identity matrix is absolutely separable, but
(somewhat surprisingly?), so are many other matrices.
N. Johnston PCP Matrices
(Pairwise) Completely Positive MatricesNecessary and Su�cient Conditions
(Absolute) Separability and Entanglement
Separability and EntanglementConnection with PCP MatricesAbsolute Separability
Absolute PPT
Very little is known about absolute separability, but (as usual) there
is a simple necessary condition for absolute separability:
De�nition
A matrix Z ∈ Mn(C)⊗Mn(C) is called absolutely PPT if UZU∗
is PPT for all unitary U ∈ Mn(C)⊗Mn(C).
Absolute PPT has a nice SDP characterization.
Absolute separability implies absolute PPT.
We don't even know if the converse holds!
N. Johnston PCP Matrices
(Pairwise) Completely Positive MatricesNecessary and Su�cient Conditions
(Absolute) Separability and Entanglement
Separability and EntanglementConnection with PCP MatricesAbsolute Separability
Absolute PPT
Very little is known about absolute separability, but (as usual) there
is a simple necessary condition for absolute separability:
De�nition
A matrix Z ∈ Mn(C)⊗Mn(C) is called absolutely PPT if UZU∗
is PPT for all unitary U ∈ Mn(C)⊗Mn(C).
Absolute PPT has a nice SDP characterization.
Absolute separability implies absolute PPT.
We don't even know if the converse holds!
N. Johnston PCP Matrices
(Pairwise) Completely Positive MatricesNecessary and Su�cient Conditions
(Absolute) Separability and Entanglement
Separability and EntanglementConnection with PCP MatricesAbsolute Separability
Absolute PPT
Very little is known about absolute separability, but (as usual) there
is a simple necessary condition for absolute separability:
De�nition
A matrix Z ∈ Mn(C)⊗Mn(C) is called absolutely PPT if UZU∗
is PPT for all unitary U ∈ Mn(C)⊗Mn(C).
Absolute PPT has a nice SDP characterization.
Absolute separability implies absolute PPT.
We don't even know if the converse holds!
N. Johnston PCP Matrices
(Pairwise) Completely Positive MatricesNecessary and Su�cient Conditions
(Absolute) Separability and Entanglement
Separability and EntanglementConnection with PCP MatricesAbsolute Separability
Absolute PPT
Very little is known about absolute separability, but (as usual) there
is a simple necessary condition for absolute separability:
De�nition
A matrix Z ∈ Mn(C)⊗Mn(C) is called absolutely PPT if UZU∗
is PPT for all unitary U ∈ Mn(C)⊗Mn(C).
Absolute PPT has a nice SDP characterization.
Absolute separability implies absolute PPT.
We don't even know if the converse holds!
N. Johnston PCP Matrices
(Pairwise) Completely Positive MatricesNecessary and Su�cient Conditions
(Absolute) Separability and Entanglement
Separability and EntanglementConnection with PCP MatricesAbsolute Separability
Absolute PPT
Very little is known about absolute separability, but (as usual) there
is a simple necessary condition for absolute separability:
De�nition
A matrix Z ∈ Mn(C)⊗Mn(C) is called absolutely PPT if UZU∗
is PPT for all unitary U ∈ Mn(C)⊗Mn(C).
Absolute PPT has a nice SDP characterization.
Absolute separability implies absolute PPT.
We don't even know if the converse holds!
N. Johnston PCP Matrices
(Pairwise) Completely Positive MatricesNecessary and Su�cient Conditions
(Absolute) Separability and Entanglement
Separability and EntanglementConnection with PCP MatricesAbsolute Separability
Absolute PPT
It is known that, for absolute PPT, we do not need to check that
each UZU∗ is PPT�there is a (�nite!) list of unitaries {Ui} suchthat Z is absolutely PPT if and only if UiZU
∗i is PPT for all i .
Question: Can we �nd an entangled but PPT UiZU∗i ?
Answer: No. Every single UiZU∗i has the form of the ZX ,Y
matrices, and our su�cient condition shows that if they are
PPT, they are separable.
The Upshot: If there is a gap between absolute separability
and absolute PPT, we have to look at some other weird
unitaries to �nd it.
N. Johnston PCP Matrices
(Pairwise) Completely Positive MatricesNecessary and Su�cient Conditions
(Absolute) Separability and Entanglement
Separability and EntanglementConnection with PCP MatricesAbsolute Separability
Absolute PPT
It is known that, for absolute PPT, we do not need to check that
each UZU∗ is PPT�there is a (�nite!) list of unitaries {Ui} suchthat Z is absolutely PPT if and only if UiZU
∗i is PPT for all i .
Question: Can we �nd an entangled but PPT UiZU∗i ?
Answer: No. Every single UiZU∗i has the form of the ZX ,Y
matrices, and our su�cient condition shows that if they are
PPT, they are separable.
The Upshot: If there is a gap between absolute separability
and absolute PPT, we have to look at some other weird
unitaries to �nd it.
N. Johnston PCP Matrices
(Pairwise) Completely Positive MatricesNecessary and Su�cient Conditions
(Absolute) Separability and Entanglement
Separability and EntanglementConnection with PCP MatricesAbsolute Separability
Absolute PPT
It is known that, for absolute PPT, we do not need to check that
each UZU∗ is PPT�there is a (�nite!) list of unitaries {Ui} suchthat Z is absolutely PPT if and only if UiZU
∗i is PPT for all i .
Question: Can we �nd an entangled but PPT UiZU∗i ?
Answer: No. Every single UiZU∗i has the form of the ZX ,Y
matrices, and our su�cient condition shows that if they are
PPT, they are separable.
The Upshot: If there is a gap between absolute separability
and absolute PPT, we have to look at some other weird
unitaries to �nd it.
N. Johnston PCP Matrices
(Pairwise) Completely Positive MatricesNecessary and Su�cient Conditions
(Absolute) Separability and Entanglement
Separability and EntanglementConnection with PCP MatricesAbsolute Separability
Absolute PPT
It is known that, for absolute PPT, we do not need to check that
each UZU∗ is PPT�there is a (�nite!) list of unitaries {Ui} suchthat Z is absolutely PPT if and only if UiZU
∗i is PPT for all i .
Question: Can we �nd an entangled but PPT UiZU∗i ?
Answer: No. Every single UiZU∗i has the form of the ZX ,Y
matrices, and our su�cient condition shows that if they are
PPT, they are separable.
The Upshot: If there is a gap between absolute separability
and absolute PPT, we have to look at some other weird
unitaries to �nd it.
N. Johnston PCP Matrices
(Pairwise) Completely Positive MatricesNecessary and Su�cient Conditions
(Absolute) Separability and Entanglement
Separability and EntanglementConnection with PCP MatricesAbsolute Separability
Thank you!
Thank-you!
N. Johnston PCP Matrices
N. Johnston and O. MacLean. Pairwise completely positive matrices andconjugate local diagonal unitary quantum states. Electronic Journal of Linear
Algebra, 35:156�180, 2019. arXiv:1807.06897 [quant-ph]
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