Pairwise Completely Positive Matrices and Quantum Entanglement · Pairwise Completely Positive...

(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

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.

De�nition

X = BBT .

non-negative.

De�nition

X = BBT .

non-negative.

De�nition

X = BBT .

non-negative.

De�nition

X = BBT .

non-negative.

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).

De�nition

X = (A� B)(A� B)∗ and Y = (A� A)(B � B)∗.

De�nition

X = (A� B)(A� B)∗ and Y = (A� A)(B � B)∗.

De�nition

X = (A� B)(A� B)∗ and Y = (A� A)(B � B)∗.

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

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:

1 2 1/21/2 1 2

2 1/2 1

Theorem

1 2 1/21/2 1 2

2 1/2 1

Theorem

1 2 1/21/2 1 2

2 1/2 1

Theorem

1 2 1/21/2 1 2

2 1/2 1

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|

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|

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

Theorem

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.

De�nition

Z =∑i

Xi ⊗ Yi .

De�nition

Z =∑i

Xi ⊗ Yi .

De�nition

Z =∑i

Xi ⊗ Yi .

De�nition

Z =∑i

Xi ⊗ Yi .

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).

Γ(X ⊗ Y ) = X ⊗ Y T .

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 |.

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 |.

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

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

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 (†).)

positive.

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.

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.

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.

De�nition

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!

Absolute PPT

De�nition

Absolute PPT

De�nition

Absolute PPT

De�nition

Absolute PPT

De�nition

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.

Absolute PPT

Thank you!

Thank-you!

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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