Positive and completely positive maps via free
additive powers of probability measures
Ion Nechita
CNRS, Laboratoire de Physique Théorique, Université de Toulouse
joint work with Benoit Collins (uOttawa) and Patrick Hayden (McGill)
Aarhus, April 5th, 2013
Quantum Information Theory, Quantum Computing
I
New branches of {Physics, Computer Science, Mathematics} dealing
with quantum information.
2 / 18
Quantum Information Theory, Quantum Computing
I
New branches of {Physics, Computer Science, Mathematics} dealing
with quantum information.
I
Quantum information = information held in a quantum physical
system.
2 / 18
Quantum Information Theory, Quantum Computing
I
New branches of {Physics, Computer Science, Mathematics} dealing
with quantum information.
I
Quantum information = information held in a quantum physical
system.
I
Basic idea: replace {0, 1} with C2 = span{|0i, |1i}, the state space
of a two-level quantum system.
2 / 18
Quantum Information Theory, Quantum Computing
I
New branches of {Physics, Computer Science, Mathematics} dealing
with quantum information.
I
Quantum information = information held in a quantum physical
system.
I
Basic idea: replace {0, 1} with C2 = span{|0i, |1i}, the state space
of a two-level quantum system.
Shows great promise:
I
2 / 18
Quantum Information Theory, Quantum Computing
I
New branches of {Physics, Computer Science, Mathematics} dealing
with quantum information.
I
Quantum information = information held in a quantum physical
system.
I
Basic idea: replace {0, 1} with C2 = span{|0i, |1i}, the state space
of a two-level quantum system.
Shows great promise:
I
1. Secure transmission of data, protocol security guaranteed by the laws
of nature
2 / 18
Quantum Information Theory, Quantum Computing
I
New branches of {Physics, Computer Science, Mathematics} dealing
with quantum information.
I
Quantum information = information held in a quantum physical
system.
I
Basic idea: replace {0, 1} with C2 = span{|0i, |1i}, the state space
of a two-level quantum system.
Shows great promise:
I
1. Secure transmission of data, protocol security guaranteed by the laws
of nature
2. Fast integer factorization ; current algorithms (RSA, etc) obsolete
2 / 18
Quantum Information Theory, Quantum Computing
I
New branches of {Physics, Computer Science, Mathematics} dealing
with quantum information.
I
Quantum information = information held in a quantum physical
system.
I
Basic idea: replace {0, 1} with C2 = span{|0i, |1i}, the state space
of a two-level quantum system.
Shows great promise:
I
1. Secure transmission of data, protocol security guaranteed by the laws
of nature
2. Fast integer factorization ; current algorithms (RSA, etc) obsolete
3. Fast database search
2 / 18
Quantum Information Theory, Quantum Computing
I
New branches of {Physics, Computer Science, Mathematics} dealing
with quantum information.
I
Quantum information = information held in a quantum physical
system.
I
Basic idea: replace {0, 1} with C2 = span{|0i, |1i}, the state space
of a two-level quantum system.
Shows great promise:
I
1. Secure transmission of data, protocol security guaranteed by the laws
of nature
2. Fast integer factorization ; current algorithms (RSA, etc) obsolete
3. Fast database search
4. Fast simulation of quantum systems
2 / 18
Entanglement in Quantum Information Theory
I
Quantum states with n degrees of freedom are described by density
matrices
ρ ∈ M1,+
= End1,+ (Cn );
n
Trρ = 1 and ρ ≥ 0.
3 / 18
Entanglement in Quantum Information Theory
I
Quantum states with n degrees of freedom are described by density
matrices
ρ ∈ M1,+
= End1,+ (Cn );
n
I
Trρ = 1 and ρ ≥ 0.
1,+
Two quantum systems: ρ12 ∈ End1,+ (Cm ⊗ Cn ) = Mmn
.
3 / 18
Entanglement in Quantum Information Theory
I
Quantum states with n degrees of freedom are described by density
matrices
ρ ∈ M1,+
= End1,+ (Cn );
n
I
I
Trρ = 1 and ρ ≥ 0.
1,+
Two quantum systems: ρ12 ∈ End1,+ (Cm ⊗ Cn ) = Mmn
.
A state ρ12 is called separable if it can be written as a convex
combination of product states
X
ρ12 ∈ SEP ⇐⇒ ρ12 =
ti ρ1 (i) ⊗ ρ2 (i),
i
where ti ≥ 0,
P
i ti
1,+
= 1, ρ1 (i) ∈ M1,+
m , ρ2 (i) ∈ Mn .
3 / 18
Entanglement in Quantum Information Theory
I
Quantum states with n degrees of freedom are described by density
matrices
ρ ∈ M1,+
= End1,+ (Cn );
n
I
I
Trρ = 1 and ρ ≥ 0.
1,+
Two quantum systems: ρ12 ∈ End1,+ (Cm ⊗ Cn ) = Mmn
.
A state ρ12 is called separable if it can be written as a convex
combination of product states
X
ρ12 ∈ SEP ⇐⇒ ρ12 =
ti ρ1 (i) ⊗ ρ2 (i),
i
I
= 1, ρ1 (i) ∈ M1,+
, ρ2 (i) ∈ M1,+
n .
1,+ m 1,+ Equivalently, SEP = conv Mm ⊗ Mn .
where ti ≥ 0,
P
i ti
3 / 18
Entanglement in Quantum Information Theory
I
Quantum states with n degrees of freedom are described by density
matrices
ρ ∈ M1,+
= End1,+ (Cn );
n
I
I
Trρ = 1 and ρ ≥ 0.
1,+
Two quantum systems: ρ12 ∈ End1,+ (Cm ⊗ Cn ) = Mmn
.
A state ρ12 is called separable if it can be written as a convex
combination of product states
X
ρ12 ∈ SEP ⇐⇒ ρ12 =
ti ρ1 (i) ⊗ ρ2 (i),
i
I
I
= 1, ρ1 (i) ∈ M1,+
, ρ2 (i) ∈ M1,+
n .
1,+ m 1,+ Equivalently, SEP = conv Mm ⊗ Mn .
where ti ≥ 0,
P
i ti
Non-separable states are called entangled.
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More on entanglement - pure states
I
Separable rank one (pure) states ρ12 = Pe⊗f = Pe ⊗ Pf .
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More on entanglement - pure states
I
I
Separable rank one (pure) states ρ12 = Pe⊗f = Pe ⊗ Pf .
Bell state or maximally entangled state ρ12 = PBell , where
1
C2 ⊗ C2 3 Bell = √ (e1 ⊗ f1 + e2 ⊗ f2 ) 6= x ⊗ y .
2
4 / 18
More on entanglement - pure states
I
I
Separable rank one (pure) states ρ12 = Pe⊗f = Pe ⊗ Pf .
Bell state or maximally entangled state ρ12 = PBell , where
1
C2 ⊗ C2 3 Bell = √ (e1 ⊗ f1 + e2 ⊗ f2 ) 6= x ⊗ y .
2
I
For rank one quantum states, entanglement can be detected and
quantified by the entropy of entanglement
min(m,n)
Eent (Px ) = H(s(x)) = −
X
si (x) log si (x),
i=1
∼ Mm×n (C) is seen as a m × n matrix and si (x)
where x ∈ Cm ⊗ Cn =
are its singular values.
4 / 18
More on entanglement - pure states
I
I
Separable rank one (pure) states ρ12 = Pe⊗f = Pe ⊗ Pf .
Bell state or maximally entangled state ρ12 = PBell , where
1
C2 ⊗ C2 3 Bell = √ (e1 ⊗ f1 + e2 ⊗ f2 ) 6= x ⊗ y .
2
I
For rank one quantum states, entanglement can be detected and
quantified by the entropy of entanglement
min(m,n)
Eent (Px ) = H(s(x)) = −
X
si (x) log si (x),
i=1
∼ Mm×n (C) is seen as a m × n matrix and si (x)
where x ∈ Cm ⊗ Cn =
are its singular values.
I
A pure state x ∈ Cm ⊗ Cn is separable ⇐⇒ Eent (Px ) = 0.
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Separability criteria
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Let A be a C ∗ algebra. A map f : Mn → A is called
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Separability criteria
I
Let A be a C ∗ algebra. A map f : Mn → A is called
I
positive if A ≥ 0 =⇒ f (A) ≥ 0;
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Separability criteria
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Let A be a C ∗ algebra. A map f : Mn → A is called
I
I
positive if A ≥ 0 =⇒ f (A) ≥ 0;
completely positive (CP) if idk ⊗ f is positive for all k ≥ 1 (k = n is
enough).
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Separability criteria
I
Let A be a C ∗ algebra. A map f : Mn → A is called
I
I
I
positive if A ≥ 0 =⇒ f (A) ≥ 0;
completely positive (CP) if idk ⊗ f is positive for all k ≥ 1 (k = n is
enough).
Let f : Mn → A be a completely positive map. Then, for every state
ρ12 ∈ M1,+
mn , one has [idm ⊗ f ](ρ12 ) ≥ 0.
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Separability criteria
I
Let A be a C ∗ algebra. A map f : Mn → A is called
I
I
I
I
positive if A ≥ 0 =⇒ f (A) ≥ 0;
completely positive (CP) if idk ⊗ f is positive for all k ≥ 1 (k = n is
enough).
Let f : Mn → A
ρ12 ∈ M1,+
mn , one
Let f : Mn → A
ρ12 ∈ M1,+
mn , one
I
I
I
be a completely positive map. Then, for every state
has [idm ⊗ f ](ρ12 ) ≥ 0.
be a positive map. Then, for every separable state
has [idm ⊗ f ](ρ12 ) ≥ 0.
P
ρ12 separable =⇒
Pρ12 = i ti ρ1 (i) ⊗ ρ2 (i).
[idm ⊗ f ](ρ12 ) = i ti ρ1 (i) ⊗ f (ρ2 (i)).
For all i, ([ρ2 (i)) ≥ 0, so [idm ⊗ f ](ρ12 ) ≥ 0.
5 / 18
Separability criteria
I
Let A be a C ∗ algebra. A map f : Mn → A is called
I
I
I
I
Let f : Mn → A
ρ12 ∈ M1,+
mn , one
Let f : Mn → A
ρ12 ∈ M1,+
mn , one
I
I
I
I
positive if A ≥ 0 =⇒ f (A) ≥ 0;
completely positive (CP) if idk ⊗ f is positive for all k ≥ 1 (k = n is
enough).
be a completely positive map. Then, for every state
has [idm ⊗ f ](ρ12 ) ≥ 0.
be a positive map. Then, for every separable state
has [idm ⊗ f ](ρ12 ) ≥ 0.
P
ρ12 separable =⇒
Pρ12 = i ti ρ1 (i) ⊗ ρ2 (i).
[idm ⊗ f ](ρ12 ) = i ti ρ1 (i) ⊗ f (ρ2 (i)).
For all i, ([ρ2 (i)) ≥ 0, so [idm ⊗ f ](ρ12 ) ≥ 0.
Hence, positive, but not CP maps f provide sufficient entanglement
criteria: if [idm ⊗ f ](ρ12 ) 0, then ρ12 is entangled.
5 / 18
Separability criteria
I
Let A be a C ∗ algebra. A map f : Mn → A is called
I
I
I
I
Let f : Mn → A
ρ12 ∈ M1,+
mn , one
Let f : Mn → A
ρ12 ∈ M1,+
mn , one
I
I
I
I
I
positive if A ≥ 0 =⇒ f (A) ≥ 0;
completely positive (CP) if idk ⊗ f is positive for all k ≥ 1 (k = n is
enough).
be a completely positive map. Then, for every state
has [idm ⊗ f ](ρ12 ) ≥ 0.
be a positive map. Then, for every separable state
has [idm ⊗ f ](ρ12 ) ≥ 0.
P
ρ12 separable =⇒
Pρ12 = i ti ρ1 (i) ⊗ ρ2 (i).
[idm ⊗ f ](ρ12 ) = i ti ρ1 (i) ⊗ f (ρ2 (i)).
For all i, ([ρ2 (i)) ≥ 0, so [idm ⊗ f ](ρ12 ) ≥ 0.
Hence, positive, but not CP maps f provide sufficient entanglement
criteria: if [idm ⊗ f ](ρ12 ) 0, then ρ12 is entangled.
Moreover, if [idm ⊗ f ](ρ12 ) ≥ 0 for all positive, but not CP maps
f : Mn → Mm , then ρ12 is separable.
5 / 18
Positive Partial Transpose matrices
I
The transposition map t : A 7→ At is positive, but not CP. Define the
convex set
PPT = {ρ12 ∈ M1,+
mn | [idm ⊗ tn ](ρ12 ) ≥ 0}.
6 / 18
Positive Partial Transpose matrices
I
The transposition map t : A 7→ At is positive, but not CP. Define the
convex set
PPT = {ρ12 ∈ M1,+
mn | [idm ⊗ tn ](ρ12 ) ≥ 0}.
I
For (m, n) ∈ {(2, 2), (2, 3)} we have SEP = PPT . In other
dimensions, the inclusion SEP ⊂ PPT is strict.
6 / 18
Positive Partial Transpose matrices
I
The transposition map t : A 7→ At is positive, but not CP. Define the
convex set
PPT = {ρ12 ∈ M1,+
mn | [idm ⊗ tn ](ρ12 ) ≥ 0}.
I
I
For (m, n) ∈ {(2, 2), (2, 3)} we have SEP = PPT . In other
dimensions, the inclusion SEP ⊂ PPT is strict.
Low dimensions are special because every positive map
f : M2 → M2/3 is decomposable:
f = g1 + g2 ◦ t,
with g1,2 completely positive. Among all decomposable maps, the
transposition criterion is the strongest.
6 / 18
The PPT criterion at work
I
Recall the Bell state ρ12 = PBell , where
1
C2 ⊗ C2 3 Bell = √ (e1 ⊗ f1 + e2 ⊗ f2 ).
2
7 / 18
The PPT criterion at work
I
Recall the Bell state ρ12 = PBell , where
1
C2 ⊗ C2 3 Bell = √ (e1 ⊗ f1 + e2 ⊗ f2 ).
2
I
Written as a matrix in M1,+
2·2

1 0
1
0 0
ρ12 = 
2 0 0
1 0
0
0
0
0

1
0 
 = 1 B11
0  2 B21
1
B12
.
B22
7 / 18
The PPT criterion at work
I
Recall the Bell state ρ12 = PBell , where
1
C2 ⊗ C2 3 Bell = √ (e1 ⊗ f1 + e2 ⊗ f2 ).
2
I
I
Written as a matrix in M1,+
2·2

1 0
1
0 0
ρ12 = 
2 0 0
1 0
0
0
0
0

1
0 
 = 1 B11
0  2 B21
1
Partial transposition: transpose each block

1
1
0
Γ

ρ12 = [id2 ⊗ t2 ](ρ12 ) = 
2 0
0
B12
.
B22
Bij :
0
0
1
0
0
1
0
0

0
0
.
0
1
7 / 18
The PPT criterion at work
I
Recall the Bell state ρ12 = PBell , where
1
C2 ⊗ C2 3 Bell = √ (e1 ⊗ f1 + e2 ⊗ f2 ).
2
I
I
I
Written as a matrix in M1,+
2·2

1 0
1
0 0
ρ12 = 
2 0 0
1 0
0
0
0
0

1
0 
 = 1 B11
0  2 B21
1
Partial transposition: transpose each block

1
1
0
Γ

ρ12 = [id2 ⊗ t2 ](ρ12 ) = 
2 0
0
B12
.
B22
Bij :
0
0
1
0
0
1
0
0

0
0
.
0
1
This matrix is no longer positive =⇒ the state is entangled.
7 / 18
Three convex sets
M1,+
mn
Imn
PPT
SEP
The inclusions SEP ⊂ PPT ⊂ M1,+
mn are generally strict.
8 / 18
Our work: two directions
I
How good is the PPT criterion ? In other words, how far is the
inclusion SEP ⊂ PPT from being an equality ?
I
I
I
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Compare volumes of sets ; random quantum states.
Threshold phenomenon: depending on the probability used to
measure volumes, either almost all states or almost none are PPT
(joint work with Aubrun, Banica).
Same phenomenon observed for SEP by Aubrun, Szarek and Ye.
The value of the thresholds are diifferent ; in some regimes, all
states belong to PPT \ SEP, so the PPT criterion fails in the worst
possible way.
9 / 18
Our work: two directions
I
How good is the PPT criterion ? In other words, how far is the
inclusion SEP ⊂ PPT from being an equality ?
I
I
I
I
I
Compare volumes of sets ; random quantum states.
Threshold phenomenon: depending on the probability used to
measure volumes, either almost all states or almost none are PPT
(joint work with Aubrun, Banica).
Same phenomenon observed for SEP by Aubrun, Szarek and Ye.
The value of the thresholds are diifferent ; in some regimes, all
states belong to PPT \ SEP, so the PPT criterion fails in the worst
possible way.
How to produce other maps which are positive but not CP and lead
to interesting separability criteria ?
9 / 18
The Choi matrix of a map
I
For any n, recall that the maximally entangled state is the
orthogonal projection onto
n
1 X
Cn ⊗ Cn 3 Bell = √
ei ⊗ ei .
n
i=1
10 / 18
The Choi matrix of a map
I
For any n, recall that the maximally entangled state is the
orthogonal projection onto
n
1 X
Cn ⊗ Cn 3 Bell = √
ei ⊗ ei .
n
i=1
I
To any map f : Mn → A, associate its Choi matrix
Cf = [idn ⊗ f ](PBell ) ∈ Mn ⊗ A.
10 / 18
The Choi matrix of a map
I
For any n, recall that the maximally entangled state is the
orthogonal projection onto
n
1 X
Cn ⊗ Cn 3 Bell = √
ei ⊗ ei .
n
i=1
I
To any map f : Mn → A, associate its Choi matrix
Cf = [idn ⊗ f ](PBell ) ∈ Mn ⊗ A.
I
Equivalently, if Eij are the matrix units in Mn , then
Cf =
n
X
i,j=1
Eij ⊗ f (Eij ).
10 / 18
The Choi matrix of a map
I
For any n, recall that the maximally entangled state is the
orthogonal projection onto
n
1 X
Cn ⊗ Cn 3 Bell = √
ei ⊗ ei .
n
i=1
I
To any map f : Mn → A, associate its Choi matrix
Cf = [idn ⊗ f ](PBell ) ∈ Mn ⊗ A.
I
Equivalently, if Eij are the matrix units in Mn , then
Cf =
n
X
i,j=1
Eij ⊗ f (Eij ).
Theorem (Choi ’72)
A map f : Mn → A is CP iff its Choi matrix Cf is positive.
10 / 18
The Choi-Jamiolkowski isomorphism
I
Recall (from now on A = Md )
Cf = [idn ⊗ f ](PBell ) =
n
X
i,j=1
Eij ⊗ f (Eij ) ∈ Mn ⊗ Md .
11 / 18
The Choi-Jamiolkowski isomorphism
I
Recall (from now on A = Md )
Cf = [idn ⊗ f ](PBell ) =
I
n
X
i,j=1
Eij ⊗ f (Eij ) ∈ Mn ⊗ Md .
The map f 7→ Cf is called the Choi-Jamiolkowski isomorphism.
11 / 18
The Choi-Jamiolkowski isomorphism
I
Recall (from now on A = Md )
Cf = [idn ⊗ f ](PBell ) =
I
I
n
X
i,j=1
Eij ⊗ f (Eij ) ∈ Mn ⊗ Md .
The map f 7→ Cf is called the Choi-Jamiolkowski isomorphism.
It sends:
1.
2.
3.
4.
All linear maps to all operators;
Hermicity preserving maps to hermitian operators;
Entanglement breaking maps to separable quantum states;
Unital maps to operators with unit left partial trace
([Tr ⊗ id]Cf = Id );
5. Trace preserving maps to operators with unit left partial trace
([id ⊗ Tr]Cf = In ).
11 / 18
Intermediate positivity notions
I
A map f : Mn → A is called k-positive if idk ⊗ f is positive.
12 / 18
Intermediate positivity notions
I
I
A map f : Mn → A is called k-positive if idk ⊗ f is positive.
A matrix C ∈ Mnd is called k-positive if hx, Cxi ≥ 0 for all vectors
x ∈ Cn ⊗ Cd of rank at most k.
12 / 18
Intermediate positivity notions
I
I
I
A map f : Mn → A is called k-positive if idk ⊗ f is positive.
A matrix C ∈ Mnd is called k-positive if hx, Cxi ≥ 0 for all vectors
x ∈ Cn ⊗ Cd of rank at most k.
In particular, C is 1-positive (or block-positive) if
∀x ∈ Cn , ∀y ∈ Cd
hx ⊗ y , C · x ⊗ y i ≥ 0.
12 / 18
Intermediate positivity notions
I
I
I
A map f : Mn → A is called k-positive if idk ⊗ f is positive.
A matrix C ∈ Mnd is called k-positive if hx, Cxi ≥ 0 for all vectors
x ∈ Cn ⊗ Cd of rank at most k.
In particular, C is 1-positive (or block-positive) if
∀x ∈ Cn , ∀y ∈ Cd
hx ⊗ y , C · x ⊗ y i ≥ 0.
Theorem
A map f : Mn → A is k-positive iff its Choi matrix Cf is k-positive. In
particular, f is positive iff Cf is block-positive.
12 / 18
Random Choi matrices
I
Let µ be a compactly supported probability measure on R. For each
d we introduce a real valued diagonal matrix Xd of Mn ⊗ Md whose
eigenvalue counting distribution converges to µ and whose extremal
eigenvalues converge to the respective extrema of the support of µ.
13 / 18
Random Choi matrices
I
Let µ be a compactly supported probability measure on R. For each
d we introduce a real valued diagonal matrix Xd of Mn ⊗ Md whose
eigenvalue counting distribution converges to µ and whose extremal
eigenvalues converge to the respective extrema of the support of µ.
I
Let Ud be a random Haar unitary matrix in the unitary group Und ,
(d)
and fµ : Mn → Md be the map whose Choi matrix is Ud Xd Ud∗ .
13 / 18
Random Choi matrices
I
Let µ be a compactly supported probability measure on R. For each
d we introduce a real valued diagonal matrix Xd of Mn ⊗ Md whose
eigenvalue counting distribution converges to µ and whose extremal
eigenvalues converge to the respective extrema of the support of µ.
I
Let Ud be a random Haar unitary matrix in the unitary group Und ,
(d)
and fµ : Mn → Md be the map whose Choi matrix is Ud Xd Ud∗ .
Theorem
Under the above assumptions, if supp(µn/k ) ⊂ (0, ∞) then, almost
(d)
surely as d → ∞, the map fµ is k-positive.
13 / 18
Random Choi matrices
I
Let µ be a compactly supported probability measure on R. For each
d we introduce a real valued diagonal matrix Xd of Mn ⊗ Md whose
eigenvalue counting distribution converges to µ and whose extremal
eigenvalues converge to the respective extrema of the support of µ.
I
Let Ud be a random Haar unitary matrix in the unitary group Und ,
(d)
and fµ : Mn → Md be the map whose Choi matrix is Ud Xd Ud∗ .
Theorem
Under the above assumptions, if supp(µn/k ) ⊂ (0, ∞) then, almost
(d)
surely as d → ∞, the map fµ is k-positive. On the other hand, if
(d)
supp(µn/k ) ∩ (−∞, 0) 6= ∅ then, almost surely as d → ∞, fµ is not
k-positive.
13 / 18
Proof ingredients
(d)
Let fµ
: Mn → Md be the map whose Choi matrix is Ud Xd Ud∗ .
Theorem
(d)
If supp(µn/k ) ⊂ (0, ∞) then, almost surely as d → ∞, the map fµ is
k-positive. If supp(µn/k ) ∩ (−∞, 0) 6= ∅ then, almost surely as d → ∞,
(d)
fµ is not k-positive.
14 / 18
Proof ingredients
(d)
Let fµ
: Mn → Md be the map whose Choi matrix is Ud Xd Ud∗ .
Theorem
(d)
If supp(µn/k ) ⊂ (0, ∞) then, almost surely as d → ∞, the map fµ is
k-positive. If supp(µn/k ) ∩ (−∞, 0) 6= ∅ then, almost surely as d → ∞,
(d)
fµ is not k-positive.
Proposition
A map f is k-positive iff for any self-adjoint projection P ∈ Mn of rank k,
the operator P ⊗ Id · Cf · P ⊗ Id is positive.
14 / 18
Proof ingredients
(d)
Let fµ
: Mn → Md be the map whose Choi matrix is Ud Xd Ud∗ .
Theorem
(d)
If supp(µn/k ) ⊂ (0, ∞) then, almost surely as d → ∞, the map fµ is
k-positive. If supp(µn/k ) ∩ (−∞, 0) 6= ∅ then, almost surely as d → ∞,
(d)
fµ is not k-positive.
Proposition
A map f is k-positive iff for any self-adjoint projection P ∈ Mn of rank k,
the operator P ⊗ Id · Cf · P ⊗ Id is positive.
Proposition (Nica and Speicher)
Let x, p be free elements in a ncps (M, τ ) and assume that p is a
selfadjoint projection of rank τ (p) = 1/t (t ≥ 1) and that x has
distribution µ. Then, the distribution of t −1 pxp inside the contracted
ncps (pMp, τ (p · p)) is µt
14 / 18
Maps associated to probability measures
I
Let µ be a compactly supported probability measure on R.
15 / 18
Maps associated to probability measures
I
Let µ be a compactly supported probability measure on R.
I
The vN algebra L∞ (R, µ), endowed with the expectation trace E is
a non-commutative probability space. Let X ∈ L∞ (R, µ) be the
identity map x 7→ x.
15 / 18
Maps associated to probability measures
I
Let µ be a compactly supported probability measure on R.
I
The vN algebra L∞ (R, µ), endowed with the expectation trace E is
a non-commutative probability space. Let X ∈ L∞ (R, µ) be the
identity map x 7→ x.
I
Consider the vN ncps free product
(M̃, tr ∗ E) = (Mn , tr) ∗ (L∞ (R, µ), E).
15 / 18
Maps associated to probability measures
I
Let µ be a compactly supported probability measure on R.
I
The vN algebra L∞ (R, µ), endowed with the expectation trace E is
a non-commutative probability space. Let X ∈ L∞ (R, µ) be the
identity map x 7→ x.
I
I
Consider the vN ncps free product
(M̃, tr ∗ E) = (Mn , tr) ∗ (L∞ (R, µ), E).
Finally, let (M, τ ) be the contracted vN ncps M = E11 M̃E11 .
15 / 18
Maps associated to probability measures
I
Let µ be a compactly supported probability measure on R.
I
The vN algebra L∞ (R, µ), endowed with the expectation trace E is
a non-commutative probability space. Let X ∈ L∞ (R, µ) be the
identity map x 7→ x.
I
I
I
Consider the vN ncps free product
(M̃, tr ∗ E) = (Mn , tr) ∗ (L∞ (R, µ), E).
Finally, let (M, τ ) be the contracted vN ncps M = E11 M̃E11 .
Define
fµ : Mn → M
Eij 7→ E1i XEj1
15 / 18
Maps associated to probability measures
I
Define
fµ : Mn → M
Eij 7→ E1i XEj1
Theorem
The map fµ is k-positive iff supp(µn/k ) ⊆ R+ .
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Example: semicircular measures
I
Let sa,σ be the semi-circle distribution of mean a and variance σ 2 ,
having support [a − 2σ, a + 2σ].
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Example: semicircular measures
I
I
Let sa,σ be the semi-circle distribution of mean a and variance σ 2 ,
having support [a − 2σ, a + 2σ].
n/k
We have sa,σ = san/k,σ√n/k , with support
p
p
n/k
supp(sa,σ ) = [an/k − 2σ n/k, an/k + 2σ n/k].
Theorem
Let n be an integer and a, σ positive parameters. The map
fa,σ : Mn → M associated to a semi-circular distribution sa,σ is k-positive
iff k ≤ 4nσ/a2 . In particular, for any n and any k < n, there exist
parameters a, σ > 0 such that the above map is k-positive but not
(k + 1)-positive.
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The End