Norm convergence of unitary random matrices
and quantum information theory
Benoı̂t Collins
uOttawa and Lyon 1
JMM, San Diego 2013
Plan
Plan
I
Convergence theorem for the output set of random quantum
channels.
Plan
I
Convergence theorem for the output set of random quantum
channels.
I
Norm convergence for unitary random matrices.
Plan
I
Convergence theorem for the output set of random quantum
channels.
I
Norm convergence for unitary random matrices.
I
Examples.
Plan
I
Convergence theorem for the output set of random quantum
channels.
I
Norm convergence for unitary random matrices.
I
Examples.
I
Joint works with S. Belinschi, C. Male, M. Fukuda, I. Nechita.
Quantum channels
I
A quantum channel Φ is a map Mn (C) → Mk (C)
Quantum channels
I
A quantum channel Φ is a map Mn (C) → Mk (C)
It is linear, completely positive and trace preserving.
Quantum channels
I
A quantum channel Φ is a map Mn (C) → Mk (C)
It is linear, completely positive and trace preserving.
I
Its Choi map is the matrix
X
CΦ =
Eij ⊗ Φ(Eij ) ∈ Mn ⊗ Mk
i,j∈{1,...,n}
Quantum channels
I
A quantum channel Φ is a map Mn (C) → Mk (C)
It is linear, completely positive and trace preserving.
I
Its Choi map is the matrix
X
CΦ =
Eij ⊗ Φ(Eij ) ∈ Mn ⊗ Mk
i,j∈{1,...,n}
Φ is CP iff CΦ is positive.
Our problem
Our problem
I
Notation: let Sn be the collection of ‘states’ on Mn , i.e. trace
1 positive operators.
Our problem
I
Notation: let Sn be the collection of ‘states’ on Mn , i.e. trace
1 positive operators. This is a convex set whose extremal
points are the rank one projections (denoted by Sne ).
Our problem
I
Notation: let Sn be the collection of ‘states’ on Mn , i.e. trace
1 positive operators. This is a convex set whose extremal
points are the rank one projections (denoted by Sne ).
I
We want to study the following sets: Φ(Sn ) and Φ(Sne ).
Our problem
I
Notation: let Sn be the collection of ‘states’ on Mn , i.e. trace
1 positive operators. This is a convex set whose extremal
points are the rank one projections (denoted by Sne ).
I
We want to study the following sets: Φ(Sn ) and Φ(Sne ).The
first one is compact convex and the second one is compact.
They are subsets of Sk .
Stinespring representation
Stinespring representation
I
Given a quantum channel Φ : Mn → Mk , there exists N and a
nonunital rank-preserving embedding i : Mn ⊂ MN ⊗ Mk
Stinespring representation
I
Given a quantum channel Φ : Mn → Mk , there exists N and a
nonunital rank-preserving embedding i : Mn ⊂ MN ⊗ Mk
such that for all x,
Φ(x) = (TrN ⊗ idk )x.
Stinespring representation
I
Given a quantum channel Φ : Mn → Mk , there exists N and a
nonunital rank-preserving embedding i : Mn ⊂ MN ⊗ Mk
such that for all x,
Φ(x) = (TrN ⊗ idk )x.
I
Φn (Sn ) and Φn (Sne ) depend only on i(1n ) =: Pn .
Random quantum channels
Random quantum channels
I
A random quantum channel is a quantum channel... chosen
at random.
Random quantum channels
I
A random quantum channel is a quantum channel... chosen
at random. We are interested in sequences Φn of such
quantum channels (k is fixed).
Random quantum channels
I
A random quantum channel is a quantum channel... chosen
at random. We are interested in sequences Φn of such
quantum channels (k is fixed).
I
Working with the Stinespring picture, for each n we fix an
N = N(n) and we choose i : Mn ⊂ MN ⊗ Mk at random
according to various distributions.
Random quantum channels
I
A random quantum channel is a quantum channel... chosen
at random. We are interested in sequences Φn of such
quantum channels (k is fixed).
I
Working with the Stinespring picture, for each n we fix an
N = N(n) and we choose i : Mn ⊂ MN ⊗ Mk at random
according to various distributions.
I
Since Φn (Sn ) and Φn (Sne ) depend only on i(1n ) = Pn , it is
enough for our purposes to study a random projection Pn of
rank n in MN ⊗ Mk
Convergence result
I
Our key assumption on the law of Pn is:
Convergence result
I
Our key assumption on the law of Pn is:
For any projection p
B ∈ Mk , ||B ⊗ 1N · Pn ||∞ converges with
probability one to f (B) as n → ∞.
Convergence result
I
I
Our key assumption on the law of Pn is:
For any projection p
B ∈ Mk , ||B ⊗ 1N · Pn ||∞ converges with
probability one to f (B) as n → ∞.
Theorem (C, Fukuda, Nechita, 2013)
There exists a convex compact set K such that Φn (Sn ) → K and
∂Φn (Sn ) → ∂K (Hausdorff distance between sets).
Convergence result
I
I
Our key assumption on the law of Pn is:
For any projection p
B ∈ Mk , ||B ⊗ 1N · Pn ||∞ converges with
probability one to f (B) as n → ∞.
Theorem (C, Fukuda, Nechita, 2013)
There exists a convex compact set K such that Φn (Sn ) → K and
∂Φn (Sn ) → ∂K (Hausdorff distance between sets).
K = {A ∈ Sk , ∀B ∈ Sk , Trk (AB) 6 f (B)}
Convergence result
I
I
Our key assumption on the law of Pn is:
For any projection p
B ∈ Mk , ||B ⊗ 1N · Pn ||∞ converges with
probability one to f (B) as n → ∞.
Theorem (C, Fukuda, Nechita, 2013)
There exists a convex compact set K such that Φn (Sn ) → K and
∂Φn (Sn ) → ∂K (Hausdorff distance between sets).
K = {A ∈ Sk , ∀B ∈ Sk , Trk (AB) 6 f (B)}
I
Replacing Φn (Sn ) by Φn (Sne ) (much smaller set) is possible if
one makes a slightly stronger assumption
Convergence result
I
I
Our key assumption on the law of Pn is:
For any projection p
B ∈ Mk , ||B ⊗ 1N · Pn ||∞ converges with
probability one to f (B) as n → ∞.
Theorem (C, Fukuda, Nechita, 2013)
There exists a convex compact set K such that Φn (Sn ) → K and
∂Φn (Sn ) → ∂K (Hausdorff distance between sets).
K = {A ∈ Sk , ∀B ∈ Sk , Trk (AB) 6 f (B)}
I
Replacing Φn (Sn ) by Φn (Sne ) (much smaller set) is possible if
one makes a slightly stronger assumption (no gap between the
first few largest eigenvalues).
First example
I
In [BCN 2012] we proved the convergence in the particular
case where Pn is a uniform random projection of rank
n ∼ tNk (t in (0, 1) is fixed).
First example
I
In [BCN 2012] we proved the convergence in the particular
case where Pn is a uniform random projection of rank
n ∼ tNk (t in (0, 1) is fixed). Here, the convergence works for
Φn (Sne ) too.
First example
I
In [BCN 2012] we proved the convergence in the particular
case where Pn is a uniform random projection of rank
n ∼ tNk (t in (0, 1) is fixed). Here, the convergence works for
Φn (Sne ) too.
I
In this case, f (B) = ||pBp||∞ where p is a rank t projection
free of B. We denote it ||B||(t) and call it the t-norm (free
compression norm).
First example
I
In [BCN 2012] we proved the convergence in the particular
case where Pn is a uniform random projection of rank
n ∼ tNk (t in (0, 1) is fixed). Here, the convergence works for
Φn (Sne ) too.
I
In this case, f (B) = ||pBp||∞ where p is a rank t projection
free of B. We denote it ||B||(t) and call it the t-norm (free
compression norm). K is a convex body with smooth
boundary under mild assumptions.
First example
I
In [BCN 2012] we proved the convergence in the particular
case where Pn is a uniform random projection of rank
n ∼ tNk (t in (0, 1) is fixed). Here, the convergence works for
Φn (Sne ) too.
I
In this case, f (B) = ||pBp||∞ where p is a rank t projection
free of B. We denote it ||B||(t) and call it the t-norm (free
compression norm). K is a convex body with smooth
boundary under mild assumptions.
I
We need new tools in RMT to construct more examples.
Asymptotic freeness for RMT
Asymptotic freeness for RMT
I
(n)
(n)
In 1992, Voiculescu proved that iid GUEs X1 , . . . , Xk
asymptotically free as n → ∞.
are
Asymptotic freeness for RMT
I
(n)
(n)
In 1992, Voiculescu proved that iid GUEs X1 , . . . , Xk are
asymptotically free as n → ∞. (i.e. the normalized trace of
any word in the matrices converges almost surely to a
quantity that can be computed with free probability).
Asymptotic freeness for RMT
(n)
(n)
I
In 1992, Voiculescu proved that iid GUEs X1 , . . . , Xk are
asymptotically free as n → ∞. (i.e. the normalized trace of
any word in the matrices converges almost surely to a
quantity that can be computed with free probability).
I
In 1998 he proved the following stronger result: if
(n)
(n)
(A1 , . . . , Ak ) is a family of n × n random matrices with an
asymptotic ∗- distribution and Un is a Haar distributed unitary
(n)
(n)
random matrix, then (A1 , . . . , Ak , Un ) has also an
asymptotic ∗- distribution (and there is asymptotic freeness).
Asymptotic freeness for RMT, ctd
I
In 2005, Haagerup and Thorbjørnsen proved that the norm of
(n)
(n)
any NC polynomial in iid GUEs X1 , . . . , Xk converges with
probability one as n → ∞ (towards where it should)
Asymptotic freeness for RMT, ctd
I
In 2005, Haagerup and Thorbjørnsen proved that the norm of
(n)
(n)
any NC polynomial in iid GUEs X1 , . . . , Xk converges with
probability one as n → ∞ (towards where it should)
I
In 2010, Male proved that if (A1 , . . . , Ak ) is a family that
satisfies strong asymptotic convergence (i.e. asymptotic
convergence plus convergence of the operator norm of any NC
polynomial),
(n)
(n)
Asymptotic freeness for RMT, ctd
I
In 2005, Haagerup and Thorbjørnsen proved that the norm of
(n)
(n)
any NC polynomial in iid GUEs X1 , . . . , Xk converges with
probability one as n → ∞ (towards where it should)
I
In 2010, Male proved that if (A1 , . . . , Ak ) is a family that
satisfies strong asymptotic convergence (i.e. asymptotic
convergence plus convergence of the operator norm of any NC
(n)
(n)
polynomial), then so does (A1 , . . . , Ak , X (n) ), where X (n) is
an independent GUE.
(n)
(n)
Norm convergence for random unitaries
Theorem (C, Male, 2011)
(n)
(n)
If (A1 , . . . , Ak ) is a family that satisfies strong asymptotic
convergence (i.e. asymptotic convergence plus convergence of the
operator norm of any NC polynomial),
Norm convergence for random unitaries
Theorem (C, Male, 2011)
(n)
(n)
If (A1 , . . . , Ak ) is a family that satisfies strong asymptotic
convergence (i.e. asymptotic convergence plus convergence of the
operator norm of any NC polynomial), then so does
(n)
(n)
(A1 , . . . , Ak , Un ), where Un is an independent Haar matrix.
Norm convergence for random unitaries
Theorem (C, Male, 2011)
(n)
(n)
If (A1 , . . . , Ak ) is a family that satisfies strong asymptotic
convergence (i.e. asymptotic convergence plus convergence of the
operator norm of any NC polynomial), then so does
(n)
(n)
(A1 , . . . , Ak , Un ), where Un is an independent Haar matrix.
Our proof builds on Camille’s proof and uses an ‘unfolding’ trick.
Norm convergence for random unitaries
Theorem (C, Male, 2011)
(n)
(n)
If (A1 , . . . , Ak ) is a family that satisfies strong asymptotic
convergence (i.e. asymptotic convergence plus convergence of the
operator norm of any NC polynomial), then so does
(n)
(n)
(A1 , . . . , Ak , Un ), where Un is an independent Haar matrix.
Our proof builds on Camille’s proof and uses an ‘unfolding’ trick.
Corollary (C, Male, 2011)
i.i.d copies of k random n × n Haar unitaries converge strongly (in
norm) towards generators of the free group factor.
Consequence: new examples
Corollary (C, Fukuda, Nechita)
(i)
Let k > 2 be an integer, Un be iid n × n Haar unitaries, and
X (i) (i)∗
Φn (x) = k −1
Un xUn .
Consequence: new examples
Corollary (C, Fukuda, Nechita)
(i)
Let k > 2 be an integer, Un be iid n × n Haar unitaries, and
X (i) (i)∗
Φn (x) = k −1
Un xUn .
Then the collection of nontrivial ordered eigenvalues of output of
all pure states converges with probability one to a deterministic set
(of Rk ).
Consequence: new examples
Corollary (C, Fukuda, Nechita)
(i)
Let k > 2 be an integer, Un be iid n × n Haar unitaries, and
X (i) (i)∗
Φn (x) = k −1
Un xUn .
Then the collection of nontrivial ordered eigenvalues of output of
all pure states converges with probability one to a deterministic set
(of Rk ).
In particular, almost surely,
lim ||Φn ||1→∞ =
n
4(k − 1)
.
k2
Consequence: new examples
Thank you!