Extremal unital completely positive normal maps

0.16 MB·English

by  Anilesh Mohari

Extremal unital completely positive normal maps
Anilesh Mohari
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1 The Institute of Mathematical Sciences,
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2 CIT Campus, Tharamani, Chennai-600113

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E-mail:[email protected]
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Date: 12th Jan, 2013
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Abstract:
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We study the convex set of unital completely positive normal map on a von-
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. Neumann algebra and find a necessary and sufficient condition for an element in the
h
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a convex set tobeextremal. Wealso dealwiththesameproblemfortheconvex subset
m
which admits a faithful normal state.
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1 Introduction:
LetM =M (IC)bethematrixalgebra over thefield ofcomplex numbers. Anycom-
n
pletely positive unital map τ [St] on M admits Stinespring’s minimal representation
τ(x) = v xv∗ where {v ∈ M : 1 ≤ k ≤ d} are linearly independent. It is quite
Pk k k k
some time now that M. D. Choi’s described a useful criteria for an element τ to be
extremal in the convex set CP of completely positive unital map. The elegant crite-
ria [Ch] says τ is an extremal element if and only if the family {v v∗ :1 ≤ k,j ≤ d}
k j
is linearly independent. Extreme points in CP is a building block to test various
physical ansatz in quantum information theory [MW].
The proof that M.D. Choi executed, uses one crucial observation that the linear
spaceL generated byelements {v∗ :1 ≤ k ≤ d}isindependentoftherepresentation
τ k
that one may choose to represent τ. Same problem with M, an arbitrary von-
Neumann algebra or more generally a C∗-algebra remains open as an interesting
problem till then. It is not hard to see that Choi’s method can not be adapted
to the general situation. On the other hand Arveson’s affine correspondence [Ar]
between the CP maps and states onM (M (C)) has little use in the general set up
d n
as correspondence uses matrix basis explicitly for M.
In this paper we invent an independent method largely generalizing Dixmi´er
version of Radon-Nykodym theorem for two normal states to two normal completely
positive maps. Main comparable mathematical result with that of Choi’s result says
that if τ,η are two unital completely positive normal maps on M and η ≤ kτ on
M , non-negative elements of M for some constant k > 0 then
+
η(x) = Xvkxtkjvj∗
k
where τ(x) = v xv∗ is the minimal representation of Stinespring and tk ∈ M′
Pk k k j
are elements determined uniquely with t = (tk) non-negative. Main application of
j
this result says that τ is extremal if and only if there exists no non-trivial solution
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with λ = (λj) with entries in M′ to
k
Xvkλkjvj∗ = 0
In case τ admits an inner representation i.e. with v ∈ M for all k, then same
k
j
holds with λ taking values in the center of M. Thus our main result in particular
k
captures classical result of Choi [Ch].
In section 3 we investigate the same problem of finding criteria for extremal
property of an element τ in CP , the convex set of unital completely positive map
φ
on M with a faithful normal invariant state φ on M. Here we took some hint from
workof[Oh]whichfollowed workof[Pa]andformulatetheproblemintheframework
of Tomita’s coupling of M and M′. Method here is as well striking different from
Landau-Streater’s adaptation of Choi’s method.
2 Extremal decomposition of a completely positive uni-
tal map:
Let A be a unital C∗-algebra. A linear map τ : A→ A is called positive if τ(x) ≥ 0
for all x ≥ 0. Such a map is automatically bounded with norm ||τ|| = ||τ(I)||. Such
a map τ is called completely positive [St] (CP) if τ ⊗ I : A ⊗ M → A ⊗ M
n n n
is positive for each n ≥ 1 where τ ⊗I is defined by (xi) → (τ(xi)) with matrix
n j j
entries (xi) are elements in A. In this paper we will only consider unital completely
j
positive maps i.e. τ(I) = I. We denote by CP(A) or simply CP for the convex
set of unital completely positive map on A. Two elements τ,τ′ are said to be (anti)
cocycle conjugate if τ ◦α = β ◦τ′ for two (anti-) automorphisms α,β on A. They
are called (anti-) conjugate if τ ◦α = α◦τ′ for an (anti-) automorphism α.
An unital completely positive map on an unital C∗ algebra A admits a minimal
Stinespring representation [St]
τ(x) = Vπ(x)V∗
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where π : A → B(Hs) is a representation of A in B(Hs) which is the Hilbert space
completion of the kernel given on the set A⊗H by
k(x⊗ζ,y⊗η) =<ζ,τ(x∗y)η >
and V∗ : H → Hs is an isometry from H into Hs defined by V∗ : ζ → I ⊗ζ such
that {π(x)V∗ζ : x ∈ A,ζ ∈ H} spans Hs. Such a minimal Stinespring’s triplet
(Hs,π,V∗) is uniquely determined modulo unitary equivalence i.e. if (Hs′,π′,V′∗)
be another triplet associated with τ, then we get U : π(x)V∗ζ → π′(x)V′∗ζ is an
unitary operator so that UVU∗ = V′ and Uπ(x)U∗ = π′(x).
Let M be a von-Neumann algebra acting on a complex separable Hilbert space
(H,< .,. >), where the inner product is linear in second variable and M be the
∗
pre-dualspaceofM. We sayτ isnormalifl.u.b.τ(x ) = τ(l.u.b.x )foranybounded
α α
increasing netx in M wherel.u.b. denotes least upperbound. We will usenotation
α
CPσ(M) for the convex set of unital completely positive normal maps on M. When
there is no room for confusion we will simply denote it by CPσ.
For a von-Neumann algebra M, it is well known that norm closed unit ball
of M is an open dense subset in the norm closed unit ball of M∗ in the weak∗
∗
topology on M∗. Further M is a proper subset unless M is finite dimensional.
∗
However M is sequentially closed [Ta] and given any bounded linear functional ω
∗
on M there is a unique element ω ∈ M so that ω = ω + ω determined by
σ ∗ σ s
||ω−ωσ|| = minω′∈M∗||ω−ω′||. The element ωs is a singular functional unless it is
the zero element. In the following we describe such a decomposition via universal
enveloping algebra of M.
Let S(A) be the convex set of states on A. The universal enveloping alge-
bra A∗∗ of a C∗-algebra i.e. the double dual A∗∗ of A is a von-Neumann alge-
bra, unitary equivalent to von-Neumann algebra MA = {π (x) : x ∈ A}′′ where
u u
π (x) = ⊕ π (x) is the direct sum of representations on H = ⊕ H
u φ∈S(A) φ u φ∈S(A) φ
and (H ,π ,ζ ) is the GNS representation associated with a state φ on A. In the
φ φ φ
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following text we identify A∗∗ with MA. Let i : A → A∗∗ ≡ MA be the inclusion
u u
map of A into A∗∗.
For a representation π :A → B(H ), we set von-Neumann algebra M = π(A)′′
π π
and the Banach space adjoint linear map by πt : M∗ → A∗. We set now linear map
π
(πt) : (M ) → A∗ by restricting πt to (M ) . Finally we set notation π˜ : A∗∗ →
∗ π ∗ π ∗
M for the Banach space adjoint map of (πt) . Similarly for a completely positive
π ∗
map τ : A → M ⊆ B(H) where M is a von-Neumann algebra acting on a Hilbert
space H, we set τ˜: A∗∗ → M for adjoint map of (τt) : M → A∗.
∗ ∗
PROPOSITION 2.1: We have the following property of universal von-Neumann
algebra A∗∗:
(a) π˜ is a linear map which is continuous with respect to weak topologies of A∗∗.
The map π˜ takes norm closed unit ball of A∗∗ onto the closed unit ball of M ;
π
(b) π˜◦i= π on A;
(c) π˜ is an unital completely positive map from A∗∗ onto M ;
π
(d) For any central element z ∈ A∗∗, π˜(z) is an element in the center of M .
π
PROOF: Statement (a) and (b) are well known property of universal enveloping
algebra of a C∗-algebra for which we refer to Lemma 2.2 in Chapter 3 of [Ta].
Statement (c) and (d) are as well known but could not find a suitable ready
reference. Here weindicate aproof. Sinceπ is a positive map and sois it’s transpose
πt. Thus the restriction (πt) is also positive. That π˜ is positive follows as (πt)
∗ ∗
is positive and onto. For n−positive property of π˜, we note that the universal
enveloping algebra over M (A) is M (A∗∗) and the canonical map i ⊗ I is the
n n n
inclusion map of M (A) into M (A∗∗). Further for a representation π : A → B(H ),
n n π
we also have π˜⊗I ◦i⊗I = π⊗I . This shows in particular that π˜ is a completely
n n n
positive map and it’s restriction on i(A) is a representation.
By Kadison-Schwarz inequality for unital completely positive map we have
π˜(x∗y) = π˜(x∗)π˜(y) for x ∈ A∗∗ and y ∈ i(A) since π˜(y∗y)= π˜(y∗)π˜(y) for y ∈ i(A).
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Taking adjoint in the above relation we also get π˜(y∗x) = π˜(y∗)π˜(x). Operator al-
gebras involved are being ∗-closed and also π˜ being onto, we get π˜(x) ∈ M M′
πT π
if x ∈ A∗∗ A∗∗′.
T
It is well known that a projection in the center of A∗∗ determines uniquely a
representation of A uptoquasi-equivalence i.e. arepresentation π is quasi-equivalent
to the sub-representation x → π (x)z for some central projection z . z is called
u π π π
support of the representation in A∗∗. For details we once more refer to Chapter 3 in
[Ta].
Now we take A = M, a von-Neumann algebra in Proposition 2.1 and π : M →
M ⊆ B(H) be the identity representation. Let z be the support projection of the
π
representation π : M → M in M∗∗ [Ta, Chapter 3]. Thus π˜(z ) is an element in
π
the center of M, where π˜ is the lift of π to the universal enveloping algebra defined
in Proposition 2.1.
We defined action of a C∗-algebra A on it’s dual A∗ by
< y,xω >=<yx,ω > < y,ωx >=<xy,ω >
where < .,. > denotes evaluation of a functional on a given element in the Banach
space. A subspace of A∗ is called A invariant if the subspace is invariant by both
left and right action. The crucial property used to define action that A is algebra.
PROPOSITION 2.2: Let M be a von-Neumann algebra with it’s pre-dual M .
∗
A closed M-invariant subspace V of M is determined unique by a projection z in
∗
the center of M by V = zM .
∗
PROOF: We refer to Theorem 2.7 in chapter 3 in [Ta].
PROPOSITION 2.3: Let M be a von-Neumann algebra with it’s pre-dual space
M and dual M∗. Then there exist a unique central projection z ∈ M∗∗, the
∗
universal enveloping von-Neumann algebra so that M = zM∗ where M∗∗ acts
∗
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natural on the Banach space M∗ given by
< y,xω >=<yx,ω > < y,ωx >=<xy,ω >
for x,y ∈ M∗∗ and ω ∈ M∗
PROOF:ItisasimpleapplicationofProposition2.2oncewetakeMinProposition
2.2 as M∗∗ so M in Proposition 2.2 is M∗.
∗
PROPOSITION 2.4: An element ω ∈ M∗ determines uniquely an element ω ∈
σ
M defined by
∗
ω (x) =< ω,zi(x) >
σ
for all x ∈ M and the map ω → ω ∈ M∗ is positive linear contractive map on
σ
the Banach space M . The element ω (x) =< ω,(1−z)i(x) > in M∗ is singular
∗ s
provided z 6= 1. The decomposition ω = ω +ω is also uniquely determined and
σ s
||ω|| = ||ω ||+||ω ||.
σ s
PROOF: We refer chapter 3 in [Ta1] for details.
Thefollowingpropositionsaysalongthesameveinnowfortheclassofcompletely
positive maps on M.
THEOREM 2.5: Given a completely positive map τ on M, there is a unique
normal completely positive map τ on M and a singular completely positive map
σ
τ such that τ = τ +τ where τ ,τ are determined uniquely by the decomposition
s σ s σ s
ωτ = (ωτ) , ωτ = (ωτ) of a normalstate ω. Further the set of normalcompletely
σ σ s s
positive maps on M is sequentially closed in the set of completely positive maps on
M and open dense set in CP with Bounded Weak topology.
PROOF: Let M∗∗ be the universal von-Neumann algebra of M and i: M→ M∗∗
be the canonical inclusion map of M into M∗∗. We write for an element ω ∈ M
∗
(ωτ) (x) =< ωτ,zi(x) >
σ
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where z is the central projection in M∗∗. The map ω → (ωτ) determines a normal
σ
positive linear map τσ on M by ωτ (x) =< ωτ,zi(x) > for all x ∈ M and ω ∈ M .
σ ∗
It also shows clearly that τ is completely positive as τ ⊗I = (τ ⊗I ) . Similarly
σ σ n n σ
we also have ωτ (x) =< ωτ,(1−z)i(x) > for all x ∈ M, ω ∈ M and the induced
s ∗
map τ is also completely positive.
s
Let τ be a sequence of such normal maps on M and it’s bounded weak limit
n
is τ i.e. ω(τ (x)) → ω(τ(x)) for all ω ∈ M and x ∈ M. Let τ be the singular
n ∗ s
part of τ. Then singular part of ωτ also converges to singular part of ωτ in weak∗
n
topology. Since each τ is normal and so is ω, we get ωτn = 0. Since this holds for
n s
all ω ∈ M , we arrive at our desired claim that τ = 0.
∗ s
We will show complement of CP is a closed set in BW topology. Let τ be a net
σ α
of completely positive singular map on Mconverges to τ in BoundedWeak topology
i.e. ωτ (x) → ωτ(x) for all x ∈ M and ω ∈ M . Let τ = τ +τ be the unique
α ∗ σ s
decomposition. Thus ωτ → ωτ +ωτ . Note that each element ωτ is a singular
α s σ α
functional i.e. an element in M⊥ and which is closed. Thus limiting element is also
∗
a functional in M⊥. This shows that ωτ = 0 for all ω ∈ M .
∗ σ ∗
Acompletely positive mapis called purely singularinshortsingular ifit’s normal
partoftheuniquedecomposition contributethezeromap. WedenotebyCP forthe
s
convex set of unital singular completely positive maps on M. One natural question
that arises at this point when an unital completely positive map on M is a convex
combination of unital normal and singular completely positive maps? For any unital
representation π, π (I) = z and π (I) = I −z and thus is a convex combination of
σ s
two such unital map if and only if z = 0 or z = I. However we have the following
observation. Furthermore any unital representation π : M → M is an extremal
element in CP. Let π = λτ + (1 − λ)τ for some τ ,τ ∈ CP and λ ∈ (0,1).
0 1 0 1
Then for a projection e, we check that π(1−e)τ (e)π(1 −e) = 0 for k = 0,1 i.e.
k
τ (e) ≤ π(e) for all k = 0,1 and λτ (e) + (1 − λ)τ = π(e). In particular π (e)
k 0 1 0
commutes with π (e) and thus isomorphic to measurable functions 0 ≤ f ,f ≤ 1 on
1 0 1
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a measure space with λf +(1−λ)f = 1. Since λ ∈ (0,1) we have f = f = 1. Now
0 1 0 1
pushing back via the isomorphism we get τ (e) = τ (e) = π(e) for all projections e
0 1
i.e. π = π = π. The proof would have been completed if π is also normal. For a
0 1
general π we consider the lifting of π,π ,π to normal completely positive map from
0 1
M∗∗ to M. Thus we have π˜ = λπ˜ +(1−λ)π˜ on M∗∗. Since π˜ is homomorphism
0 1
on i(M) and π˜ ◦i(x) = π(x), we conclude that π (x) = π˜ ◦i(x) = π˜ ◦i(x) = π(x)
k k
for all x ∈ M. Thus π is an extremal element in the set of positive unital map on
M and so is in extremal element in CP.
PROPOSITION2.6: LetMbevon-Neumannalgebraandz becentralprojection
0
intheuniversalenveloping algebra M∗∗ suchthatM = z M∗. Anelement τ ∈ CP
∗ 0
is a convex combination of two unital maps τ ∈ CP and τ ∈ CP with λ ∈ (0,1)
σ σ s s
i.e.
τ = λτ +(1−λ)τ
σ s
if andonly ifτ˜(z )isascaler whereτ˜istheuniquecompletely positive unitalnormal
0
map from M∗∗ to M such that following diagram commutes:
< ωτ,x >=<ω,τ˜(x) > (2.1)
for all x ∈ M∗∗ and ω ∈ M . Further the map τ → τ˜ is an affine one to on map.
∗
PROOF: Let τ(x) =Vπ(x)V∗ be the Stinespring’s minimal representation of τ i.e.
π : A → B(K) is unique unital representation in a Hilbert space K and V∗ : K → H
is the unique contraction ( modulo unitary equivalence ). It shows clearly
τ (x) =Vπ˜(z i(x))V∗
σ π
and
τ (x)= Vπ˜(z i(x))V∗
s π
where z is the support of π in M∗∗ and π˜ :M∗∗ → B(K) be the lifting map of the
π
representation π : M→ B(K) defined as in Proposition 2.1. By Proposition 2.1 (d),
projection π˜(z ) is an element in the center of M = {π(x) :x ∈M}′′.
π π
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Alternativedescriptionofτ andτ areasfollows. Letz bethecentralprojection
σ s 0
in M∗∗ such that z M∗ = M . Then z is the support projection of the identity
0 ∗ 0
representation π : M → M i.e. π (x) = x for all x ∈ M which is quasi-equivalent
0 0
to i : M → π (M)z. So (πt) : M → M∗ is the inclusion map and thus we have
0 u ∗ ∗
ωπ˜ ◦i = ω as an element in M∗ for any normal state ω on M.
0
We recall now
(ωτ) (x) =< ωτ,z i(x) >=<ω,τ˜(z i(x)) >
σ 0 0
for all x ∈ M and ω ∈ M . Uniqueness of the decomposition into normal and
∗
singular completely positive maps ensures that
τ (x) = τ˜(z i(x))
σ 0
So in particular we have τ˜(z ) = Vπ˜(z )V∗ ∈ M. That τ˜(z ) is a scaler if and only
0 π 0
if τ is a convex combination of two unital maps from CP and CP respectively with
σ s
λ = τ˜(z ) follows from uniqueness of the decomposition.
0
PROPOSITION 2.7: Let τ be a completely positive map on a C∗ algebra A and
η be another completely positive map on A so that for some positive constant c,
η(x) ≤ cτ(x) for all x ∈ A . Then there exists a non-negative element T′ ∈ π(A)′
+
such that
η(x) = Vπ(x)T′V∗ (2.2)
where (K,π,V∗) is the minimal Stinespring triplet associated with τ. Further for
each x ∈ M, π(x) and T′ commutes with the representation ρ : τ(A)′ → B(K)
defined by
ρ(y) : x⊗f = x⊗yf (2.3)
PROOF: Without loss of generality we assume that A is a closed ∗-subalgebra of
B(H), H be a Hilbert space. We recall Stinespring minimal representation τ(x) =
Vπ(x)V∗ where π : A → B(K) is the representation induced by the map y ⊗f →