View of Tensor products of positive maps of matrix algebras

TENSOR PRODUCTS OF POSITIVE MAPS OF MATRIX ALGEBRAS

ERLING STØRMER

Abstract

We give conditions for when tensor products of positive maps between matrix algebras are positive maps. Necessary and sufficient conditions are given.

Introduction

Even though positive linear maps appear in many situations in operator algebras and quantum information theory, main attention has so far been on completely positive maps. One reason for this is that tensor products of completely positive maps are positive, while this is false for general positive maps. In the present paper we shall consider the problem of when the tensor product of two positive maps is positive in the case when the underlying Hilbert spaces are finite dimensional.

It turns out that the problem is intimately related to symmetric mapping cones and their dual cones, see below for definitions. More specifically, ifCis a symmetric mapping cone inP (H)– the positive linear maps of the bounded operatorsB(H )onH into itself – then a mapφbelongs to the dual coneC^oof Cif if and only ifψ⊗φis positive for allψ ∈C. Indeed, it suffices to know that ψ⊗φ(p)≥0, where¹_np, n=dimH, is the density matrix for the maximally entangled state. As an application we show that ifKandLare two other finite dimensional Hilbert spaces, andψ:B(K)→B(H ),φ:B(L)→B(H ), then ψ⊗φis positive whenψisC-positive, andφisC^o-positive.

We now recall the main concepts encountered in the sequel. By amapping coneC we mean a closed subcone ofP (H )such that if φ ∈ C and α, β ∈ CP (H )– the completely positive maps inP (H), thenα◦φ∈Candφ◦β∈C. We sayC issymmetricifφ ∈C implies bothφ^∗∈C andφ^t =t◦φ◦t ∈C, whereφ^∗is the adjoint map ofφ in the Hilbert-Schmidt structure onB(H ), viz. Tr(φ(a)b) = Tr(aφ^∗(b))fora, b ∈ B(H ), andt is the transpose map onB(H )with respect to an orthonormal basise¹, . . . , enforH, and Tr is the usual trace onB(H).

Received 23 December 2010.

Ifφ:B(K)→B(H)then the functionalφonB(K)⊗B(H )defined by φ(a⊗b)=Tr(φ(a)b^t),

plays an important role in the theory. For example,φis positive if and only if φis completely positive [6]. By [7] and [8] its density matrix is the transpose C_φ^t of the Choi matrixCφ forφ, defined by

Cφ = n i,j=1

eij ⊗φ(eij)=ι⊗φ(p), whereιis the identity map andp=

ijeij ⊗eij,and(eij)is a complete set of matrix units forB(H)such thateijek =δjkei, see [1].

LetC be a mapping cone inP (H). Then its dual cone is defined by C^o= {φ∈P (H): Tr(CφCψ)≥0,∀ψ ∈C}.

IfC is symmetric, then by [9]C^ois also a symmetric mapping cone. We refer to the books [2] and [3] for the theory of completely positive maps.

Most of this work was done during a visit to Institute Mittag-Leffler (Djurs- holm, Sweden).

1. The main results

Letπ:B(H )→B(H)be defined byπ(a⊗b)=b^ta. As in [8] Lemma 10 it follows by straightforward computation that ifφ ∈P (H)then

(1) φ=Tr◦π◦(ι⊗φ^∗t).

In particular

(2) ι(x)=Tr◦π(x)=Tr(Cιx)=Tr(px).

Thus φ(x)=Tr◦π(ι⊗φ^∗t(x))

=Tr(p(ι⊗φ^∗t)(x))

=Tr(ι⊗φ^t(p)x).

Lemma1. Letψ, φ:B(H)→B(H). Then

(i) (φ◦ψ)(x) =Tr(ψ^∗⊗φ^t(p)x),∀x ∈B(H ⊗H ). (ii) ψ^∗t⊗φ(p)=ι⊗(φ◦ψ)(p).

Proof. By equations (1) and (2)

(φ◦ψ)(a⊗b)=Tr◦π(ι⊗(φ◦ψ)^∗t(a⊗b))

=Tr◦π(a⊗(ψ^∗◦φ^∗(b^t))^t)

=Tr(a(ψ^∗◦φ^∗)(b^t))

=Tr(ψ(a)φ^∗(b^t))

=Tr◦π(ψ(a)⊗φ^∗t(b))

=Tr(pψ(a)⊗φ^∗t(b))

=Tr(ψ^∗⊗φ^t(p)(a⊗b)), proving (i). Using equations (1) and (2) we also have

(φ◦ψ)(x) =Tr(p(ι⊗(φ◦ψ)^∗t(x))=Tr(ι⊗(φ◦ψ)^t(p)x).

Thus by (i) we have

ψ^∗⊗φ^t(p)=ι⊗φ^t◦ψ^t(p).

Since this holds for allφandψ, it also holds for allφ^t andψ^∗. Thus ψ^∗t ⊗φ(p)=ι⊗φ◦ψ(p),

completing the proof of the lemma.

We can now prove our main result. Note that the equivalence (i)⇔(ii) is also proved in [4].

Theorem2. Letφ ∈P (H ). LetC be a symmetric mapping cone inP (H). Then the following conditions are equivalent.

(i) φ ∈C^o– the dual cone ofC,

(ii) φ◦ψ is completely positive for allψ ∈C, (iii) ψ⊗φis positive for allψ ∈C,

(iv) ψ⊗φ(p)≥0for allψ ∈C, wherepis as before the maximal entangled state.

Proof. Clearly (iii) implies (iv). Sinceφ◦ψ is completely positive if and only ifι⊗φ◦ψ(p)≥0,by Lemma 1φ◦ψis completely positive if and only if ψ^∗t⊗φ(p)≥0. SinceC is symmetric,ψ ∈C if and only ifψ^∗t ∈C. Hence (ii) is equivalent to (iv). By [9] Thm. 2 a map belongs toC if and only if it is C-positive. Hence by [8] Thm. 1,φ ∈C^oif and only ifψ^t ◦φis completely positive for allψ ∈C. SinceC is symmetric this holds if and only ifψ◦φis completely positive for allψ ∈C, hence if and only ifφ^∗◦ψ^∗=(ψ◦φ)^∗is

completely positive for allψ^∗∈C, hence if and only ifφ^∗◦ψis completely positive for allψ ∈C. ButC^ois symmetric by [9], Thm. 1, soφ ∈C^oif and only ifφ^∗∈C^o. Thus (i)⇔(ii).

It remains to show (i) implies (iii). For this let(ei)be an orthonormal basis forH such thateijej =ei, so _n¹pis the projection onto the subspace spanned

by

iei⊗ei. Letx∈H ⊗H. Thenx=

iei⊗xi withxi ∈H. Then there isv ∈B(H)such thatvei = xi, hence 1⊗v

iei ⊗ei

= x. Letq be the projection ontoCx. Then it follows that Ad(1⊗v)(p)=λqfor someλ >0.

We have just shown that given a 1-dimensional projectionq ∈B(H )there existsv∈B(H )such that

1⊗Adv 1 np

=q.

Assuming (i)φ◦Adv∈C^o, sinceC^ois a mapping cone. By Lemma 1 ψ^∗t ⊗(φ◦Adv)(p)=ι⊗φ(Adv◦ψ)(p).

Since Adv◦ψ ∈C, by the equivalence of (i) and (ii)φ◦Adv◦ψis completely positive, hence

ι⊗φ◦Adv◦ψ(p)≥0.

Thus by the choice ofv,ψ^∗t⊗φ(q)≥0. Sinceqis an arbitrary 1-dimensional projection,ψ^∗t⊗φis positive. Again, sinceC is symmetric,ψ⊗φis positive for allψ ∈C.Thus (i) implies (iii), and the proof is complete.

Recall that a mapφ:B(K)→B(H)isC-positivefor a mapping coneC if the functionalφis positive on the cone{x∈B(K⊗H ):ι⊗α(x)≥0,∀α∈ C}. By [9], Thm. 2 or [6], Thm. 3.6, this is equivalent toφ belonging to the cone generated by maps of the formα◦βwithα∈C andβ:B(K)→B(H ) completely positive. Recall from [9], Thm. 1, that ifCis symmetric, so isC^o. Using these facts we can extend the implication (i)⇒(iii) in Theorem 2 to the following more general case.

Corollary3. LetK, LandH be finite dimensional Hilbert spaces. Let C be a symmetric mapping cone in P (H). Suppose ψ:B(K) → B(H )is C-positive, andφ:B(L)→B(H)isC^o-positive. Thenψ⊗φ:B(K⊗L)→ B(H⊗H )is positive.

Proof. By the above discussion it suffices to show the corollary forψand φof the formψ =α◦β, α∈C, β:B(K)→B(H )completely positive, and φ=γ ◦δwithγ ∈C^o, δ:B(L)→B(H)completely positive. Thus

ψ ⊗φ=(α⊗γ )◦(β⊗δ),

is positive, sinceβ⊗δis completely positive andα⊗γ is positive by Theo- rem 2. The proof is complete.

Remark. Ifψ:B(K¹)→B(H¹)isk-positive, i.e.,ψ ∈Pkin the notation of [5], andφ:B(K²)→B(H²)isk-superpositive, i.e.,φ∈SPkis of the form

iAdVi, Vi:K2→H2, then they remain the same as maps intoB(H )ifH is a Hilbert space containingH1andH2as subspaces. SinceP_k^o = SPk, see e.g. [5], it follows from Corollary 3 thatψ ⊗φis positive.

In Theorem 2 it is sometimes enough to consider only one mapψ ∈C to conclude thatφ ∈C^o. The next corollary is of this type.

Corollary4. Let ψ ∈ P (H )satisfyψ = ψ^∗ = ψ^t. Let C denote the mappng cone generated byψ. Letφ ∈ P (H). Thenφ ∈ C^o if and only if ψ⊗φis positive.

Proof. C is generated as a cone by maps of the form Adu◦ψ◦Adv, so the assumptions onψ imply thatC is a symmetric mapping cone. Since

Adu◦ψ◦Adv⊗φ =(Adu⊗ι)◦(ψ⊗φ)◦(Adv⊗ι),

and Adu⊗ιand Adv⊗ιare positive maps, it follows thatα⊗φis positive for allα ∈C if and only ifψ⊗φis positive, hence by Theorem 2,φ∈C^oif and only ifψ⊗φis positive, proving the corollary.

Remark. Theorem 2 and Corollary 4 contain well known characterizations of completely maps. Itψ =ιthen it satisfies the conditions of Corollary 4, so the mapping coneC generated byψ is the cone of completely positive maps.

Hence ifφ ∈P (H ), then by Corollary 4,φ∈C^oif and only ifι⊗φis positive, if and only ifφ∈C by definition ofC, soC^o=C. By Theorem 2 we have

Cφ =ι⊗φ(p)≥0⇔(α⊗ι)(ι⊗φ(p))≥0,∀α∈C

⇔α⊗φ(p)≥0,∀α∈C

⇔φ ∈C^o=C.

In Corollary 4 we assumedψ =ψ^∗= ψ^t. These conditions can be easily verified by checking the corresponding conditions for the Choi matrix. The next proposition is also true for self-adjoint linear maps.

Proposition5. Letφ ∈ P (H ). Thenφ = φ^∗ = φ^t if and only ifCφ is a real symmetric matrix invariant under the flipa⊗b→b⊗aonB(H )⊗B(H ).

Proof. LetJ be the conjugation onH ⊗H defined by J zei ⊗ej =zej ⊗ei, z∈^C,

whereei, . . . , enis an orthonormal basis such thateijek =δjkei. Then an easy computation shows that ifa, bare real matrices inB(H ), thenJ a⊗bJ =b⊗a, so forx ∈ B(H ⊗H ), a real matrix with respect to the basis (ei ⊗ej)for H ⊗H, thenx →J xJ is the flipF applied tox.

We haveCφ^t =C_φ^t,soφ =φ^tif and only ifCφ =C_φ^t,i.e.,Cφis symmetric.

Sinceφ∈P (H),Cφ is self-adjoint, henceCφ is symmetric if and only ifCφ

is real symmetric. Hence φ = φ^t if and only if Cφ is real symmetric. By [9], Lem. 3, Cφ^∗ = J CφJ. Henceφ = φ^∗ = φ^t if and only if Cφ is real symmetric, and by the first part of the proof,Cφ =F (Cφ), so invariant under the flip, completing the proof.

Example. A specific example of a map as in Proposition 5 is given by φ=AdV, whereV is a real symmetric matrix. Indeed, for generalV we have the formulas:

(AdV )^∗=AdV^∗, (AdV )^t =AdV ,

whereV =(aij)ifV =(aij), and AdV (x)=V xV^∗. Thus, ifV is real sym- metric, then AdV =(AdV )^∗=(AdV )^t. Furthermore, ifV is real symmetric andF the flip then

CAdV =

kl

ekl⊗AdV (ekl)=

ijkl

vkivljekl⊗eij.

Thus F (CAdV)=F

ijkl

vkivljekl⊗eij

=

ijkl

vkivljeij⊗ekl

=

ijkl

vikvjlekl⊗eij

=

ijkl

vkivljekl⊗eij

=CAdV,

where we at the third equality sign changed the roles ofiandk, andl andj, and used thatV was symmetric at the fourth. It follows thatCAdV is invariant under the flip.

REFERENCES

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