We prove a reduction theorem for capacity of positive maps of finite
dimensional C*-algebras, thus reducing the computation of capacity to the case
when the image of a nonscalar projection is never a projection.Comment: 7 page

Stormer, Erling

English

arXiv

We prove a reduction theorem for capacity of positive maps of finite dimensional C*-algebras, thus reducing the computation of capacity to the case when the image of a nonscalar projection is never a projection

Størmer, Erling

NORA - Norwegian Open Research Archives

Dept. of Math. University of OsloPure Mathematics No 28ISSN 0806–2439 October 2005A reduction theorem for capacity of positive mapsErling StørmerOctober 3, 2005AbstractWe prove a reduction theorem for capacity of positive maps of finitedimensional C∗−algebras, thus reducing the computation of capacity tothe case when the image of a nonscalar projection is never a projection.IntroductionIn quantum information theory there has been a great deal of interest in theconcept of capacity of completely positive maps. A drawback with capacity isthat it is usually quite difficult to compute, hence there is a need for developingcomputational techniques. In the present paper we shall prove a reductiontheorem for capacity which reduces its computation to the ergodic case. As aconsequence we get a partial result towards the additivity of capacity for tensorproducts.If P is a finite dimensional C∗−algebra we denote by TrP the trace on Pwhich takes the value 1 at each minimal projection. Let η denote the real func-tion η(t) = −t log t for t > 0, and η(0) = 0. Then the entropy S(a) of a positiveoperator a in P is defined by S(a) = TrP (η(a)). If M is another finite dimen-sional C∗−algebra let Φ:M → P be a positive unital linear trace preservingmap, i.e. TrP (Φ(x)) = TrM (x) for all x ∈ M. Note that we only assume Φ ispositive and not completely positive, since the latter stronger assumption is inmost cases unnecessary. Let C denote the positive operators in M with trace1. If a ∈ C letC(Φ, a) = supS(Φ(a))−∑iλiS(Φ(ai)),where the sup is over all convex combinations of operators ai ∈ C with∑i λiai =a. The capacity C(Φ) of Φ is defined byC(Φ) = supa∈CC(Φ, a).For a discussion of capacity see e.g. [2].11 The reduction theoremIf P is a finite dimensional C∗−algebra and ω is a state on P let Qω denoteits density operator in P. Then the entropy of ω (with respect to P ) is S(ω) =S(Qω). We shall need three properties of entropy, namely: it is subadditive, i.e.S(ω1+ω2) ≤ S(ω1)+S(ω2); it is concave, i.e. S(λω1+(1−λ)ω2) ≥ λS(ω1)+(1−λ)S(ω2), and if N ⊆ M ⊆ P are C∗−subalgebras then S(ω | N) ≥ S(ω | M).Our first result is taken from the book [3] and is an inequality in the oppositedirection.Lemma 1 Let M ⊆ P be finite dimensional C∗−algebras, and let e1, . . . , en beprojections in M with sum 1. Let N =⊕ni=1Ni, where Ni = eiMei. Let ω be astate on P. Then∑iω(ei)S(ω|Niω(ei)) = S(ω|N)−∑iη(ω(ei)) ≤ S(ω).Proof. Let si = ω(ei). ThenS(ω|N) =∑iS(ω(ei.ei))=∑iS(ω(ei.ei)sisi)=∑isiS(ω(ei.ei)si) + η(si)which proves the equality in the lemma.In order to prove the inequality let fk be minimal projections in P andαk > 0 such that the density operator Qω for ω is of the form Qω =∑k αkfk,so in particular∑k αk = 1. Thus S(ω) = S(Qω) =∑k η(αk). By the first partof the proof we haveS(ω|N) =∑iS(ω(ei.ei))=∑iS(∑kαkeifkei)≤∑i,kS(αkeifkei)=∑i,kαkS(eifkei) + η(αk)TrP (eifkei)=∑i,kαkη(TrP (eifkei)) + η(αk)TrP (eifkei)≤∑iη(∑kαkTrP (eifkei)) +∑kη(αk)2=∑iη(TrP (eiQωei)) + S(ω)=∑iη(ω(ei)) + S(ω),where the first inequality follows from subadditivity of S and second from con-cavity. We also used that eifkei = TrP (eifkei)p, where p is a minimal projec-tion. The proof is complete.From the definition of capacity it is clear that if Φ:M → P is as before, andN ⊆ M, then C(Φ|N) ≤ C(Φ). Our next result describes a situation when wehave equality. We shall use a result of Broise, see [5] , that if a is a self-adjointoperator in M such that Φ(a2) = Φ(a)2 then Φ(aba) = Φ(a)Φ(b)Φ(a) for allb ∈M. In particular, if e is a projection inM such that Φ(e) is a projection, thenthe above identity holds for a replaced by e. The ergodic case alluded to in theintroduction is the case when the only operators a which satisfy Φ(a2) = Φ(a)2are the scalar operators.Theorem 2 Let M,P be finite dimensional C∗−algebras. Let Φ:M → P be apositive unital trace preserving map. Suppose e1, . . . , en are projections in Mwith sum 1 such that Φ(ei) is a projection for all i. Let N =⊕eiMei. ThenC(Φ) = C(Φ|N).Proof. Clearly C(Φ) ≥ C(Φ|N). For the opposite inequality let a, am ∈ C suchthat a =∑m λmam. Let Q =⊕Φ(ei)PΦ(ei). Since Φ(eixei) = Φ(ei)Φ(x)Φ(ei)for all x ∈M,Φ(EN (x)) = EQ(Φ(x)), where EN and EQ denote the conditionalexpectations on N and Q respectively. ThusS(Φ(a)) ≤ S(EQ(Φ(a))) = S(Φ(EN (a))).Therefore by Lemma 1 applied to the states ωm defined by Qωm = Φ(am) ande1, . . . , en yields the following inequality.S(Φ(a))−∑mλmS(Φ(am))≤ S(Φ(EN (a)))−∑mλm∑iTrP (Φ(ei)Φ(am)Φ(ei))S(Φ(ei)Φ(am)Φ(ei)TrP (Φ(ei)Φ(am)Φ(ei)))= S(Φ(EN (a)))−∑mλm∑iTrP (Φ(eiamei))S(Φ(eiamei)TrP (Φ(eiamei))= S(Φ(EN (a)))−∑m,iλmTrM (eiamei)S(Φ(eiamei)TrM (eiamei))= S(Φ(EN (a)))−∑m,iµm,iS(Φ(eiamei)TrM (eiamei)),where∑m,i µm,i = 1, andeiameiTrM (eiamei)= EN ( eiameiTrM (eiamei) ) ∈ N with trace 1.Since the above inequality holds for all families (am) as aboveC(Φ, a) ≤ C(Φ|N,EN (a)).3Since this holds for all a ∈MC(Φ) = supaC(Φ, a) ≤ supaC(Φ|N,EN (a)) = C(Φ|N),proving the theorem.We can now state our main reduction theorem. Note that if the projectionsei are minimal with the property that Φ(ei) is a projection, then Φ|eiMei isergodic in the sense defined above, so the theorem is a reduction to the ergodiccase.Theorem 3 Let M,P be finite dimensional C∗−algebras and Φ:M → P apositive unital trace preserving map. Let e1, . . . , en be projections in M withsum 1 such that Φ(ei) is a projection for each i. Let Mi = eiMei and Φi =Φ|Mi:Mi → Φ(ei)PΦ(ei) be the restriction map to Mi. ThenC(Φ) = logn∑i=1eC(Φi).Proof. By Theorem 2 it suffices to consider a =∑i ai ∈ M,ai = aei ∈ Mi,where ai =∑j λjiaji with TrM (aji) = 1, aji ∈ M+i ,∑ji λji = 1. Let si =TrM (eia) = TrM (ai) = TrP (Φ(ei)Φ(a)). Then we haveS(Φ(a)) −∑jiλjiS(Φ(aji))=∑i[S(Φ(ei)Φ(a))−∑jλjiS(Φ(aji))]=∑i[S(si(1siΦ(ei)Φ(a)))− si∑jλjisiS(Φ(aji))]= −∑isi log si +∑isi[S(1siΦ(ei)Φ(a))−∑jλjisiS(Φ(aji))]We haveS(1siΦ(ei)Φ(a))−∑jλjisiS(Φ(aji)) ≤ C(Φ|Mi).ThereforeS(Φ(a)) −∑jiλjiS(Φ(aji))≤ −∑isi(log si − C(Φ|Mi))= −∑isi(log si − log C(Φ|Mi)∑k eC(Φ|Mk) ) + log∑ieC(Φ|Mi)4Since the sum∑i si(log si − log eC(Φ|Mi)Pk eC(Φ|Mk) ) is a relative entropy, it is nonneg-ative, see Lemma 4.5 in [4]. Hence we haveS(Φ(a))−∑jiλjiS(Φ(aji)) ≤ log∑ieC(Φ|Mi),Since this holds for all a we conclude that C(Φ) ≤ log∑i eC(Φ|Mi).For the converse inequality let ε > 0, and choose bi ∈M+i with TrM (bi) = 1,µji ≥ 0 with∑j µji = 1 and aji ∈ M+i with trace 1 such that∑j µjiaji = bi,andS(Φ(bi))−∑jµjiS(Φ(aji)) ≥ C(Φ|Mi)− ε.Let now si ≥ 0 have sum 1, and let ai = sibi, λji = siµji. Put a =∑i ai =∑ji λjiaji. Then by the above inequality we haveS(1siΦ(ei)Φ(ai))−∑jλjisiS(Φ(aji)) ≥ C(Φ|Mi)− ε.Thus by the computations in the beginning of the proof we haveS(Φ(a))−∑jiλjiS(Φ(aji)) ≥ −∑isi(log si − C(Φ|Mi))− ε.Hence by the same computation we did above we obtainS(Φ(a)) −∑jiλjiS(Φ(aji))≥ −∑isi(log si − log C(Φ|Mi)∑k eC(Φ|Mk) ) + log∑keC(Φ|Mk) − ε.For the value si =C(Φ|Mi)Pk C(Φ|Mk) the value of the relative entropy is 0, henceC(Φ) ≥ S(Φ(a))−∑jiλjiS(Φ(aji)) ≥ log∑keC(Φ|Mk) − ε.Since ε is arbitrary the proof is complete.A good illustration of an application of the theorem is the case when Φis a trace preserving projection map of M into itself, i.e.Φ(x) = Φ(Φ(x)) forall x ∈ M. Then the image N = Φ(M) is a Jordan subalgebra of M, andif Φ is completely positive then Φ is a conditional expectation, and Φ(M) is aC∗−algebra, see [1]. The rank of N -rankN - is the maximal number of minimalprojections in N with sum 1.Corollary 4 Let Φ:M →M be a trace preserving projection map. ThenC(Φ) = log rankΦ(M).5Proof. Let n = rankN and e1, . . . , en be minimal projections in Φ(M) with sum1. Then ekMek = Cek for all k, hence C(Φ|ekMek) = 0, so by the theoremC(Φ) = logn∑ie0 = log n.The proof is complete.The main problem concerning capacity is whether it is additive under tensorproducts, i.e. whether C(Φ ⊗ Ψ) = C(Φ) + C(Ψ) when Φ ⊗ Ψ is positive, inparticular when they are both completely positive. Our next result reduces theproblem to the case when both maps are ergodic.Corollary 5 Let M,N,P,Q be finite dimensional C∗−algebras and Φ:M → Pand Ψ:N → Q be positive unital trace preserving maps such that Φ⊗Ψ:M⊗N →P ⊗Q is positive. Let ei ∈ M and fj ∈ N be projections with sum 1 such thatΦ(ei) and Ψ(fj) are projections. LetΦi = Φ|eiMei: eiMei → Φ(ei)PΦ(ei),Ψj = Ψ|fjNfj : fjNfj → Ψ(fj)QΨ(fj).Suppose C(Φi ⊗Ψj) = C(Φi) + C(Ψj) for all i, j. ThenC(Φ⊗Ψ) = C(Φ) + C(Ψ).Proof. We apply Theorem 3 to the projections ei ⊗ fj and the correspondingmaps Φi ⊗Ψj . Thus we haveC(Φ⊗Ψ) = = log∑ijeC(Φi⊗Ψj ) = log∑ijeC(Φi)+C(Ψj)= log∑ijeC(Φi)eC(Ψj) = log∑ieC(Φi)∑jeC(Ψj)= C(Φ) + C(Ψ).The proof is complete.If Φ is completely positive and id is the identity map of N let fj be a minimalprojection for each j. Then the assumptions of the above corollary hold for theprojections 1⊗ fj . Hence we haveCorollary 6 Let M and N be finite dimensional C∗−algebras as before with Φcompletely positive. Then C(Φ⊗ id) = C(Φ) + log rankN.References[1] E. Effros and E. Størmer, Positive projections and Jordan structure inoperator algebras, Math. Scand. 45 (1979), 127–138.6[2] A. Holevo, Statistical structure of quantum theory, Lecture Notes inPhysics. Monographs 67. Springer-Verlag, Berlin, Heidelberg 2001.[3] S. Neshveyev and E. Størmer, Dynamical entropy in operator algebras,To appear.[4] M. Smorodinsky, Ergodic theory, entropy, Lecture Notes in Mathematics214. Springer-Verlag, Berlin, Heidelberg, New York 1971.[5] E. Størmer, Decomposition of positive projections on C*-algebras, Mat.Ann, 247 (1980), 21–41.Department of Mathematics, University of Oslo, 0316 Oslo, Norway.e-mail : erlings@math.uio.no7

A reduction theorem for capacity of positive maps

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A reduction theorem for capacity of positive maps

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