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Distance to boundary and minimum-error discrimination Erkka Haapasalo,1 Michal Sedlák,2,3 and Mário Ziman3,4 1Turku Centre for Quantum Physics, Department of Physics and Astronomy, University of Turku, FI-20014 Turku, Finland 2Department of Optics, Palacký University, 17. listopadu 1192/12, CZ-77146 Olomouc, Czech Republic 3RCQI, Institute of Physics, Slovak Academy of Sciences, Du´bravská cesta 9, 84511 Bratislava, Slovakia 4Faculty of Informatics, Masaryk University, Botanicka´ 68a, 60200 Brno, Czech Republic We introduce the concept of boundariness capturing the most efficient way of expressing a given element of a convex set as a probability mixture of its boundary elements. In other words, this number measures (without the need of any explicit topology) how far the given element is from the boundary. It is shown that one of the elements from the boundary can be always chosen to be an extremal element. We focus on evaluation of this quantity for quantum sets of states, channels and observables. We show that boundariness is intimately related to (semi)norms that provide 4 an operational interpretation of this quantity. In particular, the minimum error probability for 1 discrimination of a pair of quantum devices is lower bounded by the boundariness of each of them. 0 We proved that for states and observables this bound is saturated and conjectured this feature for 2 channels. The boundariness is zero for infinite-dimensional quantum objects as in this case all the elements are boundary elements. g u PACSnumbers: 3.67.-a A 9 I. INTRODUCTION contain mathematical details concerning the properties 2 of weight function, characterization of the boundary ele- ] The experimetal ability to switch randomly between ments of all considered quantum sets and numerical de- h tails of the case study. p physical apparatuses of the same type naturally endows - mathematical representatives of physical objects with a t n convex structure. This makes the convexity (and inti- a mately related concept of probability) one of the key II. CONVEX STRUCTURE AND u mathematical features of any physical theory. Even BOUNDARINESS q more, the particular “convexity flavor” plays a crucial [ roleinthedifferencesnotonlybetweenthetypesofphys- In any convex set Z we may define a convex preorder 2 ical objects, but also between the theories. For exam- ≤ . We say x ≤ y if x may appear in the convex C C v ple,theexistenceofnon-uniqueconvexdecompositionof decomposition of y with a non zero weight, i.e. there 0 densityoperatorsisthepropertydistinguishingquantum exist z ∈ Z such that y = tx+(1−t)z with 0 < t ≤ 1. 6 theory from the classical one [1]. If x ≤ y, then y has x in its convex decomposition, 4 C 7 Our goal is to study the convex structures that natu- hence, (losely speaking) y is “more” mixed than x. The . rally appear in the quantum theory and to illustrate the valueoft(optimizedoverz)canbeusedtoquantifythis 1 operationalmeaningoftheconceptsdirectlylinkedtothe relation. Namely, for any element y ∈ Z we define the 0 4 convex structure. However, most of our findings are ap- weight function ty : Z → [0,1] assigning for every x ∈ Z 1 plicable for any convex set. The main goal of this paper the supremum of possible weights t of the point x in the : is to introduce and investigate the concept of boundari- convex decomposition of y, i.e. v ness quantifying how far the individual elements of the i X convexsetarefromitsboundary. Intuitively,thebound- (cid:110) (cid:12) y−tx (cid:111) t (x)=sup 0≤t<1(cid:12)z = ∈Z . r arinessdeterminesthemostnon-uniform(binary)convex y (cid:12) 1−t a decomposition into boundary elements, hence, it quantifies how mixed the element is. We will show that this Obviously, t (y) = 1 and t (x) = 0 whenever x (cid:2) y. y y C concept is operationally related to specification of the In order to understand the geometry of the optimal z most distinguishable element (in a sense of minimum- for a given pair of elements x,y, it is useful to express errordiscriminationprobability). Forinstance,forstates the element z in the form z = y + t (y − x). As t 1−t the evaluation of boundariness coincides with the speci- increases,zmovesinthedirectionofy−xuntil(forvalue fication of the best distinghuishable state from the given t=t (x)) it leaves the set Z (see Fig. 1 for illustration). y one, hence it is proportional to trace-distance [2]. If the element z associated with t (x) is an element of y The paper is organized as follows: Section II intro- Z, then it can be identified as a boundary element of Z. duces the concept of boundariness and related results in The (algebraic) boundary ∂Z contains all elements y for general convex sets, the boundariness for quantum sets whichthereexistsxsuchthatx(cid:2) y(letusstressthisis C is evaluated in Section III and the relation to minimum- equivalentwiththedefinitionusedinRef.[3]). Hence,for error discrimination is described in Section IV. Section each boundary element y the weight function t (x) = 0 y V shortly summarizes the main results. The appendices forsomexandalsotheoppositeclaimholds,i.e.,ifthere 2 FIG. 1: Illustration of elements z and x(cid:48) emerging in the FIG. 3: Illustration of the proof of the Lemma 1 definition of the weight function t (x) and in the property y t (x(cid:48))≤t (x), respectively. y y See Fig. 3 for illustration. Straightforward calculation shows that we may write y = tx + (1 − t)z, where exists x∈Z :ty(x)=0 then y ∈∂Z. As a consequence, t−1 =st−1+(1−s)t−1. Fromthedefinitionoftheweight 1 2 ty(x)>0 ∀x∈Z for all inner points y ∈Z\∂Z. function, we have t ≤ ty(x). Since this holds for all Thismotivatesthefollowingdefinitionofboundariness 0<t <t (x )i=1,2weget( s + 1−s )−1 ≤t (x), b(y)= inf t (x) whichi conycludies the proof. (cid:4) ty(x1) ty(x2) y y x∈Z The following proposition is one of the key results of thissection. Itguaranteesthatoneoftheelementsofthe measuring how far the given element of Z is from the boundary ∂Z. Suppose x(cid:48) belongs to the line generated optimal decomposition (determining the boundariness) by x and y, i.e. x(cid:48) =y−k(y−x) (x(cid:48) =x for k =1 and can be chosen to be an extreme point of Z. It is shown x(cid:48) = y for k = 0). Then t (x(cid:48)) ≤ t (x) whenever k ≥ 1 in Appendix A that, whenever Z ⊂ Rn for some n ∈ N, y y the weight function t is continuous if (and only if) y ∈ (see Fig. 1). Hence, the infimum can be approximated y Z\∂Z. Continuityoft isstudiedintheappendicesalso again by some boundary element of Z. In other words, y in a slightly more general context. thevalueofboundarinessisdeterminedbythemostnon- uniform convex decomposition of y into boundary ele- Proposition 1 SupposethatZ ⊂Rn isconvexandcom- ments of Z, i.e. y can be, in a sense, approximated by pact set. For every y ∈ Z \∂Z there exists an extreme expressionsb(y)x+(1−b(y))z withx,z ∈∂Z. Therefore, point x∈Z such that b(y)=t (x). b(y) ≤ 1/2. See Fig. 2 for illustration of boundariness y for simple convex sets. Proof. Thecontinuityimpliesthatt acquiresitslowest y Lemma 1 Let y ∈ Z. The inverse x (cid:55)→ 1/ty(x) of the value on the compact set Z, i.e., b(y) = infx∈Zty(x) = weight function ty is convex, i.e., minx∈Zty(x). Sincey ∈Z\∂Z,wehavety(x)>0. More- over, because of the convexity of x (cid:55)→ 1/t (x) proven in 1 s 1−s y (cid:0) (cid:1) ≤ + Lemma 1, it follows that t sx +(1−s)x t (x ) t (x ) y 1 2 y 1 y 2 (cid:0) (cid:1)−1 mint (x) = max1/t (x) for all x , x ≤ y and 0≤s≤1. y y 1 2 C x∈Z x∈Z (cid:0) (cid:1)−1 = max 1/t (x) = min t (x), Proof. For every 0<t <t (x ) i=1,2 we define z = y y i y i i x∈extZ x∈extZ y−1−titi(xi−y)∈Z. Further,wedefinex=sx1+(1−s)x2 where extZ denotes the set of extreme points of Z. (cid:4) andz =uz +(1−u)z ,wherex, z ∈Z becauses∈[0,1] 1 2 Theconvexsetsappearinginquantumtheoryaretyp- and ically compact and convex subsets of Rn, meaning that s1−t1 the above proposition is applicable in our subsequent u= t1 ∈[0,1]. (1) s1−t1 +(1−s)1−t2 analysis. Itiseasytoshowthat,inthecontextofPropo- t1 t2 sition 1, for any y ∈ Z \∂Z and x ∈ Z there exists an (cid:0) (cid:1) element z ∈ ∂Z such that y = t (x)x + 1 − t (x) z. y y This, combined with Proposition 1, yields that for any y ∈ Z \∂Z there is x ∈ extZ and z ∈ ∂Z such that (cid:0) (cid:1) y =b(y)x+ 1−b(y) z when Z is a convex and compact subset of Rn. Suppose that y ∈ Z \ ∂Z, where Z ⊂ Rn is a convex and compact set. Let x ∈ extZ be an element, whoseexistenceisguaranteedbyProposition1,suchthat b(y) = t (x). If one had b(y) = 0, this would mean that y FIG.2: (Coloronline)Contourplotsofboundarinessforsim- t (x) = 0 implying that x does not appear in any con- y pleconvexsets. Letusnotethatthemaximalvalueofbound- vex decomposition of y. This yields the counterfactual ariness is not the same in all of them. result y ∈ ∂Z. Hence, b(y) > 0 for any non-boundary 3 element y ∈Z, and we see that, in the context of Propo- In Section IV we will employ this proposition to relate sition 1, b(y) = 0 if and only if y ∈ ∂Z. Compactness the concept of boundariness to error rate of minimum- is an essential requirement for this property. Consider, error discrimination in case of quantum convex sets of e.g., a convex set Z ⊂Rn that has a direction, i.e., there states, channels and observables. Shortly, the optimal is a vector v ∈ Rn and a point y ∈ Z \∂Z such that values of error probabilities are associated with the so- y+αv ∈ Z for all α > 0. Such set is not compact and calledbase norms[4,5],thussettingp(x−y)=(cid:107)x−y(cid:107) Z one easily sees that b(y)=0. in Eq. (2) we obtain an operational meaning of boundariness. Let us stress that the base norm (cid:107)x−y(cid:107) can Z Remark 1 (Evaluation of boundariness) be introduced only if certain conditions are met. In practise, it is useful to think about some numerical In particular, let us assume that the real vector space way how to evaluate the boundariness. It follows from V is equipped with a cone C ⊂V, i.e., C is a convex set the definition of boundariness that for any element y ∈Z such that αv ∈ C for any v ∈ C and α ≥ 0. Moreover, writtenasaconvexcombinationy =tx+(1−t)z withz ∈ we assume that C is pointed, i.e., C ∩(−C) = {0} and ∂Z the value of t (being t (x) in this case) provides an y generating,i.e.,C−C =V. Further,supposeZ ⊂C isa upperboundontheboundariness,hencet≡t (x)≥b(y). y base for C, i.e., Z is convex and for any v ∈C there are Supposewearegivenyandchoosesomevalueoft. Recall unique x ∈ Z and α ≥ 0 with v = αx. Especially when that for a fixed y ∈ Z and for every x ∈ Z the element x ∈ Z, there is no non-negative factor α (cid:54)= 1 such that z (x) = (y−tx)/(1−t) leaves the set Z for t = t (x). t y αx∈Z. Moreover, it follows that 0∈/ Z. Therefore, if we choose t≤b(y) implying t≤t (x), then y Let us note that all quantum convex sets are bases for z (x)∈Z for all x∈Z. However, if it happens that t> t generating cones for their ambient spaces. For example, b(y), then for some x we find t>t (x) and consequently y the set of density operators S(H) on a Hilbert space H z (x)∈/ Z. Even more, according to Proposition 1 such x t is the base for the cone of positive trace-class operators (determining the element z (x) out of Z) can be chosen t which, in turn, generates the real vector space of selfad- tobeextremal. Inconclusion, ift>b(y), thenthereexist joint trace-class operators. This is the natural ambient x∈extZ such that z (x)=(y−tx)/(1−t)∈/ Z. t spaceforS(H)ratherthantheentirespaceofselfadjoint This observation provides the basics of the numerical bounded operators, although the value for the boundari- methodweusedtotestwhetheragivenvalueoftcoincide ness of an individual state does not change if the consid- with b(y), or not. In particular, for any y we start with ered ambient space is larger than the space of selfadjoint the maximal value of t=1/2 (if we do not have a better trace-class operators. estimate) and decrease it until we reach the value of t for WheneverZ isabaseofageneratingconeinV onecan which z (x) ∈ Z for all x ∈ extZ. Equivalently, we may t define the base norm (cid:107)·(cid:107) : V → [0,∞). In particular, Z start with t=0 and increase its value until we find t for for each v ∈V which z (x)∈/ Z for some x∈extZ and ∀ε>0. t+ε (cid:107)v(cid:107) = inf {λ+µ|v =λx−µy for some x,y ∈Z} Inwhatfollowswewillformulateapropositionthatre- Z λ,µ≥0 lated relates the value of boundariness to any (bounded) seminormdefinedonthe(real)vectorspaceV containing Bydefinition(cid:107)x(cid:107) ≤1forallx∈Z,hence,accordingto Z the convex set Z. Proposition 2 Proposition 2 Consider a (semi)norm p : V → [0,∞) (cid:107)x−y(cid:107) ≤2(1−b(x)). (3) Z such that p(x)≤a for all x∈Z with some a≥0. Then (cid:0) (cid:1) (cid:0) (cid:1) If Z defines a base of a generating pointed cone in p(x−y)≤2a 1−t (x) ≤2a 1−b(y) (2) y V the weight function t (x) has a relation to Hilbert’s y projective metric. Details of this relation are discussed for all x, y ∈Z. inAppendixB.SincemembersofabaseZ canbeseenas representativesoftheprojectivespacePV,theprojective Proof. Pick x, y ∈Z. The last inequality in (2) follows metricalsodefinesawaytocompareelementsofZ which immediately from the definition of boundariness so we can be used to relate this metric to distinguishability concentrate on the first inequality. If t (x)=0 then the y measures [5]. claim is trivial and follows from the triangle inequality fortheseminorm. Letusassumethatt (x)>0andpick y t ∈ [0,t (x)). According to the definition of the weight y III. QUANTUM CONVEX SETS function, wehavez(t)=(1−t)−1(y−tx)∈Z. Itfollows (cid:0) (cid:1) that x−y =(1−t) x−z(t) yielding There are three elementary types of quantum devices: (cid:0) (cid:1) (cid:0) (cid:0) (cid:1)(cid:1) p(x−y) = (1−t)p x−z(t) ≤(1−t) p(x)+p z(t) sources (states), measurements (observables) and transformations (channels). They are represented by density ≤ 2a(1−t). operators, positive-operator valued measures, and com- As we let t to approach t (x) from below, we obtain the pletelypositivetrace-preservinglinearmaps,respectively y first inequality of (2). (cid:4) (for more details see for instance [6]). 4 A. States B. Observables Let us illustrate the concept of boundariness for the In quantum theory, the statistics of measurements is convex set of quantum states, i.e. for the set of density fully captured by quantum observables which are math- operators ematically represented by positive-operator valued measures (POVM). Any observable C with finite number of S(H )={(cid:37):(cid:37)≥O,tr[(cid:37)]=1}, d outcomes labeled as 1,...,n is represented by positive where(cid:37)≥Ostandsforthepositive-semidefinitnessofthe operators (called effects) C ,...,C ∈ L(H) such that 1 n (cid:80) operator (cid:37). Suppose that the Hilbert space H is finite C = I. Suppose the system is prepared in a state d j j dimensionalwiththedimensiond. Theboundarinessb((cid:37)) (cid:37). Then, in the measurement of C, the outcome j occurs determinesadecomposition(itneednotbeunique)ofthe with probability p =tr[(cid:37)C ]. The set of all observables j j state (cid:37) into boundary elements ξ and ζ with the fixed number n of outcomes is clearly convex. We interpret C = tA+(1−t)B as an n-outcome mea- (cid:37)=b((cid:37))ξ+(1−b((cid:37)))ζ. surement with effects C =tA +(1−t)B . j j j Adensityoperatorbelongstotheboundaryifandonlyif Let us concentrate on the finite-dimensional case H= ithasanontrivialkernel(i.e. ithas0amongitseigenval- Hd anddenotebyσ(C)theunionofalleigenvalues(spec- ues, for details see appendix C1). In other words there tra) of all effects Cj of a POVM C and denote by λmin exists vectors |ϕ(cid:105) and |ψ(cid:105) such that ξ|ϕ(cid:105) = 0 = ζ|ψ(cid:105), the smallest number in σ(C). An observable C belongs respectively. Therefore, to the boundary if and only if [3] λmin = 0; this is also proved in appendix C2. Using the same argumentation λ ≤(cid:104)ψ|(cid:37)|ψ(cid:105)=b((cid:37))(cid:104)ψ|ξ|ψ(cid:105), min as in the case of states we find that λ ≤(cid:104)ϕ|(cid:37)|ϕ(cid:105)=[1−b((cid:37))](cid:104)ϕ|ζ|ϕ(cid:105), min λ ≤b(C). (5) min where λ is the minimal eigenvalue of (cid:37). Moreover, min since (cid:104)ϕ|ζ|ϕ(cid:105) ≤ 1 and (cid:104)ψ|ξ|ψ(cid:105) ≤ 1 (because (cid:37) ≤ I) it Suppose |ψ(cid:105) is the eigenvector associated with the followsthatboundarinessisboundedinthefollowingway eigenvalue λmin of the effect Ck for some value of k ∈ {1,...,n}. Defineanextremal(andprojective)n-valued λmin ≤b((cid:37))≤1−λmin. (4) observable A (in accordance with Proposition 1) The upper bound in (4) holds trivially, because, in gen-  |ψ(cid:105)(cid:104)ψ| if j =k eral, the boundariness is smaller or equal 1/2. On the  A = I−|ψ(cid:105)(cid:104)ψ| for unique j (cid:54)=k (6) j otherside,thetightnessofthelowerbound(4)isexactly O otherwise what we are interested in. Based on our general consideration (Proposition 1) we The observable B with effects know we may choose ξ to be the extremal element, i.e. a 1 one-dimensional projection. Set ξ =|ψ(cid:105)(cid:104)ψ|, where |ψ(cid:105) is Bj = 1−λ (Cj −λminAj) the eigenvector of (cid:37) associated with the minimal eigen- min value λ . Then belongs to the boundary, because min (cid:37)=λ |ψ(cid:105)(cid:104)ψ|+(1−λ )(cid:37)−λmin|ψ(cid:105)(cid:104)ψ| (1−λmin)Bk|ψ(cid:105)=Ck|ψ(cid:105)−λminAk|ψ(cid:105)=0, min min 1−λ min hence 0 ∈ σ(B). Using these two boundary elements of istheconvexdecompositionof(cid:37)intoboundaryelements thesetofn-valuedobservableswemaywriteC=λ A+ min saturating the above lower bound, hence we have just (1−λ )B, hence the lower bound 5 can be saturated min proved the following proposition. and we can formulate the following proposition: Proposition 3 Theboundarinessofastate(cid:37)ofafinite- Proposition 4 Given an n-valued observable C of a dimensional quantum system is given by finite-dimensional quantum system, the boundariness equals b((cid:37))=λ , min where λmin is the minimal eigenvalue of the density op- b(C)=λmin, erator (cid:37). where λ is the minimal eigenvalue of all effects min Thus, the minimal eigenvalue possesses a direct opera- C1,...,Cn forming the POVM of the observable C. tional interpretation of the mixedness of the density operator. Indeed, the maximum b((cid:37)) = 1/d is achieved onlyforthemaximallymixedstate(cid:37)= 1I. Theinfinite- C. Channels d dimensional case is somewhat trivial, because, according to Proposition 12 in the appendices, all infinite- Transformation of a quantum systems over some time dimensional states are on the boundary, i.e., ∂S(H )= interval is described by a quantum channel mathemati- ∞ S(H ). Consequently, the boundariness of any state in cally represented as a trace-preserving completely posi- ∞ this case is zero. tive linear map. It is shown in the appendix C3 that 5 for infinite-dimensional quantum systems the boundary 0.25 of the set of channels coincide with the whole set of channels, hence, the boundariness (just like for states) 0.20 vanishes. Therefore, we will focus on finite-dimensional bΕp quantum systems, for which the channels can be isomor- 0.15 phically represented by so-called Choi-Jamiolkowski op- (cid:72) (cid:76) Λmin erators. Inparticular,forachannelE onad-dimensional 0.10 quantum system its Choi-Jamiolkowski operator is the unique positive operator E = (E ⊗I)(P+), where P+ = 0.05 |ψ+(cid:105)(cid:104)ψ+| and |ψ+(cid:105) = √1d(cid:80)dj=1|j(cid:105)⊗|j(cid:105). By definition, E belongs to a subset of density operators on H ⊗H 0.00 d d 0.0 0.1 0.2 0.3 0.4 0.5 satisfying the normalization tr E = 1I, where tr de- 1 d 1 p notes the partial trace over the first system (on which the channel acts). FIG. 4: (Color online) The strict difference between the Whiletheextremalityofchannelsisabitmorecompli- boundariness b (upper line) and minimal eigenvalue λ min cated than for the states, the boundary elements of the (lower line) for erasure channels is illustrated. Let us stress set of channels can be characterized in exactly the same thatthedifferenceisnotnegligibleanditismaximalforvalue p=1/4. way as for states. In fact, E is a boundary element if andonlyiftheassociatedChoi-JamiolkowskioperatorE contains zero in its spectrum (see Appendix C3). Given achannelE wemayusetheresult(4)derivedfordensity where |U(cid:105) = √1 (|u(cid:105)⊗|0(cid:105)+|u⊥(cid:105)⊗|1(cid:105)) is a maximally 2 operators to lower bound the boundariness entangled state and |u(cid:105) ≡ U|0(cid:105), |u⊥(cid:105) ≡ U|1(cid:105). Our goal is to evaluate t for which the operator G specified λmin ≤b(E), (7) in Eq. (8) describes the channel G from the boundary. This reduces to analysis of eigenvalues of (1−t)G that where λ is the minimal eigenvalue of the Choi- √ √ min reads {p,1−p,1(1−2t− D),1(1−2t+ D)}, where Jamiolkowski operator E. However, since the structures 2 2 D =(1−2p)2+4t2. Itisstraightforwardtoobservethat of extremal elements for channels and states are dif- they are all strictly positive for t < p(1−p), thus, the ferent, the tightness of the lower bound (7) does not identity t = p(1−p) defines the cases when channels G follow from the consideration of states. Surprisingly, the belong to the boundary of the set of channels indepen- following example shows that this is indeed not the case. dently of the particular choice of the unitary channel F. In conclusion, all unitary channels determine the same Case study: Erasure channels. Consider a qubit “era- value of t=p(1−p), hence, the boundariness of erasure sure” channel E transforming an arbitrary input state p channels equals b(E )=p(1−p). (cid:37) into a fixed output state ξ = p|0(cid:105)(cid:104)0|+(1−p)|1(cid:105)(cid:104)1|, p p Theexampleofaqubit“erasure”channelE illustrates 0 < p < 1/2 inducing Choi-Jamiolkowski operator p E = ξ ⊗ 1I. In order to evaluate boundariness of (seeFigure4)that,unlikeforstatesandobservables,the p p 2 boundariness of a channel E may differ from the lower the channel E , according to proposition 1, it suffices to p bound (7) given by the minimal eigenvalue of the Choi inspect convex decompositions operator E. This finding is summarized in the following E =tF +(1−t)G, (8) proposition. p where F corresponds to an extremal qubit channel, G is Proposition 5 Forqubit“erasure”channelsE with0< p a channel from the boundary. Our goal is to minimize p<1/2 the boundariness is strictly larger than the min- thevalueoft≡tEp(F)overextremalchannelsF inorder imal eigenvalue of the Choi-Jamiolkowski operator. In to determine the value of boundariness. particular, b(E )=p(1−p)>λ =p/2. p min Theextremalityconditions(linearindependenceofthe set {A†A } ) implies that extremal qubit channels can Further, we will investigate for which channels (if for j k jk beexpressedviaatmosttwoKrausoperatorsA . Conse- any) the lower bound on boundariness is tight, i.e. when j quently, the corresponding Choi-Jamiolkowski operators b(E )=λ . A trivial example is provided by channels p min are either rank-one (unitary channels), or rank-two op- from the boundary for which b(E ) = λ = 0, but are p min erators. In what follows we will discuss only the analy- there any other examples? Consider a channel E such sis of rank-one extremal channels, because it turns out that the minimal eigenvalue subspace of the associated that they are minimizing the value of weight function Choi-Jamiolkowski operator E contains a maximally en- t (F). The details concerning the analysis of rank-two tangled state. Then a decomposition with t = λ ex- Ep min extremal channels (showing they cannot give boundari- ists and it corresponds to a mixture of a unitary chan- ness) are given in Appendix D. nel (extremal element) and some other channel from the Any qubit unitary channel F(ρ) = UρU† is repre- boundary. On the other hand, if the subspace of the sented by a Choi-Jamiolkowski operator F = |U(cid:105)(cid:104)U|, minimaleigenvalueofE doesnotcontainanymaximally 6 entangledstateitisnaturaltoconjecturethatthebound- all non-unitary boundary channels F, because we found ariness will be strictly greater then λ . The following decomposition E =tF +(1−t)G with t>λ . min min proposition proves that this conjecture is valid. The above two parts of the proof show that t (F) > E λ forthechannelE oftheclaimandforanychannelF. min Proposition 6 Consider an inner element E of the set Theclaimfollowsfromtheobservationthat,accordingto of channels such that the minimal eigenvalue subspace Proposition 1, b(E) = t (F) for some (extreme) channel E of its Choi-Jamiolkowski operator E does not contain F and,especiallyforthisoptimalchannel,t (F)>λ . E min any maximally entangled state. Then its boundariness is (cid:4) strictly larger than the minimal eigenvalue, i.e. b(E) > λ . min IV. RELATION TO MINIMUM-ERROR DISCRIMINATION Proof. We split the proof into two parts. First, we prove t (F)>λ for any unitary channel F and then E min Quantum theory is known to be probabilistic, hence, we prove it for any other channel F. Let us write the individual outcomes of experiments have typically very spectral decomposition of operator E as limited (if any) operational interpretation. One example r of this type is the question of discrimination among a (cid:88) E = λ P , (9) limited number of quantum devices. In its simplest form k k the setting is the following. We are given an unknown k=1 quantum device, which is with equal prior probability where the eigenvalues λk > 0 are non-decreasing with k either A, or B (A and B are known to us). Our task is (i.e. λ1 = λmin), Pk are the projectors onto eigensub- to design an experiment in which we are allowed to use (cid:80) spaces corresponding to λk and kPk = I is the iden- the given device only once and we are asked to conclude tity operator on Hd ⊗ Hd. Since E is an inner point theidentityofthedevice. Clearly,thiscannotbedonein λ1 (cid:54)= 0. The Choi-Jamiolkowski operators associated allcasesunlesssomeimperfectionsareallowed. Thereare with unitary channels F have the form F = |ϕ(cid:105)(cid:104)ϕ|, various ways how to formulate the discrimination task. where |ϕ(cid:105) is a maximally entangled state. The assump- The most traditional [1, 2] one is aimed to minimize tion of the proposition implies that P1|ϕ(cid:105) =(cid:54) |ϕ(cid:105). In or- the average probability of error of our conclusions. Sur- der to prove that tE(F) > λmin it suffices to show that prisingly, the success is quantified by norm-induced dis- there exists t > λmin such that E −tF ≥ 0 (implying tances [7], hence, the discrimination problem provides a G = (E −tF)/(1−t) describes a quantum channel G). clear operational interpretation of these norms. We may It is useful to write express the optimal error probability of minimum-error √ √ discrimination as follows |ϕ(cid:105)= α|v(cid:105)+ 1−α|v (cid:105), (10) ⊥ 1 1 p (A,B)= (1− (cid:107)A−B(cid:107)), (12) where 0 ≤ α < 1, P |v(cid:105) = |v(cid:105) and P |v (cid:105) = 0. Define error 2 2 1 1 ⊥ a positive operator X =λ |v(cid:105)(cid:104)v|+λ |v (cid:105)(cid:104)v | and write 1 2 ⊥ ⊥ where the type of the norm (cid:107)A − B(cid:107) depends on the E−tF =E−X+X−tF. TheoperatorE−X isclearly considered problem. positive. Further, we will show that X −tF is positive Recently, it was shown in Ref. [4] that in general con- when we set t=λ λ /[λ +(λ −λ )α]>λ and as a 1 2 1 2 1 min vex settings the so-called base norms are solutions to consequenceE−tF ≥0. Bydefinition,X−tF actsnon- minimum-error discrimination problems. In particular, triviallyintwo-dimensionalsubspacespannedbyvectors it was also shown that base norms coincide with the |v(cid:105) and |v (cid:105). Within this subspace it has eigenvalues 0 ⊥ completely bounded (CB) norms in the case of quan- and λ +λ −t>0, hence, it is positive. This concludes 2 1 tum channels, states and observables, thus, according to thefirstpartoftheproofconcerningdecompositionswith Proposition 2 and Eq. (3) the following inequality holds unitary channels. Now, let us assume that the channel F is not unitary. (cid:107)A−B(cid:107) ≡(cid:107)A−B(cid:107) ≤2(1−b(A)). Z cb Since the Choi-Jamiolkowski operator F associated with In rest of this section we will illustrate that for quantum the channel F is a density operator, it follows that its structures the base norms (being completely bounded maximaleigenvalueµ ≤1(saturatedonlyforunitary max norms)andboundarinessareintimatelyrelated. Wewill channels). Set t = λ /µ . Then for non-unitary min max channels t > λ and since 0 < λ ≤ 1/d2 ≤ µ it investigate how tight the above inequalities are for par- min min max ticular quantum convex sets. follows that 0<t≤1. For all vectors |ϕ(cid:105) λ (cid:104)ϕ|E−tF|ϕ(cid:105)≥λmin− µminµmax =0, (11) A. States max and, therefore, G=(E−tF)/(1−t)≥0, too. As in the Letusstartwiththecaseofquantumstates,forwhich first part of the proof this means that t (F) > λ for the CB norm coincides with the trace-norm (see for in- E min 7 stance [4, 7]), i.e. ||A|| = tr[|A|]. Recall that the con- in definition of A) is the vector defined by the relation tr clusion of Proposition 2, when applied for states, is C |ψ(cid:105)=λ |ψ(cid:105) for some k. According to Proposition 2 k min (cid:107)(cid:37)−ξ(cid:107)tr ≤2[1−b((cid:37))]. (13) (cid:107)C−A(cid:107)≤2(1−λmin), (14) Using the absolute scalability of the norm the roles of (cid:37) where the norm (cid:107)C−A(cid:107) (the base norm = completely andξ canbeexchangedandfrom(12)and(13)itfollows bounded norm = diamond norm) can be evaluated as [4] that (cid:88) (cid:107)C−A(cid:107) = sup |tr[(cid:37)(C −A )]| j j 1 (cid:37) p ((cid:37),ξ)≥ max{b((cid:37)),b(ξ)}, j error 2 Assuming (cid:37)=|ψ(cid:105)(cid:104)ψ| we obtain i.e. the mixedness of states measured by their boundariness lower bounds the optimal error probability of dis- (cid:107)C−A(cid:107)≥1−λ +(cid:88)(cid:104)ψ|C |ψ(cid:105) min j crimination between them. Moreover, for a given state (cid:37) j(cid:54)=k we may write because (cid:104)ψ|A |ψ(cid:105) = 0 for j (cid:54)= k, (cid:104)ψ|A |ψ(cid:105) = 1 and j k 1 (cid:80) (cid:104)ψ|C |ψ(cid:105) = λ . Moreover, since (cid:104)ψ|C |ψ(cid:105) = mξinperror((cid:37),ξ)≥ 2b((cid:37)), 1−(cid:104)kψ|C |ψ(cid:105)=m1in−λ we find that forj(cid:54)=thke chojsen ob- k min servablesC,Awehave(cid:107)C−A(cid:107)≥2(1−λ ). Combining henceinterpretingtheboundarinessasthelimitingvalue min this with the lower bound (14) valid for any observable ofthebestdistinguishabilityofthestate(cid:37)fromanyother we have proven the proposition. (cid:4) state. In other words, the boundariness determines the information potential of the state as the distinguishability of states is the key figure of merit for quantum C. Channels communication protocols [8]. As before, let |ψ(cid:105) be the state for which (cid:37)|ψ(cid:105) = Forchannelstheboundarinessisnotgivenbyminimal λ |ψ(cid:105). It is straightforward to see that min eigenvalue of the Choi-Jamiolkowski operator. Actually, we are missing an analytical form of channel’s boundari- (cid:107)(cid:37)−|ψ(cid:105)(cid:104)ψ|(cid:107) =2(1−λ ). tr min ness. Hence, in general the saturation of the inequality Hence, the upper bound (13) can be saturated and we sup(cid:107)E −F(cid:107) ≤2(1−b(E)) (15) have proven the following proposition. cb F Proposition 7 For a given state (cid:37) is open and we chose to test the saturation of the bound fortheexamplesofquantumchannelsthatwestudiedin sup(cid:107)(cid:37)−ξ(cid:107)tr =2(1−b((cid:37))). Section IIIC. Let us stress that analytical expressions of ξ the completely bounded norm are rather rare, but there exist efficient numerical methods for its evaluation [9]. Inparticular,thisimpliesthatthestatesfromthebound- For the qubit “erasure” channel E that transforms an ary (with b((cid:37)) = 0) can be used as noiseless carriers of p arbitrary input state (cid:37) into a fixed output state ξ = bits of information as for each of them one can find a p p|0(cid:105)(cid:104)0|+(1−p)|1(cid:105)(cid:104)1|thecompletelyboundednorm(cid:107)E − perfectly distinguishable ”partner” state. p F(cid:107) can be expressed as cb (cid:107)E −F(cid:107) = sup (cid:107)(E −F)⊗I(|ψ(cid:105)(cid:104)ψ|)(cid:107) , (16) p cb p tr B. Observables (cid:107)ψ(cid:107)=1 For observables we may formulate an analoguous re- where I is the qubit identity channel and |ψ(cid:105) is a two sult: qubit state. Choice of F =I and (cid:112) √ Proposition 8 Suppose that C is an n-valued observ- |ψ(cid:105)= 1−p|0(cid:105)⊗|0(cid:105)+ p|1(cid:105)⊗|1(cid:105) (17) able. Then lower bounds the norm in (15) by 2(1−p(1−p)) as can sup(cid:107)C−A(cid:107)=2(1−b(C)), be seen by direct calculation. Due to the result b(Ep) = A p(1−p)fromsectionIIICthiscanbeequivalentlywritten as2(1−b(E ))≤sup (cid:107)E −F(cid:107) ,whichimpliesthatthe p F p cb where (cid:107)·(cid:107) is the base norm (identified with completely bound (15) is tight for the channel E . p bounded norm) for observables. Let us, further, consider the class of channels whose Choi operator E contains some maximally entangled Proof. We will prove that A defined in Eq. (6) yields state |φ(cid:105) in its minimal eigenvalue subspace. For these the supremum of the claim. Let us recall that |ψ(cid:105) (used channels b(E) = λ (see section IIIC). Choose F to min 8 be a unitary channel, i.e. F = |φ(cid:105)(cid:104)φ| and set |ψ(cid:105) = solely by the convex structure of the set. Recently, it √ 1/ 2(|0(cid:105)⊗|0(cid:105)+|1(cid:105)⊗|1(cid:105)) (maximally entangled state). wasshowninRef.[4]thatforthesetsofquantumstates, Then measurements and evolutions base norms coincide with so-called completely bounded norms. These norms are (cid:107)E−F(cid:107) =(cid:107)(E −F)⊗I(|ψ(cid:105)(cid:104)ψ|)(cid:107) ≤(cid:107)E −F(cid:107) , known [7, 10] to appear naturally in quantum minimum tr tr cb error discrimination tasks. As a result, this connection and direct calculation gives (cid:107)E−F(cid:107)tr = 2(1−λmin) = providesaclearoperationalinterpretationforthebound- 2(1−b(E)). Altogether, we have shown ariness as described in Section IV. Moreprecisely,ifwewanttodetermineinwhichofthe 2(1−b(E))≤sup(cid:107)E −F(cid:107)cb, (18) two known (equally likely) possibilities A or B an un- F known state (or measurement, or channel) was prepared which means that for this type of channels the bound and given to us, the probability of making an erroneous (15) is tight. conclusion exceeds one half times the boundariness for any of the elements A and B. For a generic pair of possibilities A and B this bound is not necessarily tight, however if we keep A fixed then the boundariness of A is V. SUMMARY proportional to the minimum error probability discrimination of A and the most distinguishable quantum de- Convexity is one of the main mathematical features of vicefromA. Tobeprecisethiswasshownonlyforstates modern science and it is natural to ask how the physical and observables (in which case the analytic formula for concepts and structures are interlinked with the exist- boundariness was derived), but we conjecture that this ing convex structure. Using only the convexity we in- featureholdsalsoforquantumchannels. Weverifiedthis troduced the concept of boundariness and investigated conjecture for erasure channels and the class of channels its physical meaning in statistical theories such as quan- containing a maximally entangled state in the minimum tum mechanics. Intuitively, the boundariness quantifies eigenvalue subspace of their Choi-Jamiolkoski operators. how far an element of the convex set is from its bound- In conclusion let us mention a rather intriguing ob- ary. Thedefinitionoftheboundaryisbasedsolelyonthe servation. In all the cases we have met the “optimal” convexityandnoothermathematicalstructureoftheset decompositions (determining the value of boundariness) is assumed. contain pure states, sharp observables and unitary chan- We have shown that the value of boundariness b(y) nels. In other words, only special subsets of extremal el- identifies the most non-uniform convex decomposition of ements (for observables and channels) are needed. This inner element y into a pair of boundary elements. Fur- is true for all states and for all observables. The case of ther,weshowed(Proposition1)thatforcompactconvex channelsisopen,butnocounter-exampleisknown. This sets such optimal decomposition is achieved when one of observation suggests that the concept of boundariness the boundary points is also extremal. This surprising could provide some operational meaning to sharpness of property simplifies significantly our analysis of quantum observables and unitarity of evolution. convexsetsandallowesustoevaluatethevalueofbound- ariness. In particular, we have found that, in contrast to the Acknowledgments case of states and observables, for channels the general lower bound on boundariness (b ≥ λ ) given by the min minimal eigenvalue of the Choi-Jamiolkowski represen- The authors would like to thank Teiko Heinosaari for tation is not saturated (see Section III). We illustrated stimulating this work and insightful discussions. This this feature explicitely for the class of qubit “erasure” work was supported by COST Action MP1006 and channels E mapping whole state space into a fixed state VEGA 2/0125/13 (QUICOST). E.H. acknowledges fi- p ξ =p|0(cid:105)(cid:104)0|+(1−p)|1(cid:105)(cid:104)1|(0<p<1/2). Theboundari- nancial support from the Alfred Kordelin foundation. p ness of this channel was found to be b(E ) = p(1−p) > M.S. acknowledges support by the Operational Program p λ =p/2 (Proposition 5). We showed that the satura- Education for Competitiveness - European Social Fund min tion of the bound is equivalent with existence of max- (project No. CZ.1.07/2.3.00/30.0004) of the Ministry imally entangled state in the minimal eigenvalue sub- of Education, Youth and Sports of the Czech Repub- space of the channel’s Choi-Jamiolkowski operator. Let lic. M.Z. acknowledges the support of projects GACR us stress that the boundariness vanishes for infinite di- P202/12/1142 and RAQUEL. mensional systems, because the associated convex sets contain no interior points (discussed in Appendix B). Concerning the operational meaning of boundariness, Appendix A: Properties of the weight function wefirstdemonstratedthattheboundarinesscanbeused to upper bound any (semi)norm induced distance pro- The purpose of this appendix is to prove results that viding the (semi)norm is bounded on the convex set. An are needed for Proposition 1. Let us first recall a few example of such norm is the base norm which is induced basic definitions in linear analysis. Suppose that V is a 9 real vector space. For a subset X ⊂V we denote by V X the smallest affine subspace of V containing X. For any x∈X, the linear subspace V −x is just the linear hull X ofX−x,whereweintroducedthenotationX−x≡{y− x|y ∈ X}. We say that U ⊂ V is absorbing if for every v ∈V there is α>0 such that α−1v ∈U; especially 0∈ U. The following lemma gives another characterization for the boundary of a convex set Z, which is useful for studyingthecontinuitypropertiesoftheweightfunction. FIG. 5: The scalar α extending y − x from the starting point y to the boundary coincides with the Minkowski gauge Lemma 2 Suppose that Z is a convex subset of a real P (y−x) and the decomposition of y with respect to this Z−y vector space V. An element y ∈ Z is inner, i.e., y ∈ boundarypointandxgivesthevaluet (x)oftheweightfunc- y Z \∂Z if and only if Z −y is absorbing in the subspace tion. V −y. Z of the set Z −y defines a point z(t), which determines Proof. Let us assume that y is an inner point of Z and the value of the weight function t . These considera- y suppose that v ∈ VZ −y. For simplicity, let us assume tions can be formulated mathematically as follows: Pick that v (cid:54)=0. The convexity of Z−y and the definition of t∈[0,t (x))anddefinez(t)=(1−t)−1(y−tx)∈Z. Now y VZ yield that there are d+, d− ∈Z−y and λ+, λ− ≥0, z(t)−y =t(1−t)−1(y−x)∈Z−y. Astapproachesty(x) where λ+ > 0 or λ− > 0 such that v = λ+d+ −λ−d−. from below, α(t) = (1−t)/t decreases and from this we The fact that y is an inner point implies that when d− ∈ see that (cid:0)1−ty(x)(cid:1)/ty(x) = PZ−y(y −x) or, when we Z−y then ∃q >0 such that −qd− ∈Z−y. Hence, v = denote the Minkowski gauge PZ−y : VZ −y → [0,∞) of αd, where α=λ +λ /q >0, d= λ+d +λ−(−qd )∈ Z−y by p (x)≡P (y−x), + − α + qα − y Z−y Z −y, which proves that Z −y is absorbing in V −y. Z 1 SupposenowthatZ−yisabsorbinginV −yandx∈Z, t (x)= . (A1) Z y 1+p (x) so that x − y = d ∈ Z − y. Also −d ∈ V − y and y Z because Z − y is absorbing, there is α > 0 such that According to Lemma 2 the gauge p is well defined, y −α−1d∈Z−y, i.e., y−α−1d=z ∈Z and when y ∈Z \∂Z. From the convexity of the Minkowski gauge we again see that x(cid:55)→1/t (x)=1+p (x) is con- y y 1 α y = x+ z. vex on Z whenever y ∈ Z \ ∂Z. We immediately see 1+α 1+α that, in the case of a topological vector space V, whenever y ∈ Z \∂Z, the weight function t is continuous if This means that for all x∈Z, x≤ y, i.e., y ∈/ ∂Z. (cid:4) y C and only if the Minkowski gauge p is continuous, i.e., The weight function can be associated with a function y Z − y is a neighbourhood of the origin of V − y. In called as Minkowski gauge. This connection gives more Z finite-dimensional settings, any convexabsorbing set is a insight in the properties of the weight function in the neighbourhoodoforigin(asonemayeasilycheck). Thus infinite-dimensionalcase. WhenAisanabsorbingsubset we obtain the following result needed for proving Propo- of a real vector space W, we may define a function P : A sition 1. W →R, Proposition 9 Suppose that Z ⊂ Rn for some n ∈ N. PA(w)=inf{α≥0|α−1w ∈A}, w ∈W. The weight function ty is continuous if and only if y ∈ Z\∂Z. P is called the Minkowski gauge of A. For basic prop- A erties of this function, we refer to [11]. If A is convex, Thequantumphysicalsetsofstates,POVMsandchan- then P is a convex function, and nelsareallcompact(evenintheinfinite-dimensionalcase A with respect to suitable topologies), implying that, e.g., {v ∈W |P (v)<1}⊂A⊂{v ∈W |P (v)≤1}. Proposition 1 is applicable for the sets of (finite dimen- A A sional) quantum devices. When A is an absorbing convex balanced subset, P has A many properties reminiscent to a norm, whose unit ball is A. When W is a (locally convex) topological vector Appendix B: Relation to Hilbert’s projective metric space, the Minkowski gauge P is continuous if and only A if A is a neighbourhood of the origin. TheweightfunctionisalsorelatedtotheHilbert’spro- SupposethatZ isaconvexsubsetofarealvectorspace jective metric. Suppose C ⊂ V is a pointed generating V and y ∈ Z. The basis for connecting a Minkowski cone of a real vector space V (see definition in Section gauge to the weight function ty is provided by the fol- II). We may define the functions lowing observation: Consider a vector y −x ∈ V −y, where x ∈ Z. As can be seen from Fig. (5), the sZcaling inf(v/w) = sup{λ∈R|v−λw ∈C}, factorαthatshrinksorextendsthisvectortotheborder sup(v/w) = inf{λ∈R|λw−v ∈C}, 10 v, w ∈ V. Through these functions, one can define Let us fix a minimal dilation (M,π,J) for Ψ. Let us Hilbert’s projective metric h:V ×V →[0,∞], h(v,w)= define F(Ψ) as the set of positive operators E ∈ L(M) ln(cid:0)sup(v/w)/inf(v/w)(cid:1) that can be lifted into a well- such that Eπ(a) = π(a)E for all a ∈ A and J†EJ = I. defined metric in the projective space PV; for more on The following proposition is essentially due to [14]. this subject, see [5, 12, 13] Proposition 10 SupposethatΨ∈CP(A;H)isequipped When Z is a base for C, one can easily show that, with the minimal dilation (M,π,J). The sets F(Ψ) and for x, y ∈ Z, inf(y/x) = sup{t ∈ [0,1)|y − tx ∈ C}. F(Ψ) are in one-to-one correspondence set up by Moreover, if x, y ∈Z and y−tx∈C for some t∈[0,1), then y−tx = sz for some (unique) s ≥ 0 and z ∈ Z. If Φ(a)=J†π(a)EJ, Φ∈F(Ψ), E ∈F(Ψ) (C1) s(cid:54)=1−t, then one sees that both y ∈Z and for all a∈A. 1 t s y = x+ z (B1) s+t s+t s+t Lemma 3 Suppose that Φ, Ψ ∈ CP(A;H) and fix the minimal dilation (M,π,J) for Ψ. Now Φ = Ψ if and C belongs to Z contradicting the fact that Z is a base. only if there is E ∈F(Ψ) with bounded inverse such that Hence s=1−t and Φ(a)=J†π(a)EJ for all a∈A. inf(y/x)=sup{t∈[0,1)|y−tx∈(1−t)Z}=t (x). y Proof. Case Φ = Ψ is obvious. Let us concentrate on the case Φ(cid:54)=Ψ. Similarly, the convex function x (cid:55)→ 1/t (x) is associated y Let us assume that Φ = Ψ. Because, especially, with the sup-function. C Φ ≤ Ψ, there is an operator E ∈ F(Ψ) such that C Φ(a) = J†π(a)EJ for all a ∈ A. Denote the closure √ of the range of E by M and the projection of M Appendix C: Boundary of quantum convex sets E onto this subspace by P . Since E commutes with π, E also P commutes with π, and we may define the map E The question of the boundary elements for states, ob- π : A → L(M ), π (a) = P π(a)| . Also define servablesandchannelscanbetreatedinaunifiedwayas E √ E E E ME J = EJ. Itisstraight-forwardtocheckthatthetriple all these objects can be understood as transformations E (M ,π ,J ) constitutes a minimal dilation of Φ. Since represented by completely positive linear maps. In this E E E alsoΨ≤ ΦandΦ(cid:54)=Ψ,itfollowsthatthereist∈(0,1) section, we give conditions of being on the boundary for C and Ψ(cid:48) ∈ CP(A;H) such that Φ = tΨ+(1−t)Ψ(cid:48). In all relevant quantum devices. For the sake of brevity, we other words, there is a number t ∈ (0,1) such that the characterize the boundary for all relevant quantum con- map Ψ(cid:48), vex sets in one go. This, however, necessitates the use of Heisenberg picture which is used only in this section. 1 1 Ψ(cid:48)(a)= (Φ−tΨ)= J†π(a)(E−tI)J, a∈A, Let us fix a Hilbert space H and a unital C∗-algebra 1−t 1−t A. WesaythatalinearmapΦ:A→L(H)iscompletely is completely positive or, equivalently, E ≥tI. Hence E positive (CP) if for any n = 1, 2,... and a ,..., a ∈ A 1 n has a bounded inverse. and |v (cid:105),..., |v (cid:105)∈H 1 n Suppose that E ∈ F(Ψ) is as in the first part of the n proof and E−1 ∈ L(M). From Proposition 10 it follows (cid:88) (cid:104)v |Φ(a†a )|v (cid:105)≥0. immediately that Φ ≤ Ψ. Denote E(cid:48) = P E−1| . j j k k C E ME j,k=1 We have E(cid:48) ≥0, JE†E(cid:48)JE =J†J =I and For any CP map Φ there is a Hilbert space M, a linear E(cid:48)π (a) = P E−1π(a)| =P E−1π(a)EE−1| E E ME E ME map J : H → M and a linear map π : A → L(M) such = P E−1Eπ(a)E−1| =P π(a)E−1| that π(1)=I , π(a†)=π(a)† and π(ab)=π(a)π(b) for E ME E ME M = π (a)E(cid:48) all a, b ∈ A (i.e., π is a unital *-representation of A on E M)thatconstituteaminimal Stinespring dilationforΦ. foralla∈A, sothatE(cid:48) ∈F(Φ)whenwefixthedilation This means that Φ(a) = J†π(a)J for all a ∈ A and the (M ,π ,J ) for Φ. Furthermore E E E subspaceofMgeneratedbythevectorsπ(a)J|v(cid:105),a∈A, √ √ |v(cid:105)∈H, is dense in M. J†π (a)E(cid:48)J =J†π(a) EE−1 EJ =J†π(a)J =Ψ(a) E E E In what follows, we only study unital CP maps, i.e., Φ(1 ) = I . We denote the set of all unital CP maps for all a ∈ A. According to Proposition 10 this means A H Φ : A → L(H) by CP(A;H). Since the set CP(A;H) is that Ψ≤ Φ. (cid:4) C convex, it is equipped with the preorder ≤ . We denote We denote the spectrum of an operator E ∈L(M) on C Φ= Ψ if Φ≤ Ψ and Ψ≤ Φ. For any Ψ∈CP(A;H) a Hilbert space M by sp(E). The following proposition, C C C we may define the set which is an immediate corollary of the previous lemma, characterizes the boundary elements of the set of unital F(Ψ)={Φ∈CP(A;H)|Φ≤ Ψ}. CP maps. C