Geometry of sets of quantum maps: a generic positive map acting on a high-dimensional system is not completely positive
aa r X i v : . [ qu a n t - ph ] D ec Geometry of sets of quantum maps: a generic positivemap acting on a high-dimensional system is notcompletely positive
Stanis law J. Szarek , , Elisabeth Werner , , and Karol ˙Zyczkowski , Case Western Reserve University, Cleveland, Ohio, USA Universit´e Paris VI, Paris, France Universit´e de Lille 1, Lille, France Institute of Physics, Jagiellonian University, Krak´ow, Poland Center for Theoretical Physics, Polish Academy of Sciences, Warsaw, Poland
December 21, 2007
Abstract
We investigate the set a) of positive, trace preserving maps acting on densitymatrices of size N , and a sequence of its nested subsets: the sets of maps which areb) decomposable, c) completely positive, d) extended by identity impose positivepartial transpose and e) are superpositive. Working with the Hilbert-Schmidt (Eu-clidean) measure we derive tight explicit two-sided bounds for the volumes of allfive sets. A sample consequence is the fact that, as N increases, a generic positivemap becomes not decomposable and, a fortiori , not completely positive. Due to theJamio lkowski isomorphism, the results obtained for quantum maps are closely con-nected to similar relations between the volume of the set of quantum states and thevolumes of its subsets (such as states with positive partial transpose or separablestates) or supersets. Our approach depends on systematic use of duality to derive quantitative estimates, and on various tools of classical convexity, high-dimensionalprobability and geometry of Banach spaces, some of which are not standard. Introduction
Processing of quantum information takes place in physical laboratories, but it may beconveniently described in a finite dimensional Hilbert space. The standard set of toolsof a quantum mechanician includes density operators which represent physical states. Adensity operator ρ is Hermitian, positive semi-definite and normalized. The set of den-sity operators of “size” 2 is equivalent, with respect to the Hilbert-Schmidt (Euclidean)geometry, to a three ball, usually called the Bloch ball. The set of density operators of“size” N forms an N − M N ,the space of N × N (complex) matrices.The interesting geometry of these non-trivial, high–dimensional sets attracts a lot ofrecent attention [1, 2, 3, 4, 5]. In particular one computed their Euclidean volume andhyper-area of their surface [6], and investigated properties of its boundary [7].If the dimension N of the Hilbert space H N is a composite number, the densityoperator can describe a state of a bipartite system. If such a state has the tensor productstructure, ρ = ρ A ⊗ ρ B , then it represents uncorrelated subsystems. In general, following[8], a state is called separable if it can be written as a convex combination of productstates. In the opposite case the state is called entangled and it is valuable for quantuminformation processing [9], since it may display non–classical correlations.The set M sep N of separable states forms a convex subset of positive volume of theentire set of states, which we will denote by M tot N [10]. Some estimations of the relativesize of the set of separable states were obtained in [11, 12, 13, 14, 15, 16, 17], while itsgeometry was analyzed in [18, 19, 20, 21]. Similar issues for infinite-dimensional systemswere studied in [22].Quantum information processing is inevitably related with dynamical changes of thephysical system. Transformations that are discrete in time can be described by linear quantum maps , or super-operators , Φ : M N → M N (or, more generally, Φ : M K →M N ). A map is called positive (or positivity-preserving ) if any positive (semi-definite)operator is mapped into a positive operator. A map Φ called completely positive (CP)if the extended map Φ ⊗ I k is positive for any size k of the extension. Here I k is theidentity map on M k . We will denote the cones of positive and completely positive maps(on M N ) by P N and CP N respectively, or simply by P and CP if the size of the systemis fixed or clear from the context.Conservation of probability in physical processes imposes the trace preserving (TP)property: Tr Φ( ρ ) = Tr ρ . It is a widely accepted paradigm that any physical process maybe described by a quantum operation : a completely positive, trace preserving map. (Inthe context of quantum communication, quantum operations are usually called quantumchannels .)The set CP TP N of quantum operations, which act on density operators of size N , formsa convex set of dimension N − N . Due to Jamio lkowski isomorphism [23, 24] the set2 − CP TP N can be considered as a subset of the ( N − M tot N of densityoperators acting on an extended Hilbert space, H N ⊗ H N . This useful fact contributesto our understanding of properties the set of quantum operations, but its geometry isnontrivial even in the simplest case of N = 2 [25, 26].The main aim of the present work is to derive tight two-sided bounds for the Hilbert–Schmidt (Euclidean) volume of the set CP TP N of quantum operations acting on densityoperators of size N and analogous estimates for the volume of the sets P TP N of positivetrace preserving maps, and of similar subsets of the superpositive cone SP N (see (13)and/or [27]) or the cone D N of decomposable maps (see (17)) etc. We show that, forlarge N , some subsets cover only a very small fraction of its immediate superset, while insome other cases the gap between volumes is relatively small. These bounds are relatedto (and indeed derived from, making use of the Jamio lkowski isomorphism) analogousrelations between the volumes of various subsets of the set of quantum states such asthose consisting of separable states or of states with positive partial transpose (PPT) (seethe paragraph following (15)) and their dual objects. Our methods are quite general andallow to produce tight two-sided estimates for many other sets of quantum states or ofquantum maps.The paper is organized as follows. In the next section we introduce some necessarydefinitions involving the set of trace preserving positive maps and its relevant subsets orsupersets, which will allow us to present an overview of the results obtained in this paper(summarized in Tables 2-4). Section 3 contains more definitions and various preliminaryresults. Most of those results are not new, but many of them are not well-known in thequantum information theory community. In section 4 we state precise versions of ourresults and outline their proofs. Some details of the proofs and technical results (fromall sections) are relegated to Appendices. 3 Positive and trace preserving maps: notation andoverview of results
Let Φ : M N → M N be a linear quantum map, or a super-operator. More general mapsΦ : M K → M N may also be considered and analyzed by essentially the same methods,but we choose to focus on the case K = N to limit proliferation of parameters.Let ρ ∈ M N ; the transformation ρ ′ = Φ( ρ ) can be described by ρ ′ nν = Φ nνmµ ρ mµ , (1)where we use the usual Einstein summation convention. The pair of upper indices nν defines its “row,” while the lower indices mµ determine the “column.” This agrees withthe usual linear algebra convention of representing linear maps as matrices. The relevantbasis of M N is here E ij := | e i ih e j | , i, j = 1 , . . . , N , where ( e i ) Ni =1 is an orthonormal basisof H N (which can be identified with C N ), and the mµ ’th “column” of Φ nνmµ , i.e., the N × N matrix (cid:0) Φ nνmµ (cid:1) Nn,ν =1 , is indeed Φ( E mµ ) = Φ nνmµ E nν .By appropriately reshuffling elements of Φ nνmµ we obtain another matricial represen-tation of a quantum map, the dynamical matrix D Φ [28], sometimes also called in theliterature “the Choi matrix” of Φ. The dynamical matrix is obtained as follows D mnµν := Φ nνmµ . (2)An alternative (and useful) description of the dynamical matrix is as follows D Φ := (cid:0) I N ⊗ Φ (cid:1) ρ max = N X m,µ =1 E mµ ⊗ Φ( E mµ ) , where ρ max = | ξ ih ξ | , with | ξ i = P Nm =1 e m ⊗ e m , is a maximally entangled pure state on H N ⊗ H N .We point out that the order of indices of the matrix D in (2) is different than inthe previous work [24, 26]. (The reason for this change will be elucidated in the nextparagraph.) Note that in the present notation the operation of “reshuffling,” whichconverts matrix Φ into D , corresponds to a “cyclic shift” of the four indices.It is sometimes convenient to arrange the row and column indices of D Φ ( mn and µν respectively) in the lexicographic order, thus obtaining a standard “flat” N × N matrixwith a natural block structure: the leading indices mµ indicate the position of the block4nd the second pair of indices nν refers to the position of the entry within a block. Inother words, the mµ ’th block of D Φ is Φ( E mµ ) or D Φ = (Φ( E mµ )) Nm,µ =1 , (3)an N × N block matrix with each block belonging to M N .If a super-operator Φ belongs to the positive cone P (i.e., Φ is positivity-preserving),then it also maps Hermitian matrices to Hermitian matrices. This in turn is equivalent toΦ commuting with complex conjugation † ; in what follows we will generally consider onlymaps with this property. It is easy to check that Hermiticity-preserving is equivalent tothe following relation (which has no obvious interpretation)Φ nνmµ = Φ νnµm . (4)However, expressing condition (4) in terms of the dynamical matrix we obtain D mnµν = D µνmn , which just means that D Φ is Hermitian. Thus one may describe linear Hermiticity-preserving maps on M N via Hermitian dynamical N × N matrices. The propertyof being positive can be characterized just as elegantly. A theorem of Jamio lkowski [23]states that a map Φ is positive, Φ ∈ P , if and only if the corresponding dynamical matrix D Φ is block positive . [A (square) block matrix ( M ij ) (say, with M ij ∈ M N for all i, j ) issaid to be block positive iff, for every sequence of complex scalars ξ = ( ξ j ), the N × N matrix P i,j M ij ¯ ξ i ξ j is positive semi-definite.]Arguably the most useful upshot of the dynamical matrix point of view arises in thestudy of CP maps. A theorem of Choi [29] states that a map Φ is completely positive,Φ ∈ CP , iff D Φ is positive semi-definite. Therefore, to each CP map on M N correspondsan N × N (positive semi-definite) matrix, and vice versa . In particular, the rescaleddynamical matrix D associated with a (non-zero) CP map represents a state of a bi–partite system, σ := D/ Tr D ∈ M tot N – see e.g. [23, 24], an element of the base ofthe positive semi-definite cone obtained by intersecting that cone with the hyperplane oftrace one matrices.If the dimension of the cones or other sets under consideration is relevant, we willexplicitly use a lower index, writing, e.g., CP for the set of one–qubit completely positivemaps. The trace preserving property, Tr Φ( ρ ) = Tr ρ , is equivalent to a condition for the partialtrace of the dynamical matrix X n D mnµn = δ mµ , or Tr B D = I A . (5)5herefore the compact set CP TP N of quantum operations may be defined as a common partof the affine plane representing the condition (5) and the cone of positive semi-definitedynamical matrices - see Figure 1 in section 3.In (5) and (occasionally) in what follows we use the labels A, B to distinguish betweenthe space on which the original state ρ acts, namely H A , and the space of Φ( ρ ), denoted H B . In particular, I A stands for the identity operator on H A . Since such conventions aresomewhat arbitrary (as was the ordering of indices of D ), some care needs to be exercisedwhen comparing (5) and similar formulae with other texts (such as, e.g., [26]). Let H = { M ∈ M d : Tr M = 0 } . Next, let H b = { M ∈ M d : Tr M = d / } and let H + = { M ∈ M d : Tr M ≥ } . If C ⊂ M d is a cone, we will denote by C b := C ∩ H b thecorresponding base of C . (This definition makes good sense if C ⊂ H + or, equivalently,if the d × d identity matrix I d belongs to the dual cone C ∗ (see (11)). In this casethe cones generated by C b and C coincide, perhaps after passing to closures.) We willuse the same notation for the sets of quantum maps corresponding to matrices via theChoi-Jamio lkowski isomorphism. Thus, for example, Φ : M N → M N belongs to H b iff Tr D Φ = Tr Φ( I N ) = N . (Here the identity matrix I N and its image Φ( I N ) are N × N matrices, while D Φ is a d × d matrix, with d = N ; in particular the two traceoperations take place in different dimensions.) Then P ∩ H b = P b is a base of the cone P , CP ∩ H b = CP b is a base of the cone CP , and similarly for other cones that will beintroduced later. The (real) dimension of the bases is N − M = d / is twofold.First, the condition can be rewritten as h M, e i HS = 1, where e = I d /d / is a matrixwhose Hilbert-Schmidt norm is equal to one; this allows to treat e as a distinguishedelement of cones and – at the same time – of their duals. Next, the primary objectsof our analysis are quantum maps , and the chosen normalization assures that TP (and,dually, unital; see Appendix 6.5) maps are in H b . When we are primarily interested in states , the normalization Tr M = 1 can be thought of as more natural (the distinguishedelement I d /d is the then the maximally mixed state, usually denoted by ρ ∗ ).While all the matrix spaces or spaces of maps are a priori complex, all cones ofinterest will live in fact in the real space M sa d of Hermitian matrices or in the space ofHermicity-preserving maps. We will use the same symbols H , H b etc. to denote thesmaller real (vector or affine) subspaces; this should not lead to misunderstanding.6 .4 Other cones, all sets of interest compiled in one table Analogous point of view will be employed when studying other cones of quantum mapssuch as • the cone S P of superpositive maps (also called entanglement breaking , see (13) and theparagraphs that follow) • the cone D of decomposable maps (see (17)) • the cone T of maps which extended by identity impose positive partial transpose (see(15) and the paragraphs that follow).In all cases we will identify the corresponding cone of N × N matrices and will relate invarious ways bases of the cones and their sections corresponding to the trace preservingrestriction. For easy reference, we list all objects of interest in the table below; see alsoFigures 1 and 3 in section 3. The missing definitions and unexplained relations (generallyappealing to duality) will also be clarified there.Table 1: Sets of quantum maps and the sets of quantum states associated to them viathe Jamio lkowski–Choi isomorphism, cf. (6)-(9) below. The inclusion relation holds ineach collumn, e.g. P N ⊃ D N ⊃ CP N ⊃ T N ⊃ SP N . The symbols ◦ and ⋆ in therightmost column denote sets consisting of also non-positive semi-definite matrices whichtechnically are not states (and are not readily identifiable with objects appearing in theliterature).Maps Φ : M N → M N States σ ∈ M N cones Tr D Φ = N Tr B D Φ = I N Tr σ = 1positive P N ⊃ P b N ⊃ P TP N ◦ decomposable D N ⊃ D b N ⊃ D TP N ⋆ completelypositive CP N ⊃ CP b N ⊃ CP TP N M tot N PPTinducing T N ⊃ T b N ⊃ T TP N M PPT N superpositive SP N ⊃ SP b N ⊃ SP TP N M sep N The action of the Jamio lkowski–Choi isomorphism, associating cones of maps to cones ofmatrices and their respective bases, can be summarized asΦ is positive (Φ ∈ P N ) ⇔ σ is block-positive (6)7 is completely positive (Φ ∈ CP N ) ⇔ σ is positive semi-definiteΦ ∈ CP b N ⇔ σ = 1 N D Φ ∈ M tot N (7)Φ is PPT inducing (Φ ∈ T N ) ⇔ σ ∈ PPT Φ ∈ T b N ⇔ σ = 1 N D Φ ∈ M PPT N (8)Φ is superpositive (Φ ∈ SP N ) ⇔ σ is separableΦ ∈ SP b N ⇔ σ = 1 N D Φ ∈ M sep N (9)The description of the matricial cone associated to the cone D N of decomposable mapsis largely tautological: the sum of the positive semi-definite cone and its image via thepartial transpose. We likewise have N D Φ ∈ ◦ (resp., ∈ ⋆ ) iff Φ ∈ P b N (resp., ∈ D b N ). Explicit formulae for volumes of high dimensional sets are often not very transparent(when they can be figured out at all, that is). This may be exemplified by the closedexpression for the volume of the d − M tot d , the set of of densityoperators of size d that has been computed in [6]vol (cid:0) M tot d (cid:1) = √ d (2 π ) d ( d − / Γ(1) . . . Γ( d )Γ( d ) . (10)Given the complexity of formulae such as (10), the following concept is sometimesconvenient. Given an m -dimensional set K , we define vrad( K ), the volume radius of K ,as the radius of an Euclidean ball of the same volume (and dimension) as K . Equivalently,vrad( K ) = (cid:0) vol( K ) / vol( B m ) (cid:1) /m , where B m is the unit Euclidean ball. It is fairly easy(if tedious) to verify that (10) implies a much more transparent relation vrad (cid:0) M tot d (cid:1) = e − / d − / (1 ± O ( d − )) as d → ∞ , and similar two-sided estimates valid for all d .This point of view allows to present in a compact way the gist of our results. Westart by listing, in Table 2, bounds and asymptotics for volume radii of bases of variouscones of maps acting on N –level density matrices. Observe that the bounds for volumeradii of three middle sets ( D , CP and T ) do not depend on dimensionality. On the otherhand, the volume radii of the base for the largest set P of positive maps grow as √ N ,while the volume radii of the smallest set SP of superpositive maps decrease as 1 / √ N .The base of the set of completely positive maps acting on density matrices of size N is up to a rescaling by the factor 1 /N equivalent to the set of mixed states M tot d of8able 2: Volume radii for the bases of mutually nested cones of positive maps whichact on N –level density matrices. Here r CP denotes the volume radius of the base CP b N of the set of completely positive maps. The last column characterizes the asymptoticalproperties, where r lim X := lim N →∞ vrad( X bN ) with X standing for D , CP or T . It is tacitlyassumed that the limits exist, which we do not know for X 6 = CP (the rigorous statementswould involve then lim inf or lim sup, cf. Theorem 5). The question marks “?” indicatethat we do not have asymptotic information that is more precise than the one implied bythe bounds in the middle column. It is an interesting open problem whether r lim T admitsa nontrivial (i.e., < upper bound; cf. remark (c) following Theorem 5.Sets of maps Bounds for volume radii Asymptoticspositive P √ N ≤ vrad( P b N ) ≤ √ N ?decomposable D r CP ≤ vrad( D b N ) ≤ r CP r lim D ≤ CP ≤ r CP := vrad( CP b N ) ≤ r lim CP = e − / PPT inducing T r CP ≤ vrad( T b N ) ≤ r CP r lim T ≥ super positive SP
16 1 √ N ≤ vrad( SP b N ) ≤ √ N ?dimensionality d = N , and similarly for other cones of maps – see eq. (7)-(9). Therefore,the results implicit in the last three rows of Table 2 are equivalent to the following bounds,presented in Table 3, for the volume radii of the set of quantum states and its subsets,some of which were known.Finally, we list in Table 4 the volume radii of the main objects of study in this paper:the set CP TP N of quantum operations and of other “ensembles” of trace preserving maps.Each of these sets forms a N − N cross-section of the corresponding N − CP b N etc.).Although the volume of the larger set is sometimes known (10), the cross-sections ap-pear much harder to analyze. Our approach does not aim at producing exact values (eventhough here and in the previous tables we made an effort to obtain “reasonable” valuesfor the numerical constants appearing in the formulae). Instead, we produce two-sidedestimates for the volume radius of CP TP N , which are quite tight in the asymptotic sense (asthe dimension increases) and analogous bounds for the sets of positive, decomposable,PPT–inducing and super–positive trace preserving maps. Note that these bounds aresimilar to the results for the bases of all five sets presented in Table 2, but are not theirformal consequences. 9able 3: Volume radii for the set of states M tot d of size d and its subsets M PPT d and M sep d .The latter two sets are well defined if the dimensionality d is a square of an integer. Here a ∼ b means that lim d →∞ a/b = 1, while a & b stands for lim inf d →∞ a/b ≥
12 1 √ d ≤ r tot := vrad( M tot d ) ≤ √ d r tot ∼ e − / √ d PPT states
14 1 √ d r tot ≤ r PPT := vrad( M PPT d ) ≤ √ d r tot r PPT &
12 1 √ d separable states
16 1 d ≤ vrad( M sep d ) ≤ d ?While we concentrate in this work on the study of various classes of trace preservingmaps, our approach allows deriving estimates of comparable degree of precision for othersets of quantum maps. As an illustration, we sketch in Appendix 6.5 an argument givingtight bounds for the volume of trace non-increasing (TNI) maps. An exact formula forthat volume was recently found by a different method [30] independently from the presentwork.Finally, let us point out that formula (10) is valid only in the case when the underlyingHilbert space is complex, and that our analysis focuses on the complex setting, as it isthe one that is of immediate physical interest. However, all the discussion preceding (10)can be carried out also for real Hilbert spaces, and virtually all results that follow dohave real analogues. This is because even when closed formulae are not available, themethods of geometric functional analysis allow to derive two-sided dimension free boundson volume radii and similar parameters. Accordingly, while in the real case one may beunable to precisely calculate coefficients such as e − / above, it will be generally possibleto determine the relevant quantities up to universal multiplicative constants.10able 4: Asymptotic properties of volume radii for five nested sets of trace preservingmaps. Same caveat as in Table 2 applies to the limits in the second column. Upper andlower bounds valid for all N (as in the middle columns of Tables 2 and 3) can be likewiseobtained.Sets of tracepreserving maps asymptotics of their volume radiipositive P ≤ lim N →∞ vrad( P TP N ) √ N ≤ D e − / ≤ lim N →∞ vrad( D TP N ) ≤ CP lim N →∞ vrad( CP TP N ) = e − / PPT inducing T ≤ lim N →∞ vrad( T TP N ) ≤ e − / super positive SP ≤ lim N →∞ vrad( SP TP N )1 / √ N ≤ This is immediate from Table 4: the volume radius of the set of positive trace preservingmaps acting on an N dimensional system is of order √ N , while the volume radius of thecorresponding set of decomposable trace preserving maps is O (1). Thus, for large N , thelatter set constitutes a very small part of the former one. Note that in order to comparevolumes we need to raise the ratio of the volume radii to the power N − N , which yieldsroughly N − N / , a fraction that is (strictly) subexponential in the dimension of the set.11 Known and preliminary results
Spaces of operators or matrices are endowed with the canonical Hilbert-Schmidt innerproduct structure. The Choi-Jamio lkowski isomorphisms transfers this structure to thespace of quantum maps. We define(Φ , Ψ) := h D Φ , D Ψ i HS := Tr D † Φ D Ψ . The spaces in question and the corresponding inner products are a priori complex. How-ever, if we restrict our attention to the real vector spaces of Hermicity-preserving maps Φand Hermitian matrices D Φ , which we will do in what follows, the scalar product becomesreal and we may simply write (Φ , Ψ) = Tr D Φ D Ψ . We next define a duality ∗ for cones of maps via their representation (or dynamical)matrix by C ∗ := { Ψ : M N → M N : (Φ , Ψ) ≥ ∈ C} . (11)This is a very special case of associating to a cone in a vector space the dual cone in thedual space (here M d is identified with its dual via the inner product h· , ·i HS ). Dualityfor cones of matrices and cones of maps is the same by definition.We point out that all the cones C we consider are non-degenerate, i.e., they are offull dimension in the real vector space M sa N of Hermitian matrices, or in the space oflinear maps commuting with † (equivalently, every map/matrix – Hermicity-preservingor Hermitian, as appropriate – can be written as the difference of two elements of C ) andfurther −C ∩ C = { } . Consequently, their duals are also non-degenerate.Since the cone of positive semi-definite matrices is self-dual, it follows that CP ∗ = CP . (12)The superpositive cone SP may be defined via duality SP := P ∗ . (13)By the bipolar theorem for cones (( C ∗ ) ∗ = C ), we then have SP ∗ = P . (14)12Note that the bipolar theorem for closed cones follows, for example, from the easilyverifiable identity C ∗ = −C ◦ , where ◦ is the standard polar defined by K ◦ = { x : h x, y i ≤ y ∈ K } , and from the bipolar theorem for the standard polar, i.e., fromthe equality ( K ◦ ) ◦ = K valid whenever K is a closed convex set containing 0.] Clearly SP ⊂ CP ⊂ P , see Figure 1.Figure 1: Sketch of sets of maps. a) The cone P of positive maps includes the cone CP of completely positive maps and its subcone CP containing the superpositive maps, dualto P . Trace preserving maps belong to the cross–section of the cones with an affine planeof dimension N − N (and of codimension N ), representing the condition Tr B D = I A .b) The sets of trace preserving maps in another perspective. This is a complete picturefor N = 2 since some of the cones coincide, namely P = D and T = S P . For N ≥
3, thecomplete picture is more complicated, see Figure 3.To clarify the duality relations (13), (14) and the structure of the cone SP , we recallthat Φ is positive iff D Φ is block positive, which – by definition – is equivalent to Φ( ρ ξ ) ≥ ρ ξ := | ξ ih ξ | , that is, for every rank one positive semi-definitematrix. In other words, for any ξ ∈ H A and for any η ∈ H B ,0 ≤ h Φ( | ξ ih ξ | ) η, η i HS = Tr Φ( | ξ ih ξ | ) | η ih η | = Tr D Φ ( ρ ξ ⊗ ρ η ) = h D Φ , ρ ξ ⊗ ρ η i HS where the first tracing takes place in H B (or M N ) and the other in H A ⊗ H B , or in M N (and similarly for the two Hilbert-Schmidt scalar products). This is the same assaying that D Φ belongs to the cone of matrices that is dual to the separable cone (thecone generated by all ρ ξ ⊗ ρ η = ρ ξ ⊗ η or, equivalently, by all products ρ A ⊗ ρ B of positivesemi-definite matrices). By the bipolar theorem for cones, this is equivalent to the cone { D Φ : Φ ∈ SP } being exactly the separable cone.An alternative description of SP , which justifies the “entanglement breaking” termi-nology, is as follows: Φ is superpositive iff for every k the extended quantum map Φ ⊗ I k maps positive semi-definitive matrices to (positive semi-definite) separable matrices, orstates to separable states if Φ is trace preserving.13ometimes (see, e.g., Appendix 6.2) it is useful to work with extended sets of mapssuch as the convex hulls of P TP N ∪ { } or P b N ∪ { } . For technical reasons, we find thelatter one more useful; we will denote it by P E = P E N , and similarly for other cones. Here0 denotes the “zero” map, which may be chosen as a reference point. Further, one mayconsider symmetrized sets such as CP sym = CP sym N , the convex hull of −CP b ∪ CP b , where −CP b is the symmetric image of CP b with respect to 0. (Note that CP sym is also theconvex hull of −C P E ∪ C P E , see Figure 2 below.) The advantage in using 0-symmetricsets is that, first, they often admit an interpretation as unit balls with respect to naturalnorms and, second, that symmetric convex bodies have been studied more extensivelythan general ones convex bodies.Figure 2: The set CP b of normalized quantum maps arises as a cross-section of theunbounded cone of CP maps with the hyperplane representing the condition Tr D = N .The set CP E of maps extended by the zero map is the convex hull of C P b ∪ { } , while CP sym is the symmetrized set, the convex hull of −CP b ∪ CP b . CP sym may be identifiedwith a ball in trace class norm, whose radius equals N . Analogous notation (and similaridentifications) may be employed for other sets of maps including P , SP etc., or forabstract cones.We next introduce the auxiliary cone of completely co-positive (CcP) maps C c P = { Φ : T ◦ Φ ∈ CP } , where T : M N → M N is the transposition map (which is positive, but not completelypositive for N > T := CP ∩ C c P . (15)In terms of dynamical (Choi) matrices, D T ◦ Φ is obtained from D Φ by transposing eachblock, i.e., by the partial transpose in the second system. This means that { D Φ : Φ ∈ T } is exactly PPT , the positive partial transpose cone (positive semi-definite matrices whose14artial transpose is also positive semi-definite). Since, as is easy to check, separablematrices are in
PPT , it follows that
SP ⊂ T ⊂ CP (16)For N = 2 the sets T and SP coincide, while for larger dimensions the inclusion SP ⊂ T is proper as shown in Figure 3.Figure 3: Sketch of sets of maps for N ≥
3. a) The cone P of positive maps includesa sequence of nested subcones: the cone D of decomposable maps, C P of completelypositive maps, T of maps which extended by identity impose positive partial transpose,and the cone S P of superpositive maps. b) The sequence of nested subsets of the compactset of positive trace preserving maps.Similarly to superpositive maps, there is an alternative description of T in the lan-guage of extended quantum maps: Φ ∈ T iff Φ ⊗ I k is PPT inducing for any size k of theextension, i.e., for any state ρ acting on the bipartite system its image, ρ ′ = Φ ⊗ I ( ρ ) ∈PPT . [The necessity of the latter condition follows by noticing that the partial transposeof ρ ′ equals ( T ⊗ I ) ρ ′ = ( T ◦ Φ ⊗ I ) ρ , which is positive semidefinite due to T ◦ Φ beingCP.]A quantum map Φ is called decomposable , if it may be expressed as a sum of a CPmap Ψ and a another CP map Ψ composed with the transposition T ,Φ = Ψ + T ◦ Ψ (17)or, equivalently, as a sum of a CP map and a CcP map. In other words, the cone D ofdecomposable maps is defined by D := CP + C c P (the Minkowski sum). Since the transposition preserves positivity, D ⊂ P . It is known[31, 32] that every one-qubit positive map is decomposable, so the sets P and D coincide.15owever, already for N = 3 there exist positive, non–decomposable maps [33], so D forms a proper subset of P – see Figure 3.It follows from the identity (Φ , T ◦ Ψ) = ( T ◦ Φ , Ψ) valid for all Φ , Ψ that C c P ∗ = C c P . Accordingly, the dual cone D ∗ verifies D ∗ = ( CP + C c P ) ∗ = CP ∗ ∩ C c P ∗ = CP ∩ C c P = T (18)This is a special case of the identity ( C + C ) ∗ = C ∗ ∩ C ∗ (the Minkowski sum) valid forany two convex cones C , C . It now follows by the bipolar theorem that D = T ∗ . (19)As SP ⊂ T ⊂ CP by (16), it follows by duality that
CP ⊂ D ⊂ P . We now return to the analysis of bases of cones of matrices, as defined in section 2.3.As was to be expected, natural set-theoretic and algebraic operations on cones induceanalogous operations on bases of cones. Sometimes this is trivial as in ( C ∩C ) b = C b1 ∩C b2 ,in other cases simple: ( C + C ) b = conv( C b1 ∪ C b2 ), where conv stands for the convex hull.What is more interesting and somewhat surprising is that also duality of cones carriesover to precise duality of bases in the following sense. Lemma 1
Let V be a real Hilbert space, C ⊂ V a closed convex cone and let e ∈ V be aunit vector such that e ∈ C ∩ C ∗ . Set V b := { x ∈ V : h x, e i = 1 } and let and C b = C ∩ V b and ( C ∗ ) b = C ∗ ∩ V b be the corresponding bases of C and C ∗ . Then ( C ∗ ) b := C ∗ ∩ V b = { y ∈ V b : ∀ x ∈ C b h− ( y − e ) , x − e i ≤ } . (20) In other words, if we think of V b as a vector space with the origin at e , and of C b and ( C ∗ ) b as subsets of that vector space, then ( C ∗ ) b = − ( C b ) ◦ . Recall that for abstract cones
C ⊂ V , the dual cone C ∗ is defined (cf. (11)) via C ∗ := { x ∈ V : ∀ y ∈ C h x, y i ≥ } . § h x, e i = h y, e i = 1, then h− ( y − e ) , x − e i = −h y, x i + 1 and so the conditionfrom (20) can be restated as ∀ x ∈ C b − h y, x i + 1 ≤ ⇔ ∀ x ∈ C b h y, x i ≥ . Since under our hypotheses C b generates C , the latter condition is equivalent to h y, x i ≥ x ∈ C , i.e., to y ∈ C ∗ , as required.Let us now return to our more concrete setting of V = M sa d (endowed with theHilbert-Schmidt scalar product) and e = I d /d / . Even more specifically, we will con-sider V = M sa N , identified via the Choi-Jamio lkowski isomorphism with the space ofHermicity preserving quantum maps on M N , and the cones that we defined in prior sec-tion. Note that the quantum map associated to e = I N /N is the so-called “completelydepolarising map,” which is usually denoted by Φ ∗ and whose action is described byΦ ∗ ( M ) = ( N − Tr M ) I N . The duality relations for cones (12), (13), (14) and (18), (19)combined with Lemma 1 imply now Corollary 2
We have the following duality relations for the bases of cones ( CP b ) ◦ = −CP b , ( SP b ) ◦ = −P b , ( P b ) ◦ = −SP b ( D b ) ◦ = −T b , ( T b ) ◦ = −D b , (21) where both the polarity and the negative signs refer to the vector structure in H b = { Φ :Tr D Φ = Tr Φ( I N ) = N } with Φ ∗ as the origin. In other words, we have for example D b = { Φ ∈ H b : ∀ Ψ ∈ T b ( − (Φ − Φ ∗ ) , (Ψ − Φ ∗ )) ≤ } . While the duality relations for cones described in the preceding subsection are rather wellknown, the duality for bases in the present generality appears to be a new observation.When combined with standard results from convex geometry, most notably Santal´o andinverse Santal´o inequalities [35, 36] (see below), and other tools of geometric functionalanalysis, it allows for relating volumes of bases of cones to those of the dual cones, andultimately for asymptotically precise estimates of these volumes and of volumes of thecorresponding sets of trace preserving maps.Let us also note here one immediate but interesting (and presumably known) conse-quence of the duality relations. 17 orollary 3
For each of the sets CP b N , SP b N , P b N , D b N and T b N , the Euclidean (i.e., Hilbert-Schmidt) in-radius is ( N − − / and the Euclidean out-radius is ( N − / . We observe first that, for each of the above sets, Φ ∗ is the only element that is invariantunder isometries of the set. Accordingly, it is enough to restrict attention to Hilbert-Schmidt balls centered at Φ ∗ . For CP b N , the assertion is just a reflection of the elementaryfact that M tot d contains a Hilbert-Schmidt ball of radius 1 / p d ( d −
1) centered at themaximally mixed state ρ ∗ , and that the distance from ρ ∗ to pure states is p − /d . For SP b N , it is a consequence of equality of in-radii of M tot N and M sep N (in the bivariate case)established in [13] (the out-radius of the latter is of course attained on pure separablestates). It then follows that the in- and out-radii must be the same for the intermediateset T b N . Finally, since the out-radius of K ◦ is the reciprocal of the in-radius of K (and vice versa ), we deduce the assertion for P b N and D b N via (21).It is curious to note that the statement about the out-radius of P b N is equivalent – viasimple geometric arguments – to the following fact (which a posteriori is true) If M = ( M jk ) Nj,k =1 is a block-positive matrix, then Tr ( M ) ≤ (Tr M ) .It would be nice to have a simple direct proof of the above inequality, as it would yield(via Lemma 1 and (21)) an alternate derivation of the result from [13] concerning thein-radius of the set of separable states in the bivariate case.Similarly, the best (i.e., the smallest) constant R in the inclusion CP b − Φ ∗ ⊂ R ( SP b − Φ ∗ )is the same as the best constant in P b − Φ ∗ ⊂ R ( CP b − Φ ∗ ) . It has been shown in [13] that the optimal R satisfies N / ≤ R ≤ N −
1. [Theupper bound follows just from the formulae for the inradius of SP b and the outradiusof CP b (or, equivalently, M sep , M tot ).] Again, there could be a more direct elementaryargument. Remark 4
The Euclidean inradii and outradii of CP TP N , SP TP N , P TP N , D TP N and T TP N arethe same as for the larger C b -type sets, i.e., ( N − − / and ( N − / . As pointed out in the arguments following the statement of Corollary 3, while the factthat the inradii and outradii of all sets in that Corollary are identical is nontrivial,there is no mystery about at least some of the maps (or directions) that witness them.In the language of the sets of states (i.e., matrices with trace one normalization) suchwitnesses are, for outradii pure states, and universal witnesses that work for all five sets18re pure separable states. By duality (i.e., Lemma 1), direction that witness inradii (forall sets) are obtained by reflecting a pure separable state with respect to the maximallymixed state ρ ∗ . In the language of quantum maps purity (i.e., the Choi matrix being ofrank one) corresponds to the map being of the form ρ → v † ρv (Kraus rank one), andthe trace preserving condition is then equivalent to v being unitary. If that unitary isseparable (i.e., a tensor product of two unitaries acting on the first and second system),the corresponding pure state will be separable. This means that universal witnesses ofoutradii of C b -type sets exist also in the smaller set by the trace preserving condition(5), i.e., inside the C TP -type sets. Since condition (5) defines an affine subspace, the“opposite” directions giving witnesses to the inradii also belong there.An alternative use of duality considerations involves symmetrized sets (cf. Figure2). If C ⊂ V is a cone and C b its base, we define C sym := conv( −C b ∪ C b ); the minussign referring now to the symmetric image with respect to 0. If, as earlier, e is thedistinguished point of C ∩ C ∗ defining C b and ( C ∗ ) b , then( C sym ) ◦ = ( e − C ∗ ) ∩ ( − e + C ∗ ) , (22)where the polarity has now the standard meaning (i.e., inside the entire space V and with respect to the origin). In other words, the polar of C sym is the order interval [ − e, e ],in the sense of the order induced by the cone C ∗ . The advantages of this approach isthat we find ourselves in the category of centrally symmetric convex sets, which is betterunderstood than that of general convex sets, and that frequently the object in question( C sym and its polar) have natural functional analysis interpretation as balls in naturalnormed spaces. One disadvantage is that in place of one very simple operation (symmetricimage with respect to e ) we have two elementary and manageable, but somewhat non-trivial operations (symmetrization and passing to order intervals). We postpone thediscussion of (22) and related issues to the Appendix. The classical Santal´o inequality [35] asserts that if K ⊂ R m is a 0-symmetric convex bodyand K ◦ its polar body, then vol( K ) vol( K ◦ ) ≤ (cid:0) vol (cid:0) B m (cid:1)(cid:1) or, in other wordsvrad(K) vrad(K ◦ ) ≤ . (23)Moreover, the inequality holds also for not-necessarily-symmetric convex sets after anappropriate translation, in particular if the origin is the centroid of K or of K ◦ , a conditionthat will be satisfied for all sets we will consider in what follow. Even more interestingly,there is a converse inequality [36], usually called “the inverse Santal´o inequality,”vrad(K) vrad(K ◦ ) ≥ c (24)19or some universal numerical constant c >
0, independent of the convex body K (sym-metric or not) and, most notably, of its dimension m .The inequalities (23), (24) together imply that, under some natural hypotheses (whichare verified in most of cases of interest), the volume radii of a convex body and of its polarare approximately (i.e., up to a multiplicative universal numerical constant) reciprocal.By Lemma 1, the same is true for the base of a cone and that of the dual cone. Thisobservation reduces, roughly by a factor of 2, the amount of work needed to determinethe asymptotic behavior of volume radii of, say, sets from the third column of Table 1.We note, however, that since, at present, there are no good estimates for the constant c from (24) if K is not symmetric, it is often more efficient to revisit arguments from[14, 16] which allow to estimate volume radii of polar bodies without resorting to theinverse Santal´o inequality. (An argument yielding reasonable value of c for symmetric bodies was recently given in [37].) 20 Volume estimates: precise statements and approx-imate arguments
The results stated in section 3 allow us, in combination with known facts, to determine theasymptotic orders (as N → ∞ ) for the volume radii (and hence reasonable estimates forthe volumes) of bases of all cones of quantum maps discussed up to this point. Our goalis slightly more ambitious; we want to find not just the asymptotic order of each quantity,but also establish inequalities valid in every fixed dimension and involving explicit fairlysharp numerical constants. Specifically, we will show the following Theorem 5
We have the following inequalities, valid for all N , and the following asymp-totic relations (i) ≤ vrad (cid:0) CP b N (cid:1) ≤ , lim N →∞ vrad (cid:0) CP b N (cid:1) = e − / ≈ . N / ≤ vrad (cid:0) P b N (cid:1) ≤ N / (iii) N − / ≤ vrad (cid:0) SP b N (cid:1) ≤ N − / (iv) ≤ vrad (cid:0) T b N (cid:1) vrad (cid:0) CP b N (cid:1) ≤ , e / ≤ lim inf N vrad (cid:0) T b N (cid:1) vrad (cid:0) CP b N (cid:1) (v) 1 ≤ vrad (cid:0) D b N (cid:1) vrad (cid:0) CP b N (cid:1) ≤ , lim sup N vrad (cid:0) D b N (cid:1) vrad (cid:0) CP b N (cid:1) ≤ e / Remarks : (a) Estimates on volume radii listed in Table 2 are either identical to thecorresponding inequalities stated above, or follow by the same argument.(b) Since the asymptotic orders of the volume radii of the families CP b N , T b N and D b N arethe same, we chose – for greater transparence – to compare the volume radii of the twolatter sets to that of CP b N in (iv) and (v), rather than give separate estimates for each ofthese quantities.(c) It is an interesting open problem whether there exists a universal constant α < (cid:0) T b N (cid:1) ≤ α vrad (cid:0) CP b N (cid:1) for all N > (cid:0) M PPT N (cid:1) ≤ α N vol (cid:0) M tot N (cid:1) for some α < all N > (cid:0) D b N (cid:1) and vrad (cid:0) CP b N (cid:1) . Inquiries to similar effect can be found in theliterature [10, 38].(d) It is likely that the asymptotic bound 2 e / in (v) holds actually for all N . Indeed,there is a strong numerical evidence that the estimate vrad (cid:0) CP b N (cid:1) ≥ e − / from (i) isvalid for all N and not just in the limit. (In view of the explicit character of the formula(10) this issue shouldn’t be too difficult to resolve.) Should that be the case, the next stepwould be to carefully analyze the dependence of vrad (cid:0) D b N (cid:1) on N given by the argumentspresented in this paper.Since the bases of cones, whose volume radii are described by Theorem 5, are effec-tively homothetic images, with ratio N , of the corresponding sets of trace one matrices21see Table 1 and the formulae that follow it), some of the inequalities/relations of Theo-rem 5 follow from known estimates for the volumes of various sets of states, particularly ifwe do not insist on obtaining “good” numerical constants that are included in the state-ments. For example, the estimates in statement (iii) are contained in Theorem 1 from[16]; one obtains the constants and 4 by going over the proof of that Theorem specifiedto bilateral systems. Similarly, the statement (iv) is (essentially) a version of Theorem4 from [16] which asserts that, in the present language, vrad (cid:0) M PPT N (cid:1) / vrad (cid:0) M tot N (cid:1) ≥ c for some constant c > N (the upper estimate with con-stant 1 is trivial). However, the argument from [16] yields only c = and e − / for theasymptotic lower bound.Next, the asymptotic relation in (i) follows from the explicit formula (10); see thecomments following (10). Presumably, the estimates in (i) can also be derived from (10),but there are more elementary arguments. For a simple derivation of the lower bound fromthe classical Rogers-Shephard inequality [39] see [14], section II. And here is an apparentlynew proof of the upper bound: combine the duality results of the preceding section,specifically the identification ( CP b ) ◦ = −CP b from (21), with the Santal´o inequality (23)to obtain1 ≥ vrad (cid:0) CP b (cid:1) vrad (cid:0) ( CP b ) ◦ (cid:1) = vrad (cid:0) CP b (cid:1) vrad (cid:0) − CP b (cid:1) = vrad (cid:0) CP b (cid:1) , as required. We recall that, in the context of (21), the operations ◦ and − take place inthe space H b of quantum maps verifying Tr D Φ = N , with Φ ∗ thought of as the origin;note that Φ ∗ is the centroid of CP b and so (23) with K = CP b indeed does apply in thatsetting.Arguments parallel to the last one lead to versions of the remaining statements with some universal constants. For example, the identification ( SP b ) ◦ = −P b combined withthe Santal´o inequality (23) and its inverse (24) leads to1 ≥ vrad (cid:0) SP b (cid:1) vrad (cid:0) P b (cid:1) ≥ c, where c is the (universal) constant from (24). Combining the above inequality with(iii) we obtain c N / ≤ vrad (cid:0) P b N (cid:1) ≤ N / . Similarly, 1 ≥ vrad (cid:0) T b (cid:1) vrad (cid:0) D b (cid:1) ≥ c combined with (the already shown version of) (iv) and with (i) implies vrad (cid:0) D b N (cid:1) vrad (cid:0) CP b N (cid:1) ≤ N vrad (cid:0) D b N (cid:1) vrad (cid:0) CP b N (cid:1) ≤ e / . As the constants in Theorem 5 are not meant to beoptimal, we relegate the somewhat more involved (but still based on classical facts)arguments yielding them to Appendix 6.1.The inequalities of Theorem 5 compare volumes of bases of cones, that is, sets ofmaps Φ normalized by the condition that the trace of D Φ , the corresponding Choi (or22ynamical) matrix, is N (or Tr Φ( I N ) = N ). [Of course, any other normalization –most notably Tr D Φ = 1 leading to sets of states – would work just as well for comparingvolumes provided we were consistent.] However, if we want to study quantum operations,i.e., trace-preserving quantum maps (or, similarly, unital maps), then – as explained in theprevious sections – the corresponding constraints are stronger than just normalization bytrace: in each case we are looking at an N -codimensional section of the cone as opposedto the 1-codimensional base. However, in either case the codimension is much smallerthan the dimension, which is N − N . The following technical result will imply thatthen, under relatively mild additional assumptions assuring that the base of the cone isreasonably balanced (which will be the case for all the cones we studied), the volumeradius of the section will be very close to that of the entire base. Proposition 6
Let K be a convex body in an m -dimensional Euclidean space with cen-troid at a , and let H be a k -dimensional affine subspace passing through a . Let r = r K and R = R K be the in-radius and out-radius of K . Then (cid:16) vrad( K ) R − m − km b ( m, k ) (cid:17) mk ≤ vrad( K ∩ H ) ≤ vrad( K ) r − m − km b ( m, k ) (cid:18) mk (cid:19) m ! mk , (25) where b ( m, k ) := (cid:16) vol m ( B m ) vol k ( B k ) vol m − k ( B m − k ) (cid:17) m . The proof of the Proposition is relegated to Appendix 6.3; now we explain its conse-quences. First, let us analyze the parameters that appear in (25). By Corallary 3,for all bases of cones that we consider here we have r = 1 / √ d − R = √ d − d = N . Next, we have m = d − N − k = d − d = N − N and m − k = d − N −
1, in particular mk = 1 + N = 1 + d and m − km = N +1 = d +1 .Further, the quantity b ( m, k ) = (cid:16) Γ( k/ m − k ) / m/ (cid:17) m (related to the Beta function)is easily shown to satisfy 1 / √ < b ( m, k ) < m, k it is actually1 − O ( log NN )). Similarly, 1 ≤ (cid:0) mk (cid:1) m ≤ k, m and 1 + O ( log NN ) for our values of m, k . Consequently, if vrad( K ) is subexponential in d (in our applications it is a lowpower of N , hence of d ), then vrad( K ∩ H ) / vrad( K ) → N → ∞ .This leads to Theorem 7
We have the following asymptotic relations (i) lim N →∞ vrad (cid:0) CP TP N (cid:1) = e − / (ii) ≤ lim inf N vrad (cid:0) P TP N (cid:1) N / ≤ lim sup N vrad (cid:0) P TP N (cid:1) N / ≤ ≤ lim inf N vrad (cid:0) SP TP N (cid:1) N − / ≤ lim sup N vrad (cid:0) SP TP N (cid:1) N − / ≤ e / ≤ lim inf N vrad (cid:0) T TP N (cid:1) vrad (cid:0) CP b N (cid:1) (v) lim sup N vrad (cid:0) D TP N (cid:1) vrad (cid:0) CP b N (cid:1) ≤ e / Upper and lower bounds in the spirit of Theorem 5 (i.e., valid for all N ) can be likewiseobtained.The reader may wonder why we perform our initial analysis on bases of cones ratherthan working directly with the smaller sets of trace preserving maps. The reason for thisis two-fold. First, the bases being homothetic to various sets of states, any informationabout them is at the same time more readily available and interesting by itself. Second,while we do have – as a consequence of Lemma 1 – nice duality relations between basesof cones, similar results for sets of trace preserving maps are just not true. As a demon-stration of that phenomenon we show in Appendix 6.4 that, in contrast to the bases CP b ,the sets CP TP are very far from being self-dual in the sense of (21). We derived tight explicit bounds for the effective radius (in the sense of Hilbert-Schmidtvolume), or volume radius, of the set of quantum operations acting on density matricesof size N , and for other convex sets of trace preserving maps acting such matrices suchas positive, decomposable, PPT inducing or superpositive maps. The novelty of ourapproach depends on systematic use of duality to derive quantitative estimates, and ontechnical tools, some of which are not very familiar even in convex analysis.Since the volume radii of the sets of trace preserving maps that are positive display adifferent dependendce on the dimensionality than those of the smaller set of decomposablemaps, the ratio of the volumes of the latter and the former set tends rapidly to 0 as thedimension increases. In other words, a generic positive trace preserving map is notdecomposable and, a fortiori , not completely positive. Thus we were able to prove astronger statement than the one advertised in the title of the paper. Similarly, a genericPPT inducing quantum operation (and, a fortiori , a generic quantum operation) is notsuperpositive. Analogous relations (some of which were known) exist between the sets ofstates related to those of maps via the Jamio lkowski isomorphism. Acknowledgements : We enjoyed inspiring discussions with I. Bengtsson, F. Benatti,V. Cappellini and H–J. Sommers. We acknowledge financial support from the Polish Min-istry of Science and Information Technology under the grant DFG-SFB/38/2007, from24he National Science Foundation (U.S.A.), and from the European Research ProjectsSCALA and PHD. 25
Appendices
The arguments given in the preceding section did not yield the asserted values of theconstant in part (ii), the constants e / and in part (iv), and the constants 8 and2 e / in part (v) of Theorem 5. We will now present the somewhat more involved line ofreasoning that does yield these constants.The following concepts will be helpful in our analysis. If K ⊂ R m is a convex bodycontaining the origin in its interior, one defines the gauge of K via k x k K := inf { t ≥ x ∈ tK } . Roughly, k x k K is the norm, for which K is the unit ball, except that there is no symmetryrequirement. Next, the mean width of K (or, more precisely, the mean half-width ) isdefined by w ( K ) := Z S m − k x k K ◦ dx = Z S m − max y ∈ K h x, y i dx (integration with respect to the normalized Lebesgue measure on S m − ). A classicalresult known as Urysohn’s inequality (see, e.g., [41]) asserts then thatvrad( K ) ≤ w ( K ) . (26)A companion inequality, which is even more elementary, isvrad( K ) ≥ w ( K ◦ ) − . (27)The proof of (27) is based on expressing the volume as an integral in polar coordinates andthen using twice H¨older inequality: vrad( K ) = (cid:0)R S m − k x k − mK dx (cid:1) /m ≥ R S m − k x k − K dx ≥ (cid:0)R S m − k x k K dx (cid:1) − = w ( K ◦ ) − . Applying (26) in our setting of the N − H b and for K = D b N ,we obtainvrad( D b N ) ≤ w ( D b N ) = w (conv( CP b N ∪ C c P b N )) ≤ w ( CP b N + C c P b N ) = 2 w ( CP b N ) ≤ , (28)because w ( · ) commutes with the Minkowski addition (of sets), and because w ( CP b N ) = w ( C c P b N ) ≤
2. The latter is a consequence of similar estimates for the set of all states(which is equivalent to CP b N up to a homothety), see [14, 16]. (We note that while the limit relation w ( CP b N ) → N → ∞ follows easily from well-known facts about randommatrices, the estimate valid for all N requires finer arguments such as those presented in26ppendices of [14]). Combining the above estimate with part (i) of Theorem 3 we obtainthe upper estimate in part (v) with the asserted constant 8.The same bound w ( D b N ) ≤ K = T b N )and with part (i) leads to the lower bound in part (iv).To obtain the asymptotic bounds from part (iv) and (v) with the required constants e / and 2 e / we argue similarly, but instead of the universal estimate w ( D b N ) ≤ N w ( D b N ) ≤ S m − is strongly concentrated around itsmean. In particular, if the out-radius of K is at most R , then R S m − |k x k K ◦ − w ( K ) | dx = O ( R/m / ). If K = CP b N or C c P b N , then, by Corollary 3, R = ( N − / while m = N −
1, hence
R/m / = ( N + 1) − / < N − . It is then an elementary exercise to showthat w ( D b N ) = Z S m − max (cid:16) k x k ( CP b N ) ◦ , k x k ( C c P b N ) ◦ (cid:17) dx ≤ max (cid:0) w ( CP b N ) , w ( C c P b N ) (cid:1) + O ( N − ) ≤ O ( N − ) , whence lim sup N w ( D b N ) ≤
2, as required. Universal (as opposed to asymptotic) upperbounds on vrad( D b N ) better than 4 obtained in (28) can also be derived this way, mostefficiently by converting spherical integrals to Gaussian integrals and using the Gaussianisoperimetric inequality. (This would also improve somewhat the bounds and 8 inparts (iv) and (v), but we will not pursue this direction here as the payoff doesn’t seemto justify the effort.)Finally, to obtain the lower bound on vrad( P b N ) from part (ii) of the Theorem, wenote that the upper bound 4 N − / for vrad( SP b N ) (stated in part (iii)) was de facto (see[16]) deduced from the stronger estimate w ( SP b N ) ≤ N − / . It then remains to apply(27) and the duality between P b N and SP b N . We will now analyze the polar of the symmetrized body C sym . Recall the notation ofsection 3.2: V is a real Hilbert space, C ⊂ V a closed convex cone, C ∗ the dual cone.Next, e ∈ C ∩ C ∗ is a unit vector, V b := { x ∈ V : h x, e i = 1 } is an affine subspace of V and C b = C ∩ V b is the base of the cone C . Finally, the symmetrized body is defined as C sym := conv( −C b ∪ C b ); the minus sign referring to the symmetric image with respect to0. An important point, following from classical results [39, 44] and explained in AppendixC of [14], is that under mild assumptions which are satisfied for all the cones we consider,the volume radii of C b and of C sym differ by a factor smaller than 2.27ur main assertion (equation (22) in section 3.2) is that( C sym ) ◦ = ( e − C ∗ ) ∩ ( − e + C ∗ ) , where the polarity has the standard meaning (i.e., inside the entire space V and withrespect to the origin). That is, y ∈ ( C sym ) ◦ iff both y + e and e − y are in C ∗ or, in otherwords, iff y belongs to [ − e, e ], the order interval in the sense of the order induced by thecone C ∗ . For example, if we want to investigate P b and P sym = conv (cid:0) −P b ∪ P b (cid:1) , wemay specify the framework above to C = P , obtaining( P sym ) ◦ = (cid:0) − Φ ∗ + SP (cid:1) ∩ (cid:0) Φ ∗ − SP (cid:1) . To prove the assertion, denote V − := { x ∈ V : h x, e i ≤ } (one of the half-spacesdetermined by V b ) and C E = C ∩ V − (cf. Figure 2 in section 3). Then C sym = conv( −C b ∪ C b ) = conv (cid:0) −C E ∪ C E (cid:1) . Hence, using standard rules for polar operations (see, e.g., [41]),( C sym ) ◦ = (cid:0) − C E (cid:1) ◦ ∩ (cid:0) C E (cid:1) ◦ . Next, (cid:0) C E (cid:1) ◦ = (cid:0) C ∩ V − (cid:1) ◦ = conv (cid:0) ( V − ) ◦ ∪ C ◦ (cid:1) = conv (( −∞ , · e ∪ −C ∗ ) = e − C ∗ , where the bar stands for the closure. Combining this with the preceding formula andagain using the standard rules gives( C sym ) ◦ = ( e − C ∗ ) ∩ ( − e + C ∗ )or the intersection of two cones with vertices at e and − e . Clearly this does not equal( C ∗ ) sym except in dimension 1. However, the two bodies are closely related. For example,if e is the point of symmetry of C b , then ( C ∗ ) sym is a cylinder with the base ( C ∗ ) b andthe axis [ − e, e ], while ( C sym ) ◦ is a union of two cones whose common base is ( C ∗ ) b − e ,the central section of the cylinder, and the vertices are − e and e . The two bodies onlydiffer in one dimension; if thought of as unit balls with respect to the correspondingnorms, the two norms coincide on the hyperspace V := { x ∈ V : h x, e i = 0 } andon the complementary one-dimensional space R e , but on the entire space we have inthe first case the direct sum in the ℓ ∞ sense, while in the second case in the ℓ sense.If the base C b is non-symmetric, the situation is more complicated. For example, thesection V ∩ ( C sym ) ◦ is congruent to the intersection of ( C ∗ ) b with its symmetric imagewith respect to e , but (see [42]) the volume radii of the two bodies are comparable if, forexample, e is the only point that is fixed under isometries of ( C ∗ ) b (as is the case in allour applications), or just the centroid of ( C ∗ ) b .28 .3 Proof of Proposition 2: for “balanced” cones, C b and C TP have comparable volume radius We may assume that a = 0 (otherwise consider K − a ). By hypothesis, we have then rB m ⊂ K ⊂ RB m , (29)where B m is the m -dimensional unit Euclidean ball. For a subspace E , denote by P E theorthogonal projection onto E . Then (see [42, 43]),vol m ( K ) ≤ vol k ( K ∩ H ) vol s ( P H ⊥ K ) , (30)where s = m − k and H ⊥ is the m − k -dimensional space orthogonal to the k -dimensionalsubspace H . Thereforevol m ( K )vol m ( B m ) ≤ vol k ( K ∩ H )vol k ( B k ) vol s ( P H ⊥ K )vol s ( B s ) vol k ( B k )vol s ( B s )vol m ( B m )Hence, using (29),vrad( K ) m ≤ vrad( K ∩ H ) k R s vol k ( B k ) vol s ( B s )vol m ( B m ) , which is the first inequality in (25). For the second inequality, we start with the evenmore classical result (see [44] or [45]; same notation as (30))vol m ( K ) ≥ (cid:18) mk (cid:19) − vol k ( K ∩ H ) vol s ( P H ⊥ K ) , (31)which doesn’t even require that H passes through the centroid of K . As above, this canbe rewritten in terms of volume radii as (cid:18) mk (cid:19) vrad( K ) m ≥ vrad( K ∩ H ) k r s vol k ( B k ) vol s ( B s )vol d ( B m ) , which is the second inequality in (25). CP TP N The purpose of this Appendix is to show that, in contrast to the bases CP b , the sets CP TP are very far from being self-dual in the sense of (21), that is, that the polar of CP TP insidethe space defined by the trace preserving condition (5) considered as a vector space withΦ ∗ as the origin is quite different from the reflection of CP TP with respect to Φ ∗ .29enerally, if K ⊂ R m is a convex body containing the origin in its interior and H ⊂ R m is a vector subspace, K ◦ ∩ H is always contained in the polar of K ∩ H inside H , and the discrepancy between the two (i.e., the smallest constant λ ≥ K ∩ H is contained in λ ( K ◦ ∩ H )) is the same as the discrepancy between K ∩ H and the orthogonal projection of K onto H . That discrepancy is also equal to themaximal ratio between max x ∈ K h u, x i and max y ∈ K ∩ H h u, y i (32)over nonzero vectors u ∈ H .In our case K = CP b N and K ∩ H = CP TP N . As a vector space, H may be identifiedwith maps whose dynamical matrix has partial trace equal to 0. We will argue in thelanguage of dynamical (Choi) matrices considered as “flat” block matrices. In theseterms, membership in H is equivalent to each block being of trace 0. We will choose as u the block matrix whose 11-th block is U = E − N − I N and the remaining blocks are 0.Further, we will choose as x the matrix whose 11-th block is X = N E and the remainingblocks are 0; then the scalar product corresponding to h u, x i is tr( U X ) = N −
1. On theother hand, if Y is the 11-th block of the Choi matrix of any element of CP TP N , then Y is astate and so the scalar product corresponding to h u, y i is tr( U Y ) = tr( E Y ) − N − tr Y ≤ tr Y − N − tr Y = 1 − N − . Accordingly, the discrepancy between the two maxima in (32)is at least ( N − / (1 − N − ) = N . We want to determine the asymptotic order of the volume radius of the set of all com-pletely postive, trace non increasing maps Φ : M N → M N i.e. the set CP TNI N := { Φ ∈ CP N : Tr Φ( ρ ) ≤ Tr ρ for all ρ ≥ } . As pointed out earlier, an exact formula for that volume was very recently found (in-dependently from this work and by a different method) in [30]. However, an argumentusing the approach of this paper is conceptually very simple and so we include it. Wehave
Proposition 8
We have, for all N , (cid:0) e N / (cid:1) − N ≤ vol (cid:0) CP TNI N (cid:1) vol (cid:0) CP TP N (cid:1) × vol (cid:0) { M ∈ M N : 0 ≤ M ≤ I N } (cid:1) ≤ N − N / (33)To derive estimates on vol (cid:0) CP TNI N (cid:1) from the Proposition, one needs to use the readilyavailable information on the two factors in the denominator of the middle term of (33).First, the asymptotic order of the volume radius of CP TP N was determined in Theorem30(i). Next, the set A := { M ∈ M N : 0 ≤ M ≤ I N } is a ball of radius 1 / I N /
2, and so its volume radius admits easy bounds given bythe in- and outradius: 1 / √ N / √ N /
Corollary 9 lim N →∞ vrad (cid:0) CP TNI N (cid:1) = e − / . The key point is that in order to calculate the volume radius we need to raise the volumeto the power 1 /N . Thus the factors such as ( e N / ) − N on the left hand side of (33)are inconsequential since it leads to an expression of the form 1 − O ( log NN ). For thesame reason, the effects of vol( A ) and of the b ( m, k )-type factor, which also enters thecalculation (cf. Proposition 6 and the comments following it), tend to 0 as N → ∞ .For the proof of Proposition 8 we note first that CP TNI N is canonically isometric to theset of subunital maps CP SU N := { Φ ∈ CP : Φ( I N ) ≤ I N } . In what follows we will work with the latter set. The isometry, which assigns to Φ : M N → M N the dual (in the linear algebra, or Banach space sense) map Φ ∗ , sends CP TP N to the set of unital maps CP U N := { Φ ∈ CP N : Φ( I N ) = I N } . The set CP SU N admits anatural fibration: with every M ∈ A , we may associate F M = { Φ ∈ CP : Φ( I N ) = M } ; (34)in particular F I N = CP U N . In the language of Choi (dynamical) matrices the equality from(34) translates to tr B D Φ = M , or to P j D jj = M if we think of D Φ as a block matrix D Φ = ( D jk ) = (cid:0) Φ( E jk ) (cid:1) . Since all fibers F M are parallel to the subspace N defined bytr B D Φ = 0 (or P j D jj = 0), one can express the volume as an integralvol (cid:0) CP TNI N (cid:1) = vol (cid:0) CP SU N (cid:1) = N − N / Z A vol (cid:0) F M (cid:1) dM (35)The reason for the factor N − N / is that while the fibration is naturally parametrized bythe elements of A , the projection of F M onto N ⊥ is actually the map ρ → N − tr( ρ ) M ,whose Choi matrix is N − I N ⊗ M (or a block matrix whose all diagonal blocks are M/N and off-diagonal blocks are 0). Now, the Hilbert-Schmidt norm of N − I N is N − / , andso the projection of CP SU N onto N ⊥ is isometric to N − / A .The second inequality in (33) is now an immediate consequence of (35) and thebound vol (cid:0) F M (cid:1) ≤ vol (cid:0) F I N (cid:1) = vol (cid:0) CP U N (cid:1) = vol (cid:0) CP TP N (cid:1) , valid for all M ∈ A , which31n turn follows, e.g., from F M being the image of F I N under the contraction g M : Φ( · ) → M Φ( · ) M . (On the level of dynamical matrices, the action of g M is given by D Φ → (cid:0) I N ⊗ M (cid:1) D Φ (cid:0) I N ⊗ M (cid:1) or, in the language of block matrices, by ( D jk ) → ( M D jk M ).)The fact that g M (cid:0) F I N (cid:1) ⊂ F M is obvious from the definition; surjectivity for invertible M ’s follows by considering the inverse g − M = g M − , and for singular M ’s by looking atinvertible approximants.For the first inequality in (33) we may use (31) with K = CP SU N and the section K ∩ H = CP U N . As pointed out earlier, the set P H K = P N ⊥ is then isometric to N − / A , andit remains to use the elementary bound (cid:0) mk (cid:1) = (cid:0) ms (cid:1) ≤ ( em/s ) s . A alternative argumentis to restrict the integration in (35) to { M : tI N ≤ M ≤ I N } , which is a ball in theoperator norm of radius (1 − t ) /
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