Aurelian Gheondea, and Ali Şamil Kavruk
Citation: Journal of Mathematical Physics 50, 022102 (2009); View online: https://doi.org/10.1063/1.3072683
View Table of Contents: http://aip.scitation.org/toc/jmp/50/2
Absolute continuity for operator valued completely
positive maps on C
ⴱ-algebras
Aurelian Gheondea1,a兲and Ali Şamil Kavruk2,b兲 1
Department of Mathematics, Bilkent University, Bilkent, Ankara 06800, Turkey and Institutul de Matematică al Academiei Române, C. P. 1-764, 014700 Bucureşti, Romania
2
Department of Mathematics, University of Houston, Houston, Texas 77204-3476, USA 共Received 15 October 2008; accepted 16 December 2008;
published online 11 February 2009兲
Motivated by applicability to quantum operations, quantum information, and quan-tum probability, we investigate the notion of absolute continuity for operator valued completely positive maps on Cⴱ-algebras, previously introduced by Parthasarathy 关in Athens Conference on Applied Probability and Time Series Analysis I 共Springer-Verlag, Berlin, 1996兲, pp. 34–54兴. We obtain an intrinsic definition of absolute continuity, we show that the Lebesgue decomposition defined by Parthasarathy is the maximal one among all other Lebesgue-type decompositions and that this maxi-mal Lebesgue decomposition does not depend on the jointly dominating completely positive map, we obtain more flexible formulas for calculating the maximal Le-besgue decomposition, and we point out the nonuniqueness of the LeLe-besgue de-composition as well as a sufficient condition for uniqueness. In addition, we con-sider Radon–Nikodym derivatives for absolutely continuous completely positive maps that, in general, are unbounded positive self-adjoint operators affiliated to a certain von Neumann algebra, and we obtain a spectral approximation by bounded Radon–Nikodym derivatives. An application to the existence of the infimum of two completely positive maps is indicated, and formulas in terms of Choi’s matrices for the Lebesgue decomposition of completely positive maps in matrix algebras are obtained. © 2009 American Institute of Physics. 关DOI:10.1063/1.3072683兴
I. INTRODUCTION
Modern quantum theory intensively uses the notion of operator valued completely positive maps on Cⴱ-algebras as a mathematical model for quantum operations 共e.g., see, Davies12 and Kraus24兲. In quantum probability, operator valued completely positive maps on Cⴱ-algebras play the role of transition probability mappings for quantum Markov processes共e.g., see Meyer26and Parthasarthy27兲. An essential role in the comparison of operator valued completely positive maps on Cⴱ-algebras is played by the Radon–Nikodym derivative for completely positive maps, first introduced and studied by Arveson.6Slightly more general Radon–Nikodym derivatives have been considered by Belavkin and Stazsewski,9 while Raginsky31 illustrated the importance of these results and related ideas to some special problems in quantum information theory, and Belavkin et
al.8 applied this Radon–Nikodym derivative to define and investigate a minimax fidelity for quantum channels.
From the point of view of quantum measurements, the natural setting is that of open quantum systems for which admissible devices are modeled by quantum operations, that is, completely positive maps on the Cⴱ-algebra of observables of the physical system. The cone of completely
a兲Electronic mail: aurelian@fen.bilkent.edu.tr and a.gheondea@imar.ro. b兲Electronic mail: kavruk@math.uh.edu.
50, 022102-1
positive maps defines a natural partial order relation. The main tool to deal with completely positive maps is the minimal Stinespring representation that produces a “larger” Hilbert spaceK and a representationof the Cⴱ–algebraA of observables on K.
An intrinsic deficiency of the Radon–Nikodym derivative is the requirement that the two completely positive maps should be somehow comparable, otherwise the Radon–Nikodym deriva-tive may not exist. Recalling the classical Lebesgue decomposition, which represents a measure as the superposition of the “good”共absolutely continuous兲 part and the “bad” 共singular兲 part, with respect to another measure , the next step in the comparison theory for operator valued com-pletely positive maps on Cⴱ-algebras was naturally pointing toward investigations on Lebesgue-type decompositions. This requires first the clarification of absolute continuity and singularity in this noncommutative setting. Motivated by quantum probability, this further step was done by Parthasarthy28who exhibited, for two unital completely positive maps,:A→B共H兲, where A is a Cⴱ-algebras and H is a Hilbert space, a construction inspired by the classical Lebesgue decomposition theorem that can be used in order to produce a decomposition=ac+s, where
acandsare again completely positive and they are called, respectively, the absolutely
continu-ous part and, respectively, the singular part, of with respect to . Consequently, is called
-absolutely continuous if s= 0. However, natural questions on intrinsic characterizations of absolute continuity and singularity, uniqueness of Lebesgue decompositions, as well as more flexible formulas for calculating them are left unanswered.
Further, Radon–Nikodym derivative as a bounded operator can be obtained only when com-parable completely positive maps are used, while for relatively absolutely continuous completely positive maps that are not comparable 共with respect to the natural order relation兲 the Radon– Nikodym derivative can be defined as an unbounded operator only. This raises other questions on approximation properties, e.g., in the spectral sense, of this unbounded Radon–Nikodym deriva-tive.
Our approach to the aforementioned questions, from the perspective of quantum measure-ments theory, is based on the observation that the minimal Stinespring representation of a quantum operationis a dilation to a larger ambient Hilbert spaceK, in which Radon–Nikodym derivatives can be considered as a transcription of the information carried by the quantum operations domi-nated by to quantum effects mathematically modeled by contractive positive operators on K. Thus, given two operator valued completely positive mapsand, on the same Cⴱ-algebra, what we first do is to make the dilation by means of the minimal Stinespring representation correspond-ing to a completely positive mapthat dominated bothand, and then to find Lebesgue-type decompositions for the corresponding Radon–Nikodym derivatives. In this way, we can generally reduce most of the obstructions to the known theories of Ando in Ref.3 of Lebesgue decompo-sitions for positive contractions and that of Simon33on canonical forms for non-negative quadratic forms共Jøorgensen19considered similar canonical forms for unbounded operators兲, combined with the observations of Kosaki共cf. Refs.21and22兲 on the connections between these noncommuta-tive Lebesgue-type decompositions, as well as his refinements of the Ando’s theory. However, in doing this there is a price we have to pay: the representation is living within a certain von Neumann algebra, the commutant of the representation of the Cⴱ-algebra on K, which can be rather narrow when compared to the ambient B共K兲. This constraint yields a series of additional obstructions especially in connection with uniqueness.
Following this chain of ideas, we actually show that, to a certain extent, the above mentioned results of Ando, Simon, and Kosaki can be streamlined to the case of operator valued completely positive maps on Cⴱ-algebras in such a way that the construction of the Lebesgue decomposition, as well as the underlying notion of absolute continuity employed by Parthasarthy, fitted into this general framework. Moreover, even though the relations between the existing comparison theories for non-negative bounded operators, for operator valued completely positive maps on Cⴱ-algebras, and that for non-negative quadratic forms look rather intricate, at least at the first glance, we can show that one can smoothly translate concepts and results between them once the dictionary is available. In this paper the point of view is to place the comparison theory for non-negative bounded operators in the center and derive the connections starting from here, but we also indicate
that, to a certain extent, the three noncommutative comparison theories, for completely positive maps, for non-negative bounded operators, and for non-negative quadratic forms, are equivalent. The article is organized as follows. In Sec. II we first briefly review the construction and the main results on the Radon–Nikodym derivatives for completely positive maps which enables us to apply the results within the aforementioned theories of Ando, Kosaki, and Simon. The main ingredient is the minimal Stinespring representation for operator valued completely positive maps on Cⴱalgebras. From here, it becomes natural to get the intrinsic definition of absolute continuity for operator valued completely positive maps on Cⴱ-algebras. Then we reformulate the chain rule in a more natural setting, we extend the Radon–Nikodym derivative to absolutely continuous positive definite maps, and get a spectral approximation of it by bounded Radon–Nikodym de-rivatives.
The main results are contained in Sec. III in which we first show, see Theorem 3.1, that the Lebesgue-type construction of Parthsarathy in Ref.28 produces actually the maximal Lebesgue decomposition, such that the maximality property makes it unique, and in which absolute conti-nuity and singularity are exactly the natural concepts considered in Sec. II. Then, in Theorem 3.3, we show that the maximal Lebesgue decomposition does not depend on the jointly dominating completely positive map and, hence, that more flexible formulas become available.
The question on the uniqueness of the Lebesgue decomposition for operator valued com-pletely positive maps is much more difficult than the corresponding results for operators. Some sufficient conditions for uniqueness are considered in Proposition 3.8 but here the results are less complete, at this level of generality, because they require a more detailed investigation of Le-besgue decompositions for positive operators in an ambient von Neumann algebra that may be different of B共H兲, and hence, different techniques may be needed. Some of these difficulties, especially in connection with dealing with dense operator ranges relative to von Neumann alge-bras and a still open problem of Dixmier, are apparent from the investigations of Kosaki in Ref. 22. By considering the recovering of the comparison theory for non-negative operators we reduce the question of providing examples of nonuniqueness to the known results of Ando in Ref.3, but the general uniqueness problem remains for later investigations.
In Sec. IV we illustrate the applicability of the Radon–Nikodym derivative and Lebesgue-type decomposition to the infimum problem for completely positive maps, cf. Theorem 4.2. Other applications to known special similarity problems for operator valued completely bounded maps on Cⴱ-algebras will be published elsewhere.
In Sec. V we specialize in completely positive maps in matrices and indicate how to calculate the Radon–Nikodym derivatives and Lebesgue decompositions in terms of the Choi matrices that might be interesting for applications to quantum information theory, e.g., see Ref.25. In this finite dimensional setting, a key role is played by the tracial completely positive map that is maximal in a certain sense, cf. Raginsky31 and the bibliography cited there.
Appendix A contains a review of the Radon–Nikodym derivative and Lebesgue decomposi-tions for bounded non-negative operators on a Hilbert space, where the most important ingredients are two binary operations called parallel sum and shorted operator. This Appendix, hopefully, will make our article more readable. In this operator setting, the notions of absolute continuity and singularity appear naturally, formulas to calculate the Radon–Nikodym derivative and Lebesgue decompositions are available, and questions like uniqueness of Lebesgue decompositions can be answered in more concrete terms.
Appendix B contains the technicalities related to the extension of the Radon–Nikodym de-rivatives for two relatively absolutely continuous completely positive maps as well as an attempt to a better understanding of the bad part, that is, singular part 共from a different perspective, an even more general theory of Lebesgue decompositions for “linear relations,” that offers a better understanding of the singular part, was recently considered in Ref.18兲. In this respect, it appears to be quite natural to first consider a comparison theory for non-negative quadratic forms on vector spaces and so we do. Then we show that this contains as a special case the comparison theory for completely bounded maps, and hence, it allows us to establish the connection with certain canoni-cal decompositions investigated previously by Simon33 for quadratic forms and, respectively, by
Jørgensen19 for unbounded operators, and, consequently, we get also a spectral approximation property of the Radon–Nikodym derivatives of two relatively absolutely continuous completely positive maps by bounded Radon–Nikodym derivatives, as in Theorem 2.11.
Finally, in Appendix C, we indicate the interdependencies of the three existing noncommuta-tive comparison theories by showing how the one for bounded posinoncommuta-tive operators and the one for quadratic forms can be obtained from the comparison theory of operator valued completely posi-tive maps on Cⴱ-algebras.
Radon–Nikodym derivatives for different types of non-negative forms on either von Neumann algebras or Cⴱ-algebra have been investigated for a long time and a very general theory for ⴱ-algebras has been considered by Gudder16共a good source of literature on this type of questions as well兲. For the special case of normal positive forms on von Neumann algebras, a comparison theory was performed by Kosaki.22From a certain perspective, these are special cases of operator valued completely positive maps, but we consider this as a different direction of investigation that we do not pursue in this article.
II. RADON–NIKODYM DERIVATIVES OF COMPLETELY POSITIVE MAPS A. Completely positive maps
Assume A is a unital Cⴱ-algebra and let H be a Hilbert space. A linear mapping :A
→B共H兲 is positive if共A+兲債B共H兲+, that is, it maps positive elements into positive operators
For n苸N let Mn denote the Cⴱ-algebra of n⫻n complex matrices, identified with the
Cⴱ-algebraB共Cn兲. The Cⴱ-algebraA丢Mnidentified with the Cⴱ-algebra Mn共A兲 of n⫻n matrices
with entries inA has natural norm and order relation for self-adjoint elements, induced by the embedding Mn共A兲債B共H丢Cn兲=B共Hn兲, where Hn denotes the Hilbert space direct sum of n
copies ofH. Using these considerations, a linear mapping:A→B共H兲 is completely positive if for any n苸N the mappingn=丢In:A丢Mn→B共Hn兲 is positive. Note that, with respect to the
identificationA丢Mn= Mn共A兲, the mapping nis given by
n共关aij兴i,j=1 n 兲 = 关共a ij兲兴i,j=1 n , 关aij兴i,j=1 n 苸 M n共A兲. 共2.1兲
A linear map :A→B共H兲 is called positive definite if for all n苸N, 共aj兲j=1
n 苸A, and 共h j兲j=1 n 苸H, we have
兺
i,j=1 n 具共aⴱjai兲hi,hj典 ⱖ 0. 共2.2兲Since for any共aj兲j=1
n 苸A the matrix 关a j
ⴱa
i兴i,j=1 n
is a non-negative element in Mn共A兲, ifis positive
definite then it is completely positive. Conversely, because any positive element in Mn共A兲 can be
written as a sum of elements of type关aⴱjai兴i,j=1 n
, it follows that complete positivity is the same with positive definiteness.
CP共A;H兲 denotes the set of all completely positive maps from A into B共H兲. If , 苸CP共A;H兲 one writes ⱕ if −苸CP共A;H兲; this is the natural partial order 共reflexive, antisymmetric, and transitive兲 on the cone CP共A;H兲. With respect to the partial order relation ⱕ, CP共A;H兲 is a strict convex cone.
Given 苸CP共A;H兲 we consider its minimal Stinespring representation 共;K; V兲 共cf. Stinespring34兲. Recall that K is the Hilbert space quotient completion of the algebraic tensor product of the linear spaceA丢H endowed with the inner product,
具a丢h,b丢k典=具共bⴱa兲h,k典 for all a,b 苸 A, h,k 苸 H. 共2.3兲 is defined on elementary tensors by共a兲共b丢h兲=共ab兲丢h for all a , b苸A and h苸H, and then
extended by linearity and continuity to a ⴱ-representation:A→K. Also, Vh =关1丢h兴苸K
for all h苸H, where 关a丢h兴denotes the equivalence class in the factor spaceA丢H/N, andN
rep-resentation 共;K; V兲 of is uniquely defined, modulo unitary equivalence, subject to the following conditions.
共i兲 Kis a Hilbert space and V苸B共H,K兲.
共ii兲 is aⴱ-representation of A on Ksuch that共a兲=Vⴱ共a兲V for all a苸A.
共iii兲 共A兲VH is total in K.
In caseis unital, the linear operator Vis an isometry and hence, due to the uniqueness, one usually replaces V with the canonical embedding HK. This is explains why the Stinespring representation theorem is considered to be a “dilation-type result,” and the minimal Stinespring representation is sometimes called the minimal Stinespring dilation, cf. Ref.29 and the bibliog-raphy cited there.
We record briefly the well-known example of non-negative Borel measures on compact Haus-dorff spaces, for later use.
Example 2.1: Let X be a compact Hausdorff space and let C共X兲 denote the unital Abelian
Cⴱ-algebra of complex valued continuous functions on X. If is a finite non-negative Borel measure on X then it can be viewed as a bounded linear map: C共X兲→C,
共f兲 =
冕
X
f共x兲d共x兲, f 苸 C共X兲. 共2.4兲
Consider the Hilbert spaceK= L2共X;兲 and the canonical representation
of C共X兲 on L2共X;兲
defined by 共f兲=Mf, where Mf苸B共L2共X;兲兲 is the operator of multiplication with f. Let
V:C→L2共X;兲 be the linear operator that maps each complex number z to the constant function
on X with value z. Then共;K; V兲 is the minimal Stinespring representation of, in particular,
is completely positive.
B. Radon–Nikodym derivatives
Let,苸CP共A;H兲 be such thatⱕand consider the minimal Stinespring representation 共;K; V兲 of, and similarly for . Then the identity operator J,:A丢H→A丢H has the
property that J,N債N, hence it can be factored to a linear operator J,:共A丢H兲/N→共A
丢H兲/N and then can be extended by continuity to a contractive linear operator J,
苸B共K,K兲. It is easy to see that
J,V= V, 共2.5兲
and that
J,共a兲 =共a兲J, for all a苸 A. 共2.6兲 Thus, letting
D共兲 ª J,ⴱ J, 共2.7兲
we get a contractive linear operator in B共K兲. In addition, as a consequence of 共2.6兲, D共兲 commutes with all operators共a兲 for a苸A, briefly, D共兲苸共A兲⬘ 关given a subset T of B共H兲 we writeT
⬘
=兵B苸B共H兲兩AB=BA for all A苸T 其 for the commutant of T 兴 and共a兲 = VⴱD共兲共a兲V= VⴱD共兲1/2共a兲D共兲1/2V for all a苸 A. 共2.8兲
The property 共2.8兲uniquely characterizes the operator D共兲. The operator D共兲 is called the
Radon–Nikodym derivative ofwith respect to.
It is immediate from 共2.8兲 that, for any n苸N, 共aj兲j=1
n 苸A, and 共h j兲j=1
n 苸H, the following
兺
i,j=1 n 具共aⴱjai兲hi,hj典 =冐
D共兲1/2兺
j=1 n 共aj兲Vhj冐
2 . 共2.9兲This shows that for any ,苸CP共A;H兲 with ,ⱕ, we have ⱕ if and only if D共兲 ⱕD共兲.
In addition, if ,苸CP共A;H兲 are such that ,ⱕ then for any t苸关0,1兴 the completely positive map共1−t兲+ tisⱕ and
D共共1 − t兲+ t兲 = 共1 − t兲D共兲 + tD共兲. 共2.10兲 The above considerations can be summarized in the following.
Theorem 2.2:关Arveson 共Ref.6兲兴 Let苸CP共A;H兲. The mapping哫D共兲 defined in共2.7兲,
with its inverse given by共2.8兲, is an affine and order-preserving isomorphism between the convex
and partially ordered sets共兵苸CP共A;H兲兩ⱕ其;ⱕ兲 and 共兵A苸共A兲
⬘
兩0ⱕAⱕI其;ⱕ兲.One says that uniformly dominates , and we write ⱕu, if for some t⬎0 we have ⱕt. This is a partial preorder relation共only reflexive and transitive兲. The associated equivalence relation共we can call it uniform equivalence兲 is denoted by ⯝u, that is, for,苸CP共A;H兲 we
have⯝uif and only if ⱕuⱕu. It is immediate from Theorem 2.2 the following.
Corollary 2.3: For a given苸CP共A;H兲 , the mapping哫D共兲 defined in共2.7兲, with its inverse given by 共2.8兲, is an affine and order-preserving isomorphism between the convex cones 共兵苸CP共A;H兲兩ⱕu其;ⱕ兲 and 共兵A苸共A兲⬘兩0ⱕA其;ⱕ兲.
We also record the example of the classical Radon–Nikodym derivative, from the perspective of Arveson’s Radon–Nikodym derivative.
Example 2.4: Letbe a finite non-negative Borel measure on the compact Hausdorff space
X and consider its minimal Stinespring representation共;K; V兲 as in Example 2.1. Let be another finite non-negative Borel measure on X such that, when viewing these measures as bounded linear functional on C共X兲 as in共2.4兲, we have ⱕu, that is, for some t⬎0 we have
冕
Xh共x兲d共x兲 ⱕ t
冕
X
h共x兲d共x兲, h 苸 C共X兲, 共2.11兲
equivalently, 共E兲ⱕt共E兲 for all E Borel subsets of X. Note that the commutant 共C共X兲兲
⬘
coincides with the maximal Abelian von Neumann algebraM=兵Mg兩g苸L⬁共X;兲其, where Mg isthe bounded operator of multiplication on L2共X;兲 with the function g in the Cⴱ-algebra共actually, a Wⴱ-algebra兲 L⬁共X;兲 of-essentially bounded functions on X.
By Corollary 2.3, D共兲, the Radon–Nikodym derivative of苸CP共C共X兲;C兲 with respect to
苸CP共C共X兲;C兲, is a non-negative operator in the von Neumann algebra M and hence, there exists uniquely f苸L⬁共X;兲 such that D共兲=Mf.
On the other hand, the classical Radon–Nikodym derivative d/d is originally a
-absolutely integrable function on X such that d= fd, but it is actually-essentially bounded, with the-essential supremum储f储⬁ⱕt, because of共2.11兲. It is easy to see that f = d/d -a.e., due to the uniqueness of both Radon–Nikodym derivatives.
C. Absolute continuity
Given,苸CP共A;H兲, we say thatis-absolutely continuous, and we writeⰆ, if there exists a sequencen苸CP共A;H兲 subject to the following conditions.
共acp1兲 The sequence 共n兲 is nondecreasing, that is,nⱕn+1for all n苸N.
共acp2兲 SO–limn共a兲=共a兲 for all a苸A.
共acp3兲 nⱕu for all n苸N, more precisely, for each n苸N there exists tn⬎0 such that n
ⱕtn.
This definition is inspired from the definition of absolute continuity for positive definite functions onⴱ-semigroups given by Ando and Szymański in Ref.5.
Lemma 2.5: For a given苸CP共A;H兲 , letn,苸CP共A;H兲,nⱕⱕu, andnⱕn+1for
all n苸N. Then the following are equivalent.
共i兲 SO–limn→⬁D共n兲=D共兲.
共ii兲 For all a苸A SO–limn→⬁n共a兲=共a兲 , that is,
lim
n→⬁储n共a兲h −共a兲h储 for all a 苸 A, h 苸 H.
共iii兲 For all a苸A, WO–limn→⬁n共a兲=共a兲 , that is,
lim
n→⬁具n共a兲h,k典 = 具共a兲h,k典 for all a 苸 A, h,k 苸 H.
Proof: If D共n兲 converges SO to D共兲 then, by共2.8兲, for all a苸A and h苸H, we have
lim
n→⬁储n共a兲h −共a兲h储 = limn→⬁储V
ⴱ共D
共n兲 − D共兲兲共a兲Vh储 = 0,
hence共ii兲 holds.
Since, clearly,共ii兲 implies 共iii兲, it remains to prove that 共iii兲 implies 共i兲. To see this, note first that, by considering the sequence of completely positive maps −n, it is enough to prove the
implication in case= 0. Then, by共2.9兲we get that the sequence 共D共n兲1/2兲 converges strongly
on the subspace generated by 共A兲VH, which by minimality is dense in K and, taking into account that all the operators D共n兲 are contractions, it follows by a standard argument in
operator theory that共D共n兲兲 converges strongly to 0. 䊐
In the following we use the definition of absolute continuity of non-negative operators as in the paragraph below Theorem 6.2 in Appendix A 2. From Corollary 2.3 and Lemma 2.5 one gets the following analog of Proposition 2.7 in Ref.5.
Proposition 2.6: For a given苸CP共A;H兲 , let,苸CP共A;H兲 be such that,ⱕu. Then is-absolutely continuous if and only if D共兲 is D共兲-absolutely continuous.
We also record the following basic properties of the relation of absolute continuity. Proposition 2.7: Let苸CP共A;H兲.
共i兲 If,苸CP共A;H兲 are-absolutely continuous, then the same is t+ s for all t , sⱖ0.
共ii兲 Assume that共n兲 is a sequence in CP共A;H兲 subject to the following properties.
共1兲 The sequence 共n兲 is nondecreasing.
共2兲 There exists苸CP共A;H兲 such that共a兲=SO limn→⬁n共a兲 for all a苸A.
共3兲 For all n苸N ,nis-absolutely continuous.
Thenis -absolutely continuous.
共iii兲 If⯝uthen is-absolutely continuous if and only if is-absolutely continuous.
The notion of absolute continuity for operator valued completely positive maps is a generali-zation of the notion of absolute continuity for positive measures. Among the different ways of proving this fact, one may choose the way passing through Example 2.4 and Proposition 2.6.
Proposition 2.8: Let X be a Hausdorff compact space and , be two non-negative finite Borel measures on X , considered as completely positive maps,: C共X兲→C as in共2.4兲. Then the
following assertions are equivalent.
共1兲 Ⰶ.
共2兲 For every Borel subset A債X such that 共A兲=0 it follows 共A兲=0 共that is, the classical
D. Recovering the minimal Stinespring representation
Let,苸CP共A;H兲 be such thatⱕuand, as in Sec. II C, let 共;K; V兲 be the minimal Stinespring representation for. Let D共兲 be the Radon–Nikodym derivative ofwith respect to
as in共2.7兲. Then, with respect to the decompositionK= Ker共D共兲兲丣共K両Ker共D共兲x兲兲,
V=
冋
V,1V,2
册
, 共a兲 =冋
,1共a兲 0
0 ,2共a兲
册
for all a苸 A, where the diagonal form of共a兲 follows because it commutes with D共兲. Clearly,共a兲 = 共D共兲1/2PK両Ker共D共兲兲V,2兲ⴱ,2共a兲PK両Ker共D共兲兲D共兲1/2V,2, a苸 A,
and from here it follows that the triple 共,2;K両Ker共D共兲兲; PH両Ker共D共兲兲D共兲1/2V,2兲 is a
Stinespring representation for . The minimality follows from the observation that the set
,2共A兲V,2H is total in H両Ker共D共兲兲. Thus, we have the following.
Theorem 2.9: Under the previous assumptions, the triple
共PK両Ker共D共兲兲PK両Ker共D共兲兲;K両Ker共D共兲兲;PK両Ker共D共兲兲D共兲1/2PK両Ker共D共兲兲V兲
is the minimal Stinespring representation for.
Whenandare also unital then the above theorem coincides with Theorem 4.1 in Ref.28. E. Chain rule
Let ,, beB共H兲-valued completely positive maps on the Cⴱ-algebra A such that ⱕu
ⱕu. Let us consider the minimal Stinespring representations 共;K; V兲, 共;K; V兲, and
共;K; V兲, of,, and, respectively,. By the definition of the bounded linear operators J,,
J,, and J, 共see Sec. II B兲 it follows that
J,= J,J,, hence
D共兲 = J,ⴱ J,= J,ⴱ Jⴱ,J,J,= J,ⴱ D共兲 J,. The formula共2.12兲is called the chain rule in. Ref.8.
We now consider all the Radon–Nikodym derivatives D共兲, D共兲, and D共兲 and ask for the relation between them, an analog of the chain rule for the Radon–Nikodym derivatives of functions. In order to simplify the notation, let P = PK両Ker共D共兲兲. By Theorem 2.9, we have the
following identifications:
K=K両Ker共D共兲兲, = PP, V= PD共兲1/2V. 共2.12兲
By共2.8兲, for any a苸A we have
共a兲 = VD共兲1/2共a兲D共兲1/2V= VⴱPD共兲1/2PD共兲1/2P共a兲PD共兲1/2PV
= VⴱPD共兲1/2PD共兲D共兲1/2P共a兲PV.
From these calculations and taking into account the uniqueness of the Radon–Nikodym derivative it follows:
Theorem 2.10: Let,,苸CP共A;H兲 be such thatⱕuⱕu. Then, modulo the
identifica-tion ofKas in共2.13兲, we have
D共兲 = D共兲1/2PK両Ker共D共兲兲D共兲PK両Ker共D共兲兲D共兲1/2. 共2.13兲
Actually, the chain rule in Theorem 2.10 can be obtained directly from共2.12兲, as follows. By the definition of the Radon–Nikodym derivative as in共2.7兲, it follows that J,= WD共兲1/2, where
W苸B共K,K兲 is a partial isometry with Ker共W兲=Ker共D共兲兲 and surjective 共because J, has dense range兲. But, the meaning of the identification as in 共2.12兲 is that there exists a unitary operator U苸B共K,K両Ker共D共兲兲兲 such that Wⴱ= PK両Ker共D共兲兲U. Thus, from共2.12兲, modulo
the identification as in共2.12兲, we get 共2.13兲.
F. Radon–Nikodym derivatives for absolutely continuous completely positive maps The first task of this subsection is to consider the extension of Radon–Nikodym derivatives to relatively absolutely continuous completely positive maps that will be, in general, unbounded operators. This can be defined in two ways, as shown in Ref.9and, respectively, Ref.28, and we show that, actually, these two methods produce the same definition.
We derive the main result from a comparison theory for quadratic forms, whose details are given in the Appendix B. In our opinion, we consider this as the natural framework to deal with this kind of questions. One of the advantages is that, in this way, a spectral approximation of the Radon–Nikodym derivative is also obtained: this is the second task of this subsection.
Thus, recalling that our setting is that of B共H兲-valued completely positive maps on a
Cⴱ-algebra A, consider the vector space V=A丢H, the algebraic tensor product. To any
苸CP共A;H兲 we associate the quadratic form q with
q
冉
兺
j=1 n aj丢hj冊
=兺
i,j=1 n 具共ajⴱai兲hi,hj典, 共2.14兲which is non-negative by共2.2兲. Thus, the formula共2.14兲establishes an affine and order-preserving embedding of CP共A;H兲 into Q共V兲+共see Appendix B 1 for notation兲.
To this end, we recall the following definition: given a von Neumann algebra M債B共H兲, a densely defined linear operator T inH is affiliated to M, and we write TM, if for any operator
X苸M
⬘
we have XT債TX, more precisely, Dom共T兲 is invariant under X and XTh=TXh for all h苸Dom共T兲. If T is a 共unbounded兲 self-adjoint operator in H, then TM if and only all thespectral projections of T are inM.
Also, given a closable operator T in the Hilbert space, a subspaceD債Dom共T兲 is called a core for T if the closure of T兩D coincides with the closure of T.
Theorem 2.11: Let ,苸CP共A;H兲 and let 共;K; V兲 be the minimal Stinespring
repre-sentation of . Then, is -absolutely continuous if and only if there exists a (generally
un-bounded) linear operator D共兲 , uniquely determined by the following properties.
共i兲 D共兲 is a positive selfadjoint operator in K and it is affiliated with共A兲⬘. 共ii兲 Lin兵共A兲VH其 is a core for D共兲1/2and D共兲1/2V is bounded.
共iii兲 共a兲=共Dp共q兲1/2V兲ⴱ共a兲Dp共q兲1/2V for all a苸A.
Proof: The technicalities of the proof have been deferred to Appendix B. Briefly, if for
arbitrary苸CP共A;H兲 we consider its Stinespring representation 共;K; V兲 then 共K;⌸兲 is a Hilbert space induced by q, with the definition in Appendix B 1, where
⌸
冉
兺
j=1 n aj丢hj冊
=兺
j=1 n 共aj兲hj. 共2.15兲These observations show that we can first consider the underlying comparison theory for non-negative quadratic forms as in Appendix B with the important difference that, for the case of completely positive maps, everything concerning the Radon–Nikodym derivatives should corre-spond to operators in the von Neumann algebra共A兲⬘. Having this in mind, we thus can apply
Theorem 7.13共a兲 in Appendix A. 䊐
Theorem 2.11 is essentially Corollary 4.2 in Ref. 28. Theorem 7.7 shows that the cited corollary is equivalent with the main result in Ref.9.
Since the Radon–Nikodym derivative of absolutely continuous positive definite maps is, in general, an unbounded operator, there is a natural need for its approximation with bounded Radon–Nikodym derivatives. Let 共An兲 be a sequence of 共generally, unbounded兲 positive
self-adjoint operators in a Hilbert spaceH. One says that 共An兲 converges in the strong resolvent sense
to the positive self-adjoint operator A inH if SO–limn→⬁共I+An兲−1=共I+A兲−1. The name is justified
by the known fact that, in this case, we also have SO–limn→⬁共I − An兲−1=共I − A兲−1 for all
苸C∖R 共e.g., see Corollary VIII.1.4 in Ref.20兲. The same approach as in the proof of Theorem 2.11 can be used in order to apply Theorem 7.13共b兲 and get the following.
Theorem 2.12: Let,苸CP共A;H兲 and let 共;K; V兲 be the minimal Stinespring
repre-sentation of.
Assume thatis-absolutely continuous. Then, for any sequence共n兲 of maps in CP共A;H兲
that is nondecreasing, SO–limn→⬁n共a兲=共a兲 for all a苸A, andnⱕufor all n苸N , it follows
that D共n兲 converges to D共兲 in the strong resolvent sense.
III. LEBESGUE DECOMPOSITIONS FOR COMPLETELY POSITIVE MAPS
A. The maximal Lebesgue decomposition
Letandbe two completely positive maps fromA into B共H兲.is called-singular if the
only map 苸CP共A;H兲 such that ⱕ, is 0. Note that is -singular if and only if is
-singular and, in this case, we call andmutually singular.
A decomposition=1+0, where1,0苸CP共A;H兲,1is-absolutely continuous and0
is-singular, is called a-Lebesgue decomposition of .
Theorem 3.1: Let,苸CP共A;H兲. There exists a -Lebesgue decomposition of=ac+s
such thatacis maximal among all-absolutely continuous maps苸CP共A;H兲 withⱕ.
We will get the proof of Theorem 3.1 after proving the following lemma.
Lemma 3.2: Let F苸B共K兲 be such that 0ⱕFⱕI. Then, with the notation as in共A4兲, 关F兴共I − F兲 = PH両Ker共F兲共I − F兲.
Proof: It is easy to see that
Ran共F兲 債 兵h 苸 K兩共I − F兲1/2h苸 Ran共F1/2兲其 債 H両Ker共F兲,
hence, by Theorem 6.1 it follows that PF,I−F= PK両Ker共F兲and then
关F兴共I − F兲 = 共I − F兲1/2P
K両Ker共F兲共I − F兲1/2=共I − F兲PK両Ker共F兲.
䊐
Proof of Theorem 3.1: To simplify the notation, let 共,K,V兲 be the minimal Stinespring representation for ª+. Then the Radon–Nikodym derivatives D共兲 and D共兲 in B共K兲 satisfy the following relation:
D共兲 + D共兲 = I. 共3.1兲
Define the linear mappingsac,s:A→B共H兲 by
ac共a兲 = VⴱD共兲PK両Ker共D共兲兲共a兲V and s共a兲 = VⴱPKer共D共兲兲共a兲V. 共3.2兲 Note that for any a苸A the operators 共a兲, D共兲, and PK両Ker共D共兲兲, mutually commute. It is
straightforward to check thatacandsare completely positive and that=ac+s.
Clearly,ac,sⱕ. We claim that
D共ac兲 = 关D共兲兴D共兲. 共3.3兲
关D共兲兴D共兲 = D共兲PK両Ker共D共兲兲.
Then共3.3兲follows by the uniqueness of the Radon–Nikodym derivative, as explained in Sec. II B. Now, in view of Theorem 2.2, Proposition 2.6, and the definition ofacas in共3.2兲, in order to
prove that ac is -absolutely continuous we have to prove that D共ac兲 is D共兲-absolutely
continuous. In view of共3.3兲, the latter is a consequence of Theorem 6.4 part共i兲.
In addition, again by Theorem 2.2 and Proposition 2.6, in order to prove thatacis maximal
among all-absolutely continuous苸CP共A;H兲 such thatⱕ, we have to prove that D共ac兲 is
maximal among all D共兲-absolutely continuous operators C苸B共K兲 such that CⱕD共兲. In view of共3.3兲, the latter is a consequence of Theorem 6.4 part共ii兲.
Similarly we have
D共s兲 = D共兲 − 关D共兲兴D共兲 = PKer共D共兲兲. 共3.4兲
Then, an argument as before, using Theorem 2.2, Proposition 2.6,共3.4兲, and finally Theorem 6.4,
proves thatsis-singular. 䊐
The -Lebesgue decomposition =ac+s constructed during the proof of Theorem 3.1 is
called the maximal-Lebesgue decomposition of and, clearly, it is unique with this property. A natural question with respect to this maximal Lebesgue decomposition is: to which extent does it depend on the choice of=+? Thinking in terms of quantum measurements, the choice ofand the minimal Stinespring representation, the dependency should not occur. Indeed, we can get the analog of the formulas 共3.3兲 and 共3.4兲, when is replaced by an arbitrary completely positive map such that bothandare uniformly dominated by, and then we can show that the construction of the maximal Lebesgue decomposition does not depend on the choice of.
Theorem 3.3: Let,and be in CP共A;H兲 such that uniformly dominatesand . Let
=ac+s be the -Lebesgue decomposition of defined at 共3.2兲. Then, the Radon–Nikodym
derivatives ofacands, with respect to , can be calculated as follows:
D共ac兲 = 关D共兲兴D共兲 and D共s兲 = D共兲 − 关D共兲兴D共兲. 共3.5兲
In particular, letting 共;K; V兲 be the minimal Stinespring representation for, we have ac共a兲 = Vⴱ共关D共兲兴D共兲兲共a兲V, a苸 A, 共3.6兲
and
s共a兲 = Vⴱ共D共兲 − 关D共兲兴D共兲兲共a兲V, a苸 A. 共3.7兲
Before proving this theorem we prove the following.
Lemma 3.4: If C苸B共K,H兲 has dense range and A,B苸B共H兲+, then, with the notation as in
共A1兲and共A4兲, we have
共CⴱAC兲:共CⴱBC兲 = Cⴱ共A:B兲C and 关CⴱAC兴共CⴱBC兲 = Cⴱ共关A兴B兲C.
Proof: Indeed, for arbitrary k苸K, according to共A3兲,
具Cⴱ共A:B兲Ck,k典 = 具共A:B兲Ck,Ck典 = inf兵具Af, f典 + 具B共Ck − f兲,Ck − f典兩f 苸 H其,
and, taking into account that C has dense range, this is
=inf兵具ACg,Cg典 + 具B共Ck − Cg兲,Ck − Cg典兩g 苸 K其 =inf兵具CⴱACg,g典 + 具CⴱB共Ck − Cg兲,k − g典兩g 苸 K其
=具共CⴱAC兲:共CⴱBC兲k,k典.
关CⴱAC兴共CⴱBC兲k = lim n→⬁共nC ⴱAC:CⴱBC兲k = lim n→⬁ Cⴱ共nA:B兲Ck = Cⴱlim n→⬁共nA:B兲Ck = C ⴱ共关A兴B兲Ck. 䊐
Proof of Theorem 3.3: We use the notation as in Sec. II. Let 共;K; V兲 be the minimal Stinespring representation for. To simplify the notation, let
A = D共兲, B = D共兲.
Clearly, the Radon-–Nikodym derivative of=+with respect tois D共兲+D共兲=A+B. Let
K,1= Ker共A+B兲 and K,2=K両Ker共A+B兲, subspaces in K. With respect to the decomposition
K=K1丣K2we have
A + B =
冋
0 00 C
册
, 共a兲 =冋
,1共a兲 0
0 ,2共a兲
册
for all a苸 A, and V =冋
V1 V2
册
, where C is an operator in B共K2兲+, necessarily one to one, and commutes with
,2共a兲, for all a
苸A. By Theorem 2.9, the triple 共,2;K,2; C1/2V兲 is the minimal Stinespring representation for =+. In order to simplify the notation, let F = D共兲, hence I−F=D共兲 共see the proof of Theorem 3.1兲. Also, by Lemma 3.2 we have D共ac兲=关F兴共1−F兲. Then, for all a苸A,
共a兲 = 共C1/2V ,2兲ⴱF,2共a兲共C1/2V,2兲 = V,2ⴱ C1/2FC1/2,2共a兲V,2 =
关
V1ⴱ V,2ⴱ兴
冋
0 0 0 C1/2FC1/2册冋
,1共a兲 0 0 ,2共a兲册冋
V,1 V,2册
= Vⴱ冋
0 0 0 C1/2FC1/2册
共a兲V. In a similar way, 共a兲 = Vⴱ冋
0 0 0 C1/2共1 − F兲C1/2册
共a兲V and ac共a兲 = Vⴱ冋
0 0 0 C1/2关F兴共1 − F兲C1/2册
共a兲V. By the uniqueness of the Radon–Nikodym derivative we obtain thatA =
冋
0 00 C1/2FC1/2
册
and B =冋
0 0
0 C1/2共1 − F兲C1/2
册
.Note that C1/2must also be one to one and hence, as a non-negative operator, it has dense range. Thus, by Lemma 3.4, 关A兴B =
冋冋
0 0 0 C1/2FC1/2册册冋
0 0 0 C1/2共1 − F兲C1/2册
=冋
0 0 0 关C1/2FC1/2兴共C1/2共1 − F兲C1/2兲册
=冋
0 0 0 C1/2关F兴共1 − F兲C1/2册
= D共ac兲. Since−ac=s, we also have that D共s兲=B−关A兴B.Finally, the formulas共3.6兲and共3.7兲can be obtained easily from共3.5兲, Theorem 6.4, Theorem
2.2, and Theorem 3.1. 䊐
The constructions ofac andsperformed during the proof of Theorem 3.1 can be given in
consequence of Theorem 3.3, they produce the same maximal Lebesgue decomposition. If both and are unital, then =12+12 is a unital completely positive map as well. In this case, Parthasarthy28gave the definitions as in共3.2兲and the decomposition=ac+swas called the
-Lebesgue decomposition of, whileacands were called the absolutely continuous part and,
respectively, the singular part ofwith respect to. Further, in Ref.28was called-absolutely continuous if s= 0. As a consequence of Theorems 3.1 and 3.3 it follows that this notion of
absolute continuity is the same with that introduced in Sec. II C. Corollary 3.5: Let,苸CP共A;H兲.
共a兲 is-absolutely continuous if and only ifs= 0.
共b兲 is-singular if and only ifac= 0.
Another consequence of the results we got so far refers to other characterizations of singular-ity for completely positive maps. The equivalence of共iii兲–共vi兲 is contained in Corollary 1.4.4 of Ref.6.
Corollary 3.6: With the notation as before, the following assertions are equivalent. 共i兲 is-singular.
共ii兲 is-singular.
共iii兲 The Radon–Nikodym derivative D+共兲 is a projection.
共iv兲 The Radon–Nikodym derivative D+共兲 is a projection.
共v兲 is an extremal element of the convex set 兵苸CP共A;H兲兩ⱕ+其. 共vi兲 is an extremal element of the convex set 兵苸CP共A;H兲兩ⱕ+其.
Proof: Due to the symmetry of the relation of singularity, it is sufficient to prove that共i兲, 共iii兲,
and共v兲 are mutually equivalent.
Ifis-singular, by the above corollary it follows that=sand hence, by共3.4兲it follows
that the Radon–Nikodym derivative D+共兲 is a projection.
Conversely, if the Radon–Nikodym derivative D+共兲 is a projection then, by共3.3兲, it fol-lows that D共ac兲=0 and henceac= 0, that is,is-absolutely continuous. Thus,共i兲 is equivalent with共iii兲.
In order to prove the equivalence of 共iii兲 with 共v兲, we use Theorem 2.2 and the well-known fact that, given a Hilbert spaceH, the set of extremal elements of the set 兵B苸B共H兲兩0ⱕBⱕI其 coincides with the set of orthogonal projections inH 共e.g., see Lemma 3.2 in Ref.12兲. 䊐
In view of Proposition 2.7, another consequence of Theorem 3.1 is the following.
Corollary 3.7: Let,,苸CP共A;H兲 and consider the maximal-Lebesgue decompositions
=ac+s and=ac+s.
共i兲 For any tⱖ0 we have 共t兲ac= tac. 共ii兲 ac+acⱕ共+兲ac.
共ii兲 Ifⱕthenacⱕac.
B. Uniqueness
The maximal -Lebesgue decomposition =ac+s defined at 共3.2兲is unique by its
maxi-mality property but, if the condition of maximaxi-mality is dropped, then the uniqueness may be affected. We examine first some sufficient conditions of uniqueness of Lebesgue decompositions.
Proposition 3.8:
共i兲 If,苸CP共A;H兲 andacⱕu, that is, there exists t⬎0 such thatacⱕt, thenadmits a unique-Lebesgue decomposition.
共ii兲 If苸CP共A;H兲 is such that the property of -absolute continuity is hereditary then, for
any苸CP共A;H兲 there exists a unique-Lebesgue decomposition of.
共i兲 Without loss of generality we can assume thatacⱕ. Consider the maximal-Lebesgue
decomposition of =ac+s as well as another -Lebesgue decomposition =1+0,
where1,0苸CP共A;H兲,1is-absolutely continuous and0is-singular. Then, taking into account the maximality ofacit follows that0=−1ⱖac−1ⱖ0, henceac−1is -singular. Since ac−1 is -absolutely continuous, it follows that ac=1, and hence s=0.
共ii兲 Similar to the proof of共i兲.
䊐 When comparing with the results on uniqueness of the Lebesgue decomposition in Appendix A 3, the above proposition looks very weak. The difficulty of transposing Theorem 6.5 and its Corollary 6.6 to the setting of completely positive maps comes from the fact that, in this case, the Radon–Nikodym derivatives live in the von Neumann algebra 共A兲
⬘
that can be “very small.” This is illustrated in the following.Example 3.9: Let A be the Cⴱ-algebra L⬁关0,1兴 and H=L2关0,1兴. Let ,:A→B共H兲 be defined by 共共f兲g兲共g兲=tf共t兲g共t兲 and 共共f兲g兲共g兲=共1−t兲f共t兲g共t兲, for f 苸L⬁关0,1兴 and g苸L2关0,1兴. Then both and are completely positive and += is the map 共共f兲g兲共t兲= f共t兲g共t兲, for f 苸L⬁关0,1兴 and g苸L2关0,1兴. Then the minimal Stinespring representation of is 共
;K; V兲
where, K= L2关0,1兴,
共f兲=Mf, the operator of multiplication with f苸L⬁关0,1兴, and V is the
identity operator on L2关0,1兴. We have that
共A兲
⬘
=兵Mf兩 f 苸L⬁关0,1兴其 is a maximal Abelian vonNeumann algebra inB共L2关0,1兴兲. On the other hand,is-absolutely continuous and there exists
only one Lebesgue decomposition of with respect to. However, there exists no k such that
ⱕk.
For the case ofH=C, the difficulties on the uniqueness, and an example of nonuniqueness, see Kosaki.22
Finally, we observe that when considering the complete isomorphism CP共C;H兲苹哫共1兲 苸B共H兲+, as explained at the end of Sec. III A, an application of Theorem 6.5 gives the
nonu-niqueness of the Lebesgue decomposition for some operator valued completely positive maps. IV. AN APPLICATION TO THE INFIMUM PROBLEM FOR QUANTUM OPERATIONS
LetH be a Hilbert space and B共H兲+the cone of bounded non-negative operators inH. Given
A , B苸B共H兲+, the infimum of A and B, denoted by A∧B, is the greatest lower bound of the set
兵A,B其 in the ordered set B共H兲+, that is, A∧BⱕA,B and, if C苸B共H兲+is such that CⱕA,B, then
CⱕA∧B. Questions on characterization of the existence and computation of the infimum operator
are related to the lattice properties of quantum effects, cf. Gudder.17 First, let us note that B is
A-singular if and only if A∧B=0, hence, it is expected that this kind of questions should be related
with the Radon–Nikodym derivatives and Lebesgue decompositions. The complete answer to this question, when considering only quantum effects, is given by the following theorem. For the notation on the shorted operator关A兴B see Appendix A.
Theorem 4.1: 关Ando 共Ref.4兲兴 Let A,B苸B共H兲+. Then A∧B exists if and only if the Radon–
Nikodym derivatives关A兴B and 关B兴A are comparable, that is, either 关A兴Bⱕ关B兴A or 关B兴Aⱕ关A兴B. In this case, A∧B=min兵关A兴B,关B兴A其.
For other共more restrictive兲 formulas of calculating the infimum A∧B, see also Ref.15and the bibliography cited there.
When considering open quantum systems, quantum operations take the place of the quantum effects, and in connection with their lattice properties the infimum problem for quantum operations can be stated in similar terms: for,苸CP共A;H兲 we write∧苸CP共A;H兲 共if it exists兲 for the infimum ofand, that is,∧ⱕ, and for any苸CP共A;H兲 such thatⱕ,, it follows
ⱕ∧. With this definition, is-singular if and only if∧= 0. As an application of our results on the Lebesgue decomposition, we can show that the picture for quantum operations is similar to that for quantum effects.
Theorem 4.2: Let,苸CP共A;H兲 and consider=ac+s, the-Lebesgue decomposition
and only ifacand acare comparable and, in this case,
∧= min兵ac,ac其. 共4.1兲
We need an auxiliary result.
Lemma 4.3: Let A苸B共H兲+be a contraction and letM be a von Neumann algebra in B共H兲 such that A苸M. Then the following are equivalent.
共i兲 The infimum of A and I − A with respect toM+exists.
共ii兲 The infimum of A and I − A with respect toB共H兲+exists.
共iii兲 共A兲, the spectrum of A , is contained either in 兵0其艛
关
12, 1兴or in
关0 ,
1 2兴
艛兵1其.Moreover, in the case when any of the assertions (i), (ii), or (iii) holds, then the two infima at (i) and, respectively, (ii) coincide and, letting g苸C共关0,1兴兲 be the function,
g共t兲 = min兵t,1 − t其 =
再
t, 0ⱕ t ⱕ 1/2,1 − t, 1/2, ⱕ t ⱕ 1,
冎
共4.2兲we have, by continuous functional calculus, A∧共I−A兲=g共A兲 .
Proof: The equivalence of 共ii兲 and 共iii兲 is proven in Theorem 3.1 in Ref. 15. Clearly 共ii兲 implies共i兲.
In order to prove共i兲 implies 共iii兲, we observe that the proof of the corresponding implication 共ii兲 implies 共iii兲 in Theorem 3.1 in Ref.15is done in such a way that all the constructions are kept within the von Neumann algebra generated by A, and hence inM. 䊐
Proof of Theorem 4.2: Let=+ and consider the Radon–Nikodym derivatives D共兲 and
D共兲, as in the proof of Theorem 3.1. Then 共3.1兲holds. By Theorem 2.2, it follows that ∧ exists in CP共A;H兲 if and only if D共兲∧D共兲 exists with respect to the von Neumann algebra 共A兲
⬘
. By Lemma 4.3, it follows that the latter is equivalent with the fact that D共兲∧D共兲 exists with respect to B共H兲+ and then, by Theorem 4.1, this is equivalent with the fact that theshorted operators关D共兲兴D共兲 and 关D共兲兴D共兲 are comparable. But, in view of 共3.3兲, these shorted operator are, respectively, D共ac兲 and D共ac兲. Using once again Theorem 2.2, it follows
that∧exists if and only if D共ac兲 and D共ac兲 are comparable, and共4.1兲holds. 䊐
Let us observe that, by the assertion 共iii兲 in Lemma 4.3 and with the notation as in the preceding proof, a formula for computing∧ that is more explicit than共4.1兲, can be given in terms of the infimum of the Radon–Nikodym derivatives D共兲∧D共兲.
V. THE LEBESGUE DECOMPOSITION FOR COMPLETELY POSITIVE MAPS ON MATRICES
In this section we focus on completely positive maps from Mn, the Cⴱ-algebra of n⫻n
matrices, to Mk, for which we describe the Lebesgue decomposition in terms of the Choi matrices.
This situation corresponds to applications to quantum information theory in the sense that in order to make the Lebesgue–Radon–Nikodym decomposition available to quantum information theorists this should be described in terms of the Choi’s matrix, cf. Leung.25In this final dimensional case an additional and very helpful fact is that there exists a completely positive map that uniformly majorizes all the others, namely, the tracial completely positive map. To this end, we first recall, in a slightly different formulation, known results on the structure of completely positive maps, cf.11,7,24,31and the bibliography cited there. Because of this, we will skip most of the proofs.
First note that, in this finite dimensional case, there exists only one Lebesgue decomposition, as explained in Sec. III B.
For n苸N let 兵ei共n兲其i=1n be the canonical basis ofCn. For n , k苸N we consider the matrix units
兵Ei,j共n,k兲兩i=1, ... ,n, j=1, ... ,k其傺Mn,kof size n⫻k 共as usually, the space Mn,kof n⫻k matrices is
identified withB共Ck,Cn兲兲, that is, Ei,j共n,k兲is the n⫻k matrix with all entries 0 except the 共i, j兲th entry which is 1. In case n = k, we denote simply Ei,j共n兲= Ei,j共n,n兲.
Here and in the following we use the tensor notation for rank one operators, that is, ifH and
K are Hilbert spaces and h苸H and k苸K are nontrivial vectors, then the rank 1 operator h丢¯k
苸B共K,H兲 is defined by 共h丢¯兲x=具x,k典h for all x苸H. With this notation we havek
Ei,j共n,k兲= ei共n兲丢¯e共k兲j , i = 1, . . . ,n, j = 1, . . . ,k. 共5.1兲
We also record the following direct consequences of the definitions: for all j = 1 , . . . , n and i = 1 , . . . , k we have
Ei,j共n,k兲ⴱ= E共k,n兲j,i , 共5.2兲
and if, in addition, p苸N, r=1, ... ,k, and s=1, ... ,p, then
Ei,j共n,k兲Er,s共k,p兲=␦j,rEi,s共n,p兲. 共5.3兲
We also use the lexicographic reindexing of兵Ei,j共n,k兲兩i=1, ... ,n, j=1, ... ,k其, more precisely 共E1,1共n,k兲, . . . ,E1,k共n,k兲,E2,1共n,k兲, . . . ,E2,k共n,k兲, . . . ,E共n,k兲n,1 , . . . ,En,k共n,k兲兲 = 共E1,E2, . . . ,Enk兲. 共5.4兲
An even more explicit form of this reindexing is the following:
Er= Ei,j共n,k兲 where r =共j − 1兲k + i, for all i = 1, ... ,n, j = 1, ... ,k. 共5.5兲
Lemma 5.1: The formula
共m−1兲k+i,共l−1兲k+j=具共Ei,j共n兲兲el共k兲,em共k兲典, m,l = 1, ... ,k, i, j = 1, ... ,n, 共5.6兲
and its inverse
共C兲 =
兺
r,s nk
r,sErⴱCEs, C苸 Mn, 共5.7兲
establish a linear and bijective correspondence,
B共Mn, Mk兲 苹哫 ⌽ = 关r,s兴r,s=1 nk 苸 M
nk. 共5.8兲
Given苸B共Mn, Mk兲 the matrix ⌽ as in共5.8兲is called the Choi matrix of.
Remark 5.2: With respect to the identification Cnk⯝Cn丢Ck, any matrix⌽=关 r,s兴r,s=1
nk 苸M nk
=B共Cnk兲 is identified with a linear operator ⌽苸B共Cn丢Ck兲, in such a way that the formula共5.6兲
becomes
共m−1兲k+i,共l−1兲k+j=具⌽共ej共n兲兲丢e共k兲l ,ei共n兲丢em共k兲典, m,l = 1, ... ,k, i, j = 1, ... ,n. 共5.9兲
Remark 5.3: In the correspondence in Lemma 5.1, is unital if and only if
兺
i=1 n
共m−1兲k+i,共l−1兲m+i=␦l,m for all l,m苸 兵1, ... ,k其.
In the following, to a certain extent, our arguments and calculations will be parallel to those performed by Raginsky in Sec. V.A of Ref. 31but we will go further in exploiting the explicit formulas for the Lebesgue decomposition obtained in Theorem 3.3.
Let: Mn→Mkbe the linear map defined by
共C兲 =1
ntr共C兲Ik, C苸 Mn. 共5.10兲
V:Ck→ Cn2k⯝ Cn丢Cnk⯝ Cn丢Cn丢Ck be defined by Vh =
冑
1 n冤
E1h E2h ] Enkh冥
, h苸 Ck, 共5.11兲or, equivalently, with the identificationCn2k⯝Cn丢Cn丢Ckand the reindexing defined at共5.4兲,
Vh =
兺
i=1 n兺
j=1 k Ei,j共n,k兲h丢ei共n兲丢e共k兲j . 共5.12兲We consider also the map
:Mn→ Mn2k⯝ B共Cn 2k 兲 ⯝ B共Cn 丢Cnk兲, defined by 共C兲 = C丢Ink, C苸 Mn. 共5.13兲
Lemma 5.4: With the notation as in 共5.10兲–共5.13兲, 共;Cn2k; V兲 is the minimal Stinespring
representation of, in particular, 苸CP共Mn, Mk兲.
In addition, uniformly dominates any map苸CP共Mn, Mk兲.
Lemma 5.4 can be used to obtain the following result of Kraus23 and Choi:11 Let : Mn
→Mkbe a completely positive map. Then there are n⫻k matrices V1, V2, . . . , Vmwith mⱕnk such
that
共A兲 = V1ⴱAV1+ V2ⴱAV2+ ¯ + VmⴱAVm for all A苸 Mn. 共5.14兲
We also record the following fact: ifH and K are Hilbert spaces and ⌽,⌿苸B共K兲+, then it is easy to see, from共A2兲and共A4兲, that
共IH丢⌽兲:共IH丢⌿兲 = IH丢共⌽:⌿兲, 关IH丢⌽兴共IH丢⌿兲 = IH丢关⌽兴⌿. 共5.15兲
The main result of this section is the following description of the Lebesgue decomposition for completely positive maps between matrices, in terms of Choi matrices. For the definition of the “shorted matrix”关⌿兴⌽, see Appendix A.
Theorem 5.5: The formula共5.6兲and its inverse共5.7兲establish an affine and order-preserving isomorphism,
CP共Mn, Mk兲 苹哫 ⌽ 苸 Mnk
+
. 共5.16兲
Moreover, if,苸CP共Mn, Mk兲, ⌽,⌿苸Mnk
+ are the positive nk⫻nk matrices corresponding
by共5.8兲to and, respectively,, and the positive matrices A , B苸Mnk are defined by
A =关␣i,j兴i,j=1 nk
=关⌿兴⌽, B = 关i,j兴i,j=1 nk
=⌽ − 关⌿兴⌽, 共5.17兲
then, the absolutely continuous partacand the singular partsofwith respect toare given
by the following formulas: ac共C兲 =
