V. I. Paulsen, Completely bounded maps and dilations, Amer. Math. Soc., Essex, England, 1986, ISBN 0--582--98896-9.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....For the transposition on C we have in particular k k cb = d. The cb norm has some nice features which we will use frequently. This includes its multiplicativity T 2 k cb = kT 1 k cb kT 2 k cb and the fact that kTk cb = 1 for every channel. For more properties of the cb norm we refer to [14]. 3.2 Achievable rates and capacity How can we reduce the error level kT Id k cb As an example, consider a small unitary rotation, i.e. T (X) U XU , with kT Id k cb 2kU 1Ik small. Then if we know U , it is easy to correct T by the inverse rotation, either before T , as an encoding , ....

V. I. Paulsen. Completely bounded maps and dilations. Longman Scienti c & Technical (1986). 22

....is defined as X : # X # X. Then 1 = Tr A for any positive operator A, and furthermore 1 = 1 for any density operator #. We will also need to estimate norm di#erences of quantum channels; the ideal norm for this purpose is the so called norm of complete boundedness (or CB norm, for short) [10], defined by cb : sup n#N #T # id n #, #M n (3) where n stands for the algebra of n n complex matrices. The norm on the r.h.s. of (3) is the operator norm, defined for a general linear map M : 2 ) by #M# : sup X#B(H 1 ) #X# 1 #1 #M(X)# 1 . 4) Note that the above ....

....The norm on the r.h.s. of (3) is the operator norm, defined for a general linear map M : 2 ) by #M# : sup X#B(H 1 ) #X# 1 #1 #M(X)# 1 . 4) Note that the above definition is tailored specifically for quantum channels in the Schrodinger picture; consult the monograph of Paulsen [10] for generalities. We have cb = 1 for any quantum channel T . We shall have an occasion to use some other properties of the CB norm in later sections; all we need right now is the inequality #T (A)# 1 # #T# 1 , which is obvious from definitions, and the multiplicativity of the CB norm ....

V.I. Paulsen, Completely Bounded Maps and Dilations, Longman Scientific & Technical, 1986.

....operator norm) such that (A) V 1 E )V 1 : 8) Conversely, any map of the form (8) satis es k k cb kV 1 kkV 2 k. Note that the Stinespring and the Haagerup Paulsen Wittstock theorems together imply that any CP map is automatically CB. In fact, for a CP map T , we have kTk cb = kT (1)k [27]. Also, the di erence of two CP maps is always CB. Theorem 2.1 suggests an alternative way to de ne the CB norm of a map , namely as k k cb = inffkV 1 kkV 2 kg; 9) where the in mum is taken over all possible decompositions of in the form (8) Moreover, the theorem guarantees that the in ....

.... : B(H ) B(K ) any A 2 B(H ) and any B 2 T (H ) we have the following. 1. k k cb k k cb k k cb ; 2. k cb = k k cb k k cb ; 3. k (A)k k k cb kAk; 4. k (B)k 1 k k cb kBk 1 . For proofs see, e.g. the article of Kitaev [19] or the monographs of Paulsen [27] and Pisier [29] 3 The Radon Nikodym theorem for completely positive maps In this section we review a theorem of the Radon Nikodym type that allows for a complete classi cation of all CP maps S that are completely dominated by a given CP map T . As we have already mentioned, this theorem can be ....

V.I. Paulsen, Completely Bounded Maps and Dilations (Longman Scienti c and Technical, New York, 1986).

....the operator norm, kTk : sup X2B(H ) kXk=1 kT (X)k : 2.42) 31 Unfortunately, the operator norm is rather ill behaved: it is not stable with respect to tensor products. In particular, there are some positive maps T , for which the norm k id n k will increase with n, as the following example [97] shows. Example 2.4.1 (transposition map revisited) Consider the transposition map (cf. Example 2.2.1) on the algebra M 2 , and de ne the map 2 : id 2 on M M 2 . Let F be the ip operator F = e j j for which we have kFk = 1. Now 2 (F ) which has norm 2. Thus k 2 ....

....F be the ip operator F = e j j for which we have kFk = 1. Now 2 (F ) which has norm 2. Thus k 2 k 2. A good choice then is the metric induced by the stabilized version of the operator norm (2. 42) namely the norm of complete boundedness (or cb norm for short) de ned by [97] kTk cb : sup For any operator X 2 B(H ) and any two maps S; T on B(H ) with nite cb norm (in the case of nite dimensional H , this is always true [97] we have the relations kT (X)k kTk cb kXk ; 2.44) kSTk cb kSk cb kTk cb ; 2.45) Tk cb = kSk cb kTk cb : 2.46) Furthermore, for ....

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V.I. Paulsen, Completely Bounded Maps and Dilations (Longman Scienti c & Technical, Harlow UK, 1986).

..... Viewing X # as a dual Banach space of X (for now) we define an isometric embedding J : X # # ( P u#U M n(u) # ## B(H) by setting J(f) # u#U u(f ) where H = P u#U # n(u) 2 ) 2 . J(X # ) is called the operator space dual of X , and denoted by X # . It is known (see e.g. 1] or [23]) that X # norms X : for any x # Mn (X) x = sup #x, f# f # Mn (X # ) f # 1 , The author was supported in part by the Texas Advanced Research Program 010366 163, and by the NSF grant DMS 9970369. Typeset by A M S T E X 1 2 TIMUR OIKHBERG where the M n 2 valued inner ....

....the Texas Advanced Research Program 010366 163, and by the NSF grant DMS 9970369. Typeset by A M S T E X 1 2 TIMUR OIKHBERG where the M n 2 valued inner product is defined by #a# x, b# f# = a# bf(x) for a, b # Mn , x # X and f # X # . The reader is referred to [1] 3] 7] 8] [23], or [31] for more information concerning operator spaces. A Banach space E can be embedded into B(H) in a variety of ways. The embeddings giving rise to the minimal and maximal operator space structures (introduced in [1] 3] and [6] and further investigated in [24] and [25] are especially ....

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V. Paulsen, Completely bounded maps and dilations, Longman Scientific & Technical, Harlow, England, 1986.

....# c # n (Theorem 3.3) Other possible strengthenings of the principle of local reflexivity are discussed in Section 4. Throughout the paper we shall use standard Banach space terminology which can be found, for instance, in [4] and [16] We make some peripheral remarks about operator spaces; see [23] or [27] for an introduction to that subject. We say that a Banach space X has the # approximation property (# AP) if for every finite dimensional space E ## X and # 0 there exists a finite rank map u : X # X such that u E = I E and u # # #. If X has the # AP for some #, we say X ....

....on # 2 (with its natural operator space structure) and Y is a separable operator space containing K, does there exist a bounded projection from Y onto K It is known that a completely bounded projection from Y onto K need not exist. By the Stinespring extension theorem (see e.g. Theorem 7. 3 of [23] or Theorem 3.6 of [26] this question is equivalent to the following: if Y is a separable subspace of B(# 2 ) containing K, does there exist a projection from Y onto K By writing Y = span[K, y 1 , y 2 , and cutting o# the o# diagonal parts of y 1 , y 2 , we can reformulate the ....

V. Paulsen, Completely bounded maps and dilations, Wiley, 1986.

....# c # n (Theorem 3.3) Other possible strengthenings of the principle of local reflexivity are discussed in Section 4. Throughout the paper we shall use standard Banach space terminology which can be found, for instance, in [4] and [16] We make some peripheral remarks about operator spaces; see [23] or [27] for an introduction to that subject. We say that a Banach space X has the # approximation property (# AP) if for every finite dimensional space E ## X and # 0 there exists a finite rank map u : X # X such that u E = I E and u # # #. If X has the # AP for some #, we say X ....

....on # 2 (with its natural operator space structure) and Y is a separable operator space containing K, does there exist a bounded projection from Y onto K It is known that a completely bounded projection from Y onto K need not exist. By the Stinespring extension theorem (see e.g. Theorem 7. 3 of [23] or Theorem 3.6 of [26] this question is equivalent to the following: if Y is a separable subspace of B(# 2 ) containing K, does there exist a projection from Y onto K By writing Y = span[K, y 1 , y 2 , and cutting o# the o# diagonal parts of y 1 , y 2 , we can reformulate the ....

V. Paulsen, Completely bounded maps and dilations, Wiley, 1986.

....map OE : A B is called completely contractive iff OE n is contractive for all n 2 N; a complete quotient map iff OE n is a quotient map for all n 2 N; and completely isometric iff OE n is isometric for all n 2 N. Finally, we define kOEk n : kOE n k for n 2 N and kOEk 1 : sup n2N kOEk n . See [6] for this terminology. Let A ae B (H) be a closed subalgebra with id H = 2 A. Consider the corresponding unital operator algebra A : fx Delta id H j x 2 A; 2 C g ae B (H) We show that if OE : A B is a complete isometry, then the unital extension OE : A B is also a ....

....can be viewed as operators from C d Psi x to C d Psi y, with Psi denoting the orthogonal complement. This yields a completely isometric linear representation OE : J M d Gamma1 . Since OE ffi aej I is a d Gamma 1 contractive linear map to ADJOINING A UNIT 5 M d Gamma1 , Theorem 5. 1 of [6] yields that OE ffi aej I is completely contractive. Thus aej I is completely contractive. In particular, if A is a commutative unital operator algebra, then any contractive unital representation A M 2 is completely contractive. For certain representations of function algebras, this was ....

Vern I. Paulsen, Completely bounded maps and dilations, Longman Scientific & Technical, Harlow, 1986.

....results about interpolation in M n (M d ) In this way, Theorem 4.1 and Theorem 7. 3 become special cases of results in [2] and [6] A map OE : A B between operator algebras is called completely contractive iff the induced maps OE (n) M n (A) M n (B) are contractive for all n 2 N [12]. Completely isometric maps and complete quotient maps are defined by requiring that OE (n) be isometric or a quotient map for all n 2 N, respectively. The essential step in the proof of the interpolation results is to obtain a completely isometric representation of the quotient algebra M d =I(z ....

Vern I. Paulsen, Completely bounded maps and dilations, Longman Scientific & Technical, Harlow, 1986.

....section, we establish relations between the conditions (i) W (A) W (B) and (ii) A has a dilation of the form B Omega I = B Phi B Phi Delta Delta Delta. The tool is the facility of constraint unitary dilations in conjunction with the theory of completely positive linear maps (see e.g. [4, 5, 7, 17] for general background) Let S be a subspace of M n satisfying I n 2 S, and A 2 S whenever A 2 S. A linear map OE : S B(H) is said to be positive if it maps positive elements to positive elements. OE is said to be completely positive if OE k : M k (S) M k (B(H) defined by (A ij ) 1i;jk ....

V.I. Paulsen, Completely Bounded Maps and Dilations, Longman, Harlow, England, 1986.

....# algebra. However, a subspace of a maximal operator space (we shall call it a submaximal space) need not be maximal. In this paper we investigate the subspace structure of maximal spaces. We make free use of standard operator space notation, definitions and results (see e.g. 1] 3] 5] 6] [20], or [29] We briefly recall some of these definitions below. If H is a Hilbert space, we equip it with row (column) operator space structures by identifying it with HR = B(H # , C) HC = B(C, H) Let Rn = # n 2 ) R , Cn = # n 2 ) C , R = # 2 ) R and C = # 2 ) C . Note that R can be ....

....(u) 1. This implies that w 1 : Rn # Cn # (E) # cb # n 1 4 . Proof of Theorem 5.4. By Lemma 5.5, id : MIN(# n 2 ) # (# n 2 ) cb # # n and id : # n 2 ) # MAX(# n 2 ) cb # # n. On the other hand, id : MIN(# n 2 ) # MAX(# n 2 ) cb # n (see [20] or [21] This implies (1) and (2) 3) follows from Lemma 5.5 and the fact that id : Rn Cn # Cn = # n. 4) and (5) are proved in the same manner. 6) follows from Lemmas 5.5, 5.6 and the fact that id : # n 2 ) # (# n 2 ) # cb # id : Rn Cn # Rn#Cn cb = # n. ....

V. Paulsen, Completely bounded maps and dilations, Longman Scientific & Technical, Harlow, England, 1986.

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V. I. Paulsen, Completely bounded maps and dilations, Amer. Math. Soc., Essex, England, 1986, ISBN 0--582--98896-9.

No context found.

V. I. Paulsen, Completely Bounded Maps and Dilations, Longman Scienti c and Technical, Harlow, Essex, 1986.

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V. I. Paulsen. Completely bounded maps and dilations. Longman Scienti c & Technical (1986).

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V.I. Paulsen, Completely Bounded Maps and Dilations (Longman Scientific & Technical, London, 1986).

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