M.-D. Choi. Completely positive linear maps on complex matrices. Linear Algebra and Its Applications, 10:285--290, 1975.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....See the many references in Ref. 98] for appropriate background. It has long been known that the trace preserving completely positive linear maps # over a D dimensional vector space can be placed in a one to one correspondence with density operators on a D dimensional space via the relation [79, 99, 100] # = I #( #ME (116) signifies a maximally entangled state on HD #HD , # D i# i# . 117) This is usually treated as a convenient representation theorem only, but maybe it is no mathematical accident. Perhaps there is a deep physical reason for it: The time evolution one ....

M.-D. Choi, "Completely Positive Linear Maps on Complex Matrices," Lin. Alg. App. 10, 285--290 (1975).

....lemma, linear opreators on C C m ) has been This looks suspiciously like the Stinespring dilation of a CP map and, in fact, it is (for details see, e.g. Davies proof [9, Sect. 9. 3] of the Naimark theorem) treated extensively in a variety of forms in the mathematical literature (see, e.g. [5, 17, 21, 28, 30] for a sampling of results related to positive and completely positive maps) More recently, this correspondence has been exploited fruitfully in some quantum information theoretic contexts, such as optimal cloning maps [7] optimal teleportation protocols [16] separability criteria for entangled ....

M.-D. Choi, \Completely positive linear maps on complex matrices," Linear Algebra Appl. 10, 285 (1975).

....completely positive if T (X) 1#i#N A i XA i ; A i , X M(N) 5) Choi s representation of linear operator T : M(N) M(N) is a block matrix CH(T ) i,j = T (e i e j ) Dual to T respect to the inner product X,Y = tr(XY ) is denoted as T # . Very usefull and easy Choi s result [6] states that T is completely positive i# CH(T ) is (BUDM ) Using this natural (linear) correspondence between completely positive operators and (BUDM ) we will freely transfer properties of (BUDM ) to completely positive operators . For example , a linear operator T is called separable i# ....

D.M. Choi , Completely positive linear maps on complex matrices , Linear Algebra Appl. , 10 (1975 ) , 285 - 290 .

.... quantum channels on the state of a system may be described by completely positive linear maps N , from the space B(H i ) of bounded operators on a finite dimensional input Hilbert space H i , to the space B(H o ) of bounded operators on a finite dimensional output Hilbert space H o [9] 10] 3] [11]. I will sometimes use the term quantum operation for a trace nonincreasing completely positive map. Such maps have representations in terms of linear operators A i [10] 3] A(ae) X i A i aeA y i ; 2.22) with X i A y i A i I ; 2.23) equality holds in the latter when the map is ....

....set fA i g an operator decomposition, or simply decomposition, of the operation A, and sometimes write: A fA i g (2.24) to indicate that fA i g is an operator decomposition of A. Any two decompositions of the same operation, fA i g having r operators and fB i g having s operators, are related by [11]: A i = s X j=1 m ij B j (2.25) 38 where m is the matrix of a maximal partial isometry from the complex vector space C s to C r , i.e. its columns are s orthonormal vectors in C r : X j m ij m kj = ffi ik (2.26) or in other words: mm y = I (s) 2.27) A partial isometry is a ....

M.-D. Choi, "Completely positive linear maps on complex matrices," Linear Algebra and Its Applications, vol. 10, pp. 285, 1975.

.... of a system may be described by completely positive linear maps N , from the space B(H c ) of bounded linear operators on a input SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY 2 Hilbert space H c , to the space B(H o ) of bounded linear operators on an output Hilbert space H o [25] 26] [27]. In this paper, we consider only discrete channels, which we de ne as having nite dimensional input and output Hilbert spaces (the word bounded in the speci cation of the input and output spaces is redundant in the discrete case) We will sometimes use the term quantum operation for a ....

....fA i g an operator decomposition, or simply decomposition, of the operation A, and sometimes write: A fA i g (3) to indicate that fA i g is an operator decomposition of A. Any two decompositions of the same operation, fA i g having r operators and fB i g having s r operators, are related by [27]: A i = s X j=1 m ij B j (4) where m is the matrix of a maximal partial isometry from the complex vector space C s to C r . A partial isometry is a generalization of a unitary operator, which must satisfy V V y = for some projector . Such an isometry will then also satisfy V y V = ....

M.-D. Choi, \Completely positive linear maps on complex matrices, " Linear Algebra and Its Applications, vol. 10, pp. 285, 1975.

....matrices into itself; Phi is m Gammapositive if Phi Omega I m is positive on Mn Omega Mm , where I m is the identity operator on Mm ; and Phi is completely positive if Phi is m positive for every m. The structure of completely positive maps on Mn is quite well understood (e.g. see [C, PH, BHH, LoS]) Recently, there has been interest in studying completely positive maps satisfying some special properties such as leaving invariant the trace function, or leaving invariant the identity map, e.g. see [BPS, LS] and their references) In particular, it has been shown that the structure of such ....

....matrix A 2 Mn such that Phi is of the form X 7 A ffi X. Proof. a) b) If Phi : Mn Mn is completely positive and leaves the diagonal entries invariant, then Phi(E ii ) 0 with diagonal entries equal to those of E ii . It follows that Phi(E ii ) E ii . b) c) It is known (e.g. see [C] and [PH] that Phi : Mn Mn is a completely positive map if and only if the n 2 Theta n 2 block matrix ( Phi(E ij ) 1i;jn is positive semidefinite. It follows that for any i 6= j, Phi(E ij ) a ij E ij for some a ij 2 C with a ij = a ji . Set a ii = 1 for i = 1; n, and let A = ....

M.D. Choi, Completely positive linear maps on complex matrices, Linear Algebra Appl. 10:285-290 (1975).

....2. POSITIVE AND COMPLETELY POSITIVE MAPS We start by recalling and establishing some definitions and facts related to trace ideals and to the concept of positive and completely positive maps ( for more details on trace ideals see e.g. 31, 32] and on positive and completely positive maps e.g. [33, 34, 5, 6, 35, 26]) Let H be any complex, separable Hilbert space with scalar product denoted by h ; i. By B(H) we denote the C algebra of all bounded operators on H equipped with the norm jj jj. By I2 B(H) we denote the identity map on H. For any A 2 B(H) we set jAj = A A) 1=2 2 B(H) where ....

....= I. Obviously trace preserving maps leave the set C 1;1 of density matrices invariant. If in J 1 is completely positive and trace invariant, then by (5) jj jj 1 = 1. All completely positive maps on J 1 have the Kraus representation [22] a consequence of a theorem of Stinespring [33] see also [5] for a proof in the finite dimensional case) Given such a there is an at most denumerable set of elements = f i g i2N in B(H) satisfying X i2K i i jj jj 1 I (8) for any finite subset K N such that = again with (A) X i2N i A i ; 9) which now may be an ....

M.-D. Choi, Completely positive linear maps on complex matrices, Lin. Alg. and Appl. 10 (1975), 285 -- 290.

....1 2 2 [0; The proof of 3. 4. is an application of Lemma 2. 4: oe(L A 2 Pi A0 ) ae C Gamma ( b( 0 (as b( increases ) 8 2 [0; b( 0 ( 8 2 [0; d( 6= 0 and oe(A) ae C Gamma 2 To get more information about d( we regard special positive operators (see e.g. [1]) Definition and Theorem 2.6 An operator T : H n H m is called completely positive if for all k 2 N the operator I k Omega T : H nk H mk is positive. An equivalent condition is that T allows a representation of the form T (X) X V i XV i ; where V i 2 R m Thetan : In ....

M.-D. Choi. Completely positive linear maps on complex matrices. Linear Algebra and Its Applications, 10:285--290, 1975.

....is completely positive if for every N the matrix [ Phi(A ij ) N ThetaN is positive whenever [A ij ] N ThetaN is positive. Such maps have turned out to be of crucial importance in the theory of C algebras. A fundamental structure theorem for completely positive maps on M was proved by Choi [7]. The Schur product map Phi(X) Z ffi X where Z is positive, is an example of a completely positive map. This is so because for each N the N Theta N block matrix with all entries Z is again positive. A linear map Phi on M is said to be unital if Phi(I ) I , and trace preserving if tr ....

....map in the same way as one did for L Gamma1 A . It is interesting to note that the maps L Gamma1 A and S Gamma1 F are not just positive, they are completely positive. Again one can see this in different ways. The very form of the solutions (4.2) and (4. 3) and Choi s characterisation [7] immediately give this fact. Another illuminating way of looking at this is the following. Given any matrix A and a positive integer N let e A be the direct sum of N copies of A ; i.e. e A is a block diagonal matrix with N copies of A down its diagonal. If A is positively stable then so is e A . ....

M.D. Choi, Completely positive linear maps on complex matrices, Linear Algebra Appl. 10:285--290 (1975).

....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 ....

....By Arveson s extension theorem [4, Theorem 1.2.3] OE can be extended to a completely positive linear map Phi from M 3 to B(H) Then there is an isometry V such that Phi(X) V (X Omega I)V for all X 2 M 3 ; thus A = OE(B) Phi(B) V (B Omega I)V and Condition (c) holds. As shown in [7], if H is of finite dimension, the number of copies of B needed is finite. Conversely, suppose (c) holds, i.e. A = V (B Omega I)V for some isometry V . Then OE is just the completely positive linear map X 7 V (X Omega I)V . 2 Usually, a general positive linear map is far away from ....

M.D. Choi, Completely positive linear maps on complex matrices, Linear Algebra Appl. 10 (1975), 285-290.

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M.-D. Choi. Completely positive linear maps on complex matrices. Linear Algebra and Its Applications, 10:285--290, 1975.

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M.-D. Choi. Completely positive linear maps on complex matrices. Linear Algebra and Its Applications, 10:285--290, 1975.

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