W.F. Stinespring: "Positive functions on C*-algebras", Proc.Amer.Math.Soc. 6(1955)211--216  Home/Search   Document Not in Database   Summary   Related Articles   Check

....tailor made interaction Hamiltonian, and finally the ancilla (or, more generally, a suitable subsystem) is discarded. The claim that every channel can be represented in the last two forms is a direct consequence of the fundamental structure theorem for completely positive maps, due to Stinespring [30]. We state it here in a version adapted to pure quantum systems, containing no classical components. Theorem 4 (Stinespring Theorem) Let T : Mn #M m be a completely positive linear map. Then there is a number #, and an operator V : C m # C n# C # such that T (X) V # (X# 1I # )V, 6.5) ....

W.F. Stinespring: "Positive functions on C*-algebras", Proc.Amer.Math.Soc. 6 (1955) 211--216

....state oe into M systems of the same kind, in an approximation of the M fold tensor product of the state oe. 1 Inst. f. Mathematische Physik, TU Braunschweig, Mendelssohnstr.3, 38106 Braunschweig, Germany 2 Electronic mail: R.Werner tu bs.de 1. Introduction One of the fundamental features distinguishing quantum theory from classical theories is epitomized by the No Cloning Theorem [WZ] The quantum copiers forbidden by this theorem, in much the same way as perpetual motion machines are forbidden by the Second Law of thermodynamics, are defined as follows: a copier takes one quantum ....

....as input and produces as output two systems of the same kind. If one now runs experiments in which each input is prepared according to the same density matrix, either one of the outputs is discarded, and some measurement is then performed on the remaining output, one should get the same statistical results as measured directly on the inputs, for arbitrary initial preparations and final measurements. The impossibility of cloning machines is intimately connected to other impossible tasks of quantum theory, notably joint measurement and teleportation . It is well known that there are ....

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W.F. Stinespring: "Positive functions on C*-algebras", Proc.Amer.Math.Soc. 6(1955)211--216

....with either A or B abelian, which includes all states. Completely positive operators are completely bounded, and when 1I 2 A, and P is completely positive, we have kP (1I)k = kPk = kPk cb [Pau] The fundamental structure theorem for completely positive maps is the Stinespring Dilation Theorem [Sti], stating that every completely positive P : A B(H) can be decomposed in an essentially unique way into P (A) V (A)V , where : A B(K) is a representation of A on a Hilbert space K, and V : H K is a bounded operator. Basic examples of complete contractions are differences P = P ....

W.F. Stinespring: "Positive functions on C*-algebras", Proc.Amer.Math.Soc. 6(1955) 211--216

....if the operator T (n) is positive, where T (n) acts on the n n matrices M n (A) with entries in A and is given by T (n) x ij ) i;j = T (x ij ) i;j . The operator T is called completely positive if it is n positive for all n = 1; 2; 3; This terminology goes back to Stinespring [6]. For the case A = M n (C ) all linear operators on M n (C ) are elementary operators with length at most n 2 . An example of Choi [1] says that the operator Tx = n 1)trace(x) x on M n (C ) is (n 1) positive but not n positive. It is not hard to compute that the minimal length of this ....

W. Forrest Stinespring, Positive functions on C*-algebras, Proc. Amer. Math. Soc. 6 (1955) 211-216.

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W.F. Stinespring: "Positive functions on C*-algebras", Proc.Amer.Math.Soc. 6(1955)211--216

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