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In general, and including the case of mixed states where |s| 1, the density in .NET Make USS Code 128 in .NET In general, and including the case of mixed states where |s| 1, the density

In general, and including the case of mixed states where use none none integrating togenerate none with none Customer Bar Code s. 1, the density Appendix A Fig. A.1.

Representation of the density matrix of a two-state system in terms of the Bloch sphere and Bloch vector. The three components of the Bloch vector s = {s1 , s2 , s3 } specify the density operator according to the parameterization of Eq. (A25).

The two eigenvalues are (1 . s. )/2 and the two eigenvectors are speci ed by u and u. u s s2 u s1 operator of the form of Eq. (A25) has two eigenvalues 1 1+ 2 1 1 g2 = 2 g1 = 2 2 2 s1 + s2 + s3 = 2 2 2 s1 + s2 + s3 1 [1 + s. ] , 2 1 = [1 . s. ] , 2. (A27). and its ei genvectors are determined by the two vectors u and u shown in the Bloch sphere in Fig. A.1.

For pure states, where . s. = 1, u co incides with s and its tip lies on the surface of the Bloch sphere. For a mixed state where . s. < 1, v ector u points in the same direction as s but unlike s maintains unit length so that its tip always lies on the surface of the Bloch sphere. Equation (A26) can be used to express u and u in terms of the vectors in state space..

Entangled states Let us now consider a two-particle (or two-mode) system (known also as a bipartite system) and, for simplicity, let us assume that each particle can be in either of two one-particle states . 1 or 2 . Using the notation 1 , parti none for none cle 1 in state 1, 2 , particle 1 in state 2, 1 , particle 2 in state 1, 2 , particle 2 in state 2,. (2) (2) (1) (1). we conside none for none r a pure two-particle superposition state (in general an entangled state). = C1 1 . 2. + C2 2 1. (A28). an example none for none of which is given in Eq. (A2). (We have inserted the direct product symbol here for emphasis, although we generally assume it to be understood.

A.4 Schmidt decomposition and clear none for none from context throughout this book.) Clearly, this can be extended for multiparticle (multipartite) systems. For such multipartite systems, we can de ne reduced density operators for each of the subsystems by tracing the density operator over the states of all the other systems.

In the present case, with the density operator of the two-particle system given by = . , the reduced density operator for particle 1 is (1) = Tr2 = 1 1 = C1 . 2 1 . (1) (2) (2). + 2 2. + . C2 . 2 2 . 2 .. (A29). This has t none none he form for a mixed state for particle 1 as long as Ci = 0, i = 1, 2. Similarly, for particle 2,. (2) = Tr1 = C1 . 2 1 . + . C2 . 2 2 . 2 .. (A30). Evidently, when one of the particles is considered without regard to the other, it is generally in a mixed state. Thus one may characterize the degree of entanglement according to the degree of purity of either of the subsystems. If Tr[ (2) ]2 = 1, (2) 2 the state .

is not an entangled state; but if Tr[ ] < 1 we may conclude that describes an entanglement between subsystems 1 and 2. Schmidt decomposition There is another convenien t way to approach the problem of characterizing entanglement, at least for cases where there are only two subsystems. We refer to the von Neumann entropy, which we introduce in the next section. But rst, as a preliminary, we introduce the Schmidt decomposition [1, 2].

To keep the discussion general we do not restrict the dimensions of the Hilbert spaces of the subsystems. Suppose we let {. ai , i = 1, 2, 3, . . .

} a none for none nd {. b j , j = 1, 2, 3, . . .

} form orthonormal bases for subsystems U and V respectively. If the Hilbert spaces of these systems are denoted HU and HV respectively, then any state . H = HU HV can be written as i, j=1 ci j ai b j . (A31). In these bases, the density operator for the composite system is =. = . i, j,k,l ci j ck l ai ai b j bl ai b j bl , (A32).
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