Triangle inequality in .NET Connect Data Matrix in .NET Triangle inequality

Triangle inequality use .net data matrix 2d barcode drawer tomake data matrix ecc200 with .net Customer Bar Code The subadditivit .NET data matrix barcodes y inequality in Eq. (21.

28) provided an upper bound to the joint entropy S( AB ), namely S( AB ) S( A ) + S( B ). The triangle inequality, also referred to as the Araki Lieb inequality, provides a lower bound to S( AB ), according to . S( A ) S( B ). S( AB ). (21.43).

To prove this pr operty, assume the existence of a third system R, such as the composite system AB R in a pure state. The system R is referred to as a purifying system for AB (see Appendix W). From the subadditivity inequality we nd S( A R ) S( A ) + S( R ).

(21.44). Because AB R is .net framework barcode data matrix in a pure state, we also nd that S( AB ) = S( R ) and S( A R ) = S( B ) (see earlier subsection on composite system in pure state). Substituting these last two equalities into Eq.

(21.44) we obtain S( AB ) S( B ) S( A ). Since A and B play a symmetric role, we also have S( AB ) S( A ) S( B ), and, hence, S( AB ) .

S( A ) S( B ). , which proves t Visual Studio .NET Data Matrix he property in Eq. (21.

43). It can be shown (but the proof is not to be considered here) that the equality in Eq. (21.

43) is given by the condition A R = A R , which means that the A, R information is uncorrelated and, therefore, that all possible correlation of A information with the rest of the world is exclusively with B.. 21.2 Relative, joint, and conditional entropy, and mutual information Concavity of entropy and entropy of system in random states Assume a quantum system that can be in any random mixed state i , according t Visual Studio .NET gs1 datamatrix barcode o some known probability distribution pi . Each of these random states is associated with a density operator i .

The concavity of a function f (x) corresponds to the property f (x) f ( x ). Here, I shall establish that entropy is a concave function of the densityoperator variable i , namely, S( ) S( ), or, formally: (21.45).

pi S( i ) S pi i . (21.46). Furthermore, if .net vs 2010 DataMatrix one assumes that the set of operators i has support on orthogonal subspaces (meaning that the set of all possible eigenstates form an orthonormal basis), an exact relation also exists between S( ) and S( ) , which nicely relates to the Shannon entropy of a classical source H (X ) = i pi log pi , according to: S( ) = S( ) + H (X ), or, formally: (21.47).

pi i pi S( i ) + H (X ).. (21.48). The property exp 2d Data Matrix barcode for .NET ressed in Eq. (21.

47) or Eq. (21.48) can easily be interpreted as follows: the quantum information of a system whose states are random equals the mean information, as averaged over the individual states, plus the information on the probability distribution, which is the Shannon entropy.

Note that since H (X ) is nonnegative, this property also establishes the concavity of the entropy, as expressed in Eq. (21.45) or Eq.

(21.46). In the case where the probability distribution is uniform, the Shannon entropy H (X ) is maximal, meaning that there is maximal uncertainty as to which quantum state the system is in.

Such a situation also maximizes the entropy S( ) of the system. The deterministic case where the system has only one possible quantum state corresponds to H (X ) = 0 and a minimum entropy. The demonstration of the property expressed in Eq.

(21.48) is relatively simple. Indeed, let .

ik and ik be t he eigenstates and eigenvalues for each quantum state associated with the density operator i . Hence i has diagonal matrix elements ( i )kk = ik . Next, given any i, k we observe that .

ik and pi ik are eigenvectors and eigenvalues of
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