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Unformatted text preview: Physics 137B: Quantum mechanics II, Fall 2007 Lecture on quantum statistical mechanics and entanglement This lecture studies some examples of the density matrix formalism for quantum statistical mechanics. We will see that the familiar singlet state of two particles has a property known as entanglement that is quite surprising from a classical point of view. Entanglement is one of the basic notions of quantum information, Lightning review of the density matrix formalism: The density operator is explicitly written as = N X =1 | i W h | , (1) where | i are some normalized states (not necessarily orthogonal or complete). This is now shown to reproduce the sort of statistical average discussed above. Lets take an operator A and ask about its statistical expectation. In a particular orthonormal basis, the matrix representation of is n,n = h n | | n i = N X =1 h n | ih | n i W i . (2) Now Tr A = X n,n n,n A n ,n = X n,n , h n | ih | n i W h n | A | n i . (3) We can simplify this greatly using the completeness relation for the basis | n i : completeness requires X n | n ih n | = 1 . (4) Then in the above sum, both the sums over n and n just give unity, leaving Tr A = X W h | A | i . (5) Some simple properties of the density matrix that follow from the above definition are Tr = 1 (6) and all diagonal elements are nonnegative, since the diagonal elements are just the probabilities of being in different pure states. We also showed that for a pure state,...
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