By Harrison D.M.
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Sample text
51) ˆ B] ˆ = 0, and means that there is a basis, say This is equivalent to the condition that [A, ˆ ˆ {|φk }, in which both A and B are diagonal. 52) k where the eigenvalues ak may be degenerate (that is, there may exist k and k such that ak = ak ) and similarly for bk . Thus, one way of making a simultaneous measurement of A and B is to make a measurement of Kˆ = k k|φk φk |, and from the result k determine the appropriate values ak and bk for A and B. 3 Systems and meters The standard (projective) presentation of quantum measurements is inadequate for a number of reasons.
42) λ Thus, if the state were pure at time t, and we make a measurement, but ignore the result, then in general the state at time t + T will be mixed. That is, projective measurement, unlike unitary evolution,2 is generally an entropy-increasing process unless one keeps track of the measurement results. 17). 9 Show that a projective measurement of decreases the purity Tr ρ 2 of the unconditional state unless the a-priori state ρ(t) can be diagonalized in the same basis as can ˆ . 2 Of course unitary evolution can change the entropy of a subsystem, as we will discuss in Chapter 3.
If the two are initially correlated they should be considered jointly as the system. 4 Most general formulation of quantum measurements 33 The state in Eq. 140) is unnormalized. 141) ρr (t + T ) = Or ρ(t)/℘r . 5, it is possible to define a probability operator, or effect, Eˆ r , such that, for all ρ, Tr[Or ρ] = Tr[ρ Eˆ r ]. 144) It is easy to verify that Eˆ r = j which is obviously Hermitian and positive. 145) r is the only mathematical restriction on the set of operations Or . The unconditional system state after the measurement is ρ(t + T ) = Or ρ(t) = Oρ(t).