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By Polyakov A.

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It is only for the class of efficient measurements that one can derive the following powerful theorem [Nie01, FJ01]: H [ρ(t)] ≥ ℘r H [ρr (t + T )]. 157) for arbitrary state matrices ρj and positive weights wj summing to unity. 5 The interpretation of this theorem is that, as long as no classical noise is introduced in the measurement, the a-posteriori conditional state is on average less mixed than (or just as mixed as) the a-priori state. That is, the measurement refines one’s knowledge of the system, as one would hope.

In this case the measurement operators are, in the measurement (x) basis, Mˆ p = 2−1/2 |0 0| + e−iπp |1 1| . 19 Verify that the non-selective evolution is the same under these two different measurements, and that it always turns the system into a mixture diagonal in the measurement basis. Clearly, measurement of the apparatus in the complementary basis does not collapse the system into a pure state in the measurement basis. In fact, it does not change the occupation probabilities for the measurement basis states at all.

However, this construction does not work when one considers ˆ weighted by the operator products. The correct post-measurement expectation for Aˆ B, probability for outcome r, is ˆ r = Tr Mˆ r† Aˆ Bˆ Mˆ r ρ . 122) because, in general, Mˆ r is not unitary. The correct Heisenberg formulation of measurement is as follows. The total state (of system plus apparatus) remains equal to the initial state, which is usually taken to factorize as ρtotal (t) = ρS ⊗ ρA . 124) 30 Quantum measurement theory which here is for time t, before the measurement interaction between system and apparatus.

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