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Sample text

81). 139) j for some set of operators { ˆ r,j : j }. For a given operation Or , the set { ˆ r,j : j } is not unique. For this reason it would be wrong to think of the operators ˆ r,j as measurement operators. Rather, the operation is the basic element in this theory, which takes the a-priori system state to the conditioned a-posteriori state: ρ˜r (t + T ) = Or ρ(t). 140) It is possible to be even more general by allowing the apparatus to be initially correlated with the system. We do not consider this situation because it removes an essential distinction between apparatus and system, namely that the former is in a fiducial state known to the experimenter, while the latter can be in an arbitrary state (perhaps known to a different experimenter).

This completes our formal description of quantum measurement theory. Note that the above formulae, from Eq. 139) to Eq. 147), are exact analogues of the classical formulae from Eq. 26) to Eq. 34). The most general formulation of classical measurement was achieved simply by adding back-action to Bayes’ theorem. 1. Quantum measurement theory as generalized Bayesian analysis Concept Quantum formula Bayesian formula Initial state such that Measurement result For each r define such that Pr[R = r] can be written as where Conditioned state Interpretation ρ(t), a positive operator Tr[ρ(t)] = 1 R, a random variable an operation Or ρ˜r (t + T ) = Or ρ(t) ≥ 0.

For s > t, this is no longer true. 129) r where ρr (t + T ) is the a-posteriori conditioned system state. 23 Convince yourself of this. It should be noted that these two descriptions of measurement, in terms of changing operators or changing states, have classical equivalents. They are descriptions in terms of changing system variables or changing probability distributions for these variables. 1. Specifically, we began with the ‘Heisenberg’ description, with correlations arising between system and apparatus variables, and then moved to the complementary ‘Schr¨odinger’ description with system state collapse derived using Bayes’ theorem.

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