Protective and State Measurement: A Review

In this paper, I shall summarize the general way in which the measurement process can be cast (by making use of effects and amplitude operators). Then, I shall show that there are two main problems with the state measurement: (i) how to avoid the disruptive back action on the state of the measured system during detection and (ii) how to extract a complete information from this state. In order to deal with them I shall first introduce quantum non-demolition (QND) measurement and examine the problem whether the entire probability distribution of the measured observable is not altered by a QND measurement, what would allow repeated QND measure- ments with different observables to extract the whole information from the mea- sured system. However, I shall show that this is not the case. Then, I shall deal with protective measurement as such and show that a reversible measurement is in fact not a measurement. However, by taking advantage of statistical methods (and therefore by renouncing to measure the state of a single system), we can indeed reconstruct the wave function but only by partially recovering the information con- tained in the state. Two further prices to pay are to admit the existence of negative quasi–probabilities due to the interference terms and to make use of unsharp ob- servbables for guaranteeing informational completeness.