Quantum Coherence and Decoherence

31 Recently, Aharonov, Anandan and Vaidman [1,2] showed t hat the wavefunction of a si ngle quantum system could be determined from t he resu lts of a series of "protective measurements" performed on the system. In t he protective measurement scheme, a-prio ri knowledge of the wavefundion of the system is used in order to measure t his system and protect its wavefunction from changing at the same time. Aharonov I Anandan and Vaidman argued that the protective measurement accounts for the physical reality of the wavefun ction. Yet, it seems thai one shou ld be able to measu re the wavefundion of a single system without any arprior i knowledge, if the wavefunction were rea l. According to the projection postulate, a precise measurement of a single system would always yield one of the eigenvalues of the measured observable. This eigenvalue can be used to estimate the expectation value of the observable. The uncertainty of t he measured observable could never be estimated using the sin gle measw'ement result. This uncertainty cannot be estimated even if we use the results of ad ditional measurements of t he si ngle system. After the measurement, the wavefunction of the system coll apses to the eigenstate, which corres ponds to the measured eigenvalue, and the results of additional measurements would not add any information about the initial wavefundion of the system. No~, consider a very weak measurement of the si ngle system. This meas urement would leave the wavefu ndion of the system almost unchanged. Consecutive weak measurements of the same system, t herefore, would give us some addit ional in fo rmation about the initial wavefunction of t he system. Since t he meas ured system is approximately in the same state when the different measuremen ts are performed, one may expect the statistics of the measurement results to be approximately t he same as the results of an ensemble measurement, where a single measurement is performed on each system in an ensemble of identical systems. Specifically, one may expect the statistics of t hese meas urement resu lts to enable us to estimate the uncertainties of t he measured observables with finite estimate errors, where the abili ty to estimate the uncertainty of a specific observable di stinguis hes a measurement of the wavefunction from a measurement of the observable. In this work, we show that this intuitive picture …