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Series on the foundations of natural science and technology

Determinism and Probability

Published Nov 1, 2025• 0 citations
Publisher
DOI
10.1142/9789819820139_0007

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A refinement of Shor's Algorithm for determining order is introduced, which determines a divisor of the order after any one run of a quantum computer with almost absolute certainty. The information garnered from each run is accumulated to determine the order, and for any k greater than 1, there is a guaranteed minimum positive probability that the order will be determined after at most k runs. The probability of determination of the order after at most k runs exponentially approaches a value negligibly less than one, so that the accumulated information determines the order with almost absolute certainty. The probability of determining the order after at most two runs is more than 60%, and the probability of determining the order after at most four runs is more than 90%.
A question raised by Pauli as to whether or not the probability densities in space and momentum determine the wave function is answered negatively. The assertion by Kemble that the probability density and its time derivative determine the wave function is shown to be not generally true. It is shown that measurement of the probability density and the probability current determine the wave function of a spinless particle. The measurement of the probability current is discussed. Measurements which determine the spin state and the density matrix of a mixture are also considered.
We find a probability of determining an unknown quantum state if some information about it is known.Namely, we know that the state is one of N quantum states.These states are represented as eigenstates of operator of projection of a spin on N directions, corresponding to positive eigenvalues.The probability of determining the unknown state with the smallest number of measurements and for some number of measurements n is obtained.Dependence of these probabilities on the angles between the vectors defining the N directions is studied.