Reducing the Statistical Axiom

In support of a recent argument by Deutsch (quant-ph/9906015) an arrangement of “quantum dots” and “channels” is specified which does not anticipate a unitary time evolution, yet is able to transform square-root amplitudes into an integer number of equal amplitudes and, hence, equal probabilities. Exploited are linearity of the time evolution and permutational symmetry. Further, it is postulated that none of the state vectors tend to zero at infinite times. PACS numbers: 03.65.Bz In a recent paper, Deutsch [1] provided a remarkably simple argument to infer from only “non-probabilistic” axioms of quantum mechanics that in a superposition state of the form |ψ〉 = √ m n |A〉+ √ n−m n |B〉 (1) the probabilities for detecting A or B are m/n and 1 − m/n, respectively. By contrast, this assignment of probabilities is usually regarded as an independent Statistical Axiom. Deutsch’s argument consists of two steps. First, using concepts of decision theory, one derives that in a superposition with equal amplitudes the probabilities will be equal, too. Second, one reduces the more general case (1) to the equal-amplitude case by coupling the 2-state system to an n-state system and performing a suitable unitary transformation which preserves the values of A and B: |A〉 → |A〉|0〉 U −→ √