Kolmogorov complexity upper bound of probability in computable POVM measurement (New Aspects of Theoretical Computer Science)
Kohtaro Tadaki · Institutional Repositories DataBase (IRDB) · 2003
We apply algorithmic information theory to quantum mechanics in order to shed light on an algorithmic structure which is inherent in quantum mechanics.There are two equivalent ways to define the (classical) Kolmogorov $\infty \mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}\mathrm{x}\mathrm{i}\mathrm{t}\mathrm{y}$ $K(s)$ of agiven classical finite binary string $s$ .In the standard way, $K(s)$ is defined as the length of the shortest input string for the universal self-delimiting Turing machine to output $s$ .In the other way, we first introduce the so-called universal probability $m$ , and then define $K(s)$ as $-\log_{2}m(s)$ without using the concept of program-size.We generalize the universal probability to amatrix-valued function, and identify this function with apositive operator-valued measure (POVM), which describes the statistics of outcomes in aquantum measurement in the general set- ting.Based on this identification, we study acomputable POVM measurement with countable measurement outcomes performed upon afinite dimensional quantum system.We show that, up to amultiplicative constant, $2^{-K(s)}$ is the upper bound for the probability of each measurement outcome $s$ in such aquantum measurement.In what follows, the upper bound $2^{-K(s)}$ is shown to be optimal in acertain sense.