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.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.

May 1, 20030 citations

A Kolmogorov Extension Theorem for POVMs

Roderich Tumulka

arXiv:0710.3605Letters in Mathematical PhysicsFeb 26, 200812 citations

Upper bound by Kolmogorov complexity for the probability in computable POVM measurement

Kohtaro Tadaki

We apply algorithmic information theory to quantum mechanics in order to shed light on an algorithmic structure which inheres in quantum mechanics. There are two equivalent ways to define the (classical) Kolmogorov complexity K(s) of a given 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 a matrix-valued function, and identify this function with a POVM (positive operator-valued measure). On the basis of this identification, we study a computable POVM measurement with countable measurement outcomes performed upon a finite dimensional quantum system. We show that, up to a multiplicative constant, 2^{-K(s)} is the upper bound for the probability of each measurement outcome s in such a POVM measurement. In what follows, the upper bound 2^{-K(s)} is shown to be optimal in a certain sense.We apply algorithmic information theory to quantum mechanics in order to shed light on an algorithmic structure which inheres in quantum mechanics. There are two equivalent ways to define the (classical) Kolmogorov complexity K(s) of a given 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 a matrix-valued function, and identify this function with a POVM (positive operator-valued measure). On the basis of this identification, we study a computable POVM measurement with countable measurement outcomes performed upon a finite dimensional quantum system. We show that, up to a multiplicative constant, 2^{-K(s)} is the upper bound for the probability of each measurement outcome s in such a POVM measurement. In what follows, the upper bound 2^{-K(s)} is shown to be optimal in a certain sense.

arXiv:quant-ph/0212071Dec 11, 20026 citations

Noise-Tolerant Quantum Oracles (計算機科学基礎理論の新展開研究集会報告集)

レイモンド・H・プテラ, 一雄岩間, 茂山下

数理解析研究所講究録May 1, 20030 citations

\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)

-\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.