dorsal/arxiv
View SchemaUpper bound by Kolmogorov complexity for the probability in computable POVM measurement
| Authors | Kohtaro Tadaki |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0212071 |
| URL | https://arxiv.org/abs/quant-ph/0212071 |
Abstract
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.
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"abstract": "We apply algorithmic information theory to quantum mechanics in order to shed\nlight on an algorithmic structure which inheres in quantum mechanics.\n There are two equivalent ways to define the (classical) Kolmogorov complexity\nK(s) of a given classical finite binary string s. In the standard way, K(s) is\ndefined as the length of the shortest input string for the universal\nself-delimiting Turing machine to output s. In the other way, we first\nintroduce the so-called universal probability m, and then define K(s) as -log_2\nm(s) without using the concept of program-size. We generalize the universal\nprobability to a matrix-valued function, and identify this function with a POVM\n(positive operator-valued measure). On the basis of this identification, we\nstudy a computable POVM measurement with countable measurement outcomes\nperformed upon a finite dimensional quantum system. We show that, up to a\nmultiplicative constant, 2^{-K(s)} is the upper bound for the probability of\neach measurement outcome s in such a POVM measurement. In what follows, the\nupper bound 2^{-K(s)} is shown to be optimal in a certain sense.",
"arxiv_id": "quant-ph/0212071",
"authors": [
"Kohtaro Tadaki"
],
"categories": [
"quant-ph",
"cs.CC"
],
"title": "Upper bound by Kolmogorov complexity for the probability in computable POVM measurement",
"url": "https://arxiv.org/abs/quant-ph/0212071"
},
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