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QUANTUM MEASUREMENTS AND ALGORITHMIC RANDOMNESS
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DOI：10.1017/jsl.2026.10191.png)
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Abstract
En 中文
Nies and Scholz formalized the notion of an infinite qubitstring and referred to it as a 'state'. They defined 'quantum Martin-L & ouml;f randomness' for states. We give a notion of measurement of a state in a computable basis and introduce 'quantum measurement randomness', a randomness notion for states. A state is quantum measurement random if measuring it in any computable basis yields a Martin-L & ouml;f random bitstring with probability one. Our main result is that quantum Martin-L & ouml;f randomness strictly implies quantum measurement randomness. This uses the construction of a quantum measurement random state which is not quantum Martin-L & ouml;f random. We prove two general results on which this construction relies: The first concerns Martin-L & ouml;f randomness relative to computable measures and extends a result of V. Vovk. The second is a combinatorial result about Kronecker products.
Keywords:
Martin-L & ouml
f randomness
Kolmogorov complexity
qubit
density matrix
projective measurement