Algorithmic Randomness and Kolmogorov Complexity for Qubits

Abstract

Nies and Scholz defined quantum Martin-L\"of randomness (q-MLR) for states (infinite qubitstrings). We define a notion of quantum Solovay randomness and show it to be equivalent to q-MLR using purely linear algebraic methods. Quantum Schnorr randomness is then introduced. A quantum analogue of the law of large numbers is shown to hold for quantum Schnorr random states. We introduce quantum-K, ($QK$) a measure of the descriptive complexity of density matrices using classical prefix-free Turing machines and show that the initial segments of weak Solovay random and quantum Schnorr random states are incompressible in the sense of $QK$. Several connections between Solovay randomness and $K$ carry over to those between weak Solovay randomness and $QK$. We then define $QK_C$, using computable measure machines and connect it to quantum Schnorr randomness. We then explore a notion of `measuring' a state. We formalize how `measurement' of a state induces a probability measure on the space of infinite bitstrings. A state is `measurement random' ($mR$) if the measure induced by it, under any computable basis, assigns probability one to the set of Martin-L\"of randoms. I.e., measuring a $mR$ state produces a Martin-L\"of random bitstring almost surely. While quantum-Martin-L\"of random states are $mR$, the converse fails: there is a $mR$ state, $\rho$ which is not quantum-Martin-L\"of random. In fact, something stronger is true. While $\rho$ is computable and can be easily constructed, measuring it in any computable basis yields an arithmetically random sequence with probability one. So, classical randomness can be generated from a computable state which is not quantum random. We conclude by studying the asymptotic von Neumann entropy of computable states.