Quantum Martin-Löf randomness (q-MLR) for infinite qubit sequences was introduced by Nies and Scholz [J. Math. Phys. 60(9), 092201 (2019)]. We define a notion of quantum Solovay randomness, which is equivalent to q-MLR. The proof of this goes through a purely linear algebraic result about approximating density matrices by subspaces. We then show that random states form a convex set. Martin-Löf absolute continuity is shown to be a special case of q-MLR. Quantum Schnorr randomness is introduced. A quantum analog of the law of large numbers is shown to hold for quantum Schnorr random states.

中文翻译：

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量子算法随机性

Nies和Scholz提出了无限量子位序列的量子马丁-洛夫随机性（q-MLR）[J. 数学。物理 60（9），092201（2019）]。我们定义了量子Solovay随机性的概念，它等同于q-MLR。证明是通过纯线性代数结果得出的，该结果关于用子空间近似密度矩阵。然后，我们证明随机状态形成一个凸集。Martin-Löf的绝对连续性被证明是q-MLR的特例。介绍了量子Schnorr随机性。大数定律的量子模拟被证明对量子Schnorr随机态成立。