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 $QK$. Many properties enjoyed by prefix-free Kolmogorov complexity ($K$) have analogous versions for $QK$; notably a counting condition. Several connections between Solovay randomness and $K$, including the Chaitin type characterization of Solovay randomness, carry over to those between weak Solovay randomness and $QK$. We work towards a Levin-Schnorr type characterization of weak Solovay randomness in terms of $QK$. Schnorr randomness has a Levin-Schnorr characterization using $K_C$; a version of $K$ using a computable measure machine, $C$. We similarly define $QK_C$, a version of $QK$. Quantum Schnorr randomness is shown to have a Levin-Schnorr and a Chaitin type characterization using $QK_C$. The latter implies a Chaitin type characterization of classical Schnorr randomness using $K_C$.

翻译：我们引入了量子-K(QQ$),这是用古典无前缀图灵机器测量密度矩阵描述复杂性的一种尺度,它表明弱索洛维随机和量子Schnorr随机状态的初始部分在美元意义上是不可压缩的。许多无前缀的科尔莫戈罗夫复杂状态(K$)享有的属性相似的版本,其价值为美元;特别是一个计算条件。索洛维随机性和美元之间的若干连接,包括沙丁型索洛瓦随机性特征的描述,结转到弱索洛瓦随机性与美元之间的部分。我们致力于对弱索洛瓦随机性的利文-史诺尔类型描述。Schnorr随机性具有使用美元Levin-Schnologr定性的类似版本;Schnorr Revin-Schnoronical 的版本,其价值为$C。我们同样地定义了QK$-C,该版本是Q. Qantumtum Schnorr 随机性,使用“卡纳”的卡纳克兰-卡纳(C)类型。