Connecting commutativity and classicality for multitime quantum processes

Understanding the demarcation line between classical and quantum is an important issue in modern physics. The development of such an understanding requires a clear picture of the various concurrent notions of “classicality” in quantum theory presently in use. Here we focus on the relationship between Kolmogorov consistency of measurement statistics—the foundational footing of classical stochastic processes in standard probability theory—and the commutativity (or absence thereof) of measurement operators—a concept at the core of quantum theory. Kolmogorov consistency implies that the statistics of sequential measurements on a (possibly quantum) system could be explained entirely by means of a classical stochastic process, thereby providing an operational notion of classicality. On the other hand, commutativity of measurement operators is a structural property that holds in classical physics and its breakdown is the origin of the uncertainty principle, a fundamentally quantum phenomenon. We formalize the connection between these two a priori independent notions of classicality, demonstrate that they are distinct in general and detail their implications for memoryless multitime quantum processes.

Figure 1

Schematics of physical processes considered in this article. An unknown process (turquoise) is probed at different times, here $\{t_1,t_2,t_3,t_4\}$, and the resulting joint probability distribution $\mathbb{P}(m_4,m_3,m_2,m_1)$ is recorded (a). From the four-time distribution obtained, one could compute a three-time joint distribution by marginalizing over the outcomes at a given time (b) We depict the case in which the outcomes at time $t_2$ are marginalized over, yielding the probability distribution ${\sum }_{m_2}\mathbb{P}(m_4,m_3,m_2,m_1)$ for times $t_1,t_3$, and $t_4$. In contrast, the three-time joint probability $\mathbb{P}(m_4,m_3,m_2/,m_1),$ as shown in (c), is obtained by not performing a measurement at $t_2$. In general, this is a different experiment than the one of (b) and therefore leads to a different probability distribution. However, if the process is classical—and the employed measurements are noninvasive—then the joint probabilities of the experiments depicted in (b) and (c) coincide [see Eq. (10)].

Figure 2

Multitime probing of a Markovian quantum process. A quantum process without memory on a discrete set of times $t_1,\cdots ,t_n$ can be described by an initial state $\rho$ of the system and a collection of independent CPTP maps $\left\{{\mathrm{\Lambda }}_{i:i-1}\right\}$ that act on the system alone between neighboring times (blue). At each time $t_i$, we envisage an agent probing the process and observing a measurement outcome $m_i$, with the postmeasurement state feeding forward (yellow). Such a probing is represented by a CP map ${\mathcal{K}}_i^{\left(m_i\right)}$ at each time.