Two things have long bothered me about the way we discuss quantum non-Markovianity. First, that the concept fails to exclude classical processes, so that the use of the word “quantum” seems unjustified. Second, and related to the first but somewhat more puzzling, that a mere mixture of two Markovian processes can appear non-Markovian. Such a non-Markovianity surely shouldn’t be taken too seriously. This disconnect has persisted despite years of work, subtly undermining the foundations of our understanding.

In our new paper, Causal and Noncausal Revivals of Information: A New Regime of Non-Markovianity in Quantum Stochastic Processes, just published in PRX Quantum (open access) together with Rajeev Gangwar, Kaumudibikash Goswami, Himanshu Badhani, Tanmoy Pandit, Brij Mohan, Siddhartha Das, and Manabendra Nath Bera, we finally provide a satisfying resolution. By distinguishing between causal and noncausal information revivals, we create a conceptual and operational framework that clarifies what non-Markovianity in quantum systems really means.

The Puzzle of Mixtures and Backflows

Let’s begin with the basics. A process is Markovian if its future evolution depends only on the present state, not the past. In quantum dynamics, this notion gets murky when the system is not isolated but interacts with an environment. Textbooks typically assume that the system and its environment start uncorrelated: a neat assumption, but physically unrealistic. Initial correlations are inevitable, especially in strongly coupled systems. (In fact, the first post on this blog was about this very topic.)

This led to a long line of research investigating whether quantum channels can still describe the evolution of a subsystem even when initial correlations are present. While various conditions and formalisms were proposed (notably assignment maps and quantum discord), none satisfactorily addressed the oddity that a convex combination of Markovian processes can appear non-Markovian. This is particularly troubling if we hope to construct a resource theory of non-Markovianity, where convexity is essential.

Revivals Are Not Always Backflows

Our central insight is that information revivals, which we model as increases in the mutual information between a system and an ancillary reference after an interaction with the environment, can come in two flavors: causal and noncausal. Causal revivals correspond to genuine backflows of information from the environment to the system. Noncausal revivals, however, can be explained entirely using degrees of freedom that never interacted with the system at all.

This subtle distinction allows us to separate apparent non-Markovianity (which can be faked by a clever model with hidden correlations) from genuinely quantum non-Markovianity (which requires a true causal connection).

To formalize this, we bring in tools from quantum information theory: squashed entanglement, conditional mutual information, and Petz’s theory of statistical sufficiency. We prove that if a revival can be explained without violating the data-processing inequality for some inert extension (i.e., hidden, non-interacting degrees of freedom), then it is noncausal. In contrast, causal revivals must involve a flow of information from the environment.

Why This Matters

Distinguishing causal from noncausal revivals does more than just clean up conceptual confusion. It has direct implications for experiments and applications. For instance, we provide operational conditions, depending only on system observables, to detect genuine information backflow. This opens the door to device-independent verification of quantum memory effects.

Moreover, we show that anything that is not genuine non-Markovianity, thus including noncausal revivals, forms a convex set. In other words, if you take the convex mixture of processes without genuine non-Markovianity, you get another process without genuine non-Markovianity: genuine non-Markovianity, unlike conventional non-Markovianity, cannot simply be generated by classical randomization. This is a very welcome feature: convexity is a cornerstone of resource theories, and it means that we can now meaningfully talk about genuine non-Markovianity as a resource in quantum information processing.