For a stochastic process describing observations of a dynamical system, complexity science provides systematic methods to decompose the information produced into true irreducible randomness, and that which corresponds to structure superficially disguised as random but can in fact be learned and predicted. Such methods then equip vital tools for prediction, control and inference of the system's internal structure. However, quantum analogs remain underdeveloped due to the inherent complications of invasive measurements and quantum correlations. By harnessing the Choi state representation of process tensors which encode multi-time input-output relations, we formulate a convergent measure of irreducible randomness in stationary quantum stochastic processes. This directly enables a measure of temporal correlations which lower bounds the memory resources required to replicate the process via a recurrent quantum circuit.
For a stochastic process describing observations of a dynamical system, complexity science provides systematic methods to decompose the information produced into true irreducible randomness, and that which corresponds to structure superficially disguised as random but can in fact be learned and predicted. Such methods then equip vital tools for prediction, control and inference of the system's internal structure. However, quantum analogs remain underdeveloped due to the inherent complications of invasive measurements and quantum correlations. By harnessing the Choi state representation of process tensors which encode multi-time input-output relations, we formulate a convergent measure of irreducible randomness in stationary quantum stochastic processes. This directly enables a measure of temporal correlations which lower bounds the memory resources required to replicate the process via a recurrent quantum circuit.
Abstract If an experimentalist observes a sequence of emitted quantum states via either projective or positive-operator-valued measurements, the outcomes form a time series. Individual time series are realizations of a stochastic process over the measurements’ classical outcomes. We recently showed that, in general, the resulting stochastic process is highly complex in two specific senses: (i) it is inherently unpredictable to varying degrees that depend on measurement choice and (ii) optimal prediction requires using an infinite number of temporal features. Here, we identify the mechanism underlying this complicatedness as generator nonunifilarity—the degeneracy between sequences of generator states and sequences of measurement outcomes. This makes it possible to quantitatively explore the influence that measurement choice has on a quantum process’ degrees of randomness and structural complexity using recently introduced methods from ergodic theory. Progress in this, though, requires quantitative measures of structure and memory in observed time series. And, success requires accurate and efficient estimation algorithms that overcome the requirement to explicitly represent an infinite set of predictive features. We provide these metrics and associated algorithms, using them to design informationally-optimal measurements of open quantum dynamical systems.
What makes one quantum process more complex than another? We adopt an operational approach inspired by classical complexity science, which characterises structure through the memory required for causal generation. We develop a fully-quantum extension of computational mechanics for quantum space series: Correlated sequences of quantum outputs that do not return to interact with their generator. For stationary, memoryful quantum space series admitting finite-dimensional, primitive causal models, we show that any memory-minimal causal generator can be chosen to produce outputs isometrically, without a discarded environment. The resulting canonical matrix product state representation uniquely determines the irreducible memory required for generation. In particular, the minimum memory entropy is exactly the entanglement entropy across the past-future bipartition, giving entanglement a direct operational interpretation as irreducible memory for causal generation. The same structure determines the minimum memory dimension through the corresponding Schmidt rank. Within this class, we further show that the fundamental entropy rate vanishes, that causal generation is symmetric under time reversal, and that all memoryful processes possess infinite quantum Markov order with respect to arbitrary sharp measurements. These results establish an operational framework for quantifying intrinsic quantum structure, revealing deep connections between causal generation, complexity, memory, entropy, and entanglement.
Spatial and temporal correlations in quantum noise challenge the local Markovian models commonly used in quantum information processing. We develop a unified operational framework for quantifying temporal, spatial, and total spatiotemporal correlations in general quantum processes. Using process tensors, we define these quantities through optimal distinguishability from Markovian, spatially local, and fully uncorrelated processes. Optimization over admissible probing combs ensures monotonicity under every superprocess that preserves the corresponding free set, while Choi-state functionals and restricted probes provide accessible lower bounds. We establish hierarchy and interpolation relations among the correlation measures and derive dimension-dependent upper bounds on temporal correlations transmitted by quantum memories, with tighter bounds for classical memory, together with universal ceilings imposed by the system dimension. These bounds turn certified lower estimates into witnesses of minimum memory dimension, nonclassicality under a memory-dimension constraint, and inconsistencies in the assumed process model. We illustrate the results with effective superconducting-qubit models featuring ZZ and XY interactions, exhibiting saturation of classical-memory, quantum-memory, and system-dimension ceilings. These results pave the way toward practical approaches to characterizing and addressing spatiotemporally correlated errors in quantum devices.
Multi-time quantum processes are endowed with the same richness as multipartite states, including temporal entanglement and exotic causal structures. However, experimentally probing these rich phenomena leans heavily on fast and clean mid-circuit measurements, which are rarely available. We show here how surprisingly accessible these phenomena are in nascent quantum processors even when faced with substantially limited control. We work within the limitation where only unitary control is allowed, followed by a terminating measurement. Within this setting, we first develop a witness for genuine multi-time entanglement, and then methods to bound (from top and bottom) multi-time entanglement, non-Markovianity, purity, entropy, and other correlative measures. Our tools are designed to be implemented on quantum information processors, which we proceed to demonstrate. Finally, we discuss the limitations of these methods by testing them across random multi-time processes. Conceptually, this broadens our understanding of the extent to which temporal correlations may be determined with only deterministic control. Our techniques are pertinent to generic quantum stochastic dynamical processes, with a scope ranging across condensed matter physics, quantum biology, and in-depth diagnostics of NISQ-era quantum devices.
Understanding temporal processes and their correlations in time is of paramount importance for the development of near-term technologies that operate under realistic conditions. Capturing the complete multi-time statistics that define a stochastic process lies at the heart of any proper treatment of memory effects. This is well understood in classical theory, where a hierarchy of joint probability distributions completely characterizes the process at hand. However, attempting to generalize this notion to quantum mechanics is problematic: observing realizations of a quantum process necessarily disturbs the state of the system, breaking an implicit, and crucial, assumption in the classical setting. This issue can be overcome by separating the experimental interventions from the underlying process, enabling an unambiguous description of the process itself and accounting for all possible multi-time correlations for any choice of interrogating instruments. In this paper, using a novel framework for the characterization of quantum stochastic processes, we first solve the long standing question of unambiguously describing the memory length of a quantum processes. This is achieved by constructing a quantum Markov order condition, which naturally generalizes its classical counterpart for the quantification of finite-length memory effects. As measurements are inherently invasive in quantum mechanics, one has no choice but to define Markov order with respect to the interrogating instruments that are used to probe the process at hand: different memory effects are exhibited depending on how one addresses the system, in contrast to the standard classical setting. We then fully characterize the structural constraints imposed on quantum processes with finite Markov order, shedding light on a variety of memory effects that can arise through various examples. Finally, we introduce an instrument-specific notion of memory strength that allows for a meaningful quantification of the temporal correlations between the history and the future of a process for a given choice of experimental intervention. These findings are directly relevant to both characterizing and exploiting memory effects that persist for a finite duration. In particular, immediate applications range from developing efficient compression and recovery schemes for the description of quantum processes with memory to designing coherent control protocols that efficiently perform information-theoretic tasks, amongst a plethora of others.
Quantum combs are powerful conceptual tools for capturing multi-time processes in quantum information theory, constituting the most general quantum mechanical process. But, despite their causal nature, they lack a meaningful physical connection to time -- and are, by and large, arguably incompatible with it without extra structure. The subclass of quantum combs which assumes an underlying process is described by the so-called process tensor framework, which has been successfully used to study and characterise non-Markovian open quantum systems. But, although process tensors are motivated by an underlying dynamics, it is not a priori clear how to connect them to a continuous process tensor object mathematically -- leaving an uncomfortable conceptual gap. In this work, we take a decisive step toward remedying this situation. We introduce a fully continuous process tensor framework by showing how the discrete multi-partite Choi matrix becomes a vector in bosonic Fock space, which is intrinsically and rigorously defined in the continuum. With this equipped, we lay out the core structural elements of this framework and its properties. This translation allows for an information-theoretic treatment of multi-time correlations in the continuum via the analysis of their continuous matrix product state representatives. Our work closes a gap in the quantum information literature, and opens up the opportunity for the application of many-body physics insights to our understanding of quantum stochastic processes in the continuum.