When an experimentalist measures a time series of qubits, the outcomes constitute a classical stochastic process. We show that projective measurement induces high complexity in these processes in two specific senses: They are inherently random (finite Shannon entropy rate) and they require infinite memory for optimal prediction (divergent statistical complexity). We identify nonorthogonality of the quantum states as the mechanism underlying the resulting complexities and examine the influence that measurement choice has on the randomness and structure of measured qubit processes. We introduce quantitative measures of this complexity and provide efficient algorithms for their estimation.
When an experimentalist measures a time series of qubits, the outcomes constitute a classical stochastic process. We show that projective measurement induces high complexity in these processes in two specific senses: They are inherently random (finite Shannon entropy rate) and they require infinite memory for optimal prediction (divergent statistical complexity). We identify nonorthogonality of the quantum states as the mechanism underlying the resulting complexities and examine the influence that measurement choice has on the randomness and structure of measured qubit processes. We introduce quantitative measures of this complexity and provide efficient algorithms for their estimation.
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.
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.
The impact of projective measurements on a many-body quantum state is tightly linked to its underlying entanglement structure. Critical states are particularly sensitive, as long-range entanglement allows local measurements to have global consequences. This has been extensively studied in the context of critical states described by a conformal field theory (CFT), where measurements can alter critical properties in post-measurement quantities ('measurement-altered criticality') and induce long-range entanglement with universal features ('measurement-induced entanglement'). By contrast, little is known about the role of measurements on critical states with quenched disorder, which admit no CFT description. Here, we develop a theory of measurements on one-dimensional critical states governed by infinite-randomness fixed points (IRFPs), focusing on two closely related examples: the random XXZ chain and the random transverse-field Ising model. We show that the measurement-induced entanglement decays as a power-law with universal exponent (3−5)/2 in both models, and that the minimum number of measurements required to establish this entanglement obeys a universal scaling form. When only a finite density of measurements is made, we show that the critical properties of the state are remarkably robust: `measurement-altered criticality' is absent, with measurements either preserving the critical exponents, or destroying the criticality entirely. These results establish IRFPs as an analytically tractable arena for measurements on critical states, complementary to the CFT case. The outcome randomness that necessitates replica methods in clean systems becomes trivial at infinite randomness, with the universal response governed by the statistics induced by the quenched disorder.
The generation of high-quality random numbers is a cornerstone of numerous fields, including cryptography, simulation, and statistical modeling. Classical methods for random number generation often rely on deterministic algorithms that, despite their complexity, ultimately produce predictable sequences. This paper proposes a novel approach to random number generation that harnesses the unique properties of quantum mechanics, specifically quantum measurement. We argue that classical algorithms are fundamentally limited in their ability to produce true randomness. Our approach utilizes quantum measurement processes, projecting a qubit into a superposition state, to generate random numbers. The resulting sequences exhibit statistical properties superior to those attainable through classical methods, offering a potentially significant advancement in the field of randomness generation. We detail the theoretical underpinnings, outlining the core mechanism and highlighting the key advantages. This work establishes a framework for leveraging quantum phenomena to achieve truly random number generation. ---
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.
Understanding the emergence of the thermodynamic arrow of time in microscopic systems is of fundamental importance, particularly given that unitary evolution preserves time-reversal symmetry. While projective measurements introduce temporal irreversibility, identifying this asymmetry from single evolution trajectories in the presence of stochastic fluctuations presents a considerable challenge. Here, we harness machine learning to identify the arrow of time from individual trajectories generated by a programmable ten-qubit quantum processor based on a nitrogen-vacancy center in diamond. We implement quantum circuits that realize unitary evolutions where heat flows from hotter to colder subsystems and their time-reversed counterparts. Projective measurements inserted in these processes induce entropy production, and their outcomes constitute the evolution trajectory. We demonstrate that an unsupervised clustering algorithm autonomously divides the experimental trajectories into two distinct groups without prior knowledge, while a convolutional neural network identifies the temporal direction of these trajectories with approximately 92% accuracy. In addition, we show that a diffusion-based generative model reproduces essential signatures of directional energy flow and entropy production. Our results establish machine learning as a powerful tool for uncovering underlying physical processes from complex experimental data, advancing the interface between quantum thermodynamics and artificial intelligence.