Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2104.02467 · Quantum

Temporal correlations in the simplest measurement sequences

Lucas B. Vieira, Costantino Budroni

Published Jan 18, 2022• 15 citations
arXivHTMLPDF

Abstract

We investigate temporal correlations in the simplest measurement scenario, i.e., that of a physical system on which the same measurement is performed at different times, producing a sequence of dichotomic outcomes. The resource for generating such sequences is the internal dimension, or memory , of the system. We characterize the minimum memory requirements for sequences to be obtained deterministically, and numerically investigate the probabilistic behavior below this memory threshold, in both classical and quantum scenarios. A particular class of sequences is found to offer an upper-bound for all other sequences, which suggests a nontrivial universal upper-bound of 1/e for the classical probability of realization of any sequence below this memory threshold. We further present evidence that no such nontrivial bound exists in the quantum case.

DOI
10.22331/q-2022-01-18-623

Related papers

Memory cost of temporal correlations

Costantino Budroni, Gabriel Fagundes, Matthias Kleinmann

Abstract A possible notion of nonclassicality for single systems can be defined on the basis of the notion of memory cost of classically simulating probabilities observed in a temporal sequence of measurements. We further explore this idea in a theory-independent framework, namely, from the perspective of general probability theories (GPTs), which includes classical and quantum theory as special examples. Under the assumption that each system has a finite memory capacity, identified with the maximal number of states perfectly distinguishable with a single measurement, we investigate what are the temporal correlations achievable with different theories, namely, classical, quantum, and GPTs beyond quantum mechanics. Already for the simplest nontrivial scenario, we derive inequalities able to distinguish temporal correlations where the underlying system is classical, quantum, or more general.

arXiv:1902.06517•New Journal of Physics•Aug 19, 2019•25 citations

Structure of temporal correlations of a qubit

Jannik Hoffmann, Cornelia Spee, Otfried Gühne +1 more

In quantum mechanics, spatial correlations arising from measurements at separated particles are well studied. This is not the case, however, for the temporal correlations arising from a single quantum system subjected to a sequence of generalized measurements. We first characterize the polytope of temporal quantum correlations coming from the most general measurements. We then show that if the dimension of the quantum system is bounded, only a subset of the most general correlations can be realized and identify the correlations in the simplest scenario that can not be reached by two-dimensional systems. This leads to a temporal inequality for a dimension test, and we discuss a possible implementation using nitrogen-vacancy centers in diamond.

arXiv:1712.01234•New Journal of Physics•Oct 15, 2018•40 citations

Witnessing environment dimension through temporal correlations

Lucas B. Vieira, Simon Milz, Giuseppe Vitagliano +1 more

We introduce a framework to compute upper bounds for temporal correlations achievable in open quantum system dynamics, obtained by repeated measurements on the system. As these correlations arise by virtue of the environment acting as a memory resource, such bounds are witnesses for the minimal dimension of an effective environment compatible with the observed statistics. These witnesses are derived from a hierarchy of semidefinite programs with guaranteed asymptotic convergence. We compute non-trivial bounds for various sequences involving a qubit system and a qubit environment, and compare the results to the best known quantum strategies producing the same outcome sequences. Our results provide a numerically tractable method to determine bounds on multi-time probability distributions in open quantum system dynamics and allow for the witnessing of effective environment dimensions through probing of the system alone.

arXiv:2305.19175•Quantum•Jan 10, 2024•2 citations

Memory cost for simulating all quantum correlations from the Peres–Mermin scenario

Gabriel Fagundes, Matthias Kleinmann

Abstract Sequences of compatible quantum measurements can be contextual and any simulation with a classical model conforming with the quantum predictions needs to use internal memory. Kleinmann et al (2011 New J. Phys . 13 113011) showed that simulating sequences from the Peres–Mermin scenario requires at least three different internal states in order not to contradict the deterministic predictions of quantum theory. We extend this analysis to probabilistic quantum predictions and ask how much memory is required to simulate correlations generated for sequences of compatible observables by any quantum state. We find that even in this comprehensive approach only three internal states are required for a perfect simulation of quantum correlations in the Peres–Mermin scenario.

arXiv:1611.07515•Journal of Physics A Mathematical and Theoretical•Jun 21, 2017•23 citations

Simulating extremal temporal correlations

Cornelia Spee, Costantino Budroni, Otfried Gühne

Abstract The correlations arising from sequential measurements on a single quantum system form a polytope. This is defined by the arrow-of-time (AoT) constraints, meaning that future choices of measurement settings cannot influence past outcomes. We discuss the resources needed to simulate the extreme points of the AoT polytope, where resources are quantified in terms of the minimal dimension, or ‘internal memory’ of the physical system. First, we analyze the equivalence classes of the extreme points under symmetries. Second, we characterize the minimal dimension necessary to obtain a given extreme point of the AoT polytope, including a lower scaling bound in the asymptotic limit of long sequences. Finally, we present a general method to derive dimension-sensitive temporal inequalities for longer sequences, based on inequalities for shorter ones, and investigate their robustness to imperfections.

arXiv:2004.14854•New Journal of Physics•Sep 15, 2020•13 citations

Causal memory and structural complexity of correlated quantum sequences

Thomas J. Elliott

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.

quant-pharXiv:2610.06519•Oct 6, 2026•0 citations
We investigate temporal correlations in the simplest measurement scenario, i.e., that of a physical system on which the same measurement is performed at different times, producing a sequence of dichotomic outcomes. The resource for generating such sequences is the internal dimension, or memory , of the system. We characterize the minimum memory requirements for sequences to be obtained deterministically, and numerically investigate the probabilistic behavior below this memory threshold, in both classical and quantum scenarios. A particular class of sequences is found to offer an upper-bound for all other sequences, which suggests a nontrivial universal upper-bound of 1/e for the classical probability of realization of any sequence below this memory threshold. We further present evidence that no such nontrivial bound exists in the quantum case.
Abstract A possible notion of nonclassicality for single systems can be defined on the basis of the notion of memory cost of classically simulating probabilities observed in a temporal sequence of measurements. We further explore this idea in a theory-independent framework, namely, from the perspective of general probability theories (GPTs), which includes classical and quantum theory as special examples. Under the assumption that each system has a finite memory capacity, identified with the maximal number of states perfectly distinguishable with a single measurement, we investigate what are the temporal correlations achievable with different theories, namely, classical, quantum, and GPTs beyond quantum mechanics. Already for the simplest nontrivial scenario, we derive inequalities able to distinguish temporal correlations where the underlying system is classical, quantum, or more general.
In quantum mechanics, spatial correlations arising from measurements at separated particles are well studied. This is not the case, however, for the temporal correlations arising from a single quantum system subjected to a sequence of generalized measurements. We first characterize the polytope of temporal quantum correlations coming from the most general measurements. We then show that if the dimension of the quantum system is bounded, only a subset of the most general correlations can be realized and identify the correlations in the simplest scenario that can not be reached by two-dimensional systems. This leads to a temporal inequality for a dimension test, and we discuss a possible implementation using nitrogen-vacancy centers in diamond.
We introduce a framework to compute upper bounds for temporal correlations achievable in open quantum system dynamics, obtained by repeated measurements on the system. As these correlations arise by virtue of the environment acting as a memory resource, such bounds are witnesses for the minimal dimension of an effective environment compatible with the observed statistics. These witnesses are derived from a hierarchy of semidefinite programs with guaranteed asymptotic convergence. We compute non-trivial bounds for various sequences involving a qubit system and a qubit environment, and compare the results to the best known quantum strategies producing the same outcome sequences. Our results provide a numerically tractable method to determine bounds on multi-time probability distributions in open quantum system dynamics and allow for the witnessing of effective environment dimensions through probing of the system alone.
Abstract Sequences of compatible quantum measurements can be contextual and any simulation with a classical model conforming with the quantum predictions needs to use internal memory. Kleinmann et al (2011 New J. Phys . 13 113011) showed that simulating sequences from the Peres–Mermin scenario requires at least three different internal states in order not to contradict the deterministic predictions of quantum theory. We extend this analysis to probabilistic quantum predictions and ask how much memory is required to simulate correlations generated for sequences of compatible observables by any quantum state. We find that even in this comprehensive approach only three internal states are required for a perfect simulation of quantum correlations in the Peres–Mermin scenario.
Abstract The correlations arising from sequential measurements on a single quantum system form a polytope. This is defined by the arrow-of-time (AoT) constraints, meaning that future choices of measurement settings cannot influence past outcomes. We discuss the resources needed to simulate the extreme points of the AoT polytope, where resources are quantified in terms of the minimal dimension, or ‘internal memory’ of the physical system. First, we analyze the equivalence classes of the extreme points under symmetries. Second, we characterize the minimal dimension necessary to obtain a given extreme point of the AoT polytope, including a lower scaling bound in the asymptotic limit of long sequences. Finally, we present a general method to derive dimension-sensitive temporal inequalities for longer sequences, based on inequalities for shorter ones, and investigate their robustness to imperfections.
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.