Temporal correlations in the simplest measurement sequences

TL;DR

This paper explores the minimal memory needed for a physical system to generate specific measurement sequences over time, comparing classical and quantum systems and revealing potential universal bounds.

Contribution

It characterizes the minimal memory requirements for deterministic sequences and investigates probabilistic behaviors, proposing a universal bound in classical systems and contrasting it with quantum cases.

Findings

Classical systems have a nontrivial universal upper-bound of 1/e for sequence realization below the memory threshold.

Quantum systems do not seem to have such a universal bound.

A specific class of sequences sets an upper limit for all sequences in the classical case.

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.