Expectation values of measurement operators, interpreted as measurement probabilities, arise frequently throughout quantum information sciences. When quantum states are randomly distributed, their expectation values are also randomly distributed. In this work, with the goal of understanding nonunitary dynamics, we generalize previous derivations for distributions of expectation values [Campos et al., Phys. Lett. A 377, 1854 (2013)] to the case of sets of measurement operators and random mixed quantum states within variable sized environments. Using combinatorics approaches, we derive expressions for their moments. We proceed to construct empirical distributions of simulated Haar random brickwork quantum circuits with local depolarizing noise, and compare their form to a proposed effective global-depolarizing-like model with variable effective noise scales and environment dimensions. The fitted effective distributions reproduce peak behavior across circuit depths, noise scales, and system sizes, while deviations in the distribution tails arise from local noise effects. The fit effective model parameters are also shown to vary smoothly and consistently with circuit depth and noise scale. Finally, sets of nonsymmetric measurement operators are shown to exhibit distinct multimodal distributions relative to unimodal distributions for symmetric measurement operators, opening up questions about their simulability.

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• Quantum circuits
• Quantum correlations in quantum information
• Quantum information theory