In quantum mechanics, quantum observables, like position, momentum, energy, and spin, are in general non-commutative, meaning that their measurement order may affect the outcome. As a result, it is generally impossible to measure them simultaneously. Consequently, their joint probability distributions cannot be defined, limiting a complete understanding of quantum systems.

There have been many attempts to address this limitation through their quantum analogues called quasi-joint probability distributions, or quasi-probability distributions for short, which extend probability distributions to the negative or the complex domain, where probabilities can take negative or complex values. However, research on quasi-probability distributions of generic quantum systems beyond the position-momentum pair remains limited.

Addressing this gap, a new study published in Progress of Theoretical and Experimental Physics investigated the properties of all possible quasi-probability distributions in finite-state systems by utilizing the general framework of quasi-probability distributions developed by Lee and Tsutsui. The study especially assessed their properties in two and three-state systems for arbitrary combinations of quantum observables based on two criteria.

The first criterion evaluates the distinguishability of quantum states by quasi-probability distributions using the minimum number of observables. On the other hand, the second criterion assesses their affinity to traditional quantum distributions. This refers to quasi-probability distributions with non-zero values beyond the possible observable values, thus deviating from traditional distributions.

The study found that the Kirkwood-Dirac distribution is particularly favorable over others. Specifically, it requires only two, instead of three, observables to distinguish all quantum states in two-state systems. Their imaginary nature plays a key role in this regard. Moreover, this distribution is the only one that satisfies the second criterion, facilitating an intuitive explanation of quantum systems.

These results suggest that, contrary to previous belief, complex-valued, instead of real, quasi-probability distributions, can be key to understanding the probabilistic nature of the quantum world.

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