Against probability: A quantum state is more than a list of probability distributions

TL;DR

This paper argues that representing a quantum state solely by outcome probabilities for measurements cannot preserve all essential structural features, highlighting a fundamental tension in quantum state representations.

Contribution

It reveals an unavoidable tension between topological robustness and the preservation of subsystem structure in probability-based quantum state representations.

Findings

Probability representations cannot be both topologically robust and subsystem-structure preserving.

The tension impacts the operational interpretability of quantum states.

Highlights limitations of probabilistic frameworks in quantum foundations.

Abstract

The state $\rho$ of a quantum system can be represented by a vector $\mathbf{P}_{\mathcal{M}}(\rho)$ of outcome probabilities for a set of measurements $\mathcal{M}$. Such representations appear throughout physics, for example, in quantum field theory via correlation functions and in quantum foundations within generalized probabilistic frameworks. In this work, we identify an unavoidable tension: to enable operationally meaningful statements, the map ${\rho \mapsto \mathbf{P}_{\mathcal{M}}(\rho)}$ must be topologically robust $\unicode{x2013}$ preserving the notion of closeness between states. Yet, a probability representation that is topologically robust cannot simultaneously retain other essential structure, such as the subsystem structure.