Why state of quantum system is fully defined by density matrix

TL;DR

This paper demonstrates that the density matrix fully characterizes a quantum system's state by showing measurement probabilities depend solely on it, deriving the Born rule without assuming traditional measurement formalism.

Contribution

It provides a formal proof that the density matrix contains all information about a quantum state without relying on standard measurement postulates.

Findings

Measurement outcomes depend linearly on the density matrix.

Born rule is derivable from the properties of the density matrix.

The approach is operational, not assuming specific measurement formalism.

Abstract

We show that probabilities of results of all possible measurements performing on a quantum system depend on the system's state only through its density matrix. Therefore all experimentally available information about the state contains in the density matrix. In this study, we do not postulate that measurements obey some given formalism (such as observables, positive-operator valued measures, etc.), and do not use Born rule. The process of measurement is considered in a fully operational manner---as an interaction of a measured system with some black-box apparatus. The key point of our approach is the proof that, for improper mixtures, the expected value of any measurement depends linearly on the reduced density function. Such a proof is achieved by considering appropriate thought experiments. We demonstrate that Born rule can be derived as a natural consequence of our results.