Quantum Mechanics According to Martians: Density Matrix Theory

Abstract

We discuss the density matrix approach introduced by John von Neumann to quantum mechanics. We illustrate how it is applied in the calculation of expectation values and Born probabilities. The postulates of quantum mechanics, according to the Copenhagen interpretation, are summarized from the vantage point of the density operator framework. We learn how density operators are used to discern coherent, or pure, state ensembles from statistical mixtures of pure states. It is shown that, for entangled states, the traced density operator to a lower dimensional Hilbert space, results in reduced density operators that describe a mixed state. The Schmidt decomposition theorem and the Schmidt number, which measures the degree of entanglement of quantum states, is introduced and discussed. We define and illustrate the concept of von Neumann entropy.