Chapter 8 Density matrices

About density matrices, and how they help to solve the problem introduced by entangled states, as well as how they let us talk about mixtures and subsystems. Also a first look at the partial trace.

We cannot always assign a definite state vector to a quantum system. It may be that the system is part of a composite system that is in an entangled state, or it may be that our knowledge of the preparation of a particular system is insufficient to determine its state — for example, someone may prepare a particle in one of the states |\psi_1\rangle, |\psi_2\rangle, \ldots, |\psi_n\rangle, with (respective) probabilities p_1, p_2, \ldots, p_n, and then give it to us without telling us which state |\psi_k\rangle it’s actually in. Nevertheless, in either case we are able to make statistical predictions about the outcomes of measurements performed on the system using a more general description of quantum states.

We have already mentioned that the existence of entangled states leads to an obvious question: if we cannot attribute a state vectors to an individual quantum system, then how should we describe its quantum state? In this chapter we will introduce an alternate description of quantum states that can be applied both to a composite system and to any of its subsystems. Our new mathematical tool is called a density operator.153 We will start with the density operator as a description of the mixture of quantum states, and will then discuss the partial trace, which is a unique operation that takes care of the reduction of a density operator of a composite system to density operators of its components.

  1. If we choose a particular basis, operators become matrices. Throughout this book we use both terms (density operators and density matrices) pretty interchangeably.↩︎