Density Operator and Quantum States

Riccardo Manenti, Mário Motta

Abstract The first part of the chapter is devoted to the discussion of density operators. We first introduce the definition of a density operator and the unitary evolution of a quantum state. We then examine the measurement of a quantum system using the density operator approach. We present the density operator associated with a qubit and a harmonic oscillator at thermal equilibrium. The second part of the chapter focuses on bipartite systems. We show that the quantum state of a subsystem is given by the partial trace of the density operator of the global system. Finally, we explain the concept of purification of a density operator

Alice Flarend, Robert Hilborn

Abstract Probability density operators are a generalization of the concept of quantum states and provide a framework for describing mixed quantum states, which are weighted mixtures of pure quantum states. These mixed states are essential in describing how a quantum system interacts with its environment, a critical issue in quantum computing and quantum information science. The chapter also covers the time dependence of density operators and their relationship to quantum measurements.

Steven M. Moore

Z. Ficek, Mohamed Ridza Wahiddin

In Chapter 4, we introduced the concept of the photon number representation of the electromagnetic (EM) field that is based on the number states, the eigenstates of the Hamiltonian of the EM field. However, this is not the only possible representation, in fact not always the most convenient. In this chapter, we introduce the concept of phase space representations of a quantum system which is in a state described by the density operator ρˆ. The density operator of a given system encodes classical as well as non-classical (quantum) properties of the system. How to distinguish these two kinds of properties is of basic importance in quantum optics. Therefore, we shall address this issue by using representations of the density operator that are based on the parametric space of complex eigenvalues of the annihilation operator aˆ in a coherent state |α〉. We first determine what we mean by the density operator of a quantum system and discuss its basic properties. Next, we introduce different representations of the density operator in terms of coherent state projectors |α〉〈α|. The representations show that a state of the EM field or of an arbitrary quantum system may be regarded as a mixture of coherent states. Along the way, we will learn how to find the P , Q and Wigner representations of the density operator, what are their properties, how to interpret the properties and how to calculate relations between the different representations. Most importantly, certain field states exhibit non- classical features and these non-classical features can be manifested in the phase space representations. Accordingly, we will learn how the representations are very convenient tools to describe quantum states of simple systems.