Category: 5. Mixed States, Open Systems, and the Density Operator

The notion of Schmidt decomposition immediately leads to a converse construction known as purification: given a density matrix ρ A for a mixed state of a system A, one can construct a supersystem AB of which it is a subsystem, such that |ψ AB i is a pure state, and ρ A = Tr B…

There is another sense in which density operators are a useful way to describe nature. In general, it is impossible to isolate the system of interest from some parts of its environment. We then have to regard our system as a subsystem of a larger system: “system + environment”. If the large system in a…

We now have an alternate formulation of quantum mechanics, in terms of density operators instead of state vectors, that is good for open systems as well. Let’s go through the axioms of quantum mechanics framed in this language. 5.2.1 States and observables Postulate 1. Quantum State: The state of a quantum system is described by…

Often the density operator is the primary descriptor of a state. The de- composition in terms of component states ρ = X i p i |iihi|, is not always unique. For a pure state, it must be obvious that there is only one such decompo- sition, and this can be proved from the definitions: Theorem…

The representation of a single qubit state on the Bloch sphere can be ex- tended to the density operator. The Bloch sphere is parametrized by spherical angles or in terms of the Bloch vector of Equation 4.2, which characterizes the polarization of the state. Can we use this kind of description for a mixed state?…

A given density operator could represent a pure or a mixed state. If the system is pure, then the state is a ray in Hilbert space, and the density operator can be expressed as ρ = |ψihψ|, for some |ψi. Such a density matrix satisfies ρ 2 = |ψihψ|ψihψ| = ρ, (5.17) Tr(ρ 2 )…

The density operator on a Hilbert space, defined by Equation 5.4 satisfies the following properties: 1. ˆρ is Hermitian. Proof: ˆρ † = X n p ∗ n |ψ n i † hψ n | † = X n p n |ψ n ihψ n | = ˆρ. (5.11) 2. ˆρ is non-negative, that is,…

By “state” of a system, we mean a collection of all possible knowledge we can gather about the system, which is practically achieved by studying the distribution of outcomes of measurements made on the system. In the case of pure states, these outcomes together are described by a ray in Hilbert space. Consider measuring an…

The formalism for quantum systems developed so far applies to what are called pure states. A system in a pure state is completely specified by the state vector. A complete set of experimental tests will determine the system state fully: we have maximal knowledge of the system. For example, for a spin system, we can…