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

A mixed state cannot be described by a wave function, it requires a density matrix for its description. A pure entangled state of the whole system can be described by a wave function, but the states of its subsystems cannot be described by wave functions and are mixed states.

The difference between mixed and pure states in general has to do with whether correlations are due to entanglement. Mixed states are classical combinations (e.g. no Bell inequalities, no ‘spooky action at a distance’, no interference), whereas correlations in a pure state are due to entanglement.

The density of a substance is the relationship between the mass of the substance and how much space it takes up (volume). The mass of atoms, their size, and how they are arranged determine the density of a substance. Density equals the mass of the substance divided by its volume; D = m/v.

Density is an intensive property because there is a narrow range of densities across the samples. No matter what the initial mass was, densities were essentially the same. Since intensive properties do not depend on the amount of material, the data indicate that density is an intensive property of matter.

The density operator formalism is a generalization of the Pure State QM we have used so far. The expectation value of an operator is defined (with respect to state |ψ〉) as: The interpretation is the average of the results of many measurements of the observable A on a system prepared in state |ψ〉.

The density of states is a central concept in the development and application of RRKM theory. The density of states of a classical system is the number of states of that system per unit energy, expressed as a function of energy. This quantity may be formulated as a phase space integral in several ways.

A Density Operator is defined as an operator in quantum mechanics that encapsulates all the relevant physical information of a system. It is represented by a mathematical expression involving probabilities and the inner products of eigenstates.

In summary, the density operator is a positive semi-definite, self-adjoint operator with trace equal to unity. The eigenvalues of a positive semi-definite operator are real numbers greater than or equal to zero; thus, the operator must have a positive trace and a non-negative determinant

A mixed quantum state corresponds to a probabilistic mixture of pure states; however, different distributions of pure states can generate equivalent (i.e., physically indistinguishable) mixed states. A mixture of quantum states is again a quantum state.

The density operator has the property Tr(ρ2) ≤ 1, with equality if one of the prior probabilities is 1 and all the rest 0: ρ = |ϕn〉〈ϕn |; the density operator is then a projection operator.