Quantum Mechanics (1) · Problem 4.6

The density matrix formulation

Physics — Quantum Mechanics · density-matrix-entropy-two-state

Problem

Let us try to improve our understanding of the discussions on the density matrix formalism, and the connections with ``information'' or ``entropy'' that we have made. Thus, we consider a simple ``two-state'' system. Let ρ\rho be any general density matrix operating on the two-dimensional Hilbert space of this system. (a) Calculate the entropy, s=−Tr(ρln⁡ρ)s = -\mathrm{Tr}(\rho \ln \rho) corresponding to this density matrix. Express your result in terms of a single real parameter. Make sure the interpretation of this parameter is clear, as well as its range. (b) Make a graph of the entropy as a function of the parameter. What is the entropy for a pure state? Interpret your graph in terms of knowledge about a system taken from an ensemble with density matrix ρ\rho. (c) Consider a system with ensemble ρ\rho a mixture of two ensembles ρ1\rho_1, ρ2\rho_2: ρ=θρ1+(1−θ)ρ2,0≤θ≤1\rho = \theta \rho_1 + (1-\theta)\rho_2, \qquad 0 \leq \theta \leq 1 As an example, suppose ρ1=12(1001),andρ2=12(1111),\rho_1 = \frac{1}{2}\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \quad \text{and} \quad \rho_2 = \frac{1}{2}\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}, in some basis. Prove that von Neumann's mixing theorem holds for this example: s(ρ)≥θs(ρ1)+(1−θ)s(ρ2),s(\rho) \geq \theta s(\rho_1) + (1-\theta) s(\rho_2), with equality iff θ=0\theta = 0 or θ=1\theta = 1.

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