Thermostatics

Abstract

States are by definition (III: 2.2.18) normed, positive linear functionals on an algebra A of observables. If the dimension of the underlying space is finite, A = B(ℂⁿ), then all linear functionals are of the form A ∋ a → Tr ρa ≡ (ρ|a), ρ ∈ B(ℂⁿ), and B(ℂⁿ) is its own dual space. The inequality of (1.4.18;4),

$$|\left( {\rho |a} \right)| \leqslant \left\| a \right\|{\left\| \rho \right\|_1},{\left\| \rho \right\|_1} = Tr{\left( {{\rho ^*}\rho } \right)^{1/2}}$$

(2.1)

then holds, and is optimal in the sense that

$$\mathop {\sup }\limits_{{{\left\| \rho \right\|}_1} = 1} |\left( {\rho |a} \right)| = \left\| a \right\|,\mathop {\sup }\limits_{\left\| a \right\| = 1} |\left( {\rho |a} \right)| = {\left\| \rho \right\|_1}.$$

(2.2)

The heuristic concepts of purer and more chaotic states can be made mathematically precise with reference to a lattice structure of the classes of equivalent density matrices.