Pure and Mixed States

Abstract

We present a review on the notion of pure states and mixtures as mathematical concepts that apply for both classical and quantum physical theories, as well as for any other theory depending on statistical description. Here, states will be presented as expectation values on suitable algebras of observables, in a manner intended for the non-specialist reader; accordingly, basic literature on the subject will be provided. Examples will be exposed together with a discussion on their meanings and implications. An example will be shown where a pure quantum state converges to a classical mixture of particles as Planck’s constant tends to 0.

Appendix 1: Proof of Theorem 5

Here, we follow [15] closely. First, take a non-null positive linear functional $\omega ^{\prime }$ such that, $\forall A \in \mathfrak {A}$, $\omega ^{\prime }(A^{*}A) \leqslant \omega (A^{*}A)$. By the Cauchy-Schwartz inequality (6),

$$ \begin{array}{@{}rcl@{}} |\omega^{\prime}(A^{*}B)|^{2} &\leqslant \omega^{\prime}(A^{*}A)\omega^{\prime}(B^{*}B) \leqslant \omega(A^{*}A)\omega(B^{*}B) \\ &= \|\pi(A){\varOmega}\|^{2}\|\pi(B){\varOmega}\|^{2}, \end{array} $$

which implies, by Riesz representation theorem (see e.g. [1]), the existence of a positive operator $T \in {\mathcal{L}}({\mathcal{H}})$ such that $\left \langle \pi (A){\varOmega },T\pi (B){\varOmega }\right \rangle = \omega ^{\prime }(A^{*}B)$ (remark that the sesquilinear form $(\pi (A){\varOmega },\pi (B){\varOmega }) \longmapsto \omega ^{\prime }(A^{*}B)$ may be extended to the whole ${\mathcal{H}}$, and thus the domain of T, only because Ω is cyclic). Taking $A,B,C \in \mathfrak {A}$ arbitrary:

$$ \begin{array}{@{}rcl@{}} \left\langle\pi(A){\varOmega},T\pi(B)\pi(C){\varOmega}\right\rangle &= \omega^{\prime}(A^{*}BC) = \omega^{\prime}\left( (B^{*}A)^{*}C\right) \\ &= \left\langle \pi(A){\varOmega},\pi(B)T\pi(C){\varOmega}\right\rangle, \end{array} $$

which implies that [T,π(B)] = 0 for any B.

As known in representation theory (as a consequence of Schur’s Lemma, see e.g. [19]), if π is an irreducible representation, any self-adjoint operator commuting with it must be a multiple of the identity and vice versa. It happens that, if for some $\lambda \in \mathbb {C}$, then $\omega ^{\prime } = \lambda \omega$. Let us show an implication of this fact, that ω is a mixture if and only if π is reducible, which is enough for the theorem’s statement.

Indeed, if ω is a mixture, one may find $\omega ^{\prime }$ such that $\omega ^{\prime }(A^{*}A) \leqslant \omega (A^{*}A)$ which is not a multiple of ω: in this case, there is a scalar σ ∈ (0, 1) and states ω₁ and ω₂ (none of them multiples of ω) such that ω = σω₁ + (1 − σ)ω₂, so just take $\omega ^{\prime } = \sigma \omega _{1}$. The corresponding T will not be a multiple of the identity, hence the reducibility of π.

Conversely, supposing that π is reducible, one may find a self-adjoint $S \in \mathfrak {A}$ commuting with every π(B) and not being a multiple of . As a consequence, any non-trivial spectral projector P of S will also commute with π, not be a multiple of the identity, and further satisfy . Define the functional $\omega ^{\prime }(A) = \left \langle {\varOmega }, P\pi (A){\varOmega }\right \rangle$ and remark that it is positive, not a multiple of ω, and that

This implies that ω is a non-trivial mixture ω = λω₁ + (1 − λ)ω₂, with $\lambda = \|\omega ^{\prime }\| \in (0, 1)$ and states $\omega _{1} = \frac {1}{\|\omega ^{\prime }\|} \omega ^{\prime }$ and $\omega _{2} = \frac {1}{1-\|\omega ^{\prime }\|}(\omega - \omega ^{\prime })$ (see Remark 1 for a quick justification that ∥ω₂∥ = 1).