In physics, recall that a state of a physical system is (in the Bayesian interpretation) a specification of the information that one might have about the system (typically relative to some fixed background information). States form (at least) a poset where ψ≤ϕ\psi \leq \phi means that ϕ\phi contains all of the information of ψ\psi (and possibly more). A pure state is a maximal element of this poset: a state that specifies as much information as possible about the system. A mixed state is a state that is not pure.

Definitions

Fairly generally, a physical system has a complex C*-algebra 𝒜\mathcal{A} of observables (or more generally a unital star algebra) and a state is a positive-semidefinite linear function ρ:𝒜→ℂ\rho\colon \mathcal{A} \to \mathbb{C} such that ρ(1)=1\rho(1) = 1. We might write ρ\rho as a convex-linear combination of two other states:

(1)ρ=aσ+bτ, \rho = a \sigma + b \tau ,

where necessarily 0≤a,b≤10 \leq a, b \leq 1 and a+b=1a + b = 1.

The state ρ\rho is pure if, whenever (1) holds for σ≠τ\sigma \neq \tau, then either a=0a = 0 (hence b=1b = 1 and ρ=τ\rho = \tau) or a=1a = 1 (hence b=0b = 0 and ρ=σ\rho = \sigma); conversely, ρ\rho is mixed if we ever have (1) for σ≠τ\sigma \neq \tau and 0<a<10 \lt a \lt 1 (hence 0<b<10 \lt b \lt 1 and ρ≠σ,τ\rho \neq \sigma, \tau).

Really, this definition makes sense as long as the states form a convex space.

To define when one state gives at least as much information as another (the partial order from the Idea section), let ρ≤σ\rho \leq \sigma mean that the mutual information? I(ρ,σ)I(\rho,\sigma) equals the entropy H(ρ)H(\rho), or equivalently that the conditional entropy? H(ρ|σ)H(\rho|\sigma) is zero. (In the classical case, this partial order is attributed to Shannon (1953), which I have not read, by Li & Chong (2011), which I have only skimmed.) The maximal elements under this partial order should be precisely the pure states, but the direct definition of pure states is much simpler.

I need to check whether these are equivalent on any C *C^*-algebra. —Toby

Special cases

If AA is the algebra of continuous complex-valued functions on some compactum XX, then the pure states on AA correspond precisely to the points in XX; so pure states here are the states of classical mechanics (at least for a compact phase space). The mixed states, however, correspond more generally to Radon probability measures on XX, with the pure states as the Dirac delta measures.

On the other hand, if AA is the algebra of all bounded operators on some Hilbert space HH, then the pure states on AA correspond precisely to the rays in HH, as is usual in quantum mechanics. The mixed states, however, correspond more generally to density matrices on HH, with the pure states those matrices of the form |ψ⟩⟨ψ|{|\psi\rangle}{\langle\psi|} for some unit vector |ψ⟩{|\psi\rangle}.

Classical versus quantum

In any case, a pure state is a state of maximal information, while a mixed state is a state with less than maximal information. In the classical case, we may say that a pure state is a state of complete information, but this does not work in the quantum case; from the perspective of the information-theoretic or Bayesian interpretation of quantum physics, this inability to have complete information, even when having maximal information, is the key feature of quantum physics that distinguishes it from classical physics.

mixed quantum state

quantum probability theory – observables and states

References

For comprehensive references see those at

quantum probability theory

Textbook accounts:

Klaas Landsman, Sections 1.3 and 2.3 of: Foundations of quantum theory – From classical concepts to Operator algebras, Springer Open 2017 (pdf)