nLab pure state

Idea

A pure state is a state on a star-algebra which is an extremal point in the convex set of all states.

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 $\psi \leq \phi$ means that $\phi$ contains all of the information of $\psi$ (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 $\mathcal{A}$ of observables (or more generally a unital star algebra) and a state is a positive-semidefinite linear function $\rho\colon \mathcal{A} \to \mathbb{C}$ such that $\rho(1) = 1$. We might write $\rho$ as a convex-linear combination of two other states:

(1)$$\rho = a \sigma + b \tau ,$$

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

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

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 $\rho \leq \sigma$ mean that the mutual information? $I(\rho,\sigma)$ equals the entropy $H(\rho)$, or equivalently that the conditional entropy? $H(\rho|\sigma)$ 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 $C^*$-algebra. —Toby

Special cases

If $A$ is the algebra of continuous complex-valued functions on some compactum $X$, then the pure states on $A$ correspond precisely to the points in $X$; 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 $X$, with the pure states as the Dirac delta measures.

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

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.

Related concepts

• mixed quantum state

quantum probability theory – observables and states

• states

  • classical state

  • quantum state

    • space of states (in geometric quantization)

    • state on a star-algebra, quasi-state

    • qbit, Bell state

      dimer, tensor network state

    • quantum state preparation

    • probability amplitude, quantum fluctuation

    • pure state

      • wave function

      • bra-ket

      • Bell state

    • quantum superposition, quantum interference

    • quantum entanglement

    • quantum measurement

      wave function collapse

      Born rule

      deferred measurement principle

      quantum reader monad

      measurement problem

    • superselection sector

    • mixed state, density matrix

      entanglement entropy

      holographic entanglement entropy

    • coherent quantum state

  • ground state, excited state

  • quasi-free state

  • Fock space, second quantization

  • vacuum, vacuum state

    • Hadamard state

    • vacuum diagram

    • vacuum expectation value, vacuum amplitude, vacuum fluctuation

    • vacuum energy

    • vacuum polarization

    • interacting vacuum

    • thermal vacuum, KMS state

    • vacuum stability

      • false vacuum, tachyon, Coleman-De Luccia instanton

    • theta vacuum

    • perturbative string theory vacuum

      • non-geometric string theory vacuum

      • landscape of string theory vacua

  • entangled state

  • tensor network state

    • matrix product state

    • tree tensor network state

• observables

  • quantum observable, beable

  • algebra of observables, star-algebra

  • Bohr topos

  • quantum operator (in geometric quantization)

  • quantum operation, quantum effect, effect algebra

  • in quantum field theory

    • local observable

    • polynomial observable

      • linear observable

        • field observable

    • regular observable

    • microcausal observable

    • normal-ordered product, time-ordered products, retarded product

    • Wick algebra

    • scattering amplitude

    • interacting field algebra of observables, Bogoliubov's formula

• GNS construction

• theorems

  • order-theoretic structure in quantum mechanics

    • Gleason's theorem

    • Alfsen-Shultz theorem

    • Harding-Döring-Hamhalter theorem

    • Kochen-Specker theorem

    • Nuiten's lemma

  • Wigner's theorem

  • no-cloning theorem

  • Bell's theorem

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)

See also:

• Quantiki, Pure states

Not really references on this subject, but ones referred to in the text:

• Claude Shannon, The lattice theory of information, IEEE Transactions on Information Theory 1, 105–107 (1953)

• Hua Li and Edwin Chong, On a connection between information and group lattices, Entropy 13, 683–708 (2011) (mdpi:1099-4300/13/3/683)