nLab state

Idea

A state of a system in physics is a description of everything that is true (or known, or relevant, depending on the circumstances) about the physical system.

In the Bayesian interpretation of physics, the state of a system is not a property of reality but instead indicates an observer's knowledge about the system. A pure state gives maximal information about the system (which amounts to complete information in classical mechanics but not generally in quantum mechanics), while a mixed state is more general. A mixed state can be decomposed into a probability distribution on the space of pure states, although this decomposition is unique only for classical systems. In a frequentist interpretation of probability, a mixed state can describe only a statistical ensemble of systems; the real world is in one (generally unknown) pure state (possibly with additional hidden variables in the quantum case, depending on the interpretation of quantum physics).

States in the Schrödinger picture describe the state of the system at any given time and are subject to time evolution?, while in the Heisenberg picture a single state describes the entire history of the system (with time evolution applying to the observables instead).

In statistical physics, a microstate? is a complete description of all of the particles in a system, while a macrostate? considers only the macroscopic properties known to thermodynamics.

Definitions

The precise mathematical notion of state depends on what mathematical formalization of mechanics is used.

Algebraic definition

Quite generally, both in classical physics as well as in quantum physics, one may define states as assignments of expectation values to observables in an algebra of observables. This is the definition used in quantum probability theory (which subsumes ordinary probability theory). See at

• state in AQFT and operator algebra

for details.

In classical mechanics

In classical Lagrangian mechanics, a pure state is a point in the state space? of the system, giving all of the (generalised) positions? and velocities. In classical Hamiltonian mechanics, a pure state is a point in the phase space of the system, giving the positions and momenta. In either case, a mixed state is a probability distribution on the space of pure states.

More generally, a classical state is a linear function $\rho\colon A \to \mathbb{R}$ on the Poisson algebra $A$ underlying the classical mechanical system which satisfies positivity and normalization.

In geometric quantization

• space of states (in geometric quantization)

In Hilbert-space quantum mechanics

In quantum mechanics given by a Hilbert space $H$, a pure state is a ray in $H$, which we often call the Hilbert space of states. Strictly speaking, the space of states is not $H$ but $(H \setminus \{0\})/\mathbb{C}$, or equivalently $S(H)/\mathrm{U}(1)$. A mixed state is then a density matrix on $H$.

In AQFT

In AQFT, a quantum mechanical system is given by a $C^*$-algebra $A$, and a quantum state is usually defined as a linear function $\rho\colon A \to \mathbb{C}$ which satisfies positivity and normalization; see states in AQFT and operator algebra.

Arguably, the correct notion of state to use is that of quasi-state; every state gives rise to a unique quasi-state, but not conversely. However, when either classical mechanics or Hilbert-space quantum mechanics is formulated in AQFT, every quasi-state is a state (at least if the Hilbert space is not of very low dimension, by Gleason's theorem). See also the Idea-section at Bohr topos for a discussion of this point.

In FQFT

In the FQFT formulation of quantum field theory, a physical system is given by a cobordism representation

$$Bord_n^S \to \mathcal{C} \,.$$

In this formulation the (n-1)-morphism in $\mathcal{C}$ assigned to an $(n-1)$-dimensional manifold $\Sigma_{n-1}$ is the space of states over that manifold. A state is accordingly a generalized element of this object.

Pure and mixed states

In statistical physics, 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.

See pure state.

Examples

Here are some toy examples of spaces of states.

For an impossible system, the space of states is empty; for a trivial system (with a unique way to be), the space of states is the point. This unique state is pure.

For a classical bit, a system with two distinct ways to be, the space of states is a line segment; a state is given by a real number $t$ with $0 \leq t \leq 1$. This $t$ is the probability that the system is in the first state, with $1 - t$ the probability that it is in the second; the two pure states correspond to $t = 0$ and $t = 1$. This space naturally comes equipped with a metric structure in which its length is actually $2$; it’s the line segment from $(0,1)$ to $(1,0)$ in the Lebesgue space $l^1(2)$.

For a quantum bit, a qubit, a state is given by a self-adjoint $2$-by-$2$ complex matrix

$$\begin{pmatrix} a & b - \mathrm{i} c \\ b + \mathrm{i} c & d \end{pmatrix}$$

with unit trace (so $d = 1 - a$) and nonnegative determinant (so $a^2 + b^2 + c^2 \leq a$). However, the metric structure from viewing this as a subset of $\mathbb{R}^3$ is misleading, so we use $w \coloneqq a - d = 2 a - 1$ instead, along with $u \coloneqq 2 b$ and $v \coloneqq 2 c$ to keep the same normalization. Then the density matrix may be written

$$\frac 1 2 \begin{pmatrix} 1 + w & u - \mathrm{i} v \\ u + \mathrm{i} v & 1 - w \end{pmatrix} = \frac 1 2 (\mathrm{I} + u \sigma_x + v \sigma_y + w \sigma_z) ,$$

where the $\sigma$s are the Pauli matrices. This already has unit trace; the inequality for the determinant is

$$u^2 + v^2 + w^2 \leq 1 .$$

The pure states are those satisfying $u^2 + v^2 + w^2 = 1$, forming a unit sphere, known in this context as the Bloch sphere? (so the full space of all states is the Bloch ball).

Related concepts

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

duality between $\;$algebra and geometry

$\phantom{A}$geometry$\phantom{A}$	$\phantom{A}$category$\phantom{A}$	$\phantom{A}$dual category$\phantom{A}$	$\phantom{A}$algebra$\phantom{A}$
$\phantom{A}$topology$\phantom{A}$	$\phantom{A}$$\phantom{NC}TopSpaces_{H,cpt}$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/Gelfand-Kolmogorov+theorem">Gelfand-Kolmogorov</a>}}{\hookrightarrow} Alg^{op}_{\mathbb{R}}$$\phantom{A}$	$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$topology$\phantom{A}$	$\phantom{A}$$\phantom{NC}TopSpaces_{H,cpt}$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a class="existingWikiWord" href="https://ncatlab.org/nlab/show/Gelfand+duality">Gelfand duality</a>}}{\simeq} TopAlg^{op}_{C^\ast, comm}$$\phantom{A}$	$\phantom{A}$comm. C-star-algebra$\phantom{A}$
$\phantom{A}$noncomm. topology$\phantom{A}$	$\phantom{A}$$NCTopSpaces_{H,cpt}$$\phantom{A}$	$\phantom{A}$$\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} TopAlg^{op}_{C^\ast}$$\phantom{A}$	$\phantom{A}$general C-star-algebra$\phantom{A}$
$\phantom{A}$algebraic geometry$\phantom{A}$	$\phantom{A}$$\phantom{NC}Schemes_{Aff}$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/affine+scheme#AffineSchemesFullSubcategoryOfOppositeOfRings">almost by def.</a>}}{\simeq} \phantom{Top}Alg^{op}$$\phantom{A}$	$\phantom{A} \phantom{A}$ $\phantom{A}$commutative ring$\phantom{A}$
$\phantom{A}$noncomm. algebraic$\phantom{A}$ $\phantom{A}$geometry$\phantom{A}$	$\phantom{A}$$NCSchemes_{Aff}$$\phantom{A}$	$\phantom{A}$$\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} \phantom{Top}Alg^{op}_{fin, red}$$\phantom{A}$	$\phantom{A}$fin. gen. $\phantom{A}$associative algebra$\phantom{A}$$\phantom{A}$
$\phantom{A}$differential geometry$\phantom{A}$	$\phantom{A}$$SmoothManifolds$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras">Milnor's exercise</a>}}{\hookrightarrow} \phantom{Top}Alg^{op}_{comm}$$\phantom{A}$	$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$supergeometry$\phantom{A}$	$\phantom{A}$$\array{SuperSpaces_{Cart} \\ \\ \mathbb{R}^{n\vert q}}$$\phantom{A}$	$\phantom{A}$$\array{ \overset{\phantom{\text{Milnor's exercise}}}{\hookrightarrow} & Alg^{op}_{\mathbb{Z}_2 \phantom{AAAA}} \\ \mapsto & C^\infty(\mathbb{R}^n) \otimes \wedge^\bullet \mathbb{R}^q }$$\phantom{A}$	$\phantom{A}$supercommutative$\phantom{A}$ $\phantom{A}$superalgebra$\phantom{A}$
$\phantom{A}$formal higher$\phantom{A}$ $\phantom{A}$supergeometry$\phantom{A}$ $\phantom{A}$(super Lie theory)$\phantom{A}$	$\phantom{A}\array{ Super L_\infty Alg_{fin} \\ \mathfrak{g} }\phantom{A}$	$\phantom{A}\array{ \overset{ \phantom{A}\text{<a href="https://ncatlab.org/nlab/show/L-infinity-algebra#ReformulationInTermsOfSemifreeDGAlgebra">Lada-Markl</a>}\phantom{A} }{\hookrightarrow} & sdgcAlg^{op} \\ \mapsto & CE(\mathfrak{g}) }\phantom{A}$	$\phantom{A}$differential graded-commutative$\phantom{A}$ $\phantom{A}$superalgebra $\phantom{A}$ (“FDAs”)

in physics:

$\phantom{A}$algebra$\phantom{A}$	$\phantom{A}$geometry$\phantom{A}$
$\phantom{A}$Poisson algebra$\phantom{A}$	$\phantom{A}$Poisson manifold$\phantom{A}$
$\phantom{A}$deformation quantization$\phantom{A}$	$\phantom{A}$geometric quantization$\phantom{A}$
$\phantom{A}$algebra of observables	$\phantom{A}$space of states$\phantom{A}$
$\phantom{A}$Heisenberg picture	$\phantom{A}$Schrödinger picture$\phantom{A}$
$\phantom{A}$AQFT$\phantom{A}$	$\phantom{A}$FQFT$\phantom{A}$
$\phantom{A}$higher algebra$\phantom{A}$	$\phantom{A}$higher geometry$\phantom{A}$
$\phantom{A}$Poisson n-algebra$\phantom{A}$	$\phantom{A}$n-plectic manifold$\phantom{A}$
$\phantom{A}$En-algebras$\phantom{A}$	$\phantom{A}$higher symplectic geometry$\phantom{A}$
$\phantom{A}$BD-BV quantization$\phantom{A}$	$\phantom{A}$higher geometric quantization$\phantom{A}$
$\phantom{A}$factorization algebra of observables$\phantom{A}$	$\phantom{A}$extended quantum field theory$\phantom{A}$
$\phantom{A}$factorization homology$\phantom{A}$	$\phantom{A}$cobordism representation$\phantom{A}$