nLab state on a star-algebra

This entry describes a concrete formalization of the general notion of state in the context of quantum probability theory and algebraic quantum field theory and operator algebra. For other conceptualizations of states see there.

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Idea

The concept of state on a star-algebra is the formalization of the general idea of states from the point of view of quantum probability theory and algebraic quantum theory.

In order to motivate the definition from more traditional formulations in physics, recall that there a state $\langle - \rangle$ is the information that allows to assign to each observable $A$ the expectation value $\langle A\rangle$ that this observable has when the physical system is assumed to be in that state.

Often this is formalized in the Schrödinger picture where a Hilbert space of states $\mathcal{H}$ is taken as primary, and the observables are represented as suitable linear operators $A$ on $\mathcal{H}$. Then for $\psi \in \mathcal{H}$ a state (pure state) the expectation value of $A$ in this state is the inner product $\langle \psi \vert A \vert \psi \rangle \coloneqq (\psi, A \psi)$. This defines a linear function

$$\langle \psi \vert - \vert \psi \rangle \;\colon\; \mathcal{A} \longrightarrow \mathbb{C}$$

on the algebra of observables $\mathcal{A}$, satisfying some extra properties.

Conversely, in the Heisenberg picture one may take the “abstract” algebra of observables as primary (i.e. not necessarily manifested as an operator algebra), and declare that a state is any linear functional

$$\langle - \rangle \;\colon\; \mathcal{A} \longrightarrow \mathbb{C}$$

which is positive in that $\langle A^\ast A\rangle \geq 0$ and normalized in that $\langle 1\rangle = 1$. Under suitable conditions a Hilbert space of states may be (re-)constructed from this a posteriori via the GNS construction.

Traditionally this definition is considered for algebras of observables which are C*-algebras (as usually required for non-perturbative quantum field theory, see e.g. Fredenhagen (2003) Sec. 2, Landsman (2017) Def. 2.4), but the definition makes sense generally for plain star-algebras (Meyer (1995), I.1.1), such as for instance for the formal power series algebras that appear in perturbative quantum field theory (e.g. Bordemann-Waldmann (1996) Def. 1, Fredenhagen & Rejzner (2012) Def. 2.4, Khavkine & Moretti (2015) Def. 6, Dütsch (2018) Def. 2.11).

Of course, the notion of states depends only on the $\ast$-algebra’s underlying partially ordered complex vector space (see here) and hence makes sense in the generality of any partially ordered vector space, in which case one refers to them also as positive linear functionals or similar (e.g. Murphy (1990) §3.3).

The perspective that states are normalized positive linear functionals on the algebra of observables is implicit in traditional perturbative quantum field theory, where it is encoded in the 2-point function corresponding to a vacuum state or more generally a quasi-free quantum state (the Hadamard propagator). The perspective is made explicit in algebraic quantum field theory (see e.g. Fredenhagen 03, section 2) and for star-algebras of observables that are not necessarily C*-algebras in perturbative algebraic quantum field theory (e.g. Bordemann-Waldmann 96, Fredenhagen-Rejzner 12, def. 2.4, Khavkine-Moretti 15, def. 6, Dütsch 18, def. 2.11).

Definition

For unital star algebras

Definition

(state on a unital star algebra)

Let $\mathcal{A}$ be a unital star-algebra over the complex numbers $\mathbb{C}$. A state on $\mathcal{A}$ is a linear function

$$\rho \;\colon\; \mathcal{A} \longrightarrow \mathbb{C}$$

such that

1. (positivity) for all $A \in \mathcal{A}$ the value of $\rho$ on the product $A^\ast A$ is

   1. real$\,\rho(A^\ast A) \in \mathbb{R} \hookrightarrow \mathbb{C}$

   2. as such non-negative:

   (1)$$\rho(A^\ast A) \geq 0$$

2. (normalization)

   (2)$$\rho(\mathbf{1}) = 1$$

   for $\mathbf{1} \in \mathcal{A}$ the unit in the algebra.

(e.g. Meyer 95, I.1.1, Bordemann-Waldmann 96, Fredenhagen-Rejzner 12, def. 2.4, Khavkine-Moretti 15, def. 6, Landsman 2017, Def. 2.4)

Remark

(probability theoretic interpretation of state on a star-algebra)

A star algebra $\mathcal{A}$ equipped with a state is also called a quantum probability space, at least when $\mathcal{A}$ is in fact a von Neumann algebra.

Remark

(states form a convex set)

For $\mathcal{A}$ a unital star-algebra, the set of states on $\mathcal{A}$ according to def. is naturally a convex set: For $\rho_1, \rho_2 \colon \mathcal{A} \to \mathbb{C}$ two states then for every $p \in [0,1] \subset \mathbb{R}$ also the linear combination

$$\array{ \mathcal{A} &\overset{p \rho_1 + (1-p) \rho_2}{\longrightarrow}& \mathbb{C} \\ A &\mapsto& p \rho_1(A) + (1-p) \rho_2(A) }$$

is a state.

Definition

(pure state)

A state $\rho \colon \mathcal{A} \to \mathbb{C}$ on a unital star-algebra (def. ) is called a pure state if it is extremal in the convex set of all states (remark ) in that an identification

$$\rho = p \rho_1 + (1-p) \rho_2$$

for $p \in (0,1)$ implies that $\rho_1 = \rho_2$ (hence $= \rho$).

For $C^\ast$-Algebras

The following discusses states specifically on C*-algebras.

Definition

An element $A$ of an (abstract) $C^*$-algebra is called positive if it is self-adjoint and its spectrum is contained in $[0, \infinity)$. We write $A \ge 0$ and say that the set of all positive operators is the positive cone (of a given $C^*$-algebra).

Remark

This definition is motivated by the Hilbert space situation, where an operator $A \in \mathcal{B} (\mathcal{H})$ is called positive if for every vector $x \in \mathcal{H}$ the inequality $\langle x, A x \rangle \ge 0$ holds. If the abstract $C^*$-algebra of the definition above is represented on a Hilbert space, then we see that by functional calculus we can define a self adjoint operator $B$ by $B \coloneqq f(A)$ with $f(t) := t^{1/2}$ and get $\langle x, A x \rangle = \langle B x, B x \rangle \ge 0$. This shows that the positive elements of the abstract algebra, if represented on a Hilbert space, become positive operators as defined here in the Hilbert space setting.

Definition

A linear functional $\rho$ on an $C^*$-algebra is positive if $A \ge 0$ implies that $\rho(A) \ge 0$.

A state of a unital $C^*$-algebra is linear functional $\rho$ such that $\rho$ is positive and $\rho(1) = 1$.

Though the mathematical notion of state is already close to what physicists have in mind, they usually restrict the set of states further and consider normal states only. We let $\mathcal{R}$ be an $C^*$-algebra and $\pi$ an representation of $\mathcal{R}$ on a Hilbert space $\mathcal{H}$.

Definition/Theorem

A normal state $\rho$ is a state that satisfies one of the following equivalent conditions:

• $\rho$ is weak-operator continuous on the unit ball of $\pi(\mathcal{R})$.

• $\rho$ is strong-operator continuous on the unit ball $\pi(\mathcal{R})$.

• $\rho$ is ultra-weak continuous.

• There is an operator $A$ of trace class of $\mathcal{H}$ with $tr(A) = 1$ such that $\rho(\pi(R)) = tr(A \pi(R))$ for all $R \in \mathcal{R}$.

This appears as KadisonRingrose, def. 7.1.11, theorem 7.1.12

Remark

This list is not complete, there are more commonly used equivalent characterizations of normal states.

The last one is most frequently used by physicists, in that context the operator $A$ is also called a density matrix or density operator.

Sometimes the observables of a system are described by an abstract $C^*$-algebra, in this case an important notion is the folium:

Definition

The folium of a representation $\pi$ of an $C^*$-algebra $\mathcal{R}$ on a Hilbert space is the set of normal states of $\pi(\mathcal{R})$.

Definition

A state $\rho$ of a representation is called a vector state if there is a $x \in \mathcal{H}$ such that $\rho(\pi(R)) = \langle \pi(R)x, x \rangle$ for all $R \in \mathcal{R}$.

Theorem

Normal states are vector states if $\mathcal{R}$ is a von Neumann algebra with a separating vector. More precisely: Let $\mathcal{R}$ be a von Neumman algebra acting on a Hilbert space $\mathcal{H}$, let $\rho$ be a normal state of $\mathcal{R}$ and $x \in \mathcal{H}$ be a separating vector for $\mathcal{R}$, then there is a $y \in \mathcal{H}$ such that $\rho(R) = \langle Ry, y \rangle$ for all $R \in \mathcal{R}$.

This appears as KadisonRingrose, theorem 7.2.3.

The set of states of an $C^*$-algebra is sometimes called the state space.

The state space is non-empty (define a state on the subalgebra $\mathbb{C} 1$ and extend it to the whole $C^*$-algebra via the Hahn-Banach theorem), convex and weak$^*$-compact, so it has extreme points. By the Krein-Milman theorem? (see Wikipedia: Krein-Milman theorem) it is the weak$^*$-closure of its extreme points.

Definition

A pure state is a state that is an extreme point of the state space.

The term “pure” originates from the notion of entanglement, a pure state is not a mixture of two distinct other states.

Examples

Example

(classical probability measure as state on measurable functions)
For $\Omega$ a locally compact Hausdorff space equipped with a compatible structure of a classical probability space, hence a measure space which normalized total measure $\int_\Omega d\mu = 1$, let $\mathcal{A} \coloneqq C_0(\Omega)$ be the algebra of continuous function with values in the complex numbers and vanishing at infinity, regarded as a star algebra by pointwise complex conjugation. Then forming the expectation value with respect to $\mu$ defines a state (def. ):

$$\array{ C_0(\Omega) &\overset{\langle - \rangle_\mu}{\longrightarrow}& \mathbb{C} \\ A &\mapsto& \int_\Omega A \, d\mu }$$

(e.g. Landsman 2017, p. 16-17)

Example

(elements of a Hilbert space as pure states on bounded operators)

Let $\mathcal{H}$ be a complex separable Hilbert space with inner product $\langle -,-\rangle$ and let $\mathcal{A} \coloneqq \mathcal{B}(\mathcal{H})$ be the algebra of bounded operators, regarded as a star algebra under forming adjoint operators. Then for every element $\psi \in \mathcal{H}$ of unit norm $\langle \psi,\psi\rangle = 1$ there is the state (def. ) given by

$$\array{ \mathcal{B}(\mathcal{H}) &\overset{\langle -\rangle_\psi}{\longrightarrow}& \mathbb{C} \\ A &\mapsto& \langle \psi \vert\, A \, \vert \psi \rangle &\coloneqq& \langle \psi, A \psi \rangle }$$

These are pure states (def. ).

More general states in this case are given by density matrices.

Example

(states on group algebras are unitary representations)
Let $G$ be a discrete group and $\mathbb{C}[G]$ its group algebra regarded as a complex star-algebra under the combined operation of inversion of group elements and complex conjugation of the coefficients.

Then each state $\rho \,\colon\,\mathbb{C}[G] \longrightarrow \mathbb{C}$ (Def. ) arises from a unitary representation $\widehat{(\text{-})} \,\colon\, \longrightarrow U(\mathscr{H})$ on a Hilbert space $(\mathscr{H}, \langle -\vert-\rangle)$ with a cyclic vector $\psi \in \mathscr{H}$ as

$$\rho(g) \;=\; \big\langle \psi \big\vert \widehat{g} \cdot \psi \big\rangle \,.$$

(With suitable adjustments, this statement generalizes from discrete to topological groups.)

This is due to Gelfand & Raikov (1943) (2), Naimark (1956) §30 Thm. 1 (reviewed at eom) further generalization in Saworotnow (1970), (1972), textbook account in Dixmier (1977) Thm. 13.4.5 (ii).

Properties

Closure properties

Example

(mixtures, convex combinations)

For $k \in \mathbb{N}_+$, let

• $\big( \rho_i \colon \mathcal{A} \to \mathbb{C} \big)_{i = 1}^k$

  be a $k$-tuple of states;

• $\big( p_i \in \mathbb{R}_{\geq 0} \big)_{i = 1}^k$, $\underoverset{i = 1}{k}{\sum} p_i \;=\; 1$

  be a probability distribution on the finite set $\{1, \cdots, k\}$

then the convex combination

$$\underoverset{i = 1}{k}{\sum} p_i \cdot \rho_i \;\;\; \in \; \mathcal{A}^\ast$$

is another state on the star-algebra $\mathcal{A}$.

Proposition

(operator-state correspondence)

For $\rho \;\colon\; \mathcal{A} \to \mathbb{C}$ a state, with a non-null observable $O \in \mathcal{A}$, $\rho(O^\ast O) \neq 0$, then also

(3)$$\rho_O \;\colon\; A \;\mapsto\; \tfrac{1}{ \rho(O^\ast O) } \cdot \rho\big( O^\ast \cdot A \cdot O \big)$$

is a state.

Proof

To check positivity (1), we compute for any $A \in \mathcal{A}$ as follows:

$$\begin{aligned} \rho_O \big( A^\ast \cdot A \big) & = \tfrac{1}{\rho(O^\ast O)} \cdot \rho \big( O^\ast \cdot ( A^\ast \cdot A ) \cdot O \big) \\ & = \tfrac{1}{\rho(O^\ast O)} \cdot \rho \big( ( A \cdot O )^\ast \cdot ( A \cdot O ) \big) \\ & \geq 0 \,, \end{aligned}$$

where the first step is the definition (3) the second step uses the anti-homomorphisms-property of the star-involution, and the last step follows by the assumed positivivity of $\rho$.

To check normalization (2), we observe that:

$$\begin{aligned} \rho_O(\mathbf{1}) & = \tfrac{1}{\rho(O^\ast O)} \cdot \rho \big( O^\ast \cdot \mathbf{1} \cdot O \big) \\ & = \tfrac{1}{\rho(O^\ast O)} \cdot \rho \big( O^\ast \cdot O \big) \\ & = 1 \,. \end{aligned}$$

Multiplicative states

Recall that a state $\rho \,\colon\, \mathcal{A} \xrightarrow{\;} \mathbb{C}$ is generally only required to be a linear map on the underlying vector space of the algebra $\mathcal{A}$ (subject only to the positivity constraint), not necessarily an algebra homomorphism.

Those states which do respect the algebra product, in that $\rho(A \cdot B) = \rho(A)\, \rho(B)$ for $A,B \in \mathcal{A}$, are called multiplicative.

Proposition

A multiplicative state is necessarily a pure state.

(e.g. Zhu 1993, Ex. 13.3; Warner 2010, Lem. 7.20)

Proposition

If the star-algebra $\mathcal{A}$ is commutative, then the pure states on it are precisely the multiplicative states.

(e.g. Zhu 1993, Ex. 13.4; Warner 2010, Lem. 7.21)

Proof

The idea is that for commutative $\mathcal{A}$ Gelfand duality applies and shows that a multiplicative state is equivalently the operation of point-evaluation on the Gelfand spectrum.

Fell’s theorem

See at Fell's theorem.

Gleason’s theorem

See at Gleason's theorem.

Related concepts

• GNS construction

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

Exposition

• Paul-André Meyer, Section I.1.1 in: Quantum Probability for Probabilists, Lecture Notes in Mathematics 1538, Springer (1995) [doi:10.1007/BFb0084701]

• Jonathan Gleason, The $C^*$-algebraic formalism of quantum mechanics (2009) [pdf, pdf]

• Jonathan Gleason, From Classical to Quantum: The $F^\ast$-Algebraic Approach, contribution to VIGRE REU 2011, Chicago (2011) [pdf, pdf]

Within monographs:

• Ola Bratteli, Derek W. Robinson, §2.3 in: Operator Algebras and Quantum Statistical Mechanics – vol 1: $C^\ast$- and $W^\ast$-Algebras. Symmetry Groups. Decomposition of States., Springer (1979, 1987, 2002) [doi:10.1007/978-3-662-02520-8]

• Nikolay Bogolyubov, A. A. Logunov, A. I. Oksak, I. T. Todorov, G. G. Gould, p. 234 in: Algebra of Observables and State Space [doi:10.1007/978-94-009-0491-0_6], Chapter in: General principles of quantum field theory, Mathematical Physics and Applied Mathematics 10, Kluwer (1990) [doi:10.1007/978-94-009-0491-0]

• Kehe Zhu, Section 13 of: An Introduction to Operator Algebras, CRC Press (1993) [ISBN:9780849378751]

• Garth Warner, States, §7 in: $C^\ast$-Algebras, EPrint Collection, University of Washington (2010) [hdl:1773/16302, pdf, pdf]

Discussion under the name “positive linear functionals”:

• James D. Steward, Positive definite functions and generalizations, an historical survey, The Rocky Mountain Journal of Mathematics 6 3 (1976) 409-434 [jstor:44236118]

• Jacques Dixmier, Chapter 2 of: $C^\ast$-algebras, North Holland (1977) [ch2:pdf, ch13:pdf]

• Gerard Murphy, §3.3 in: $C^\ast$-algebras and Operator Theory, Academic Press (1990) [doi:10.1016/C2009-0-22289-6]

• Wikipedia, Positive linear functional

• eom, Positive-definite function

With an eye towards density matrices and their entropy:

• Paolo Facchi, Giovanni Gramegna, Arturo Konderak, around (6) in: Entropy of quantum states (arXiv:2104.12611, spire:1860877)

With an eye towards Gelfand-Tsetlin algebras and in the generality of conditional expectation values:

• Anatoly Vershik, Def. 1 in: Gel’fand-Tsetlin algebras, expectations, inverse limits, Fourier analysis, pp. 619 in: Pavel Etingof, Vladimir S. Retakh, Isadore Singer (eds.) The Unity of Mathematics In Honor of the Ninetieth Birthday of I. M. Gelfand, (arXiv:math/0503140, ISBN:978-0-8176-4467-3)

Original articles

• Rudolf Haag, Daniel Kastler, equation (3) in: An algebraic approach to quantum field theory, Journal of Mathematical Physics, 5 (1964) 848-861 [doi:10.1063/1.1704187, spire:9124]

• Richard Kadison, John Ringrose, Fundamentals of the theory of operator algebras, AMS (1991)

• Martin Bordemann, Stefan Waldmann, Formal GNS Construction and States in Deformation Quantization, Commun. Math. Phys. (1998) 195: 549. (arXiv:q-alg/9607019, doi:10.1007/s002200050402)

• Klaus Fredenhagen, section 2 of Algebraische Quantenfeldtheorie, lecture notes, 2003 (pdf)

• Hans Halvorson, Michael Müger, def. 1.11 in Algebraic Quantum Field Theory (arXiv:math-ph/0602036)

• Klaus Fredenhagen, Katarzyna Rejzner, definition 2.4 in Perturbative algebraic quantum field theory, In Mathematical Aspects of Quantum Field Theories, Springer 2016 (arXiv:1208.1428)

• Igor Khavkine, Valter Moretti, Algebraic QFT in Curved Spacetime and quasifree Hadamard states: an introduction, Chapter 5 in Romeo Brunetti et al. (eds.) Advances in Algebraic Quantum Field Theory, Springer, 2015 (arXiv:1412.5945)

• Katarzyna Rejzner, section 2.1.2 of Perturbative Algebraic Quantum Field Theory, Mathematical Physics Studies, Springer 2016 (web)

• Klaus Fredenhagen, Falk Lindner, Construction of KMS States in Perturbative QFT and Renormalized Hamiltonian Dynamics, Communications in Mathematical Physics Volume 332, Issue 3, pp 895-932, 2014-12-01 (arXiv:1306.6519)

• Klaas Landsman, around def. 2.4 in: Foundations of quantum theory – From classical concepts to Operator algebras, Springer Open (2017) [doi:10.1007/978-3-319-51777-3, pdf]

• Michael Dütsch, section 2.5 of From classical field theory to perturbative quantum field theory, 2018

• Nicolò Drago, Valter Moretti, The notion of observable and the moment problem for $\ast$-algebras and their GNS representations (doi:1903.07496, spire:1725528)

For more references see at operator algebra.

The case of group algebras

The characterization of states on group algebras:

• И. М. Гельфанд, Д. А. Райков, Неприводимые унитарные представления локально бикомпактных групп, Матем. сб., 13(55):2–3 (1943) 301–316 [mathnet pdf]

  
  

  Israel Gelfand, Dmitri Raikov, Irreducible unitary representations of locally bicompact groups, Recueil Mathématique. N.S., 13(55) 2–3 (1943) 301–316 [mathnet:eng/sm6181]

• Mark A. Naimark, Normed Rings, Moscow (1956)

• Parfeny P. Saworotnow, Representation of a topological group on a Hilbert module, Duke Math. J. 37 1 (1970) 145-150 [doi:10.1215/S0012-7094-70-03720-8]

• Parfeny P. Saworotnow, Generalized Positive Linear Functionals on a Banach Algebra with an Involution, Proceedings of the AMS, 31 1 (1972) 299-304 [jstor:2038564]

• Steward (1976) §4

• Dixmier (1977) Thm. 13.4.6

• eom, Positive-definite function

• Wikipedia, Positive-definite function on a group