4.2 Normal states

Normal states are a crucial subset of states in von Neumann algebras, characterized by specific continuity properties. They play a fundamental role in operator algebras and quantum theory, providing insights into the structure of these mathematical objects.

These states exhibit continuity with respect to the ultraweak topology and correspond to elements of the predual. Normal states can be represented as vector states or density operators, and are closely related to the Radon-Nikodym theorem for von Neumann algebras.

Definition of normal states

• Normal states form a crucial subset of states in von Neumann algebras characterized by specific continuity properties
• These states play a fundamental role in the study of operator algebras and quantum theory
• Understanding normal states provides insights into the structure and properties of von Neumann algebras

Continuity properties

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• Normal states exhibit continuity with respect to the ultraweak topology on a von Neumann algebra
• Defined as states that are ultraweakly continuous on the unit ball of the von Neumann algebra
• Preserve limits of increasing nets of positive operators (monotone continuity)
• Can be extended uniquely from a von Neumann algebra to its ultraweak completion

Relation to predual

• Normal states correspond bijectively to elements of the predual of a von Neumann algebra
• Every normal state ω on a von Neumann algebra M can be represented as ω(x) = Tr(ρx) for some trace-class operator ρ
• The set of normal states forms a weak*-dense subset of the state space of a von Neumann algebra
• Normal states are precisely those states that can be represented as countable convex combinations of vector states

Characterizations of normal states

Vector states

• Vector states are a special class of normal states defined by unit vectors in the Hilbert space
• For a unit vector ξ in a Hilbert space H, the vector state ω_ξ is defined as ω_ξ(x) = <ξ, xξ> for all x in the von Neumann algebra
• Every normal state can be approximated by finite convex combinations of vector states
• Vector states form a spanning set for the predual of a von Neumann algebra

Density operators

• Normal states on B(H) (bounded operators on a Hilbert space) correspond to density operators
• A density operator ρ is a positive trace-class operator with trace equal to 1
• The normal state associated with a density operator ρ is given by ω_ρ(x) = Tr(ρx) for all x in B(H)
• Pure normal states correspond to rank-one projection operators

Radon-Nikodym theorem

• The Radon-Nikodym theorem for von Neumann algebras characterizes normal positive linear functionals
• States that any normal positive linear functional φ on a von Neumann algebra M can be written as φ(x) = ω(hx) for some positive operator h affiliated with M
• Provides a generalization of the classical Radon-Nikodym theorem to the non-commutative setting
• Allows for the comparison and absolute continuity of normal states

Properties of normal states

Weak* continuity

• Normal states are weak*-continuous on the unit ball of a von Neumann algebra
• Weak*-continuity ensures that normal states respect limits of bounded nets in the weak* topology
• This property distinguishes normal states from singular states
• Allows for the extension of normal states to the ultraweak completion of the von Neumann algebra

Normality vs singularity

• Normal states and singular states form a complementary pair in the state space of a von Neumann algebra
• Every state can be uniquely decomposed into a normal part and a singular part
• Singular states vanish on all finite-rank projections in B(H)
• The Jordan decomposition theorem extends this dichotomy to general bounded linear functionals on von Neumann algebras

Normal state space

Structure and topology

• The set of normal states forms a convex subset of the state space of a von Neumann algebra
• Equipped with the weak* topology inherited from the state space
• Normal state space is weak*-compact if and only if the von Neumann algebra is finite-dimensional
• For infinite-dimensional von Neumann algebras, the normal state space is weak*-dense in the state space

Convexity properties

• The set of normal states is a face in the state space of a von Neumann algebra
• Extreme points of the normal state space correspond to pure normal states
• Krein-Milman theorem applies, allowing any normal state to be approximated by convex combinations of pure normal states
• The normal state space is a Choquet simplex for abelian von Neumann algebras

Representations induced by normal states

GNS construction

• The GNS (Gelfand-Naimark-Segal) construction associates a cyclic representation to each state on a C*-algebra
• For normal states on von Neumann algebras, the GNS representation is always normal (i.e., ultraweakly continuous)
• The GNS Hilbert space for a normal state ω can be identified with the completion of M with respect to the inner product <x,y>_ω = ω(y*x)
• The GNS representation for a normal state is spatial (i.e., can be realized on a concrete Hilbert space)

Standard form

• Every von Neumann algebra admits a standard form representation
• In the standard form, the von Neumann algebra acts on a Hilbert space equipped with a conjugation operator J and a self-dual cone P
• Normal states correspond to vectors in the positive cone P
• The standard form provides a unified framework for studying normal states and their properties

Normal states on factors

Type I factors

• Type I factors are isomorphic to B(H) for some Hilbert space H
• Normal states on Type I factors correspond bijectively to density operators
• Pure normal states on Type I factors are vector states
• The normal state space of a Type I factor is isomorphic to the space of trace-class operators with trace 1

Type II and III factors

• Normal states on Type II and III factors exhibit more complex behavior than in the Type I case
• Type II_1 factors admit a unique normal tracial state
• Type III factors do not admit any normal tracial states
• The structure of normal states on Type II and III factors is closely related to the modular theory of Tomita-Takesaki

Normal states and von Neumann algebras

Commutant theorem

• The commutant theorem states that for a von Neumann algebra M acting on a Hilbert space H, (M')' = M
• Normal states play a crucial role in the proof of the commutant theorem
• The theorem implies that every von Neumann algebra is generated by its projections
• Provides a powerful tool for studying the structure of von Neumann algebras through their normal states

Kaplansky density theorem

• The Kaplansky density theorem states that the unit ball of a C*-subalgebra A of B(H) is strongly dense in the unit ball of its double commutant A''
• Implies that normal states on A'' are completely determined by their restriction to A
• Allows for the approximation of elements in a von Neumann algebra by elements from a strongly dense C*-subalgebra
• Plays a crucial role in the theory of operator algebras and their representations

Applications of normal states

Quantum statistical mechanics

• Normal states describe equilibrium states in quantum statistical mechanics
• KMS (Kubo-Martin-Schwinger) states, which model thermal equilibrium, are normal states on von Neumann algebras
• The Gibbs state, a fundamental concept in statistical mechanics, is a normal state on B(H)
• Normal states provide a framework for studying phase transitions and thermodynamic limits

Quantum information theory

• Normal states represent physical states in quantum information theory
• Quantum channels, which model information transmission, preserve normality of states
• Entanglement and quantum correlations can be studied using normal states on tensor products of von Neumann algebras
• Quantum error correction and quantum cryptography rely on properties of normal states

Normal states vs other state types

Normal vs singular states

• Normal states and singular states form a complementary pair in the state space
• Every state can be uniquely decomposed into a normal part and a singular part
• Normal states are continuous with respect to the ultraweak topology, while singular states are not
• The distinction between normal and singular states is fundamental in the classification of von Neumann algebras

Normal vs vector states

• Vector states form a subset of normal states
• Every normal state can be approximated by finite convex combinations of vector states
• Not all normal states are vector states (mixed states in quantum mechanics)
• The relationship between normal and vector states is crucial in the study of representations of von Neumann algebras

Approximation of normal states

Finite rank approximations

• Normal states on B(H) can be approximated by finite rank operators in the trace norm
• This approximation is the basis for many computational methods in quantum mechanics
• Allows for the study of infinite-dimensional systems through finite-dimensional approximations
• Connects the theory of normal states to matrix analysis and linear algebra

Ultraweakly dense subalgebras

• Normal states on a von Neumann algebra M are determined by their values on any ultraweakly dense subalgebra
• Allows for the study of normal states through more tractable subalgebras (C*-algebras)
• Provides a link between the theory of von Neumann algebras and C*-algebras
• Crucial in the development of non-commutative integration theory and quantum probability