State (functional analysis)
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In the mathematical field of functional analysis, a state of an operator system is a positive linear functional of norm 1. States in functional analysis generalize the notion of density matrices in quantum mechanics, which represent quantum states, both mixed states and pure states. Density matrices in turn generalize state vectors, which only represent pure states. For M an operator system in a C*-algebra A with identity, the set of all states of M, sometimes denoted by S(M), is convex, weak-* closed in the Banach dual space M * . Thus the set of all states of M with the weak-* topology forms a compact Hausdorff space, known as the state space of M. In the C*-algebraic formulation of quantum mechanics, states in this previous sense correspond to physical states, i.e. mappings from physical observables (self-adjoint elements of the C*-algebra) to their expected measurement outcome (real number).
In the mathematical field of functional analysis, a state of an operator system is a positive linear functional of norm 1. States in functional analysis generalize the notion of density matrices in quantum mechanics, which represent quantum states, both mixed states and pure states. Density matrices in turn generalize state vectors, which only represent pure states. For M an operator system in a C*-algebra A with identity, the set of all states of M, sometimes denoted by S(M), is convex, weak-* closed in the Banach dual space M * . Thus the set of all states of M with the weak-* topology forms a compact Hausdorff space, known as the state space of M. In the C*-algebraic formulation of quantum mechanics, states in this previous sense correspond to physical states, i.e. mappings from physical observables (self-adjoint elements of the C*-algebra) to their expected measurement outcome (real number).
Some important classes of states
Pure states
By the Krein-Milman theorem, the state space of M has extreme points. The extreme points of the state space are termed pure states and other states are known as mixed states.
Vector states
For a Hilbert space H and a vector x in H, the formula ω x (T) := ⟨Tx,x⟩ (for T in B(H)) defines a positive linear functional on B(H). Since ω x (1)=||x|| 2 , ω x is a state if ||x||=1. If A is a C*-subalgebra of B(H) and M an operator system in A, then the restriction of ω x to M defines a positive linear functional on M. The states of M that arise in this manner, from unit vectors in H, are termed vector states of M.
Faithful states
A state \tau is faithful, if \tau(a^* a) = 0 implies a = 0 .
Normal states
A state \tau is called normal, iff for every monotone, increasing net H_\alpha of operators with least upper bound H , \tau(H_\alpha)\; converges to \tau(H)\; .
Tracial states
A tracial state is a state \tau such that \tau(AB)=\tau (BA)\;. For any separable C*-algebra, the set of tracial states is a Choquet simplex.
Factorial states
A factorial state of a C*-algebra A is a state such that the commutant of the corresponding GNS representation of A is a factor.
See also
Quantum state Gelfand–Naimark–Segal construction Quantum mechanics Quantum state Density matrix
References
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