In
functional analysis, a state of an
operator system is a
positive linear functional In mathematics, more specifically in functional analysis, a positive linear functional on an ordered vector space (V, \leq) is a linear functional f on V so that for all positive element (ordered group), positive elements v \in V, that is v \geq 0, ...
of
norm 1. States in functional analysis
generalize the notion of
density matrices in quantum mechanics, which represent
quantum state
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution i ...
s, both . 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).
Jordan decomposition
States can be viewed as noncommutative generalizations of
probability measures. By
Gelfand representation In mathematics, the Gelfand representation in functional analysis (named after I. M. Gelfand) is either of two things:
* a way of representing commutative Banach algebras as algebras of continuous functions;
* the fact that for commutative C*-algeb ...
, every commutative C*-algebra ''A'' is of the form ''C''
0(''X'') for some locally compact Hausdorff ''X''. In this case, ''S''(''A'') consists of positive
Radon measure
In mathematics (specifically in measure theory), a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space ''X'' that is finite on all compact sets, outer regular on all Borel ...
s on ''X'', and the are the evaluation functionals on ''X''.
More generally, the
GNS construction shows that every state is, after choosing a suitable representation, a
vector state.
A bounded linear functional on a C*-algebra ''A'' is said to be self-adjoint if it is real-valued on the self-adjoint elements of ''A''. Self-adjoint functionals are noncommutative analogues of
signed measure
In mathematics, signed measure is a generalization of the concept of (positive) measure by allowing the set function to take negative values.
Definition
There are two slightly different concepts of a signed measure, depending on whether or not ...
s.
The
Jordan decomposition in measure theory says that every signed measure can be expressed as the difference of two positive measures supported on disjoint sets. This can be extended to the noncommutative setting.
It follows from the above decomposition that ''A*'' is the linear span of states.
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 equation ω
''x''(''A'') := ⟨''Ax'',''x''⟩ (for ''A'' 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''.
Normal states
A state
is called normal, iff for every monotone, increasing
net
Net or net may refer to:
Mathematics and physics
* Net (mathematics), a filter-like topological generalization of a sequence
* Net, a linear system of divisors of dimension 2
* Net (polyhedron), an arrangement of polygons that can be folded up ...
of operators with least upper bound
,
converges to
.
Tracial states
A tracial state is a state
such that
:
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
Factor, a Latin word meaning "who/which acts", may refer to:
Commerce
* Factor (agent), a person who acts for, notably a mercantile and colonial agent
* Factor (Scotland), a person or firm managing a Scottish estate
* Factors of production, suc ...
.
See also
*
Quantum state
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution i ...
*
Gelfand–Naimark–Segal construction
In functional analysis, a discipline within mathematics, given a C*-algebra ''A'', the Gelfand–Naimark–Segal construction establishes a correspondence between cyclic *-representations of ''A'' and certain linear functionals on ''A'' (called '' ...
*
Quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistr ...
**
Quantum state
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution i ...
**
Density matrix
In quantum mechanics, a density matrix (or density operator) is a matrix that describes the quantum state of a physical system. It allows for the calculation of the probabilities of the outcomes of any measurement performed upon this system, using ...
References
*
{{Ordered topological vector spaces
Functional analysis
C*-algebras