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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
(specifically
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrice ...
, operator theory, and
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defi ...
) as well as physics, a
linear operator In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
A acting on an
inner product space In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
is called positive-semidefinite (or ''non-negative'') if, for every x \in \mathop(A), \langle Ax, x\rangle \in \mathbb and \langle Ax, x\rangle \geq 0, where \mathop(A) is the
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function ** Natural domain of a partial function **Domain of holomorphy of a function * ...
of A. Positive-semidefinite operators are denoted as A\ge 0. The operator is said to be positive-definite, and written A>0, if \langle Ax,x\rangle>0, for all x\in\mathop(A) \setminus \. In physics (specifically
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 chemistry, ...
), such operators 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, via the
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 ...
formalism.


Cauchy–Schwarz inequality

If A \geq 0, then :\left, \langle Ax,y\rangle \^2 \leq \langle Ax,x\rangle \langle Ay,y\rangle. Indeed, let \varepsilon > 0. Applying Cauchy–Schwarz inequality to the inner product : (x,y)_\varepsilon \stackrel\ \langle (A+\varepsilon\cdot \mathbf)x,y\rangle as \varepsilon \downarrow 0 proves the claim. It follows that \mathopA \perp \mathopA. If A is defined everywhere, and \langle Ax,x\rangle = 0, then Ax = 0.


On ''H'', if ''A'' ≥ 0 then ''A'' is symmetric

Without loss of generality, let the inner product \langle \cdot, \cdot \rangle be anti-linear on the ''first'' argument and linear on the second. (If the reverse is true, then we work with \langle x,y\rangle_\text \stackrel \ \langle y,x\rangle instead). For x,y \in \mathopA, the
polarization identity In linear algebra, a branch of mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space. If a norm arises from an inner product the ...
: \begin \langle Ax,y\rangle = \frac( & \langle A(x+y),x+y\rangle - \langle A(x-y),x-y\rangle \\ mm& - i\langle A(x+iy),x+iy\rangle + i\langle A(x-iy),x-iy\rangle) \end and the fact that \langle Ax,x\rangle = \langle x,Ax\rangle, for positive operators, show that \langle Ax,y\rangle = \langle x,Ay\rangle, so A is symmetric. In contrast with the complex case, a positive-semidefinite operator on a real Hilbert space H_\mathbb may not be symmetric. As a counterexample, define A : \mathbb^2 \to \mathbb^2 to be an operator of rotation by an acute angle \varphi \in ( -\pi/2,\pi/2). Then \langle Ax,x \rangle = \, Ax\, \, x\, \cos\varphi > 0, but A^* = A^ \neq A, so A is not symmetric.


If ''A'' ≥ 0 and Dom ''A'' = ''H'', then ''A'' is self-adjoint and bounded

The symmetry of A implies that \mathopA \subseteq \mathopA^* and A = A^*, _. For A to be self-adjoint, it is necessary that \mathopA = \mathopA^*. In our case, the equality of domains holds because H_\mathbb = \mathopA \subseteq \mathopA^*, so A is indeed self-adjoint. The fact that A is bounded now follows from the
Hellinger–Toeplitz theorem In functional analysis, a branch of mathematics, the Hellinger–Toeplitz theorem states that an everywhere-defined symmetric operator on a Hilbert space with inner product \langle \cdot , \cdot \rangle is bounded. By definition, an operator ...
. This property does not hold on H_\mathbb.


Order in self-adjoint operators on ''H''

A natural ordering of self-adjoint operators arises from the definition of positive operators. Define B \geq A if the following hold: # A and B are self-adjoint # B - A \geq 0 It can be seen that a similar result as the
Monotone convergence theorem In the mathematical field of real analysis, the monotone convergence theorem is any of a number of related theorems proving the convergence of monotonic sequences (sequences that are non-decreasing or non-increasing) that are also bounded. Infor ...
holds for monotone increasing, bounded, self-adjoint operators on Hilbert spaces.Eidelman, Yuli, Vitali D. Milman, and Antonis Tsolomitis. 2004. Functional analysis: an introduction. Providence (R.I.): American mathematical Society.


Application to physics: quantum states

The definition of a quantum system includes a complex
separable Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natur ...
H_\mathbb and a set \cal S of positive trace-class
operators Operator may refer to: Mathematics * A symbol indicating a mathematical operation * Logical operator or logical connective in mathematical logic * Operator (mathematics), mapping that acts on elements of a space to produce elements of another sp ...
\rho on H_\mathbb for which \mathop\rho = 1. The set \cal S is ''the set of states''. Every \rho \in is called a ''state'' or a ''density operator''. For \psi \in H_\mathbb, where \, \psi\, = 1, the operator P_\psi of projection onto the span of \psi is called a ''pure state''. (Since each pure state is identifiable with a unit vector \psi \in H_\mathbb, some sources define pure states to be unit elements from H_\mathbb). States that are not pure are called ''mixed''.


References

* {{Citation , last1=Conway , first1=John , title=Functional Analysis: An Introduction , publisher=
Springer Verlag Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 i ...
, isbn=0-387-97245-5 , year=1990 Operator theory