NationMaster - Encyclopedia: Density matrix

A density matrix is a self-adjoint (or Hermitian) positive-semidefinite matrix, (possibly infinite dimensional), of trace one, that describes the statistical state of a quantum system. The formalism was introduced by John von Neumann (according to other sources independently by Lev Landau and Felix Bloch) in 1927. For the square matrix section, see square matrix. ... In mathematics, an element x of a star-algebra is self-adjoint if the involution acts trivially upon it. ... A number of mathematical entities are named Hermitian, after the mathematician Charles Hermite: Hermitian matrix Hermitian operator Hermitian adjoint Hermitian form Hermitian metric See also: self-adjoint This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... In linear algebra, the positive-definite matrices are (in several ways) analogous to the positive real numbers. ... In mathematics, a bounded linear operator A over a Hilbert space H is said to be in the trace class if for some (and hence all) orthonormal bases {ek}k of H the sum of positive terms is finite. ... Fig. ... John von Neumann (Hungarian Margittai Neumann JÃ¡nos Lajos) (born December 28, 1903 in Budapest, Austria-Hungary; died February 8, 1957 in Washington D.C., United States) was a Hungarian-born mathematician and polymath who made contributions to quantum physics, functional analysis, set theory, topology, economics, computer science, numerical analysis... Lev Davidovich Landau Lev Davidovich Landau (Russian language: Ð›ÐµÌÐ² Ð”Ð°Ð²Ð¸ÌÐ´Ð¾Ð²Ð¸Ñ‡ Ð›Ð°Ð½Ð´Ð°ÌÑƒ) (January 22, 1908 â€“ April 1, 1968) was a prominent Soviet physicist, who made fundamental contributions to many areas of theoretical physics. ... Felix Bloch. ...

It is the quantum-mechanical analogue to a phase-space probability measure (probability distribution of position and momentum) in classical statistical mechanics. The need for a statistical description via density matrices arises when one considers either an ensemble of systems, or one system when its preparation history is uncertain. For other senses of this term, see phase space (disambiguation). ... In mathematics, a probability space is a set S, together with a σ-algebra X on S and a measure P on that σ-algebra such that P(S) = 1. ...

Situations in which a density matrix is used include: a quantum system in thermal equilibrium (at finite temperatures); nonequilibrium time-evolution that starts out of a mixed equilibrium state; and entanglement between two subsystems, where each individual system must be described, via the partial trace operation, by a density matrix even though the complete system may be in a pure state; and in analysis of quantum decoherence. See also quantum statistical mechanics. It has been suggested that Quantum coherence be merged into this article or section. ... In linear algebra and functional analysis, the partial trace is a generalization of the trace. ... In quantum mechanics, quantum decoherence is the mechanism by which quantum systems interact with their environments to exhibit probabilistically additive behavior - a feature of classical physics - and give the appearance of wavefunction collapse. ... Quantum statistical mechanics is the study of statistical ensembles of quantum mechanical systems. ...

A density operator is an operator corresponding to a density matrix under some orthonormal basis. Thus it is a non-negative, self-adjoint, trace class operator of trace one. In mathematics, an operator is a function that performs some sort of operation on a number, variable, or function. ... In mathematics, an orthonormal basis of an inner product space V(i. ... A negative number is a number that is less than zero, such as −3. ... On a finite-dimensional inner product space, a self-adjoint operator is one that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose. ... In mathematics, a bounded linear operator A over a Hilbert space H is said to be in the trace class if for some (and hence all) orthonormal bases {ek}k of H the sum of positive terms is finite. ...

The need for a statistical description

By quantum mechanics, the state vector ψ^[1] of a system completely determines the statistical behavior of an observable O. This means that if O is represented by an operator A on the Hilbert space H of the system, then for any real-valued function F^[2] defined on the real numbers, the expectation value of F(O) ^[3] is the quantity The mathematical formulation of quantum mechanics is the body of mathematical formalisms which permits a rigorous description of quantum mechanics. ... Quite literally, quantum state describes the state of a quantum system. ... In physics, particularly in quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical operations. ... In mathematics, a Hilbert space is a real or complex vector space with a positive definite sesquilinear form, that is complete under its norm. ...

in Dirac notation. Bra-ket notation is the standard notation for describing quantum states in the theory of quantum mechanics. ...

Now consider the example of a "mixed quantum system" prepared by statistically combining two different pure states φ, ψ each with probability 1/2. The preparation process for such a system consists in tossing an unbiased coin and using the preparation process for φ or for ψ depending on whether the toss outcome is heads or tails.

It is not hard to show that the statistical properties of the observable O for the system prepared in such a mixed state are completely determined. However, there is no vector ξ which determines this statistical behavior in the sense that the expectation value of F(O) is

Nevertheless: there is a unique operator ρ such that the expectation value can be written as

The operator ρ is the density operator of the mixed system. A simple calculation shows that for the example mentioned above:

Formulation

For a finite dimensional Hilbert space, the most general density matrix is of the form

where the coefficients p_j are non-negative and add up to one. This represents a statistical mixture of pure states. If the given system is closed, then one can think of a mixed state as representing a single system with an uncertain preparation history, as explicitly detailed above; or we can regard the mixed state as representing an ensemble of systems, i.e. large number of copies of the system in question, where p_j is the proportion of the ensemble being in the state $|psi_j rang$ . An ensemble is described by a pure state if every copy of the system in that ensemble is in the same state, i.e. it is a pure ensemble. In mathematical physics, especially as introduced into statistical mechanics and thermodynamics by J. Willard Gibbs in 1878, an ensemble (also statistical ensemble or thermodynamic ensemble) is an idealization consisting of a large number of mental copies (possibly infinitely many) of a system, considered all at once, each of which represents...

It must be noted, however, that if the system is not closed, then it is simply not correct to claim that it has some definite but unknown state vector, as the density operator may record physical entanglements to other systems.

Example Consider a quantum ensemble of size N with occupancy numbers n₁, n₂,...,n_k corresponding to the orthonormal states |1>,...,|k>, respectively, where n₁+...+n_k = N, and, thus, the coefficients p_j = n_j /N. For a pure ensemble, where all N particles are in state |i>, we have n_j = 0, for all j ≠ i, from which we recover the corresponding density matrix ρ = |i >< i|.

However the density matrix of a mixed state does not capture all the information about a mixture; in particular, the coefficients p_j and the kets ψ_j are not recoverable from the matrix ρ without additional information. This non-uniqueness implies that different ensembles or mixtures may correspond to the same density matrix. Such equivalent ensembles or mixtures cannot be distinguished by measurement of observables alone. This equivalence can be characterized precisely. Two ensembles ψ, ψ' define the same density operator if and only if there is a matrix U with

i.e, U is unitary and such that In government, see Unitary state In mathematics, see Unitary matrix Unitary operator Unitary group Unitary representation This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ...

This is simply a restatement of the following fact from linear algebra: for two square matrices M and N, M M^* = N N^* if and only if M = NU for some unitary U. (See square root of a matrix for more details.) Thus there is an unitary freedom in the ket mixture or ensemble that gives the same density operator. However if the kets in the mixture are orthonormal then the original probabilities p_j are recoverable as the eigenvalues of the density matrix. The square root of a matrix A is a matrix B such that the matrix product B B is equal to A. Numerical method Given a square matrix A, one way to find its square root is the Denman-Beavers square root iteration, described below: Given matrix A, and the... In linear algebra, two vectors v and w are said to be orthonormal if they are both orthogonal (according to a given inner product) and normalized. ...

In operator language, a density operator is a positive semidefinite, hermitian operator acting on the state space of trace 1. A density operator describes a pure state if it is a rank one projection. Equivalently, a density operator ρ is a pure state if and only if In mathematics, a definite bilinear form B is one for which B(v,v) has a fixed sign (positive or negative) when it is not 0. ... A number of mathematical entities are named Hermitian, after the mathematician Charles Hermite: Hermitian matrix Hermitian operator Hermitian adjoint Hermitian form Hermitian metric See also: self-adjoint This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ...

i.e. the state is idempotent. This is true regardless of whether H is finite dimensional or not. In mathematics, an idempotent element is an element which, intuitively, leaves something unchanged. ...

Geometrically, when the state is not expressible as a convex combination of other states, it is a pure state. The family of mixed states is a convex set and a state is pure if it is an extremal point of that set. A convex combination is a linear combination of data points (which can be vectors or scalars) where all coefficients are positive and sum up to 1. ... A convex set in light blue, and its extreme points in red. ...

It follows from the spectral theorem for compact self-adjoint operators that every mixed state is an infinite convex combination of pure states. This representation is not unique. Furthermore, a theorem of Andrew Gleason states that certain functions defined on the family of projections and taking values in [0,1] (which can be regarded as quantum analogues of probability measures) are determined by unique mixed states. See quantum logic for more details. In functional analysis, compact operators on Hilbert spaces are a direct extension of matrices: in the Hilbert spaces, they are the precisely the closure of finite rank operators in the uniform operator topology. ... Andrew Mattei Gleason is an American mathematician. ... In mathematical physics and quantum mechanics, quantum logic can be regarded as a kind of propositional logic suitable for understanding the apparent anomalies regarding quantum measurement, most notably those concerning composition of measurement operations of complementary variables. ...

Measurement

Let A be an observable of the system, and suppose the ensemble (copies of the system in question, or one system in an uncertain state) is prepared in mixed state. The corresponding density operator is:

The average value of the measurement can be calculated by extending from the case of vectorial pure states:

where $P_i = |a_i rang lang a_i|$ , the corresponding density operator after the measurement is given by:

Entropy

The von Neumann entropy

S

of a mixture can be expressed in terms of the probabilities

p i

or in terms of the trace and logarithm of the density matrix

ρ

: Quantum statistical mechanics is the study of statistical ensembles of quantum mechanical systems. ... Look up Trace in Wiktionary, the free dictionary. ... Logarithms to various bases: is to base e, is to base 10, and is to base 1. ...

This entropy can increase but never decrease^[4]^[5] with a measurement. The entropy of a pure state is zero, while that of a proper mixture always greater than zero. Therefore a pure state may be converted into a mixture by a measurement, but a proper mixture can never be converted into a pure state. Thus the act of measurement induces a fundamental irreversible change on the density matrix; this is analogous to the "collapse" of the state vector, or wavefunction collapse. Movie Poster for Irréversible Irréversible (2002, France) is a film written, directed, edited, and photographed by Gaspar Noé. It is considered to be one of the most controversial and disturbing films ever made, due to its explicit on-camera depiction of rape and a vengeful murder. ... In certain interpretations of quantum mechanics, wavefunction collapse is one of two processes by which quantum systems apparently evolve according to the laws of quantum mechanics. ...

C*-algebraic formulation of states

It is now generally accepted that the description of quantum mechanics in which all self-adjoint operators represent observables is untenable^[6]^[7]. For this reason, observables are identified to elements of an abstract C*-algebra A (that is one without a distinguished representation as an algebra of operators) and states are positive linear functionals on A. Note that by using the GNS construction, we can recover Hilbert spaces which realize A as an algebra of operators. C*-algebras are an important area of research in functional analysis. ... In functional analysis, a state on a C*-algebra is a positive linear functional of norm 1. ... In linear algebra, a branch of mathematics, a linear functional or linear form is a linear function from a vector space to its field of scalars. ... In functional analysis, given a C*-algebra A, the GNS construction establishes a correspondence between cyclic *-representations of A and certain linear functionals on A called states. ...

Geometrically, a pure state on a C*-algebra A is a state which is an extreme point of the set of all states on A. By properties of the GNS construction these states correspond to irreducible representations of A. In mathematics, the term irreducible is used in several ways. ...

The states of the C*-algebra of compact operators K(H) correspond exactly to the density operators and therefore the pure states of K(H) are exactly the pure states in the sense of quantum mechanics.

The C*-algebraic formulation can be seen to include both classical and quantum systems. When the system is classical, the algebra of observables become an abelian C*-algebra. In that case the states become probability measures, as noted in the introduction.

Notes and references

The headquarters of the Cambridge University Press, in Trumpington Street, Cambridge. ... Hugh Everett III (November 11, 1930 â€“ July 19, 1982) was an American physicist who first proposed the many-worlds interpretation of quantum physics, which he called his relative state formulation. ... The Princeton University Press is a publishing house, a division of Princeton University, that is highly respected in academic publishing. ...

Contents

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