In
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, ...
, a density matrix (or density operator) is a matrix that describes the
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 in ...
of a physical system. It allows for the calculation of the probabilities of the outcomes of any
measurement
Measurement is the quantification of attributes of an object or event, which can be used to compare with other objects or events.
In other words, measurement is a process of determining how large or small a physical quantity is as compared ...
performed upon this system, using the
Born rule
The Born rule (also called Born's rule) is a key postulate of quantum mechanics which gives the probability that a measurement of a quantum system will yield a given result. In its simplest form, it states that the probability density of findi ...
. It is a generalization of the more usual state vectors or wavefunctions: while those can only represent
pure 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 in t ...
s, density matrices can also represent ''mixed states''. Mixed states arise in quantum mechanics in two different situations: first when the preparation of the system is not fully known, and thus one must deal with a
statistical ensemble
In physics, specifically statistical mechanics, an ensemble (also statistical ensemble) is an idealization consisting of a large number of virtual copies (sometimes infinitely many) of a system, considered all at once, each of which represents a ...
of possible preparations, and second when one wants to describe a physical system which is
entangled with another, without describing their combined state.
Density matrices are thus crucial tools in areas of quantum mechanics that deal with mixed states, such as
quantum statistical mechanics
Quantum statistical mechanics is statistical mechanics applied to quantum mechanical systems. In quantum mechanics a statistical ensemble (probability distribution over possible quantum states) is described by a density operator ''S'', which is ...
,
open quantum system In physics, an open quantum system is a quantum-mechanical system that interacts with an external quantum system, which is known as the ''environment'' or a ''bath''. In general, these interactions significantly change the dynamics of the system an ...
s,
quantum decoherence, and
quantum information
Quantum information is the information of the state of a quantum system. It is the basic entity of study in quantum information theory, and can be manipulated using quantum information processing techniques. Quantum information refers to both th ...
.
Definition and motivation
The density matrix is a representation of 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 mapping V \to W between two vector spaces that pre ...
called the density operator. The density matrix is obtained from the density operator by choice of
basis
Basis may refer to:
Finance and accounting
* Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
* Basis trading, a trading strategy consisting ...
in the underlying space. In practice, the terms ''density matrix'' and ''density operator'' are often used interchangeably.
In operator language, a density operator for a system is a
positive semi-definite,
Hermitian {{Short description, none
Numerous things are named after the French mathematician Charles Hermite (1822–1901):
Hermite
* Cubic Hermite spline, a type of third-degree spline
* Gauss–Hermite quadrature, an extension of Gaussian quadrature m ...
operator of
trace
Trace may refer to:
Arts and entertainment Music
* ''Trace'' (Son Volt album), 1995
* ''Trace'' (Died Pretty album), 1993
* Trace (band), a Dutch progressive rock band
* ''The Trace'' (album)
Other uses in arts and entertainment
* ''Trace'' ...
one acting on the
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 natural ...
of the system.
This definition can be motivated by considering a situation where a pure state
is prepared with probability
, known as an ''ensemble''. The probability of obtaining
projective measurement result
when using
projector
A projector or image projector is an optical device that projects an image (or moving images) onto a surface, commonly a projection screen. Most projectors create an image by shining a light through a small transparent lens, but some newer types ...
s
is given by
:
which makes the density operator, defined as
:
a convenient representation for the state of this ensemble. It is easy to check that this operator is positive semi-definite, Hermitian, and has trace one. Conversely, it follows from the
spectral theorem
In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix (mathematics), matrix can be Diagonalizable matrix, diagonalized (that is, represented as a diagonal matrix i ...
that every operator with these properties can be written as
for some states
and coefficients
that are non-negative and add up to one.
However, this representation will not be unique, as shown by the
Schrödinger–HJW theorem
In quantum information theory and quantum optics, the Schrödinger–HJW theorem is a result about the realization of a mixed state of a quantum system as an ensemble of pure quantum states and the relation between the corresponding purifications ...
.
Another motivation for the definition of density operators comes from considering local measurements on entangled states. Let
be a pure entangled state in the composite Hilbert space
. The probability of obtaining measurement result
when measuring projectors
on the Hilbert space
alone is given by
:
where
denotes the
partial trace
In linear algebra and functional analysis, the partial trace is a generalization of the trace. Whereas the trace is a scalar valued function on operators, the partial trace is an operator-valued function. The partial trace has applications in q ...
over the Hilbert space
. This makes the operator
:
a convenient tool to calculate the probabilities of these local measurements. It is known as the
reduced density matrix
Reduction, reduced, or reduce may refer to:
Science and technology Chemistry
* Reduction (chemistry), part of a reduction-oxidation (redox) reaction in which atoms have their oxidation state changed.
** Organic redox reaction, a redox react ...
of
on subsystem 1. It is easy to check that this operator has all the properties of a density operator. Conversely, the
Schrödinger–HJW theorem
In quantum information theory and quantum optics, the Schrödinger–HJW theorem is a result about the realization of a mixed state of a quantum system as an ensemble of pure quantum states and the relation between the corresponding purifications ...
implies that all density operators can be written as
for some state
.
Pure and mixed states
A pure quantum state is a state that can not be written as a probabilistic mixture, or
convex combination
In convex geometry and vector algebra, a convex combination is a linear combination of points (which can be vectors, scalars, or more generally points in an affine space) where all coefficients are non-negative and sum to 1. In other word ...
, of other quantum states.
There are several equivalent characterizations of pure states in the language of density operators.
A density operator represents a pure state if and only if:
* it can be written as an
outer product of a state vector
with itself, that is,
* it is a
projection
Projection, projections or projective may refer to:
Physics
* Projection (physics), the action/process of light, heat, or sound reflecting from a surface to another in a different direction
* The display of images by a projector
Optics, graphic ...
, in particular of
rank
Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as:
Level or position in a hierarchical organization
* Academic rank
* Diplomatic rank
* Hierarchy
* ...
one.
* it is
idempotent
Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...
, that is
* it has
purity
Purity may refer to:
Books
* ''Pureza'' (novel), a 1937 Brazilian novel by José Lins do Rego
* ''Purity'' (novel), a 2015 novel by Jonathan Franzen
** ''Purity'' (TV series), a TV series based on the novel
*''Purity'', a 2012 novel by Jackson P ...
one, that is,
It is important to emphasize the difference between a probabilistic mixture of quantum states and their
superposition. If a physical system is prepared to be either in state
or
, with equal probability, it can be described by the mixed state
:
where
and
are assumed orthogonal and of dimension 2, for simplicity. On the other hand, a quantum superposition of these two states with equal
probability amplitude
In quantum mechanics, a probability amplitude is a complex number used for describing the behaviour of systems. The modulus squared of this quantity represents a probability density.
Probability amplitudes provide a relationship between the quan ...
s results in the pure state
with density matrix
:
Unlike the probabilistic mixture, this superposition can display
quantum interference
In physics, interference is a phenomenon in which two waves combine by adding their displacement together at every single point in space and time, to form a resultant wave of greater, lower, or the same amplitude. Constructive and destructive ...
.
![Bloch sphere](https://upload.wikimedia.org/wikipedia/commons/6/6b/Bloch_sphere.svg)
Geometrically, the set of density operators is a
convex set
In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex r ...
, and the pure states are the
extremal point
In mathematics, an extreme point of a convex set S in a real or complex vector space is a point in S which does not lie in any open line segment joining two points of S. In linear programming problems, an extreme point is also called vertex ...
s of that set. The simplest case is that of a two-dimensional Hilbert space, known as a
qubit
In quantum computing, a qubit () or quantum bit is a basic unit of quantum information—the quantum version of the classic binary bit physically realized with a two-state device. A qubit is a two-state (or two-level) quantum-mechanical system, ...
. An arbitrary state for a qubit can be written as a linear combination of the
Pauli matrices
In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used in ...
, which together with the identity matrix provide a basis for
self-adjoint matrices:
:
where the real numbers
are the coordinates of a point within the
unit ball
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a discrete piece of action (or beat) in a theatrical presentation
Music
* ''Unit'' (a ...
and
:
Points with
represent pure states, while mixed states are represented by points in the interior. This is known as the
Bloch sphere
In quantum quantum mechanics, mechanics and Quantum computing, computing, the Bloch sphere is a geometrical representation of the pure state space of a two-level system, two-level quantum mechanical system (qubit), named after the physicist Felix ...
picture of qubit state space.
Example: light polarization
![vertical polarization](https://upload.wikimedia.org/wikipedia/commons/6/6d/Vertical_polarization.svg)
An example of pure and mixed states is
light polarization
Polarization ( also polarisation) is a property applying to transverse waves that specifies the geometrical orientation of the oscillations. In a transverse wave, the direction of the oscillation is perpendicular to the direction of motion of t ...
. An individual
photon
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they always ...
can be described as having right or left
circular polarization
In electrodynamics, circular polarization of an electromagnetic wave is a polarization state in which, at each point, the electromagnetic field of the wave has a constant magnitude and is rotating at a constant rate in a plane perpendicular to t ...
, described by the orthogonal quantum states
and
or a
superposition of the two: it can be in any state
(with
), corresponding to
linear
Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear r ...
,
circular
Circular may refer to:
* The shape of a circle
* ''Circular'' (album), a 2006 album by Spanish singer Vega
* Circular letter (disambiguation)
** Flyer (pamphlet), a form of advertisement
* Circular reasoning, a type of logical fallacy
* Circula ...
, or
elliptical polarization
In electrodynamics, elliptical polarization is the polarization of electromagnetic radiation such that the tip of the electric field vector describes an ellipse in any fixed plane intersecting, and normal to, the direction of propagation. An el ...
. Consider now a vertically polarized photon, described by the state
. If we pass it through a
circular polarizer
A polarizer or polariser is an optical filter that lets light waves of a specific polarization pass through while blocking light waves of other polarizations. It can filter a beam of light of undefined or mixed polarization into a beam of wel ...
which allows either only
polarized light, or only
polarized light, half of the photons are absorbed in both cases. This may make it ''seem'' like half of the photons are in state
and the other half in state
, but this is not correct: if we pass
through a
linear polarizer
A polarizer or polariser is an optical filter that lets light waves of a specific polarization pass through while blocking light waves of other polarizations. It can filter a beam of light of undefined or mixed polarization into a beam of well ...
there's no absorption whatsoever, but if we pass either state
or
half of the photons are absorbed.
Unpolarized light Unpolarized light is light with a random, time-varying polarization.
Natural light, like most other common sources of visible light, produced independently by a large number of atoms or molecules whose emissions are uncorrelated.
This term is so ...
(such as the light from an
incandescent light bulb
An incandescent light bulb, incandescent lamp or incandescent light globe is an electric light with a wire filament heated until it glows. The filament is enclosed in a glass bulb with a vacuum or inert gas to protect the filament from oxid ...
) cannot be described as ''any'' state of the form
(linear, circular, or elliptical polarization). Unlike polarized light, it passes through a polarizer with 50% intensity loss whatever the orientation of the polarizer; and it cannot be made polarized by passing it through any
wave plate
A waveplate or retarder is an optics, optical device that alters the Polarization (waves), polarization state of a light wave travelling through it. Two common types of waveplates are the ''half-wave plate'', which shifts the polarization directio ...
. However, unpolarized light ''can'' be described as a statistical ensemble, e. g. as each photon having either
polarization or
polarization with probability 1/2. The same behavior would occur if each photon had either vertical polarization
or horizontal polarization
with probability 1/2. These two ensembles are completely indistinguishable experimentally, and therefore they are considered the same mixed state. For this example of unpolarized light, the density operator equals
:
There are also other ways to generate unpolarized light: one possibility is to introduce uncertainty in the preparation of the photon, for example, passing it through a
birefringent crystal with a rough surface, so that slightly different parts of the light beam acquire different polarizations. Another possibility is using entangled states: a radioactive decay can emit two photons traveling in opposite directions, in the quantum state
. The joint state of the two photons ''together'' is pure, but the density matrix for each photon individually, found by taking the partial trace of the joint density matrix, is completely mixed.
Equivalent ensembles and purifications
A given density operator does not uniquely determine which ensemble of pure states gives rise to it; in general there are infinitely many different ensembles generating the same density matrix. Those cannot be distinguished by any measurement. The equivalent ensembles can be completely characterized: let
be an ensemble. Then for any complex matrix
such that
(a
partial isometry In functional analysis a partial isometry is a linear map between Hilbert spaces such that it is an isometry on the orthogonal complement of its kernel.
The orthogonal complement of its kernel is called the initial subspace and its range is called ...
), the ensemble
defined by
:
will give rise to the same density operator, and all equivalent ensembles are of this form.
A closely related fact is that a given density operator has infinitely many different
purifications, which are pure states that generate the density operator when a partial trace is taken. Let
:
be the density operator generated by the ensemble
, with states
not necessarily orthogonal. Then for all partial isometries
we have that
:
is a purification of
, where
is an orthogonal basis, and furthermore all purifications of
are of this form.
Measurement
Let
be an
observable
In physics, an observable is a physical quantity that can be measured. Examples include position and momentum. In systems governed by classical mechanics, it is a real-valued "function" on the set of all possible system states. In quantum ph ...
of the system, and suppose the ensemble is in a mixed state such that each of the pure states
occurs with probability
. Then the corresponding density operator equals
:
The
expectation value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...
of the
measurement
Measurement is the quantification of attributes of an object or event, which can be used to compare with other objects or events.
In other words, measurement is a process of determining how large or small a physical quantity is as compared ...
can be calculated by extending from the case of pure states:
:
where
denotes
trace
Trace may refer to:
Arts and entertainment Music
* ''Trace'' (Son Volt album), 1995
* ''Trace'' (Died Pretty album), 1993
* Trace (band), a Dutch progressive rock band
* ''The Trace'' (album)
Other uses in arts and entertainment
* ''Trace'' ...
. Thus, the familiar expression
for pure states is replaced by
:
for mixed states.
Moreover, if
has spectral resolution
:
where
is the
projection operator
In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if it wer ...
into the
eigenspace
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
corresponding to eigenvalue
, the post-measurement density operator is given by
:
when outcome ''i'' is obtained. In the case where the measurement result is not known the ensemble is instead described by
:
If one assumes that the probabilities of measurement outcomes are linear functions of the projectors
, then they must be given by the trace of the projector with a density operator.
Gleason's theorem
In mathematical physics, Gleason's theorem shows that the rule one uses to calculate probabilities in quantum physics, the Born rule, can be derived from the usual mathematical representation of measurements in quantum physics together with the ...
shows that in Hilbert spaces of dimension 3 or larger the assumption of linearity can be replaced with an assumption of
non-contextuality. This restriction on the dimension can be removed by assuming non-contextuality for
POVM
In functional analysis and quantum measurement theory, a positive operator-valued measure (POVM) is a measure whose values are positive semi-definite operators on a Hilbert space. POVMs are a generalisation of projection-valued measures (PVM) and ...
s as well, but this has been criticized as physically unmotivated.
Entropy
The
von Neumann entropy
In physics, the von Neumann entropy, named after John von Neumann, is an extension of the concept of Gibbs entropy from classical statistical mechanics to quantum statistical mechanics. For a quantum-mechanical system described by a density matrix ...
of a mixture can be expressed in terms of the eigenvalues of
or in terms of the
trace
Trace may refer to:
Arts and entertainment Music
* ''Trace'' (Son Volt album), 1995
* ''Trace'' (Died Pretty album), 1993
* Trace (band), a Dutch progressive rock band
* ''The Trace'' (album)
Other uses in arts and entertainment
* ''Trace'' ...
and
logarithm
In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 o ...
of the density operator
. Since
is a positive semi-definite operator, it has a
spectral decomposition such that
, where
are orthonormal vectors,
, and
. Then the entropy of a quantum system with density matrix
is
:
This definition implies that the von Neumann entropy of any pure state is zero.
If
are states that have support on orthogonal subspaces, then the von Neumann entropy of a convex combination of these states,
:
is given by the von Neumann entropies of the states
and the
Shannon entropy
Shannon may refer to:
People
* Shannon (given name)
* Shannon (surname)
* Shannon (American singer), stage name of singer Shannon Brenda Greene (born 1958)
* Shannon (South Korean singer), British-South Korean singer and actress Shannon Arrum W ...
of the probability distribution
:
:
When the states
do not have orthogonal supports, the sum on the right-hand side is strictly greater than the von Neumann entropy of the convex combination
.
Given a density operator
and a projective measurement as in the previous section, the state
defined by the convex combination
:
which can be interpreted as the state produced by performing the measurement but not recording which outcome occurred,
has a von Neumann entropy larger than that of
, except if
. It is however possible for the
produced by a ''generalized'' measurement, or
POVM
In functional analysis and quantum measurement theory, a positive operator-valued measure (POVM) is a measure whose values are positive semi-definite operators on a Hilbert space. POVMs are a generalisation of projection-valued measures (PVM) and ...
, to have a lower von Neumann entropy than
.
[.]
The von Neumann equation for time evolution
Just as the
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
describes how pure states evolve in time, the von Neumann equation (also known as the Liouville–von Neumann equation) describes how a density operator evolves in time. The von Neumann equation dictates that
:
where the brackets denote a
commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.
Group theory
The commutator of two elements, a ...
.
This equation only holds when the density operator is taken to be in the
Schrödinger picture
In physics, the Schrödinger picture is a formulation of quantum mechanics in which the state vectors evolve in time, but the operators (observables and others) are mostly constant with respect to time (an exception is the Hamiltonian which may ...
, even though this equation seems at first look to emulate the Heisenberg equation of motion in the
Heisenberg picture
In physics, the Heisenberg picture (also called the Heisenberg representation) is a formulation (largely due to Werner Heisenberg in 1925) of quantum mechanics in which the operators (observables and others) incorporate a dependency on time, but ...
, with a crucial sign difference:
:
where
is some ''Heisenberg picture'' operator; but in this picture the density matrix is ''not time-dependent'', and the relative sign ensures that the time derivative of the expected value
comes out ''the same as in the Schrödinger picture''.
[
If the Hamiltonian is time-independent, the von Neumann equation can be easily solved to yield
:
For a more general Hamiltonian, if is the wavefunction propagator over some interval, then the time evolution of the density matrix over that same interval is given by
:
]
Wigner functions and classical analogies
The density matrix operator may also be realized in phase space
In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usually ...
. Under the Wigner map, the density matrix transforms into the equivalent Wigner function,
:
The equation for the time evolution of the Wigner function, known as Moyal equation, is then the Wigner-transform of the above von Neumann equation,
:
where is the Hamiltonian, and is the Moyal bracket
In physics, the Moyal bracket is the suitably normalized antisymmetrization of the phase-space star product.
The Moyal bracket was developed in about 1940 by José Enrique Moyal, but Moyal only succeeded in publishing his work in 1949 after a le ...
, the transform of the quantum commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.
Group theory
The commutator of two elements, a ...
.
The evolution equation for the Wigner function is then analogous to that of its classical limit, the Liouville equation
:''For Liouville's equation in dynamical systems, see Liouville's theorem (Hamiltonian).''
: ''For Liouville's equation in quantum mechanics, see Von Neumann equation.''
: ''For Liouville's equation in Euclidean space, see Liouville–Bratu–Gel ...
of classical physics
Classical physics is a group of physics theories that predate modern, more complete, or more widely applicable theories. If a currently accepted theory is considered to be modern, and its introduction represented a major paradigm shift, then the ...
. In the limit of vanishing Planck's constant , reduces to the classical Liouville probability density function in phase space
In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usually ...
.
Example applications
Density matrices are a basic tool of quantum mechanics, and appear at least occasionally in almost any type of quantum-mechanical calculation. Some specific examples where density matrices are especially helpful and common are as follows:
* Statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
uses density matrices, most prominently to express the idea that a system is prepared at a nonzero temperature. Constructing a density matrix using a canonical ensemble
In statistical mechanics, a canonical ensemble is the statistical ensemble that represents the possible states of a mechanical system in thermal equilibrium with a heat bath at a fixed temperature. The system can exchange energy with the heat b ...
gives a result of the form , where is the inverse temperature and is the system's Hamiltonian. The normalization condition that the trace of be equal to 1 defines the partition function to be . If the number of particles involved in the system is itself not certain, then a grand canonical ensemble
In statistical mechanics, the grand canonical ensemble (also known as the macrocanonical ensemble) is the statistical ensemble that is used to represent the possible states of a mechanical system of particles that are in thermodynamic equilibrium ...
can be applied, where the states summed over to make the density matrix are drawn from a Fock space
The Fock space is an algebraic construction used in quantum mechanics to construct the quantum states space of a variable or unknown number of identical particles from a single particle Hilbert space . It is named after V. A. Fock who first intr ...
.
* Quantum decoherence theory typically involves non-isolated quantum systems developing entanglement with other systems, including measurement apparatuses. Density matrices make it much easier to describe the process and calculate its consequences. Quantum decoherence explains why a system interacting with an environment transitions from being a pure state, exhibiting superpositions, to a mixed state, an incoherent combination of classical alternatives. This transition is fundamentally reversible, as the combined state of system and environment is still pure, but for all practical purposes irreversible, as the environment is a very large and complex quantum system, and it is not feasible to reverse their interaction. Decoherence is thus very important for explaining the classical limit
The classical limit or correspondence limit is the ability of a physical theory to approximate or "recover" classical mechanics when considered over special values of its parameters. The classical limit is used with physical theories that predict n ...
of quantum mechanics, but cannot explain wave function collapse, as all classical alternatives are still present in the mixed state, and wave function collapse selects only one of them.
* Similarly, in quantum computation
Quantum computing is a type of computation whose operations can harness the phenomena of quantum mechanics, such as superposition, interference, and entanglement. Devices that perform quantum computations are known as quantum computers. Though ...
, quantum information theory
Quantum information is the information of the quantum state, state of a quantum system. It is the basic entity of study in quantum information theory, and can be manipulated using quantum information processing techniques. Quantum information re ...
, open quantum system In physics, an open quantum system is a quantum-mechanical system that interacts with an external quantum system, which is known as the ''environment'' or a ''bath''. In general, these interactions significantly change the dynamics of the system an ...
s, and other fields where state preparation is noisy and decoherence can occur, density matrices are frequently used. Noise is often modelled via a depolarizing channel or an amplitude damping channel In the theory of quantum communication, an amplitude damping channel is a quantum channel that models physical processes such as spontaneous emission. A natural process by which this channel can occur is a spin chain through which a number of spin ...
. Quantum tomography
Quantum tomography or quantum state tomography is the process by which a quantum state is reconstructed using measurements on an ensemble of identical quantum states. The source of these states may be any device or system which prepares quantum st ...
is a process by which, given a set of data representing the results of quantum measurements, a density matrix consistent with those measurement results is computed.
* When analyzing a system with many electrons, such as an atom
Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons.
Every solid, liquid, gas, and ...
or molecule
A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bioch ...
, an imperfect but useful first approximation is to treat the electrons as uncorrelated
In probability theory and statistics, two real-valued random variables, X, Y, are said to be uncorrelated if their covariance, \operatorname ,Y= \operatorname Y- \operatorname \operatorname /math>, is zero. If two variables are uncorrelated, there ...
or each having an independent single-particle wavefunction. This is the usual starting point when building the Slater determinant
In quantum mechanics, a Slater determinant is an expression that describes the wave function of a multi-fermionic system. It satisfies anti-symmetry requirements, and consequently the Pauli principle, by changing sign upon exchange of two elect ...
in the Hartree–Fock method. If there are electrons filling the single-particle wavefunctions , then the collection of electrons together can be characterized by a density matrix .
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. For this reason, observables are identified with elements of an abstract C*-algebra
In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of continuous ...
''A'' (that is one without a distinguished representation as an algebra of operators) and states are positive linear functional
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers).
If is a vector space over a field , the ...
s on ''A''. However, by using the GNS construction
GNS may refer to:
Places
* Binaka Airport, in Gunung Sitoli, Nias Island, Indonesia
* Gainesville station (Georgia), an Amtrak station in Georgia, United States
Companies and organizations
* Gesellschaft für Nuklear-Service, a German nuclear ...
, we can recover Hilbert spaces which realize ''A'' as a subalgebra of operators.
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 representation
In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper nontrivial subrepresentation (\rho, _W,W ...
s of ''A''.
The states of the C*-algebra of compact operator
In functional analysis, a branch of mathematics, a compact operator is a linear operator T: X \to Y, where X,Y are normed vector spaces, with the property that T maps bounded subsets of X to relatively compact subsets of Y (subsets with compact c ...
s ''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.
History
The formalism of density operators and matrices was introduced in 1927 by John von Neumann
John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest cove ...
and independently, but less systematically, by Lev Landau
Lev Davidovich Landau (russian: Лев Дави́дович Ланда́у; 22 January 1908 – 1 April 1968) was a Soviet- Azerbaijani physicist of Jewish descent who made fundamental contributions to many areas of theoretical physics.
His a ...
and later in 1946 by Felix Bloch
Felix Bloch (23 October 1905 – 10 September 1983) was a Swiss-American physicist and Nobel physics laureate who worked mainly in the U.S. He and Edward Mills Purcell were awarded the 1952 Nobel Prize for Physics for "their development of ne ...
. Von Neumann introduced the density matrix in order to develop both quantum statistical mechanics and a theory of quantum measurements. The name density matrix itself relates to its classical correspondence to a phase-space
In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usuall ...
probability measure (probability distribution of position and momentum) in classical statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
, which was introduced by Wigner in 1932.
In contrast, the motivation that inspired Landau was the impossibility of describing a subsystem of a composite quantum system by a state vector.
See also
* Atomic electron transition
Atomic electron transition is a change (or jump) of an electron from one energy level to another within an atom or artificial atom. It appears discontinuous as the electron "jumps" from one quantized energy level to another, typically in a few na ...
* Density functional theory
Density-functional theory (DFT) is a computational quantum mechanical modelling method used in physics, chemistry and materials science to investigate the electronic structure (or nuclear structure) (principally the ground state) of many-body ...
* Green–Kubo relations
The Green–Kubo relations ( Melville S. Green 1954, Ryogo Kubo 1957) give the exact mathematical expression for transport coefficients \gamma in terms of integrals of time correlation functions:
:\gamma = \int_0^\infty \left\langle \dot(t) \dot ...
* Green's function (many-body theory)
In many-body theory, the term Green's function (or Green function) is sometimes used interchangeably with correlation function, but refers specifically to correlators of field operators or creation and annihilation operators.
The name comes from ...
* Lindblad equation
In quantum mechanics, the Gorini–Kossakowski–Sudarshan–Lindblad equation (GKSL equation, named after Vittorio Gorini, Andrzej Kossakowski, George Sudarshan and Göran Lindblad), master equation in Lindblad form, quantum Liouvillian, or Li ...
* Wigner quasi-probability distribution
The Wigner quasiprobability distribution (also called the Wigner function or the Wigner–Ville distribution, after Eugene Wigner and :fr:Jean Ville, Jean-André Ville) is a quasiprobability distribution. It was introduced by Eugene Wigner in 193 ...
Notes and references
{{Quantum mechanics topics
Functional analysis
Quantum information science
Statistical mechanics
Lev Landau