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{{Quantum book backlink|Mathematical structure and systems}}
'''Density matrix''' is an operator used in [[Physics:quantum mechanics|quantum mechanics]] to describe the state of a quantum system. It provides a unified formalism for both ''pure states'', represented by state vectors, and ''mixed states'', represented by statistical ensembles of state vectors.<ref>John von Neumann, ''Mathematical Foundations of Quantum Mechanics'', Princeton University Press, 1955.</ref><ref>R. Shankar, ''Principles of Quantum Mechanics'', 2nd ed., Springer, 1994.</ref> The density matrix is important in quantum statistical mechanics, quantum measurement theory, and the theory of [[Physics:open quantum system|open quantum systems]], where a subsystem is generally not described by a single wavefunction.<ref>H.-P. Breuer and F. Petruccione, ''The Theory of Open Quantum Systems'', Oxford University Press, 2002.</ref>
[[File:Density_matrix_pure_vs_mixed_states.png|right|500px]]

== Definition ==

A quantum state may be represented by a density matrix <math>\rho</math>, which is a linear operator acting on the system's [[Hilbert space]]. For a pure state <math>|\psi\rangle</math>, the density matrix is

<math>\rho = |\psi\rangle\langle\psi|.</math><ref>J. J. Sakurai and Jim Napolitano, ''Modern Quantum Mechanics'', 2nd ed., Addison-Wesley, 2011.</ref>

More generally, for an ensemble of states <math>\{|\psi_i\rangle\}</math> occurring with probabilities <math>p_i</math>, the density matrix is

<math>\rho = \sum_i p_i |\psi_i\rangle\langle\psi_i|,</math>

with

<math>p_i \ge 0, \qquad \sum_i p_i = 1.</math><ref>Michael A. Nielsen and Isaac L. Chuang, ''Quantum Computation and Quantum Information'', Cambridge University Press, 2000.</ref>

Thus the density matrix extends the usual state-vector formalism to cases where there is classical uncertainty about which pure state has been prepared.

== Properties ==

A density matrix <math>\rho</math> satisfies the following conditions:<ref>Nielsen and Chuang, ''Quantum Computation and Quantum Information'', Cambridge University Press, 2000.</ref><ref>Breuer and Petruccione, ''The Theory of Open Quantum Systems'', Oxford University Press, 2002.</ref>

# '''Hermiticity'''
#: <math>\rho = \rho^\dagger</math>
# '''Unit trace'''
#: <math>\mathrm{Tr}(\rho)=1</math>
# '''Positive semidefiniteness'''
#: <math>\rho \ge 0</math>

These conditions are not only necessary but also sufficient: any operator satisfying them is a valid density matrix.<ref>Karl Blum, ''Density Matrix Theory and Applications'', 3rd ed., Springer, 2012.</ref>

For a pure state, the density matrix is idempotent:

<math>\rho^2 = \rho.</math>

Equivalently,

<math>\mathrm{Tr}(\rho^2)=1.</math>

For a mixed state,

<math>\mathrm{Tr}(\rho^2) < 1.</math><ref>Sakurai and Napolitano, ''Modern Quantum Mechanics'', 2nd ed., Addison-Wesley, 2011.</ref>

The quantity <math>\mathrm{Tr}(\rho^2)</math> is called the '''purity''' of the state.

== Matrix representation ==

If <math>\{|n\rangle\}</math> is an orthonormal basis, the density matrix may be written in components as

<math>\rho_{mn} = \langle m|\rho|n\rangle.</math>

For a two-level system with

<math>|\psi\rangle = \alpha|0\rangle + \beta|1\rangle,</math>

the corresponding pure-state density matrix is

<math>\rho =
\begin{pmatrix}
|\alpha|^2 & \alpha\beta^* \\
\alpha^*\beta & |\beta|^2
\end{pmatrix}.</math><ref>Shankar, ''Principles of Quantum Mechanics'', 2nd ed., Springer, 1994.</ref>

The diagonal elements represent populations in the chosen basis, while the off-diagonal elements represent quantum coherences.<ref>Breuer and Petruccione, ''The Theory of Open Quantum Systems'', Oxford University Press, 2002.</ref>

== Expectation values ==

The expectation value of an observable represented by an operator <math>A</math> is given by

<math>\langle A\rangle = \mathrm{Tr}(\rho A).</math><ref>von Neumann, ''Mathematical Foundations of Quantum Mechanics'', Princeton University Press, 1955.</ref><ref>Sakurai and Napolitano, ''Modern Quantum Mechanics'', 2nd ed., Addison-Wesley, 2011.</ref>

This formula applies to both pure and mixed states, which is one reason the density matrix formalism is so useful.

== Reduced density matrix ==

For a composite system with Hilbert space <math>\mathcal{H}_A \otimes \mathcal{H}_B</math>, the state of subsystem <math>A</math> is described by the '''reduced density matrix'''

<math>\rho_A = \mathrm{Tr}_B(\rho_{AB}),</math>

where <math>\mathrm{Tr}_B</math> denotes the [[partial trace]] over subsystem <math>B</math>.<ref>Nielsen and Chuang, ''Quantum Computation and Quantum Information'', Cambridge University Press, 2000.</ref><ref>Breuer and Petruccione, ''The Theory of Open Quantum Systems'', Oxford University Press, 2002.</ref>

Even if the total system is in a pure state, the reduced density matrix of a subsystem may be mixed. This feature is central to the study of [[quantum entanglement]], decoherence, and open-system dynamics.<ref>Wojciech H. Zurek, "Decoherence, einselection, and the quantum origins of the classical," ''Reviews of Modern Physics'' 75, 715-775 (2003).</ref>

== Time evolution ==

For a closed quantum system, the density matrix evolves according to the [[Superoperator|von Neumann equation]]

<math>i\hbar \frac{d\rho}{dt} = [H,\rho],</math>

where <math>H</math> is the Hamiltonian operator.<ref>von Neumann, ''Mathematical Foundations of Quantum Mechanics'', Princeton University Press, 1955.</ref><ref>Blum, ''Density Matrix Theory and Applications'', 3rd ed., Springer, 2012.</ref>

This is the density-matrix analogue of the [[Schrödinger equation]]. For open systems interacting with an environment, the evolution is more general and is often described by a [[Physics:Lindbladian|Lindbladian]] or other quantum master equations.<ref>G. Lindblad, "On the generators of quantum dynamical semigroups," ''Communications in Mathematical Physics'' 48, 119-130 (1976).</ref><ref>Breuer and Petruccione, ''The Theory of Open Quantum Systems'', Oxford University Press, 2002.</ref>

== Physical significance ==

The density matrix formalism is indispensable when:

* the preparation procedure produces an ensemble rather than a definite pure state;
* only a subsystem of a larger entangled system is considered;
* decoherence suppresses phase relations in a preferred basis;
* thermal equilibrium states are studied in quantum statistical mechanics.<ref>Blum, ''Density Matrix Theory and Applications'', 3rd ed., Springer, 2012.</ref><ref>Zurek, "Decoherence, einselection, and the quantum origins of the classical," ''Reviews of Modern Physics'' 75, 715-775 (2003).</ref>

In these contexts, the density matrix provides a more general and physically realistic description than a single wavefunction.

==See also==
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}
* [[Partial trace]]
* [[Physics:Coherence|Quantum coherence]]
* [[Physics:Open quantum system|Open quantum system]]
* [[Quantum entanglement]]

{{Author|Harold Foppele}}

== References ==
{{reflist|3}}

{{Sourceattribution|Quantum Density matrix|1}}

Physics:Quantum Density matrix - Revision history

imported>WikiHarold: Repair Quantum Collection B backlink template

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