Physics:Quantum Density matrix - Revision history

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	'''Density matrix''' ~~is a Book I topic in the Quantum Collection. Density~~ matrix is an operator used in 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. The density matrix is important in quantum statistical mechanics, quantum measurement theory, and the theory of open quantum systems, where a subsystem is generally not described by a single wavefunction. Density matrix is an operator used in 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.		'''Density matrix''' density matrix is an operator used in 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. The density matrix is important in quantum statistical mechanics, quantum measurement theory, and the theory of open quantum systems, where a subsystem is generally not described by a single wavefunction. Density matrix is an operator used in 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. Thus the density matrix extends the usual state-vector formalism to cases where there is classical uncertainty about which pure state has been prepared.
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	<math>\rho_A = \mathrm{Tr}_B(\rho_{AB}),</math>		<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>		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>		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 ==		== Time evolution ==

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	'''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>~~		'''Density matrix''' is a Book I topic in the Quantum Collection. Density matrix is an operator used in 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. The density matrix is important in quantum statistical mechanics, quantum measurement theory, and the theory of open quantum systems, where a subsystem is generally not described by a single wavefunction. Density matrix is an operator used in 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.
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	== Definition ==		== 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		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>		<math>\rho = \|\psi\rangle\langle\psi\|.</math><ref>J. J. Sakurai and Jim Napolitano, ''Modern Quantum Mechanics'', 2nd ed., Addison-Wesley, 2011.</ref>
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	== Time evolution ==		== Time evolution ==

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

	<math>i\hbar \frac{d\rho}{dt} = [H,\rho],</math>		<math>i\hbar \frac{d\rho}{dt} = [H,\rho],</math>
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	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>		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>		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 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 ==		== Physical significance ==
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	==See also==		==See also==
	{{#invoke:PhysicsQC\|tocHeadingAndList\|Physics:Quantum basics/See also}}		{{#invoke:PhysicsQC\|tocHeadingAndList\|Physics:Quantum basics/See also}}
	* ~~[[Partial~~ trace]]		* partial trace
	* ~~[[Physics:Coherence\|~~Quantum coherence]]		* Quantum coherence
	* ~~[[Physics:~~Open quantum system~~\|Open quantum system]]~~		* Open quantum system
	* ~~[[Quantum~~ entanglement]]		* quantum entanglement

	{{Author\|Harold Foppele}}		{{Author\|Harold Foppele}}

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