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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.

Definition

A quantum state may be represented by a density matrix $ρ$ , which is a linear operator acting on the system's Hilbert space. For a pure state $| ψ ⟩$ , the density matrix is

$ρ = | ψ ⟩ ⟨ ψ | .$ ^[1]

More generally, for an ensemble of states ${| ψ_{i} ⟩}$ occurring with probabilities $p_{i}$ , the density matrix is

$ρ = \sum_{i} p_{i} | ψ_{i} ⟩ ⟨ ψ_{i} |,$

with

$p_{i} \geq 0, \sum_{i} p_{i} = 1 .$ ^[2]

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 $ρ$ satisfies the following conditions:^[3]^[4]

Hermiticity
$ρ = ρ^{†}$
Unit trace
$T r (ρ) = 1$
Positive semidefiniteness
$ρ \geq 0$

These conditions are not only necessary but also sufficient: any operator satisfying them is a valid density matrix.^[5]

For a pure state, the density matrix is idempotent:

$ρ^{2} = ρ .$

Equivalently,

$T r (ρ^{2}) = 1 .$

For a mixed state,

$T r (ρ^{2}) < 1 .$ ^[6]

The quantity $T r (ρ^{2})$ is called the purity of the state.

Matrix representation

If ${| n ⟩}$ is an orthonormal basis, the density matrix may be written in components as

$ρ_{m n} = ⟨ m | ρ | n ⟩ .$

For a two-level system with

$| ψ ⟩ = α | 0 ⟩ + β | 1 ⟩,$

the corresponding pure-state density matrix is

$ρ = (\begin{matrix} | α |^{2} & α β^{*} \\ α^{*} β & | β |^{2} \end{matrix}) .$ ^[7]

The diagonal elements represent populations in the chosen basis, while the off-diagonal elements represent quantum coherences.^[8]

Expectation values

The expectation value of an observable represented by an operator $A$ is given by

$⟨ A ⟩ = T r (ρ A) .$ ^[9]^[10]

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 $ℋ_{A} \otimes ℋ_{B}$ , the state of subsystem $A$ is described by the reduced density matrix

$ρ_{A} = {T r}_{B} (ρ_{A B}),$

where ${T r}_{B}$ denotes the partial trace over subsystem $B$ .^[11]^[12]

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.^[13]

Time evolution

For a closed quantum system, the density matrix evolves according to the von Neumann equation

$i ℏ \frac{d ρ}{d t} = [H, ρ],$

where $H$ is the Hamiltonian operator.^[14]^[15]

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.^[16]^[17]

@@ Line 1: / Line 1: @@
+{{Short description|Quantum Collection topic on Quantum Density matrix}}
 {{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>
+{{Quantum article nav|previous=Physics:Quantum measurement problem|previous label=Measurement problem|next=Physics:Quantum Exactly solvable quantum systems|next label=Exactly solvable quantum systems}}
-[[File:Density_matrix_pure_vs_mixed_states.png|right|500px]]
+<div style="display:flex; gap:24px; align-items:flex-start; max-width:1200px;">
+<div style="width:280px;">
+__TOC__
+</div>
+<div style="flex:1; line-height:1.45; color:#006b45; column-count:2; column-gap:32px; column-rule:1px solid #b8d8c8;">
+'''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.
+</div>
+<div style="width:300px;">
+[[File:Density_matrix_pure_vs_mixed_states.png|thumb|280px|Quantum Density matrix.]]
+</div>
+</div>
 == 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>
@@ Line 80: / Line 97: @@
 <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 ==
-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>
@@ Line 92: / Line 109: @@
 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 ==
@@ Line 107: / Line 124: @@
 ==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}}
@@ Line 117: / Line 134: @@
 {{reflist|3}}
-{{Sourceattribution|Quantum Density matrix|1}}
+{{Sourceattribution|Physics:Quantum Density matrix|1}}

Physics:Quantum Density matrix: Difference between revisions

Latest revision as of 12:34, 20 May 2026

Definition

Properties

Matrix representation

Expectation values

Reduced density matrix

Time evolution

Physical significance

See also

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