# Density matrix
A **density matrix** (or density operator) is a Hermitian, positive-semidefinite operator with unit trace that represents the most general description of a quantum state, covering both pure states and statistical mixtures of states in a single mathematical object.[1](https://unseel.com/physics/density-matrix) Where a state vector \( \ket{\psi} \) describes a single system whose state is known exactly, the density matrix \( \hat{\rho} \) also handles ensembles of systems prepared in different states, and subsystems of entangled composite systems, for which no single state vector exists.[2](https://www.lancaster.ac.uk/staff/schomeru/lecturenotes/Quantum%20Information%20Processing/S1.html)
| Key fact | Detail |
|---|---|
| Defining properties | Hermitian, positive semidefinite (\( \rho \geq 0 \)), and trace one, \( \operatorname{Tr}(\rho) = 1 \)[3](https://interactivetextbooks.tudelft.nl/mqp-v2/operators.html) • [4](https://qiskit.qotlabs.org/learning/courses/general-formulation-of-quantum-information/density-matrices/density-matrix-basics) |
| Mixture form | \( \hat{\rho} = \sum_i P_i \ket{\psi_i}\bra{\psi_i} \), where a fraction \( P_i \) of an ensemble is in state \( \ket{\psi_i} \) and \( \sum_i P_i = 1 \)[2](https://www.lancaster.ac.uk/staff/schomeru/lecturenotes/Quantum%20Information%20Processing/S1.html) |
| Expectation values | \( \langle A \rangle_\rho = \operatorname{tr}(\hat{A}\hat{\rho}) \) for any observable \( \hat{A} \)[2](https://www.lancaster.ac.uk/staff/schomeru/lecturenotes/Quantum%20Information%20Processing/S1.html) |
| Purity | \( \mathcal{P} = \operatorname{tr}(\hat{\rho}^2) \); equals 1 for a pure state and \( \sum_n p_n^2 < 1 \) for a mixed state[2](https://www.lancaster.ac.uk/staff/schomeru/lecturenotes/Quantum%20Information%20Processing/S1.html) |
| Maximally mixed state | \( \hat{\rho} = \frac{1}{\mathcal{N}}\hat{I} \) in a Hilbert space of dimension \( \mathcal{N} \), with purity \( \mathcal{P} = 1/\mathcal{N} \)[2](https://www.lancaster.ac.uk/staff/schomeru/lecturenotes/Quantum%20Information%20Processing/S1.html) |
| Reduced state | \( \hat{\rho}_A = \operatorname{Tr}_B(\hat{\rho}) \), obtained by a partial trace over the other subsystem; the partial trace loses information about the full state[5](https://etneil.github.io/grad_qm_lec_notes/density.html) |
| Time evolution | \( \frac{d}{dt}\hat{\rho} = \frac{i}{\hbar}[\hat{\rho}, \hat{H}] \), with solution \( \hat{\rho}(t) = \hat{U}(t,t_0)\hat{\rho}(t_0)\hat{U}^\dagger(t,t_0) \)[2](https://www.lancaster.ac.uk/staff/schomeru/lecturenotes/Quantum%20Information%20Processing/S1.html) |
| Origin | Introduced independently by John von Neumann and Lev Landau in 1927[1](https://unseel.com/physics/density-matrix) |
## Definition and motivation
An ensemble is a large collection of physically identical quantum systems that may nevertheless be described by different states. When all members are in the same state the ensemble is pure; otherwise it is mixed. A mixed ensemble cannot be described by a single quantum state, but a statistical operator \( \hat{\rho} \), the density matrix, allows all expectation values in the ensemble to be calculated.[2](https://www.lancaster.ac.uk/staff/schomeru/lecturenotes/Quantum%20Information%20Processing/S1.html)
For a pure state the density matrix is the outer product
\[ \hat{\rho} = \ket{\psi}\bra{\psi}. \]
In a finite \( n \)-dimensional [Hilbert space](https://www.edgepedia.org/hilbert-space), this is an \( n \times 1 \) column vector times a \( 1 \times n \) row vector, giving an \( n \times n \) matrix.[6](https://www.maths.dur.ac.uk/users/kasper.peeters/qcomp/2024/density_matrix.html) For a mixed ensemble with fractions \( P_i \) in normalized states \( \ket{\psi_i} \),
\[ \hat{\rho} = \sum_i P_i \ket{\psi_i}\bra{\psi_i}. \]
Equivalently, a density operator on a Hilbert space is any operator that is Hermitian, positive semidefinite, and has trace 1; a density matrix is its matrix in some chosen basis.[3](https://interactivetextbooks.tudelft.nl/mqp-v2/operators.html) In its eigenrepresentation \( \hat{\rho} = \sum_n p_n \ket{n}\bra{n} \), all eigenvalues are nonnegative and sum to one; these \( p_n \) coincide with the ensemble fractions \( P_i \) only when the states \( \ket{\psi_i} \) are mutually orthogonal.[2](https://www.lancaster.ac.uk/staff/schomeru/lecturenotes/Quantum%20Information%20Processing/S1.html)
The matrix entries have a direct reading: the diagonal entries give the probabilities for each classical state to appear in a standard basis measurement, while the off-diagonal entries describe the degree of quantum superposition between the corresponding states and their relative phase.[4](https://qiskit.qotlabs.org/learning/courses/general-formulation-of-quantum-information/density-matrices/density-matrix-basics)
## Pure and mixed states
For a pure ensemble one eigenvalue equals 1 and all others are 0, so \( \hat{\rho} = \ket{n}\bra{n} \) is a projection operator satisfying \( \hat{\rho}^2 = \hat{\rho} \). It follows that \( \operatorname{tr}(\hat{\rho}^2) = 1 \) for a pure state, while for a mixed state \( \operatorname{tr}(\hat{\rho}^2) = \sum_n p_n^2 < 1 \). The quantity \( \mathcal{P} = \operatorname{tr}(\hat{\rho}^2) \), called the purity, therefore distinguishes pure from mixed states; the maximally mixed state \( \hat{\rho} = \frac{1}{\mathcal{N}}\hat{I} \) has the minimum purity \( 1/\mathcal{N} \).[2](https://www.lancaster.ac.uk/staff/schomeru/lecturenotes/Quantum%20Information%20Processing/S1.html) On a finite-dimensional Hilbert space, a density operator is pure precisely when it satisfies \( \rho^2 = \rho \), and every density operator can be written as a convex combination \( \rho = \sum_i p_i \ket{e_i}\bra{e_i} \) of pure ones.[3](https://interactivetextbooks.tudelft.nl/mqp-v2/operators.html)
The map from state vectors to density matrices discards information: any overall phase disappears, so \( e^{i\alpha}\ket{\psi} \) and \( \ket{\psi} \) have the same density matrix.[5](https://etneil.github.io/grad_qm_lec_notes/density.html) Conversely, different ensembles of pure states can yield the same density matrix, so the decomposition of a mixed state into an ensemble is not unique.[7](https://www.cs.cmu.edu/~yangke/thesis/proposal.pdf)
## Expectation values and measurement
The trace, defined in an orthonormal basis as \( \operatorname{tr}\hat{B} = \sum_n \bra{n}\hat{B}\ket{n} \), converts the density matrix into measurable quantities. Expectation values are
\[ \langle A \rangle_\rho = \operatorname{tr}(\hat{A}\hat{\rho}), \]
and for a measurement with projector \( \hat{P}_a \), the Born-rule probability of outcome \( a \) is \( \operatorname{tr}(\hat{\rho}\hat{P}_a) \).[2](https://www.lancaster.ac.uk/staff/schomeru/lecturenotes/Quantum%20Information%20Processing/S1.html) • [5](https://etneil.github.io/grad_qm_lec_notes/density.html) For an observable \( \hat{Q} \) with eigenvalues \( q_\mu \) and eigenstates \( \ket{\mu} \), the outcome probability is \( P_\mu = \bra{\mu}\hat{\rho}\ket{\mu} \), and \( \langle Q \rangle = \operatorname{tr}[\hat{Q}\hat{\rho}] \); the trace is basis-independent.[8](https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Quantum_Mechanics_III_%28Chong%29/03%3A_Quantum_Entanglement/3.06%3A_Density_Operators) For pure states the formalism reduces exactly to the state-vector rules, and there is a one-to-one mapping between \( \ket{\psi} \) and \( \ket{\psi}\bra{\psi} \).[6](https://www.maths.dur.ac.uk/users/kasper.peeters/qcomp/2024/density_matrix.html)
## Reduced density matrices and entanglement
For a composite system with Hilbert space \( \mathscr{H} = \mathscr{H}_A \otimes \mathscr{H}_B \), the state of subsystem A alone is described by the reduced density operator
\[ \hat{\rho}_A = \operatorname{Tr}_B[\hat{\rho}], \]
where the partial trace runs over the \( \mathscr{H}_B \) part of the space.[8](https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Quantum_Mechanics_III_%28Chong%29/03%3A_Quantum_Entanglement/3.06%3A_Density_Operators) The reduced density matrix is itself a valid density matrix, but the partial trace loses information: for a general operator \( \hat{O} \) on the product space, the reduced matrices do not suffice to compute \( \langle \hat{O} \rangle \). They do suffice for measurements on one subsystem alone, since \( \langle \hat{A} \otimes \hat{1} \rangle = \operatorname{tr}(\hat{\rho}_A \hat{A}) \).[5](https://etneil.github.io/grad_qm_lec_notes/density.html)
A pure state of the joint system can reduce to a mixed state of a subsystem. For the spin singlet, tracing out one spin gives \( \hat{\rho}_A = \frac{1}{2}\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \), a mixed matrix with purity \( \operatorname{tr}(\hat{\rho}_A^2) = 1/2 \). This matches the physics of the EPR pair: either party alone measures 50% odds of spin-up or spin-down along any axis, an unpolarized state.[5](https://etneil.github.io/grad_qm_lec_notes/density.html)
## Time evolution
Under unitary evolution generated by a Hamiltonian \( \hat{H} \), the density matrix obeys
\[ \frac{d}{dt}\hat{\rho} = \frac{i}{\hbar}[\hat{\rho}, \hat{H}], \]
with general solution \( \hat{\rho}(t) = \hat{U}(t,t_0)\hat{\rho}(t_0)\hat{U}^\dagger(t,t_0) \), where \( \hat{U} \) is the unitary time-evolution operator. This mirrors the transformation rule \( \hat{\rho} \to \hat{U}\hat{\rho}\hat{U}^\dagger \) that corresponds to \( \ket{\psi} \to \hat{U}\ket{\psi} \) for pure states.[2](https://www.lancaster.ac.uk/staff/schomeru/lecturenotes/Quantum%20Information%20Processing/S1.html) • [6](https://www.maths.dur.ac.uk/users/kasper.peeters/qcomp/2024/density_matrix.html)
## History and role
The density matrix was introduced independently by [John von Neumann](https://www.edgepedia.org/john-von-neumann) and [Lev Landau](https://www.edgepedia.org/lev-landau) in 1927. Von Neumann's 1932 book *Mathematical Foundations of Quantum Mechanics* gave quantum mechanics its rigorous mathematical framework, and the density operator entered it as the statistical operator used for quantum statistical mechanics.[1](https://unseel.com/physics/density-matrix) • [9](https://citeseerx.ist.psu.edu/document?doi=8cab594773d97319af3e31289452a0a726a554d0&repid=rep1&type=pdf) Today the formalism is the standard language of open quantum systems, decoherence, and quantum information theory, where density operators describe mixed states combining quantum states with classical uncertainty.[1](https://unseel.com/physics/density-matrix) • [3](https://interactivetextbooks.tudelft.nl/mqp-v2/operators.html)
## References
1. [The Density Matrix, Explained, Equation & Interactive, Unseel](https://unseel.com/physics/density-matrix)
2. [I.5 Density matrix, PHYS483 Quantum Information Processing Lecture Notes, Lancaster University](https://www.lancaster.ac.uk/staff/schomeru/lecturenotes/Quantum%20Information%20Processing/S1.html)
3. [9. Operators on Hilbert spaces, Mathematics for Quantum Physics, TU Delft](https://interactivetextbooks.tudelft.nl/mqp-v2/operators.html)
4. [Density matrix basics, IBM Quantum Documentation](https://qiskit.qotlabs.org/learning/courses/general-formulation-of-quantum-information/density-matrices/density-matrix-basics)
5. [18 Density matrix formalism, Graduate Quantum Mechanics Lecture Notes](https://etneil.github.io/grad_qm_lec_notes/density.html)
6. [QI III: Measurement and uncertainty, Quantum Computation and Information, Durham University](https://www.maths.dur.ac.uk/users/kasper.peeters/qcomp/2024/density_matrix.html)
7. [On the Communication Complexity of Correlation (thesis proposal), Carnegie Mellon University](https://www.cs.cmu.edu/~yangke/thesis/proposal.pdf)
8. [3.6 Density Operators, Quantum Mechanics III (Chong), Physics LibreTexts](https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Quantum_Mechanics_III_%28Chong%29/03%3A_Quantum_Entanglement/3.06%3A_Density_Operators)
9. [John von Neumann and the entropy S (on the introduction of the density matrix in the 1932 Mathematische Grundlagen der Quantenmechanik), CiteSeerX](https://citeseerx.ist.psu.edu/document?doi=8cab594773d97319af3e31289452a0a726a554d0&repid=rep1&type=pdf)
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*Topic: Encyclopedia › Physical world and mathematics › Physics › Quantum physics › Quantum mechanics › Quantum formalism and states › Quantum states and wave functions › Mixed and entangled states › Density matrix formalism*
*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*
*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*
License: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license