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{{Main|Density matrix}}
A ''pure quantum state'' is a state which can be described by a single ket vector, as described above. A ''mixed quantum state'' is a [[statistical ensemble]] of pure states (see [[quantum statistical mechanics]]).<ref name="peres" />{{Rp|page=73}}
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]] of possible preparations; and second, when one wants to describe a physical system which is [[Quantum entanglement|entangled]] with another, as its state cannot be described by a pure state. In the first case, there could theoretically be another person who knows the full history of the system, and therefore describe the same system as a pure state; in this case, the density matrix is simply used to represent the limited knowledge of a quantum state. In the second case, however, the existence of quantum entanglement theoretically prevents the existence of complete knowledge about the subsystem, and it's impossible for any person to describe the subsystem of an entangled pair as a pure state.
Mixed states inevitably arise from pure states when, for a composite quantum system <math>H_1 \otimes H_2</math> with an [[quantum entanglement|entangled]] state on it, the part <math>H_2</math> is inaccessible to the observer.<ref name="peres" />{{Rp|pages=121-122}} The state of the part <math>H_1</math> is expressed then as the [[partial trace]] over <math>H_2</math>.
A mixed state ''cannot'' be described with a single ket vector.<ref>{{Cite book |last=Zwiebach |first=Barton |author-link=Barton Zwiebach |title=Mastering Quantum Mechanics: Essentials, Theory, and Applications |date=2022 |publisher=[[MIT Press]] |isbn=978-0-262-04613-8 |location=Cambridge, Mass}}</ref>{{Rp|pages=691-692}} Instead, it is described by its associated ''density matrix'' (or ''density operator''), usually denoted ''ρ''.
The density matrix describing a mixed state is defined to be an operator of the form
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{{anchor|expectation}}The rules for measurement in quantum mechanics are particularly simple to state in terms of density matrices. For example, the ensemble average ([[expectation value (quantum mechanics)|expectation value]]) of a measurement corresponding to an observable {{math|''A''}} is given by
<math display="block">\langle A \rangle = \sum_s p_s \langle \psi_s | A | \psi_s \rangle = \sum_s \sum_i p_s a_i | \langle \alpha_i | \psi_s \rangle |^2 = \operatorname{tr}(\rho A)</math>
where <math>|\alpha_i\rangle</math> and <math>a_i</math> are eigenkets and eigenvalues, respectively, for the operator {{math|''A''}}, and "{{math|tr}}" denotes trace.<ref name="peres" />{{Rp|page=73}} It is important to note that two types of averaging are occurring, one being a weighted quantum superposition over the basis kets <math>|\psi_s\rangle</math> of the pure states, and the other being a statistical (said ''incoherent'') average with the probabilities {{math|''p<sub>s</sub>''}} of those states.
According to [[Eugene Wigner]],<ref>{{cite book | author=Eugene Wigner | author-link=Eugene Wigner | contribution=Remarks on the mind-body question | pages=284–302 | editor=I.J. Good | title=The Scientist Speculates | location=London | publisher=Heinemann | year=1962 | contribution-url=http://www.phys.uu.nl/igg/jos/foundQM/wigner.pdf }}{{Dead link|date=May 2024 |bot=InternetArchiveBot |fix-attempted=yes }} Footnote 13 on p.180</ref> the concept of mixture was put forward by [[Lev Landau]].<ref>{{cite journal | author=Lev Landau |title=Das Dämpfungsproblem in der Wellenmechanik (The Damping Problem in Wave Mechanics)| journal=Zeitschrift für Physik | volume=45 | issue=5–6 |pages=430–441 | year=1927 |doi=10.1007/bf01343064 |bibcode = 1927ZPhy...45..430L |s2cid=125732617}} English translation reprinted in: {{cite book | editor=D. Ter Haar | title=Collected papers of L.D. Landau | location=Oxford | publisher=Pergamon Press | year=1965 }} p.8–18</ref><ref name="Landau (1965)">{{cite book | author1=Lev Landau |author2= Evgeny Lifshitz | author1-link=Lev Landau| author2-link=Evgeny Lifshitz |title=Quantum Mechanics — Non-Relativistic Theory | location=London | publisher=Pergamon Press | series=Course of Theoretical Physics | volume=3 | url=https://archive.org/download/QuantumMechanics_104/LandauLifshitz-QuantumMechanics_text.pdf | edition=2nd | year=1965 }}</ref>{{rp|38–41}}
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