Editing Density matrix (section)

== Measurement ==
Let <math>A</math> be an [[observable]] of the system, and suppose the ensemble is in a mixed state such that each of the pure states <math>\textstyle |\psi_j\rangle</math> occurs with probability <math>p_j</math>. Then the corresponding density operator equals
: <math>\rho = \sum_j p_j |\psi_j \rangle \langle \psi_j|.</math>

The [[Expectation value (quantum mechanics)|expectation value]] of the [[Measurement in quantum mechanics|measurement]] can be calculated by extending from the case of pure states:
: <math> \langle A \rangle = \sum_j p_j \langle \psi_j|A|\psi_j \rangle = \sum_j p_j \operatorname{tr}\left(|\psi_j \rangle \langle \psi_j|A \right) =   \operatorname{tr}\left(\sum_j p_j |\psi_j \rangle \langle \psi_j|A\right) = \operatorname{tr}(\rho A),</math>
where <math>\operatorname{tr}</math> denotes [[trace (linear algebra)|trace]]. Thus, the familiar expression <math>\langle A\rangle=\langle\psi|A|\psi\rangle</math> for pure states is replaced by
: <math> \langle A \rangle = \operatorname{tr}( \rho A)</math>
for mixed states.<ref name=":0" />{{Rp|73}}

Moreover, if <math>A</math> has spectral resolution
: <math>A = \sum _i a_i P_i,</math>
where <math>P_i</math> is the [[projection operator]] into the [[eigenspace]] corresponding to eigenvalue <math>a_i</math>, the post-measurement density operator is given by<ref>{{cite journal|last=Lüders|first=Gerhart|author-link=Gerhart Lüders|year=1950|title=Über die Zustandsänderung durch den Messprozeß|journal=[[Annalen der Physik]]|volume=443|issue=5–8 |page=322|doi=10.1002/andp.19504430510|bibcode=1950AnP...443..322L }} Translated by K. A. Kirkpatrick as {{Cite journal|last=Lüders|first=Gerhart|author-link=Gerhart Lüders|date=2006-04-03|title=Concerning the state-change due to the measurement process|journal=[[Annalen der Physik]]|volume=15|issue=9|pages=663&ndash;670|arxiv=quant-ph/0403007|bibcode=2006AnP...518..663L|doi=10.1002/andp.200610207|s2cid=119103479}}</ref><ref>{{Citation|last1=Busch|first1=Paul|title=Lüders Rule|date=2009|work=Compendium of Quantum Physics|pages=356–358|editor-last=Greenberger|editor-first=Daniel|publisher=Springer Berlin Heidelberg|language=en|doi=10.1007/978-3-540-70626-7_110|isbn=978-3-540-70622-9|last2=Lahti|first2=Pekka|author-link=Paul Busch (physicist)|editor2-last=Hentschel|editor2-first=Klaus|editor3-last=Weinert|editor3-first=Friedel}}</ref>
: <math>\rho_i' = \frac{P_i \rho P_i}{\operatorname{tr}\left[\rho P_i\right]}</math>
when outcome ''i'' is obtained. In the case where the measurement result is not known the ensemble is instead described by
: <math>\; \rho ' = \sum_i P_i \rho P_i.</math>

If one assumes that the probabilities of measurement outcomes are linear functions of the projectors <math>P_i</math>, then they must be given by the trace of the projector with a density operator. [[Gleason's theorem]] shows that in Hilbert spaces of dimension 3 or larger the assumption of linearity can be replaced with an assumption of [[quantum contextuality|non-contextuality]].<ref>{{cite journal|first=Andrew M.|author-link=Andrew M. Gleason|year = 1957|title = Measures on the closed subspaces of a Hilbert space|url = http://www.iumj.indiana.edu/IUMJ/FULLTEXT/1957/6/56050|journal = [[Indiana University Mathematics Journal]]|volume = 6|issue=4|pages = 885–893|doi=10.1512/iumj.1957.6.56050|mr=0096113|last = Gleason|doi-access = free}}</ref> This restriction on the dimension can be removed by assuming non-contextuality for [[POVM]]s as well,<ref>{{Cite journal|last=Busch|first=Paul|author-link=Paul Busch (physicist) |date=2003|title=Quantum States and Generalized Observables: A Simple Proof of Gleason's Theorem|journal=[[Physical Review Letters]]|volume=91|issue=12|pages=120403|arxiv=quant-ph/9909073|doi=10.1103/PhysRevLett.91.120403|pmid=14525351|bibcode=2003PhRvL..91l0403B|s2cid=2168715}}</ref><ref>{{Cite journal|last1=Caves|first1=Carlton M.|author-link=Carlton M. Caves|last2=Fuchs|first2=Christopher A.|last3=Manne|first3=Kiran K.|last4=Renes|first4=Joseph M.|date=2004|title=Gleason-Type Derivations of the Quantum Probability Rule for Generalized Measurements|journal=[[Foundations of Physics]]|volume=34|issue=2|pages=193–209|arxiv=quant-ph/0306179|doi=10.1023/B:FOOP.0000019581.00318.a5|bibcode=2004FoPh...34..193C|s2cid=18132256}}</ref> but this has been criticized as physically unmotivated.<ref>{{cite journal |author1=Andrzej Grudka |author2=Paweł Kurzyński |title=Is There Contextuality for a Single Qubit? |journal=Physical Review Letters |date=2008 |volume=100 |issue=16 |page=160401 |doi=10.1103/PhysRevLett.100.160401|pmid=18518167 |arxiv=0705.0181|bibcode=2008PhRvL.100p0401G |s2cid=13251108 }}</ref>