Editing Density matrix (section)

== Definition and motivation ==
The density matrix is a representation of a [[linear operator]] called the '''density operator'''. The density matrix is obtained from the density operator by a choice of an [[orthonormal]] [[basis (linear algebra)|basis]] in the underlying space.<ref>{{cite book | last=Ballentine | first=Leslie | title=Compendium of Quantum Physics | chapter=Density Matrix | publisher=Springer Berlin Heidelberg | publication-place=Berlin, Heidelberg | year=2009 | isbn=978-3-540-70622-9 | doi=10.1007/978-3-540-70626-7_51 | page=166}}</ref> In practice, the terms ''density matrix'' and ''density operator'' are often used interchangeably.

Pick a basis with states <math>|0\rangle</math>, <math>|1\rangle</math> in a two-dimensional [[Hilbert space]], then the density operator is represented by the matrix
<math display="block">
(\rho_{ij}) = \left( \begin{matrix} 
\rho_{00} & \rho_{01} \\
\rho_{10} & \rho_{11}
\end{matrix} \right)
= \left( \begin{matrix} 
p_{0} & \rho_{01} \\
\rho^*_{01} & p_{1}
\end{matrix} \right)
</math> 
where the diagonal elements are [[real number]]s that sum to one (also called populations of the two states <math>|0\rangle</math>, <math>|1\rangle</math>). 
The off-diagonal elements are [[complex conjugate]]s of each other (also called coherences); they are restricted in magnitude by the requirement that <math>(\rho_{ij})</math> be a [[positive semi-definite matrix|positive semi-definite operator]], see below.

A density operator is a [[positive-definite matrix|positive semi-definite]], [[self-adjoint operator]] of [[trace class operator|trace]] one acting on the [[Hilbert space]] of the system.<ref name=fano1957>{{cite journal |doi=10.1103/RevModPhys.29.74 |title=Description of States in Quantum Mechanics by Density Matrix and Operator Techniques |journal=Reviews of Modern Physics |volume=29 |issue=1 |pages=74–93 |year=1957 |last1=Fano |first1=U. |bibcode=1957RvMP...29...74F }}</ref><ref>{{Cite book|last=Holevo |first=Alexander S. |author-link=Alexander Holevo |title=Statistical Structure of Quantum Theory |publisher=Springer |series=Lecture Notes in Physics |year=2001 |isbn=3-540-42082-7|oclc=318268606}}</ref><ref name=Hall2013pp419-440>{{cite book |doi=10.1007/978-1-4614-7116-5_19 |chapter=Systems and Subsystems, Multiple Particles |title=Quantum Theory for Mathematicians |volume=267 |pages=419–440 |series=Graduate Texts in Mathematics |year=2013 |last1=Hall |first1=Brian C. |isbn=978-1-4614-7115-8 }}</ref> This definition can be motivated by considering a situation where some pure states <math>|\psi_j\rangle</math> (which are not necessarily orthogonal) are prepared with probability <math>p_j</math> each.<ref>{{cite book | last1=Cohen-Tannoudji | first1=Claude | last2=Diu | first2=Bernard | last3=Laloë | first3=Franck | title=Quantum Mechanics, Volume 1 | publisher=John Wiley & Sons | publication-place=Weinheim, Germany | date=2019 | isbn=978-3-527-34553-3|pages=301–303}}.</ref> This is known as an ''ensemble'' of pure states. The probability of obtaining [[Measurement in quantum mechanics#Projective measurement|projective measurement]] result <math>m</math> when using [[projection operator|projector]]s <math>\Pi_m</math> is given by<ref name="mikeandike" />{{rp|p=99}}
<math display="block"> p(m) = \sum_j p_j \left\langle \psi_j\right| \Pi_m \left|\psi_j\right\rangle = \operatorname{tr} \left[ \Pi_m \left ( \sum_j p_j \left|\psi_j\right\rangle \left\langle \psi_j\right|\right) \right],</math>
which makes the '''density operator''', defined as 
<math display="block">\rho = \sum_j p_j \left|\psi_j \right\rangle \left\langle \psi_j\right|, </math>
a convenient representation for the state of this ensemble. It is easy to check that this operator is positive semi-definite, self-adjoint, and has trace one. Conversely, it follows from the [[spectral theorem]] that every operator with these properties can be written as <math display="inline"> \sum_j p_j \left|\psi_j\right\rangle \left\langle \psi_j\right|</math> for some states <math>\left|\psi_j\right\rangle</math> and coefficients <math>p_j</math> that are non-negative and add up to one.<ref name=davidson>{{cite book| last=Davidson| first=Ernest Roy| title=Reduced Density Matrices in Quantum Chemistry| year=1976| publisher=[[Academic Press]], London}}</ref><ref name="mikeandike" />{{rp|p=102}} However, this representation will not be unique, as shown by the [[Schrödinger–HJW theorem]].

Another motivation for the definition of density operators comes from considering local measurements on entangled states. Let <math>|\Psi\rangle</math> be a pure entangled state in the composite Hilbert space <math> \mathcal{H}_1\otimes\mathcal{H}_2</math>. The probability of obtaining measurement result <math>m</math> when measuring projectors <math>\Pi_m</math> on the Hilbert space <math>\mathcal{H}_1</math> alone is given by<ref name="mikeandike" />{{rp|p=107}}
<math display="block"> p(m) = \left\langle \Psi\right| \left(\Pi_m \otimes I\right) \left|\Psi\right\rangle = \operatorname{tr} \left[ \Pi_m \left ( \operatorname{tr}_2 \left|\Psi\right\rangle \left\langle \Psi\right| \right)  \right],</math>
where <math> \operatorname{tr}_2 </math> denotes the [[partial trace]] over the Hilbert space <math>\mathcal{H}_2</math>. This makes the operator 
<math display="block">\rho = \operatorname{tr}_2 \left|\Psi\right\rangle\left\langle \Psi\right| </math>
a convenient tool to calculate the probabilities of these local measurements. It is known as the [[reduced density matrix]] of <math>|\Psi\rangle</math> on subsystem 1. It is easy to check that this operator has all the properties of a density operator. Conversely, the [[Schrödinger–HJW theorem]] implies that all density operators can be written as <math>\operatorname{tr}_2 \left|\Psi\right\rangle \left\langle \Psi\right|</math> for some state <math>\left|\Psi\right\rangle </math>.