Editing Density matrix (section)

== Equivalent ensembles and purifications ==
{{main|Schrödinger–HJW theorem}}
A given density operator does not uniquely determine which ensemble of pure states gives rise to it; in general there are infinitely many different ensembles generating the same density matrix.<ref>{{Cite journal|last=Kirkpatrick |first=K. A. |date=February 2006 |title=The Schrödinger-HJW Theorem |journal=[[Foundations of Physics Letters]] |volume=19 |issue=1 |pages=95–102 |doi=10.1007/s10702-006-1852-1 |issn=0894-9875 |arxiv=quant-ph/0305068|bibcode=2006FoPhL..19...95K |s2cid=15995449 }}</ref> Those cannot be distinguished by any measurement.<ref>{{Cite journal|last=Ochs|first=Wilhelm|date=1981-11-01|title=Some comments on the concept of state in quantum mechanics|url=https://doi.org/10.1007/BF00211375|journal=[[Erkenntnis]]|language=en|volume=16|issue=3|pages=339–356|doi=10.1007/BF00211375|s2cid=119980948|issn=1572-8420}}</ref> The equivalent ensembles can be completely characterized: let <math>\{p_j,|\psi_j\rangle\}</math> be an ensemble. Then for any complex matrix <math>U</math> such that <math>U^\dagger U = I</math> (a [[partial isometry]]), the ensemble <math>\{q_i,|\varphi_i\rangle\}</math> defined by
: <math>\sqrt{q_i} \left| \varphi_i \right\rangle  = \sum_j U_{ij} \sqrt{p_j} \left| \psi_j  \right\rangle </math>
will give rise to the same density operator, and all equivalent ensembles are of this form.

A closely related fact is that a given density operator has infinitely many different [[Purification of quantum state|purifications]], which are pure states that generate the density operator when a partial trace is taken. Let 
: <math>\rho = \sum_j p_j |\psi_j \rangle \langle \psi_j| </math>
be the density operator generated by the ensemble <math>\{p_j,|\psi_j\rangle\}</math>, with states <math>|\psi_j\rangle</math> not necessarily orthogonal. Then for all partial isometries <math>U</math> we have that
: <math> |\Psi\rangle = \sum_j \sqrt{p_j} |\psi_j \rangle U |a_j\rangle </math>
is a purification of <math>\rho</math>, where <math>|a_j\rangle</math> is an orthogonal basis, and furthermore all purifications of <math>\rho</math> are of this form.