Editing Density matrix (section)

=== Example: light polarization ===
[[File:vertical polarization.svg|right|thumb|200px|The incandescent light bulb{{nbsp}}(1) emits completely random polarized photons{{nbsp}}(2) with mixed state density matrix:<br />
<div class="center"><math>\begin{bmatrix}
 0.5 & 0  \\
 0 & 0.5
\end{bmatrix}</math><span style="vertical-align:bottom">.</span></div>{{paragraph}}
After passing through vertical plane polarizer{{nbsp}}(3), the remaining photons are all vertically polarized{{nbsp}}(4) and have pure state density matrix:<br />
<div class="center"><math>\begin{bmatrix}
 1 & 0  \\
 0 & 0
\end{bmatrix}
</math><span style="vertical-align:bottom">.</span></div>]]
An example of pure and mixed states is [[Photon polarization|light polarization]]. An individual [[photon]]
can be described as having right or left [[circular polarization]], described by the orthogonal quantum states <math>|\mathrm{R}\rangle</math> and <math>|\mathrm{L}\rangle</math> or a [[Quantum superposition|superposition]] of the two: it can be in any state <math>\alpha|\mathrm{R}\rangle+\beta|\mathrm{L}\rangle</math> (with <math>|\alpha|^2+|\beta|^2=1</math>), corresponding to [[linear polarization|linear]], [[circular polarization|circular]], or [[elliptical polarization]]. Consider now a vertically polarized photon, described by the state <math>|\mathrm{V}\rangle = (|\mathrm{R}\rangle+|\mathrm{L}\rangle)/\sqrt{2}</math>. If we pass it through a [[circular polarizer]] that allows either only <math>|\mathrm{R}\rangle</math> polarized light, or only <math>|\mathrm{L}\rangle</math> polarized light, half of the photons are absorbed in both cases. This may make it ''seem'' like half of the photons are in state <math>|\mathrm{R}\rangle</math> and the other half in state <math>|\mathrm{L}\rangle</math>, but this is not correct: if we pass <math>(|\mathrm{R}\rangle+|\mathrm{L}\rangle)/\sqrt{2}</math> through a [[linear polarizer]] there is no absorption whatsoever, but if we pass either state <math>|\mathrm{R}\rangle</math> or <math>|\mathrm{L}\rangle</math> half of the photons are absorbed.

[[Unpolarized light]] (such as the light from an [[incandescent light bulb]]) cannot be described as ''any'' state of the form <math>\alpha|\mathrm{R}\rangle+\beta|\mathrm{L}\rangle</math> (linear, circular, or elliptical polarization). Unlike polarized light, it passes through a polarizer with 50% intensity loss whatever the orientation of the polarizer; and it cannot be made polarized by passing it through any [[wave plate]]. However, unpolarized light ''can'' be described as a statistical ensemble, e. g. as each photon having either <math>|\mathrm{R}\rangle</math> polarization or <math>|\mathrm{L}\rangle</math> polarization with probability 1/2. The same behavior would occur if each photon had either vertical polarization <math>| \mathrm{V}\rangle </math> or horizontal polarization <math>| \mathrm{H} \rangle </math> with probability 1/2. These two ensembles are completely indistinguishable experimentally, and therefore they are considered the same mixed state. For this example of unpolarized light, the density operator equals<ref name=":0" />{{Rp|75}}
: <math>\rho = \frac{1}{2} |\mathrm{R}\rangle \langle \mathrm{R}| + \frac{1}{2}|\mathrm{L}\rangle \langle \mathrm{L}| = \frac{1}{2} |\mathrm{H}\rangle \langle \mathrm{H}| + \frac{1}{2}|\mathrm{V}\rangle \langle \mathrm{V}| = \frac12\begin{pmatrix} 1 & 0 \\ 0 & 1\end{pmatrix}.</math>

There are also other ways to generate unpolarized light: one possibility is to introduce uncertainty in the preparation of the photon, for example, passing it through a [[birefringent crystal]] with a rough surface, so that slightly different parts of the light beam acquire different polarizations. Another possibility is using entangled states: a radioactive decay can emit two photons traveling in opposite directions, in the quantum state <math>(|\mathrm{R},\mathrm{L}\rangle+|\mathrm{L},\mathrm{R}\rangle)/\sqrt{2}</math>. The joint state of the two photons ''together'' is pure, but the density matrix for each photon individually, found by taking the partial trace of the joint density matrix, is completely mixed.<ref name="mikeandike" />{{Rp|106}}