When one or more particles are quantum entangled by say their spin property, do their other measurable properties (e.g., momentum, polarization, whatever?) become entangled as well?


No, other properties do not necessarily become entangled.

Example: You can take twin photons from type-II parametric down conversion. These two photons are entangled in energy and momentum, but not in polarization.

For more information about this example, see: Physical Review A 50, 5122 (free version available here).

As described in the article, terms containing energy and momentum of the twin photons are not factorizable whereas the part describing the polarization of the photons is factorized, like

$$ \left|\psi\right>=\left|0\right>+\sum_{k,k'}g(\omega_{ok})\delta(\omega_{ok}+\omega_{ek'}-\omega_p)h(L\Delta_{kk'})\hat a^\dagger_{ok}\hat a^\dagger_{ek}\left|0\right> $$

The delta function and $h(x)$ result in entanglement of energy and momentum respectively, where the part determining the polarization is written in a product of the two polarizations ($o$ for ordinary and $e$ for extraordinary polarization).

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    $\begingroup$ this answer is nice, but the article you cite is behind paywall. Can you provide some accessible reference, or quote the relevant parts of the article, or just expand a little on your argument? $\endgroup$ – glS Feb 21 '15 at 18:38
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    $\begingroup$ A free pdf of the article can be found here (for now). $\endgroup$ – Mark Mitchison Feb 21 '15 at 18:51
  • $\begingroup$ Doesn't there exists types of entanglements, where more $2$ properties are simultaneously entangled?For example,hyperentangled states or entanglement duality "iopscience.iop.org/1367-2630/16/8/08301". $\endgroup$ – WInterfell Mar 22 '15 at 12:02

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