Why is the determinant of a mixed state density matrix always positive?(adsbygoogle = window.adsbygoogle || []).push({});

In the specific case of a 2-dimensional Hilbert space, the density matrix (as well as any other hermitian matrix) can be expressed as

[tex]\rho=\frac 1 2 (I+\vec r\cdot\vec \sigma)[/tex]

so its determinant is

[tex]\det\rho=\frac 1 4(1-\vec r^2)[/itex]

We have [itex]|\vec r|=1[/itex] if and only if we're dealing with a pure state, so we seem to need the condition [itex]\det\rho\geq 0[/itex] to see that the set of mixed states is the interior of the sphere rather than the exterior.

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# Determinant of density matrix

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