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For pure states one usually uses the bra-ket Notation and then uses language e.g. "the state $|\psi>$..."

Is it also common to say similarly for mixed states, which are usually written as a density operator? E.g. "the mixed state $\rho$..."? Or is this not used in favor of the language use as e.g. "the density matrix $\rho$ of this mixed state...$ (without declaring a symbol for the state).

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    $\begingroup$ I've often heard people say "the state $\rho$". Generally physicists try to omit needless words. $\endgroup$
    – knzhou
    Commented Mar 18, 2022 at 19:24
  • $\begingroup$ I mean, if you're in an introductory course they might be careful about it to make sure everybody is following. But afterwards people know what you mean if you use any version. $\endgroup$
    – knzhou
    Commented Mar 18, 2022 at 19:25

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Yes, it's common. From a technical perspective, it's far more accurate to call $\rho$ a state than it is to call $|\psi\rangle$ a state because for any nonzero complex number $\lambda$, $|\psi\rangle$ and $\lambda |\psi\rangle$ correspond to precisely the same (pure) state. In other words, pure states should not be identified with specific ket vectors but rather with equivalence classes of vectors, and this distinction is sometimes quite important. There is no such ambiguity with density matrices.

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