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Can every idempotent density operator be written in the form $$\sum_ic_i|i\rangle\langle i|,$$ where $\{|i\rangle\}$ is an orthonormal set and $\{c_i\}$ is a set of coefficients (Edit: Without assuming that the vector space on which the density operator operates is finite dimensional)?

It seems like people are assuming that when proving that a density operator is idempotent if and only if it represents a pure state (for example here: Necessary and sufficient conditions for a pure state).

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    $\begingroup$ Density operators are Hermitian, so.... $\endgroup$ Commented May 24, 2020 at 15:34
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    $\begingroup$ Think about what "diagonalizing a matrix" means, and what the comment above mine is saying $\endgroup$ Commented May 24, 2020 at 15:56

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By definition, density operators are self-adjoint.

In the case of a finite dimensional vector space, every self-adjoint operator has an orthonormal set of eigenvectors (https://en.wikipedia.org/wiki/Spectral_theorem).

Alternatively (instead of assuming a finite dimensional vector space), we can prove the statement for compact operators (see exercise 6 in chapter 19 of [1]).

[1] Brian Hall, Quantum Theory for Mathematicians

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