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Supposing $x$ is a density matrix. I know purity formula is $$P = \mathrm{tr}(x^2)$$ But I have doubt about calculating $x^2$.

Is it $x\cdot x$ or $x\cdot x^\dagger$?

Can you give me a reference for your answer?

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    $\begingroup$ Just a remark, not only you can calculate tr(x^2) for x state, it is also possible measure this quantity with the SWAP-test. $\endgroup$
    – R.W
    Jan 8, 2023 at 8:14

1 Answer 1

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Squaring matrices works the same way as squaring numbers, i.e. you multiply the matrix by itself. Formally, $A^2=A\cdot A$. However, every density matrix $\rho$ is Hermitian, i.e. $\rho=\rho^\dagger$, so $\rho^2 = \rho\cdot\rho = \rho\cdot\rho^\dagger$, anyway.

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