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I am working a two dimensional Hilbert space with basis {$|0\rangle, |1\rangle$} and I am trying to show that the density matrix is characterized by 3 real numbers and show that these three numbers are the expected values of the Pauli matrices.

I understand that a density matrix is given by $M_{ij} = \langle i|\rho|j\rangle$ where $|i\rangle$ is an orthonormal basis of $H$ and $\rho$ is the density operator. How do I go about showing this?

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To give you a hint: Without losing generality you can use the given standard-basis. What is the dimension of the density matrix? How many complex numbers do you need to determine this matrix? Has the density matrix some special properties, you can use to get more information on these complex number and reduce them to three real numbers? What are the expectation values of the pauli matrices (as a function of these three real numbers)?

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