# Optimal measurement by projectors for density operators!

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Theorem: Let $\{\rho_i,1\leq i\leq m\}$ be a quantum state ensemble
consisting of linearly independent density operators
$\rho_i$ with prior probabilities $p_i$. Then the optimal measurement
is a von Neumann measurement with measurement operators
$\{\Pi_i=\mathcal{P}_{\mathcal{S_i}},1\leq i\leq m\}$ where $\mathcal{P}_{\mathcal{S_i}}$ is an orthogonal projection
onto an $r_i$-dimensional subspace $\mathcal{S_i}$ of $\mathcal{H}$ (*Hilbert space* of density operators) with $r_i= rank(\rho_i)$ and $\mathcal{P}_{\mathcal{S_i}}\mathcal{P}_{\mathcal{S_j}}=\delta^i_j\mathcal{P}_{\mathcal{S_i}}$ (orthonormality of operators).

This is a theorem in Quantum Information which solely uses Linear Algebra for states represented as density operators. For my case, each $p_i=\frac{1}{4}$ and all the four $\rho_i$ are linearly independent. Now do I find the $\mathcal{P}_{\mathcal{S_i}}$ for each of the density matrix? Will it be just the sum of the projectors built from eigen-vectors of that density matrix? I did this but then found that the orthonormality is not followed because two of my density matrices have some common eigen-vectors (2 eigenvectors are common out of 4 for two such density matrices). How do we construct $\mathcal{P}_{\mathcal{S_i}}$ for each $\rho_i$?

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