# The meaning of $p_{i}$ and $\rho^{i}$ as probabilities and densities in Quantum Mechanics

The question I have concerns the actual meanings of $p_{i}$ and $\rho^{i}$ Now $p_{i}:Meas_{I} \times D(H) \rightarrow [0,1]$, so for a particular set of Measurement matrices M and Density matrices $\rho$ we have that $(\textbf{M},\rho) \mapsto tr(M^{\dagger}M \rho)$. First, I don't really understand what is being done here. I see that we are taking a measurement on a Density matrix and that will give us some result - namely our measurement. So $p_{i}$ is basically just the probability that a particular measurement matrix (maybe by some sort of alighned magnetic field?) acting upon the probability density of a quantum system that we will end up actually seeing this system? So $p_{i}$ is the probability of measuring $\rho$? How am I doing herer? I think (hope) that I am pretty on target, but I am much less sure about why $(\textbf{M},\rho) \mapsto tr(M^{\dagger}M \rho)$ So is $tr(M^{\dagger}M \rho) \in [0,1]$? Thus $p_i$ signifies the probability of measuring some quantum state $\rho$ in that particular state when acted on by each Matrix $M$ in a family of measurement matrices $\textbf{M}$.

So that is just the first part (I thank you for your time and any comments). The second part is about $\rho_{i}$. I see that $\rho_{i}:Meas_{I} \times D(H) \rightarrow D(H)$ The only thing I can think of concerning this is that $\rho_{i}$ signifies a new density matrix (maybe - and will turn out to be the case) due to the act of measurement. Is this correct? Furthermore according to my readings $(\textbf{M},\rho) \mapsto \frac{1}{p_{i}(\textbf{M},\rho)}M_{i} \rho M_{i}^{\dagger}$, where $p_{i}(\textbf{M},\rho)$ is undefined if $\rho^{i}(\textbf{M},\rho) = 0$. The best I can make of this is that I would have $(\textbf{M},\rho) \mapsto \frac{1}{p_i}M_{i} \rho M_{i}^{\dagger}$ but don't know what is going on with the $M_{i} \rho M_{i}^{\dagger}$.

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