Equivalence classes in a Hilbert space I'm reading something about quantum information/quantum computing theory, and I've run into a wall. I know what is meant by an equivalence class and how something can be partitioned into equivalence classes, but I need help on the following two questions:


*

*How can a partitioning of a Hilbert space be naturally realized?

*How a density operator can be seen as a equivalence class representing a range of different possible ensembles?
For the 1. I have no clue, while for the 2. I was thinking it has something to do with the equation for the expectation of some observable $\hat{A}$, $\langle \hat{A }\rangle =tr (\rho \hat{A})$, since the trace is cyclically invariant and so a unitary transformation $|\psi '\rangle \rightarrow \hat{U } |\psi \rangle$, $\hat{A'}\rightarrow \hat{U^{-1}}\hat{A}\hat{U}$, because
$$\langle \hat{A' }\rangle=tr (\rho ' \hat{A'})=tr(\hat{U^{-1}}\rho\hat{U} \hat{U^{-1}}\hat{A}\hat{U})=tr(\hat{U^{-1}}\rho \hat{A}\hat{U})=tr(\rho \hat{A}\hat{U}\hat{U^{-1}})= \langle \hat{A }\rangle$$
But the same argument works for expectation calculated via Ehrenfest's theorem.
I have looked everywhere and found nothing.
 A: In the context of physics, there are "natural" equivalence relations motivated by the following notion: mathematical objects that determine the same physics should be considered equivalent.  These equivalence relations lead to partitions of the sets on which they are defined.
Equipped with this idea, let's examine the two points you mention:


*

*Let $\mathcal H$ be a Hilbert space.  The non-zero elements of this space can be viewed as states of a quantum system.  Two such states that differ by a nonzero complex factor should be considered equivalent because they determine the same physics (e.g. yield the same transition probabilities).  As a result, there is a physically natural equivalence relation on $\mathcal H$ defined as follows: a nonzero vector $|\psi_1\rangle$ is said to be equivalent to another nonzero vector $|\psi_2\rangle$ provided there exists a nonzero complex number $c$ for which
\begin{align}
    |\psi_1\rangle = c|\psi_2\rangle.
\end{align}
The equivalence classes determined by this relation are called rays, and the set of all such equivalence classes is called the projective Hilbert space determined by $\mathcal H$.  This set is often denoted $P(\mathcal H)$.

*Let $p = (p_1, p_2, \dots)$ be a sequence of non-negative real numbers whose sum is $1$, and let $\Psi = (|\psi_1\rangle, |\psi_2\rangle, \dots)$ be a sequence of vectors of unit length in a Hilbert space $\mathcal H$.  A pair $\mathscr E = (p,\Psi)$ of such sequences is called an ensemble.  This mathematical definition can be thought of as corresponding to $N\gg 1$ quantum systems such that $N_k = p_k N$ of them are prepared in the pure state $|\psi_k\rangle$.  Therefore, each $p_k$ can be thought of as the probability that one of the $N$ systems is prepared in the pure state $|\psi_k\rangle$.  To every ensemble $\mathscr E$, we can associate a density operator as follows:
\begin{align}
    \rho_\mathscr E = \sum_kp_k|\psi_k\rangle\langle\psi_k|.
\end{align}
We say that an ensemble $\mathscr E_1$ is equivalent to an ensemble $\mathscr E_2$ provided they determine the same density operator:
\begin{align}
    \rho_{\mathscr E_1} = \rho_{\mathscr E_2}.
\end{align}
The idea behind this definition is that the density operator associated with each ensemble determines all of the physics associated with it (e.g. ensemble averages of observables), so from a physical point of view, ensembles yielding the same density operator shouldn't be considered distinct.
