What does it mean for a density matrix to "act on a Hilbert space $\mathcal{H}"$?

Question

For a Hilbert space $\mathcal{H}_A$, I have seen the phrase

density matrices acting on $\mathcal{H}_A$

multiple times, e.g. here.

It is clear to me that if $\mathcal{H}_A$ has finite Hilbert dimension $n$, then this makes sense mathematically, because a density matrix $\rho$ can be written as $\rho \in \mathbb{C}^{n \times n}$ and elements $\phi$ of $\mathcal{H}_A$ can be written as $\phi \in \mathbb{C}^n$, so I can write down $\rho \phi \in \mathbb{C}^n$.

However, it is unclear what this means? The density matrix $\rho$ describes a (possibly mixed) state of a quantum system. But I can also interpret $\phi$ as a single state vector, describing a quantum system in a pure state.

So, what does $\rho\phi$ refer to (where $\rho \in \mathbb{C}^{n \times n}$ is a density matrix, and $\phi \in \mathbb{C}^n$ is an element of $\mathcal{H}_A$)? Can I interpret it? How can a density matrix (i.e., the representation of a state) act on states (on single state vectors)? Why do we interpret density matrices (which represent states) as operators?

Answer 1

It is common that one refers to a density matrix (or, equivalently, a density operator) $\rho$ as acting on a particular space $\mathcal{H}$. This serves to establish the "type" of $\rho$ in computer science parlance. In particular, when there are multiple spaces under consideration, it may be helpful for a reader to know that $\rho$ corresponds specifically to whatever abstract physical system is described by the space $\mathcal{H}$.

Referring to $\rho$ as acting on a space $\mathcal{H}$ also makes perfect sense, as the question points out, because $\rho$ can be viewed as a linear map from $\mathcal{H}$ to itself. The properties of $\rho$ that relate to its action as a linear mapping of this form are important and say a lot about the state. For example, the eigenvalues of $\rho$ describe the randomness or uncertainty inherent to that state.

However, this does not mean that if $\phi\in\mathcal{H}$ is a unit vector that describes a pure state, then $\rho\phi$ should have an interpretation. Such a vector might show up in a proof or calculation -- for example, the quantity $\langle \phi | \rho | \phi\rangle$, which is the inner product between $\phi$ and $\rho\phi$, is a commonly encountered quantity known as the fidelity (or squared fidelity) between the states represented by $\rho$ and $\phi$ -- but in my view the vector $\rho\phi$ is just a vector and does not have a natural or fundamental physical interpretation.

Answer 2

By "acting of $\mathcal{H}_A$" I believe you mean "acting on $\mathcal{H}_A$", which is what is written in the section you provided a link to.

What it means is "acting on a state in $\mathcal{H}_A$".

"I do not know what it means for a state to act on $\phi$."

You defined $\phi$ as a state. States do not act on states. Operators (such as $\rho$) act on states. The section you provided a link to, does not mention "acting on a state $\phi$" at any point.