# Ways of measuring open quantum systems

• I
Couchyam
TL;DR Summary
How many ways can a quantum subsystem be measured (without its complement?)
At the heart of the theory of open quantum systems is the idea that the measurement statistics of many-body systems can be expressed in terms of a reduced density matrix, obtained by tracing over degrees of freedom that are irrelevant to the system of interest.
In general, given a pure state ##|\psi\rangle## in a Hilbert space that is a tensor product of two subsystems, ##\mathcal H = \mathcal H_1\otimes \mathcal H_2##, where ##\mathcal H_1## has orthonormal basis ##|e_a\rangle## and ##\mathcal H_2## basis ##|e_b\rangle##, the reduced density matrix can be defined by
\begin{align*}
\rho_{1,aa'} = \sum_{b} \psi_{ab}\psi^*_{a'b},\quad \psi_{ab}|e_a\rangle|e_b\rangle \equiv |\psi\rangle
\end{align*}
It is easy to check that if ##A## is an operator that acts on the first system alone (i.e. ##A = A_1\otimes \mathbb I_2##, where ##\mathbb I_2## is the identity on ##\mathcal H_2##), then ##\langle A\rangle = \tr(\rho A)##.
The dynamics of the density matrix ##\rho## is induced by whatever Hamiltonian acts on ##|\psi\rangle \in \mathcal H_1\otimes \mathcal H_2##. My question is essentially whether ##\rho## can be thought of as having physical significance beyond a convenient mathematical construct for interpreting quantum statistics in certain experiments. As I understand it, the Hilbert space ##\mathcal H_2## is typically modeled as a large reservoir with a continuum of states, such as a photon/massless gauge boson, or the phonons in a crystal lattice, and this limits the extent to which memory effects are significant, although in the case of a photon field, memory effects could be significant in, for example, a system consisting of particles suspended in a laser cavity or opto-mechanical setup (i.e. where entangled photons could reflect back toward the 'matter' system of interest.) Is it generally possible to "choose" which part of a many-body system can be measured, or are there non-trivial fundamental constraints on what measurements can actually be performed? For example, would it be possible to choose to measure the state of a single spin in a coherently coupled spin network, or would the photon that transmits the information inevitably be entangled with other parts of the system? Apologies if this question isn't worded especially clearly.