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- 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.

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.