TITLE: Generalisation of the trace cyclic theorem to partial traces.
QUESTION [8 upvotes]: The trace-cyclic theorem says that linear operators commute (or cycle) within the trace, that is
$${\rm Tr}(XY) = {\rm Tr}(YX).$$
Now if $X$ and $Y$ operate on a product space $H_A \times H_B  \times ...$, it is also possible to take the partial trace over any combination of those spaces.  But for various reasons, the cyclic theorem will not hold in that case.
$${\rm Tr_A}(XY) \ne {\rm Tr_A}(YX)  \hspace{4em} \mbox{(except special cases).}$$
My general question is whether there is still some constraint on the commuted trace.  That is, does the trace-cyclic theorem generalise in some weakened form.
My specific question involves lots of details:

We are working in complex Hilbert space (quantum).
$X =   \left|R\rangle\langle{E}\right|$ is rank-1 i.e. an outer product of arbitrarily entangled Hilbert-space vectors.
$Y = U$ is unitary, but it freely entangles the subspaces.
There can be more than two subspaces and it is given that
$$
   {\rm Tr_{\bar S}}(XU) = 0.
 $$

Where $\bar{S}$ means "all subspaces other than $S$", and $S$ is arbitrary.  That is, $X$ is chosen such that when you trace over $XU$ and leave any one subspace alive, you always end up with zero.  So what I want to know is, whether:
$$
   {\rm Tr_{\bar S}}(UX) = 0.
 $$
If this is true, I would like to know why.  If it isn't true, I would like to know if any related thing is true.

REPLY [2 votes]: For partial trace a cyclicity property holds on the system you trace out. More formally, the following weaker cyclicity property holds (I'm assuming finite-dimensional Hilbert spaces here):
$$\operatorname{tr}_A\left[(X_A Y_A\otimes I_B)Z_{AB}\right] = \operatorname{tr}_A\left[(Y_A\otimes I_B)Z_{AB}(X_A\otimes I_B)\right],$$
where $I_B$ denotes the identity operator on the Hilbert space $\mathcal{H}_B$. To verify this identity, expand $Z_{AB}$ in some product operator basis (e.g., matrix units), and use the defining relation $\operatorname{tr}_A(R_A\otimes S_B) = \operatorname{tr}(R_A)S_B$ of the partial trace together with linearity of the trace.