The question asked is:
What is the Shannon channel capacity $C$ that is naturally associated to the two-spin quantum Hamiltonian $H = \boldsymbol{L\cdot S}$?
This question arises with a view toward providing a well-posed and concrete instantiation of Chris Ferrie's recent question titled Decoherence and measurement in NMR. It is influenced too by the guiding intuition of Anil Shaji and Carlton Caves' Qubit metrology and decoherence (arXiv:0705.1002) that "To make the analysis [of quantum limits] meaningful we introduce resources."
And finally, it is reasonable to hope that so simple and natural a question might have a rigorous answer that is simple and natural too---but to the best of my (imperfect) knowledge, no such answer is given in the literature.
Definitions
Let Alice measure-and-control by arbitrary local operations a spin-$j_\text{S}$ particle on a local Hilbert space $\mathcal{S}$ having $\dim \mathcal{S} = 2j_\text{S}+1$, upon which spin operators $\{S_1,S_2,S_3\}$ are defined satisfying $[S_1,S_2] = i S_3$ as usual.
Similarly let Bob measure-and-control by arbitrary local operations a spin-$j_\text{L}$ particle on local Hilbert space $\mathcal{L}$ having $\dim \mathcal{L} = 2j_\text{L}+1$ upon which spin operators $\{L_1,L_2,L_3\}$ are defined satisfying $[L_1,L_2] = i L_3$ as usual.
Let the sole dynamical interaction between the spins — and thus the primary resource constraint acting upon the communication channel — be the Hamiltonian $H = \boldsymbol{L\cdot S}$ defined on the product space $\mathcal{S}\otimes \mathcal{L}$. Further allow Bob to communicate information to Alice by a classical communication channel of unbounded capacity, but let Alice have no channel of communication to Bob, other than the channel that is naturally induced by $H$.
Then the question asked amounts to this: what is the maximal Shannon information rate $C(j_\text{S},j_\text{L})$ (in bits-per-second) at which Alice can communicate (classical) information to Bob over the quantum channel induced by $H$?
Narrative
In practical effect, this question asks for rigorous and preferably tight bounds on the channel capacity associated to single-spin microscopy. The sample-spin $S$ can be regarded as a sample spin that can be modulated in any desired fashion, and the receiver-spin $L$ can be regarded variously as a tuned circuit, a micromechanical resonator, or ferromagnetic resonator, as shown below:
The analysis of the PNAS survey Spin Microscopy's Heritage, Achievements, and Prospects (2009) can be readily extended to yield the following conjectured asymptotic form:
$$\lim_{j_\text{S}\ll j_\text{L}} C(j_\text{S},j_\text{L})=\frac{j_\text{S}\,(j_\text{L})^{1/2}}{(2\pi)^{1/2}\log 2}$$
Note in particular that the dimensionality of Bob's receiver-spin Hilbert space $\mathcal{L}$ is $\mathcal{O}(\,j_\text{L})$; thus a Hilbert-space having exponentially large dimension is not associated to Bob's receiver. However it is perfectly admissible for Alice and Bob to (for example) collaborate in squeezing their respective spin states; in particular the question is phrased such that Alice may receive real-time instruction of unbounded complexity from Bob in doing so.
Preferred Form of the Answer
A closed-form answer giving a tight bound $C(j_\text{S},j_\text{L})$ is preferred, however a demonstration that (e.g.) $\mathcal{O}(C)$ is given by some closed asymptotic expression (as above) is acceptable.
It would also be very interesting, both from a fundamental physics point-of-view and from a medical research point-of-view, to have a better appreciation of whether the above conjectured capacity bound on spin imaging and spectroscopy can be substantially improved by any means whatsoever.
This post has been migrated from (A51.SE)
Q&A (4871)
