Just to connect the dots: You can take an POVM defined by a set of positive operators $\{E_i\}$ and then define Kraus operators $K_i = E_i^{1/2}$, where "1/2" denotes the unique positive square root of a positive operator. These Kraus operators define the channel.

Thanks for the thoughtful reply. The application we have in mind is proving a general $\hbar\to 0$ limit of the Lindblad equation in quantum mechanics; we are specifically trying to be as general as possible. I think the problem in the case of no boundary and C^k functions is already quite interesting and broadly applicable, and indeed I would characterize non-C^2 functions as unphysical exceptions.

Is there a modern self-contained treatment? Everyone cites the Pfeuty and Lieb-Schultz-Mattis papers, but LSM just introduced the method, applied to different models, and Pfeuty just cites LSM for the method and basically writes down the answer without many details.

@Thriveth The effective range of electromagnetism is limited by the net-zero charge density. There's a 1/r^2 law for isolated charges, but it falls off faster for net-zero charge distribution (e.g., 1/r^3 for the dipole moment, or even faster for higher moments). This is "screening", and it means effective finite range. But gravity is unscreened because there's no negative mass.

@knzhou Sorry, you're right, I absolutely should have written "'local' conserved quantities", with quotes around 'local' to emphasize that the appropriate notion of locality is unclear (e.g., finite spatial radius vs quasi-local spatial decay.). You may also be interested in Dan Ranard's observation that the projectors $O_i=|i\rangle\langle i|$ are linearly independent but not algebraically/functionally independent.

Expanding on SuperCiocia's comment, one might say a quantum lattice system is "partially integrable" if it has an extensive number of conserved quantities that are independent in the appropriate sense and "fully integrable" if their simultaneous eigenspaces are 1-dimensional and hence give a full orthonormal basis of labeled states for the Hilbert space.

If we trivially extend a Kadison map to all Hermitian operators using $\rho' = \mathrm{Tr}[\rho_+](\rho_+/\mathrm{Tr}[\rho_+])' - \mathrm{Tr}[\rho_-](\rho_-/\mathrm{Tr}[\rho_-])'$, where $\rho_\pm$ are the positive and negative parts of $\rho$, then is the condition that the map "preserves the convex structure" equivalent to saying the map is linear? If so, then a Kadison map is just a bijective positive trace-preserving (PTP) map, right? And then Kadison's theorem is the statement that the only bijective PTP maps are unitary? (Maybe some of that breaks down in finite dimensions?)

Is this just the statement that the transformation preserves the Hilbert-space inner product but not the inner product on spinors?

What I think is going on is that some fluid properties of helium-4 (which I don't understand) sets a characteristic length scale much smaller than 1 mm, and that within each region of that size roughly 7% of the atoms are in the same mode of that size. Indeed, the "characteristic lambda transition temperature" for superfluid helium-4 is a relatively balmy 2.17 Kelvin. Need an expert to comment.