ct.category Theory – Algebra Trousers $ M_n $ as a special symmetrical Frobenius algebra in the shape of a dagger

I'm looking at the paper Categorical quantum mechanics II: classical-quantum interaction by Coecke and Kissinger (arxiv link), and I'm having trouble with one aspect in particular.

Throughout the document, quantum wires are defined as "doubled" wires, whereas the classical world is represented by single wires. In particular, quantum spiders are simply doubled spiders. My understanding of this is that we use the canonical $ dagger $-Algebra of Frobenius on $ A otimes A ^ * $given $ dagger $-Algebras of Frobenius on $ A $ and $ A ^ * $, to generate quantum spiders on systems of the form $ A otimes A ^ * $. This seems sufficient to treat the classical and quantum operations "on the same footing", ie each object in our category may have an associated Frobenius algebra, from those specified on the ground objects, and if a process is doubled, it is quantum. Coding / decoding cards allow conversion between the two. I do not understand that there is a real difference between what happens at the classical and quantum levels, if it is that the quantum is doubled – that we thus interpret a particular morphism as a simple "thick / doubled" quantum wire or two classic wires is out of place.

However, the definition 3.20 indicates that there is a second canonical algebraic structure of Frobenius on any object of the form. $ A otimes A ^ * $, namely the "pants" algebra $ M_n $. In addition, the following paragraph – and my readings on the $ CP ^ * $ construction – seems to suggest that the correct insertion of fully positive quantum processes into the category $ CP ^ * $ mixed classical / quantum processes is actually given by considering quantum processes as acting on objects of the form $ A otimes A ^ * $ with the algebra of the pants as the associated Frobenius algebra – while each component $ A $ or $ A ^ * $ may have associated a completely independent Frobenius algebra structure. If this is correct, does this imply that there are actually two Frobenius algebra structures associated with $ A otimes A ^ * $ – the algebra of the pants, and the other canonical "doubled" Frobenius algebra, described above? Does this mean that the algebra of Frobenius "doubled" on $ A otimes A ^ * $ actually represents classical communication, but the algebra of pants on the same object is quantum communication?

I must have been pretty confused here, because these two "understandings" can not be correct!

Why does longitude correspond to Frobenius in arithmetic topology and other strange phenomena?

I'm trying to address Morishita's book Nodes and Premiums to learn a little more about arithmetic topology, but some analogies surprise me. I know that correspondence has to be addressed with a grain of salt, but some parts are so fundamental that I would like to understand them better.

  1. In the table (3.3) on page 50 of his book, Morishita writes that longitude, called $ beta $, should correspond to a Frobenius lift and meridian, called $ alpha $, corresponds to a taming inertia generator (both as elements of the maximum taming quotient of the absolute Galois group of a local field). It calls "longitude" a path that bypasses a hole in the boundary of a tubular neighborhood of the node and "meridian" the edge of a disc that is a "cross section" of the tube. If the knot was the denouement, this neighborhood would be the complete torus $ S ^ 1 times D ^ 2 $: in that case, $ alpha = partial D ^ 2 $ and $ beta = S ^ 1 $. This analogy does not support my view that inertia acts as a monodromy, which is "turning around holes", but I tried to pursue it. Then (page 63, after Theorem 5.1), he describes the analog of decomposition groups for an uninterpreted node. K $: he says that this group should be generated by "a circulating loop" K $"which I think is just the image of $ alpha $. Then I'm completely lost, because I would expect Frobenius to generate decomposition groups in un-simplified situations …
  2. In chapter 11, an experimental analogy with the Iwasawa theory is suggested. Nevertheless, it seems to me that something is odd, because the typical Iwasawa theory concerns a very wild branch, whereas the same table (3.3) on page 50 seems to indicate that there is no wild topological inertia. So the Galois group of $ X_ infty / X_K $, or $ X_K $ is a node complement and $ X_ infty $ is a $ mathbb {Z} $-covering $ X_K $, in a sense, resembles an "infinitely thinly branched cover" (which has no arithmetic analog) rather than a $ mathbb {Z} _p $-extension. Am I missing something? In the same sense, he has a small parenthesis between p. 144 and p. 145 where he writes that "[Ensupposantqu'a[assumingthata[ensupposantqu'un[assumingthata$ mathbb {Z} _p $-extension be branched to a first only, and this first be totally ramified]is a hypothesis analogous to the case of knot. "Why is it so? Or the fact that one $ mathbb {Z} $-cover must be branched to a single node, and the fact that there can not be a small, unbranched layer below seems obvious to me (at least if the basic variety is not $ S ^ 3 $, other $ pi_1 (S ^ 3) = $ 0 should say there is no unassembled extension).

matrices – Frobenius normal form of a doubly stochastic matrix

Yes $ A in M_n ( mathbb {C}) $then $ A $ is called reducible if there is a permutation matrix $ P $ such as
P ^ top A P =
begin {bmatrix}
A_ {11} & A_ {12} \
0 & A_ {22}
end {bmatrix}, $$

in which $ A_ {11} $ and $ A_ {22} $ are square matrices of at least one order. Yes $ A $ is not reducible, so $ A $ is called irreducible. Note that with this definition, each matrix one by one is irreducible. A matrix $ A $ is irreducible if and only if its Led graph or digraph is strongly connected.

He is known (see, for example, Brualdi and Ryser [Theorem 3.2.4; MR1130611]) that if $ A in M_n ( mathbb {C}) $then $ A $ is irreducible or there is a permutation matrix $ P $ such as
begin {equation}
P ^ top A P =
begin {bmatrix}
A_ {11} & cdots & A_ {1k} \
& ddots & vdots \
& & A_ {kk}
end {bmatrix}, tag {1} label {fnf}
end {equation}

in which the matrices $ A_ {11}, dots, A_ {kk} $ are square and irreducible matrices. The matrix in eqref {fnf} is called Frobenius or irreducible normal form (of $ A $) (FNF) and it's not unique. However, the blocks $ A_ {11}, dots, A_ {kk} $ are unique until the similarity of permutation.

A matrix $ A $ is non-negative ($ A $ 0 ge) if $ a_ {ij} ge $ 0, $ 1 the i, j the n $. A non-negative matrix is stochastic if $ Ae = e $ and stochastic doubling if, in addition, $ A ^ top e = e $ (that is to say., $ A $ has rows and columns equal to one).

In a 1965 article, Perfect and Mirsky [MR0175917] indicate, without proof, that $ A $ is a doubly stochastic matrix, so each FNF of $ A $ is of the form
begin {bmatrix}
A_ {11} & & \
& ddots & \
& & A_ {kk}
end {bmatrix},

that is, each doubly stochastic matrix is ‚Äč‚Äčeither irreducible or is permutationally similar to a direct sum of irreducible and doubly stochastic matrices. After giving the result Perfect State and Mirsky:

"This result is almost certainly known.
derives very easily from the definitions, we omit the details of the
evidence. "- p.38.

This result is easy to establish for matrices of less than or equal to four, but does not seem obvious or seems to flow from the definitions above.

Question 1: Is this result obvious or is there a simple proof of this fact?

Question 2: Does anyone know of a reference?

Distance induced by the Frobenius norm between two multivariate normals

There is a standard on covariance matrices defined by the Frobenious standard. http://mathworld.wolfram.com/FrobeniusNorm.html
Can this be used to define a valid distance between two multivariate Gaussian distributions? Something like $ d ^ {2} left ( mathcal {N} left ( mu_ {1}, Sigma_ {1} right), mathcal {N} left ( mu_ {2}, Sigma_ {2 } right) right) = || mu_ {1} – mu_ {2} || _ {F} ^ {2} + || Sigma_ {1} – Sigma_ {2} || _ {F} ^ {2} $. I am looking for references / articles.