Can the $0$-norm represent determinism? In Scott Aaronson's Quantum Computing since Democritus, he presents classical probability theory as based on the $1$-norm, and QM as based on the $2$-norm. 

Call $\{v_1,\ldots,v_N\}$ a unit vector in the $p$-norm if $|v_1|^{\ p}+\cdots+|v_N|^{\ p}=1$

The slide below is from a presentation of his.



Now, he seems to claim that the $1$-norm and the $2$-norm are the only options here, although earlier in the book he allowed for three choices:

(1) determinism, (2) classical probabilities, or (3) quantum mechanics.

My question is: Would it be natural to let determinism be simply based on the $0$-norm? Is this problematic? Wouldn't the definition $0^0=0$ (for this purpose) and $p_i\in\{0,1\}$ suffice?
If so, what would be the equivalent to "probability vector" and "amplitude vector" be called?
("Indicator vector"??)
 A: Yes, all of this works out. You just get finite sets and functions between them. 
Incidentally, if you want an even more abstract approach to probability that unifies classical and quantum probability into one framework, check out noncommutative probability. I've written about this here and the specialization to the cases Aaronson describes is given here. 
A: Another way to view the 0-norm or determinism condition is as the intersection of the 1- and 2-norm conditions. In that respect it is not a different type of theory, only a special case, and it can be considered a matter of convention whether you count it as distinct given the other two. 
Positivity assumptions are crucial in limiting the set of possibilities to two or three.
A: I remember reading Scott Aarronsons Post on this topic. However, I must say this
(1) Its unclear what the norm is in classical probability is to begin with; and what is meant by classical probability. 
(2) its unclear to me what Quantum probability is; and what its distinction (other than as a matter of formal details is). 
One can see Narens "2014, Probabilistic Lattices",  "Probability: an Examination of Qualitative and Logical Foundations" and for more formal work, his books 1985, "Abstract Measurement Theory" and Theories of Meaninfulness  see https://books.google.com.au/books?id=Bh23CgAAQBAJ&pg=PA186&lpg=PA186&dq=Narens+2014&source=bl&ots=K3Ny5Uv_sm&sig=YDJrE28W3uncAJEt2Cs2vXSepTo&hl=en&sa=X&ved=0ahUKEwifqPWjjNrTAhUFGZQKHbMsAssQ6AEIQTAF#v=onepage&q=Narens%202014&f=false
(3) And that is because its unclear to me what Probability in general is. I begin to suspect, that Quantum probability (or rather just probability) if it to ever work at all, must act in an odd, logic based albeit somewhat contextual fashion such as in Quantum mechanics. 
Whilst I do not reject the formal division, I reject the division in terminology. It often licenses one to think that some well understood concept, works differently in the Quantum Realm. When it has never been understood to begin with, seen its inception by the drunken gambler Cardano. 
In contrast to the axioms (of  a formal form). There are differences, but Q probability (much as I hate to use this term, or the term classical probability) roughly subscribes to the same axioms and axiom schemata.
But there are differences of course, but this is the case, under any different interpretation or truth condition of chance; whether intended for QM or classical cases or both, or neither. Otherwise one would begin to suspect, that nothing different is being said
.
 Just as much as, the measure theoretical interpretation, is an interpretation, or frequentism is; even if the former, is as much of an non-interpretation, as a inter-pretation can be
A: I hope E T Jaynes ("Probability: the logic of Science, 2003") 
is not right when he says, that were I to alive in 1000 years hence, I would find that no progress has been made whatso-ever; not that his approach, logico/objective Bayesian approach, is going to help, with regard to QM or non QM matters though
Either by making some sense of what it (probability) meant to begin with, or by throwing it ,that is the entire concept of probability/plausibility,aka 'graded modal logic' whether qualitative or numerical, ontic, prescriptive or descriptive,  out of the history books and the dictionary and out of the academy.
As I think Simon Saunders said, in one of his contributions to the compendium in his co-edited, 2012,See(https://www.amazon.com/Many-Worlds-Everett-Quantum-Reality/dp/0199655502) (or rather chance?)
I say chance/probability, because as I see it, accompanying any (nearly any) formalism, reconstruction or interpretation of Quantum mechanics, is an account of probability. 
By this is meant, the interpretation, or the formalism, or some combination in between. Its not yet clear that Quantum probability is any different. partly because its unclear whether we ever understood what probability means to begin with.  QM can consider it to be roughly just another interpretation whose truth conditions are different.
