Can the $0$-norm represent determinism?

Question

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"??)

Answer 1

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.

Answer 2

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.

Answer 3

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

Answer 4

I do not think that 'classical probability' as commonly (stress non-) understood, for instance works at all, if that sentence has any sense. Perhaps, quantum mechanics is trying to tell us this. With its stronger tie with logic, if anything, quantum probability is less paradoxical.**

To permit me to make an analogy with another 'probabilistic concept'.

Much as with Simpsons paradox, there is just a lot more of this damn concept (of probability) involved in quantum mechanics, and that is perhaps why, on the whole, quantum mechanics is deemed harder to make sense of, then say classical mechanics was.

'Probability' got us into this mess (QM), or the great 'Quantum muddle', as Karl Popper called it.And somehow, making sense of 'probability' is going to have get us out of it. Whether that is, by making sense of probability, or coming to the final realization that the entire concept, was non-sensical to begin with. And the concept of probability be thrown out of the history books, dictionary and academy see S Saunders similar comment in [][1]

History will say which.. Perhaps?.

By this I mean just what it means for an event to be more/less or equal in probability value to another without any use of any other, non logical,non justified/justifiable, if not irrational, posits. Such as any form of the principal of indifference/non contextuality, which amount to circularity. Asserting that, which one wishes to prove.

Robin Giles had an idea. One needs to real all of his obscure works to get his at it, its scattered throughout his work.See for example, the work of Robin Giles for instance his [<1992 paper, "A non classical theory of subjective expected utility theory>."][2]

His odd theory involved a decision theory, a theory of probability and quantum probability in general, that in some sense analyed equi-probability and logic as of the liar paradox re-ified arguably (not just a probabilistic solution). But rather, what it means for two events, to be equi-probable in general.He even tried to give an analysis of twice as with this, on the single case, that he only alludes to.

Using monotonic-ity alone, or some other such thing.

And I think that is how probability must be analyzed, if is to make any coherent non circular sense at all. Despite this idea being incoherent in itself, if at possible in any sense!

The chance values, do not have to be what they are, as analytic, or 'a priori'. But that does not make the concept, non-a priori.

It is just that the values may not be. Presumably, if at all it would be a form of Nomic logic such as in certain physical conceptions of modal logic/conditional logic or QM which subscribe to physical laws and has nothing to with the so called logical or classical or objective Bayesian view of probability which have no, or little to no, connection to Logic as ordinarily conceived at all.

What other non-circular, justifiable grounds, is there but necessarily truth value equivalence? See Frank Arntzenius's draft manuscript on this and otherwise, See [][7] and -see <[Chance, Credence and Circles F Cariani,https://philpapers.org/rec/CARCCA-14>][8].

See https://philpapers.org/s/Robin%20Giles See [https://philpapers.org/s/Robin%20Giles][3]

See also R Giles his 1970 paper "Foundations of Quantum Mechanics", and his earlier works: and

In fact Robin Giles, alludes to this as well see "a Logic of Subjective Belief" below.

If interested, One needs to real all of his obscure works to get the general idea. Which he only alludes to here and there .https://philpapers.org/s/Robin%20Giles See [https://philpapers.org/s/Robin%20Giles][3]

See the links to Robin Giles work on non-classical probability,. non-classical logic, quantum statistic, fuzzy logic and decision theorey, as well as quantum mechanics . See the references, below if interested:

See [<"Introduction to a Logic of Assertions.Robin Giles - 1990 - In Kyburg Henry E., Loui Ronald P. & Carlson Greg N. (eds.), Knowledge Representation and Defeasible Reasoning. Kluwer Academic Publishers. pp. 361--385">][4]

[5] Giles, Robin, **A generalization of the theory of subjective probability and expected utility, Synthese 90, No.2, 301-343 (1992). ZBL0759.60005.

Giles, Robin, A logic for subjective belief, Found. Probab. Theor., stat. Inference, stat. Theor. Sci., Vol. I, Proc. int. Res. Colloq., London/Canada 1973, 41-72 (1976). ZBL0323.02043. _Ferm

Fermller, Christian G.; Roschger, Christoph, From games to truth functions: a generalization of Giles's game, Stud. Log. 102, No. 2, 389-410 (2014). ZBL1329.03060.

Fermuller, Christian G., Revisiting Giles's game. Reconciling fuzzy logic and supervaluation, Majer, Ondrej (ed.) et al., Games: Unifying logic, language, and philosophy. Berlin: Springer (ISBN 978-1-4020-9373-9/hbk; 978-1-4020-9374-6/e-book). Logic, Epistemology, and the Unity of Science 15, 209-227 (2009). ZBL1167.03019.

Fermuller, Christian G.; Metcalfe, George_, Giles's game and the proof theory of {\L}ukasiewicz logic, Stud. Log. 92, No. 1, 27-61 (2009). ZBL1185.03041.

See D Luce, 'in Particular a multiplicative Power relation amongst Three Variables https://www.google.com.au/search?q=Duncan+Luce+multiplicative+power+relations&ie=utf-8&oe=utf-8&client=firefox-b&gfe_rd=cr&ei=mCkNWaqlKajr8Af4irq4BA

Aerts, Diederik, Quantum axiomatics, Engesser, Kurt (ed.) et al., Handbook of quantum logic and quantum structures. Quantum logic. With a foreword by Anatolij Dvure\v censkij. Amsterdam: Elsevier/North-Holland (ISBN 978-0-444-52869-8/hbk). 79-126 (2009). ZBL1273.81015.

Aerts, Diederik; Coecke, Bob, The creation-discovery-view: Towards a possible explanation of quantum reality, Dalla Chiara, Maria Luisa (ed.) et al., Language, quantum, music. Papers of the 10th international congress of logic, methodology and philosophy of science, Florence, Italy, August 1995. Dordrecht: Kluwer Academic Publishers. Synth. Libr. 281, 105-116 (1999).

ZBL0988.81009.Adamson, Alan; Giles, Robin, A game-based formal system for \L$_\infty$, Stud. Log. 38, 49-73 (1979). ZBL0417.03008.

*[*A Generalization of the Theory of Subjective Probability and Expected Utility.Robin Giles - 1992 - Synthese 90 (2):301 - 343.**

Introduction to a Logic of Assertions.Robin Giles - 1990 - In Kyburg Henry E., Loui Ronald P. & Carlson Greg N. (eds.), Knowledge Representation and Defeasible Reasoning. Kluwer Academic Publishers. pp. 361--385

A Non-Classical Logic for Physics.Robin Giles - 1974 - Studia Logica 33 (4):397 - 415" * *-The Concept of a Proposition in Classical and Quantum Physics.Robin Giles - 1979 Studia Logica 38 (4):337 - 353**.][6]

/e-book).%20Logic,%20Epistemology,%20and%20the%20Unity%20of%20Science%2015,%20209-227%20(2009).%20ZBL1167.03019.%20%20%3C/cite%3EFermuller,%20Christian%20G.;%20Metcalfe,%20George_,%20Giles's%20game%20and%20the%20proof%20theory%20of%20%7B%5CL%7Dukasiewicz%20logic,%20Stud.%20Log.%2092,%20No.%201,%2027-61%20(2009).%20ZBL1185.03041.%3C/cite%3E

Answer 5

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.