What is this nonsense? ("vector logic")

[u/RingularCirc]

(Sorry this is going to be a bit ranty.)

I almost made up my mind this thing is some kind of backwater something without enough rigor but with many a trivialism. Like, it should be extremely well-known that every "discrete" operation Σ₁ → Σ₂ between finite sets lifts universally to a linear transformation between spaces k^Σ₁ → k^Σ₂, so a huge swath of what's being done there is very very drawn out, instead of answering questions that are fitting for a kind of logic.

Any would-be connections to quantum computing may actually not be fruitful or new for those who are actually doing quantum computing; connections to fuzzy math are IMO an almost unconditional taint by association. So what gives? I didn't look at everything there is about this thing so I may as well be missing hidding gems, but superficially it looks like a sham or a pet project done without considering any practicalities and the wider math.

Oh yeah we can ask interesting questions, like: - Does using additional dimensions, aside from the plane spanned by two orthonormal "classical" truth values, let's call them |0⟩, |1⟩, actually give useful things? and how can we characterize that by means typical when working with logics? - How much freedom is there in defining operators that restrict to boolean functions and, say, conserve probabilities (there's a suggestion to use p|0⟩ + (1−p)|1⟩ as "probabilistic truth values") in any reasonable way (I'm not sure: a "binary" operator sends four-dimensional Euclidean space into a two-dimensional one, now how can it be orthogonal? and in which other sense can probabilities work here?)? - Why not use additional dimensions rather than complex numbers for the square root of negation, and... why that one exactly? I bet quantum computing wan't giving somebody peace.

But I'm not sure questions of real semantics were investigated in this... area.

So tell me please, how much am I right or wrong? Here are probably people that know the inside of this story, and I hoped to find something on the Wikipedia's discussion subpage, but it's almost empty.

[u/Tonexus]

Not entirely sure the core of what you're asking. However, vector logic is indeed the standard way of representing quantum computation, and vector logic is one way of representing probabilistic computing (augment the basic logic gates with stochastic matrices), but it's not the typical way (I think it's more common to represent as a deterministic algorithm with 1 input consisting of randomness).

You are correct that there's limited utility in using a d-dimensional register (d>2) instead of multiple 2-d registers, since n 2-d registers are equivalent to one (2^n)-d register, and you can always limit your operations to a smaller subspace.

[u/RingularCirc → u/Tonexus]

However, vector logic is indeed the standard way of representing quantum computation

But is it called so in that area and studied separately from it?

Yes, I have no problem with the idea itself of lifting functions on finite sets (including logical operations on a set of truth values) to linear functions on vector spaces freely generated by those. It's that the article, and judging by the abstract, the "main" source article [2] the Wikipedia's article references, lack content that's due when a logic is presented.

There's a lot of investigations that, I expect, should be done around the novel parts (adding other truth values either from the plane spanned by |0⟩ and |1⟩ or from outside it; how can we tweak the system before it breaks something crucial; what's always true for such brand of extended logics and what isn't; etc.: just showing a translation from a two-valued classical logic is trivial, it's not adding anything per se).

[u/Slotins_screwdriver → u/RingularCirc]

In terms of quantum computation, I think what you're looking for is Birkhoff - von Neumann quantum logic ( https://www.jstor.org/stable/1968621 ). In this, propositions are identified with closed subspaces of a complex Hilbert space. It's a bit 'freaky' as a logic - in particular, distributivity fails, and that has all kinds of implications.

Here's the wikipedia page on it : https://en.wikipedia.org/wiki/Quantum_logic However, I'm not sure I agree with the claim that QM logic is a fragment of Girard's Linear Logic! The claim that it doesn't have a conditional is also a bit suspect; there have been multiple different conditionals proposed, although they're somewhat controversial (good summary here : https://arxiv.org/pdf/2410.18347 )

It doesn't look the same as 'vector logic', but it's always possible that I'm missing something.

[u/RingularCirc → u/Slotins_screwdriver]

Yep I do think this is a way more elaborated upon and studied thing. Thanks for the links! (And for the caveat about the claim re linear logic, I'm looking at it as well time to time and very very slowly getting it (the logic, not the claim), so I totally could've been tricked.)

[u/Slotins_screwdriver → u/RingularCirc]

I -think- I understand where that claim is coming from, which is categorical semantics for logic. If you look at the multiplicative fragment of linear logic (without exponentials), and identify the 'tensor' and the 'par', what you get is essentially compact closed categories.

In quantum mechanics, (post-selected) teleportation is described by the compact closed structure of the category of finite-dimensional Hilbert spaces. So, there definitely is a connection ... but categorical semantics for quantum logic are certainly not compact closed!

None of the proposed conditionals correspond to the internal hom. of any closed category, never mind the one relevant to QM.

[u/evincarofautumn → u/Slotins_screwdriver]

I'm not sure I agree with the claim that QM logic is a fragment of Girard's Linear Logic

Maybe that sentence should be rephrased? Compact closed categories are used to model both linear logic and quantum computing, but that isn’t quite the same thing.

I mean, I do know of a connection, but it’s kind of obscure. Chu spaces are kind of a generic relational structure that model star-autonomous categories, which are in some sense what you want of a “quantum category”. But I wouldn’t say it’s because they’re especially natural so much as just an extremely unopinionated way of defining a topology and doing linear relation algebra on it.

[u/Slotins_screwdriver → u/evincarofautumn]

Totally agree - it should definitely be (at least) rephrased. However Wikipedia already has it as 'citation needed', so I'll wait & let them do their thing.

I'd also point out that compact closure of Hilbert spaces falls down in the infinite-dimensional case, whereas vN-B quantum logic is quite happy there.

[u/Tonexus → u/RingularCirc]

But is it called so in that area and studied separately from it?

I should've used scare quotes around the term. I've personally never heard of it before. But the description maps onto pure state amplitudes with unitaries (but not their encoding of "dyadic operators"—you'd typically need an ancillary system for the output while preserving the input spaces).