The current page for quantum circuit is rather empty. I would like to add at least some elementary stuff (Hadamard gates and teleportation circuits and no-cloning etc.) but I don’t know of an nPOV approach to this. Should I add it anyway, at least as a motivation/idea section, or is it not relevant if it’s not presented in the nPOV?

]]>New stub Weyl functional calculus redirecting also Weyl quantization. I would like to see ref.

- Lars Hörmander,
*The weyl calculus of pseudo-differential operators*, Comm. Pure Appl. Math.**32**, 3, 359–443, May 1979, doi,

but have no access to it (can anybody help?). I also added a sentence at Idea section of functional calculus reflecting that the previous definition there is not fitting functional calculi in the context of quantization, including Weyl’s case. One should do this generality discussion more carefully. the previous definition said that the functional calculus needs to be a homomorphism (from ordinary functions to operator functions). This is true for the functional calculus described in the entry, but not for the wider usage of the phrase like in Weyl functional calculus. Maybe we can resolve this in a better way.

]]>Wheeler wanted to reduce physics to geometry in an even more fundamental way than the ADM reformulation of general relativity with a dynamic geometry whose **curvature changes with time. *

Link: https://en.wikipedia.org/wiki/Geometrodynamics

If there is a way to verify that curvature is a reality, then we may have laid the first piece of evidence for Wheeler’s Quantum Geometrodynamics.

Nicholas Ibrahim Hosein, 2016.

]]>Hi,

I have seen lots of physics going up in recent months and so I thought I would share what I have been working on. The following is an attempt to make categorical structures look super primitive.

If we take a light switch to embody an entire category, we could take the light switch to be a set with two elements and the morphisms are all endofunctions. Let’s say, for fun, that we define the endofunctor for the monad as:

flip switch up $\rightarrow$ light turns on

flip switch down $\rightarrow$ light turns off

flip switch $\rightarrow$ light toggles

do nothing to switch $\rightarrow$ light does nothing

This looks like the identity endofunctor. Now, this endofunctor, in my mind, is deeply fundamental as it is used to test a causal relation between things like the light and the light-switch. The monad is nothing but the identity monad and so, I think, the algebra is nothing but an identity element. (I already asked at mathematics stack). One normally looks at this kind of thing as passing a signal from one system to another and this then goes up to information theory. If you have read my post correctly, though, you will see that I am trying to lift that whole idea up to where we talk only about morphisms and causal structure as opposed to systems of state and the information that encode them. It was a let down to find that the algebra was this trivial for such an important bit of behaviour, one that every physicist working in a lab will use every day.

Can anyone take this thinking and get the first non-trivial algebra (it should be TINY!!!) and keep the spirit of “behaviours in a laboratory”? The co-algebra is also interesting.

If anyone is wondering where this is coming from, consider the fact that one can construct a TQFT entirely within FDHilb by replacing the usual category of cobordisms with the internal category of comonoids. Thus, the background becomes the internal category of classical structures. The category of internal comonoids is defined with axioms that look like the copying and deleting of information. If you read this post carefully you will see that I am abstracting this idea to replace the category of internal comonoids with just comonads.

]]>Hi,

If you haven’t looked at these yet, please have a look. They are calling it the biggest discovery in quantum mechanics for decades.

The reason I am posting this is that I would like to begin a discussion focusing on classical versus constructive (lack of excluded middle) mathematics as a means of dissecting the issue of realism in quantum mechanics. The articles listed are probing a very deep problem of realism in quantum mechanics. Does the wavefunction reflect our knowledge of a system or does it reflect reality itself. They use the language of ontic versus epistemic states and I will explain those.

Ontic states are variables which we think of as concerning an underlying reality and epistemic states are like probability distributions which reflect the information we have about a system. Were I to flip a coin and cover it and ask you to tell me about this coin, you would say it is 50 % heads and 50% tails. In my case, I peaked at the coin and the state that I would use to describe the coin is 100% heads, so the probability distribution just reflects the information we have been given and does not reflect reality. I could also have told you that the coin has landed on its side and is in the “Quails” state. You would then update your probability distribution to include a third state and this process begins again when I reflip the coin and ask you again what you think the state is.

The reason I am posting this here, is that this community seems familiar with the following research in the foundations of mathematics. In it I discovered the following interesting point:

Bifurcation of notions

On the other hand, differences in axiomatization or definition that make no difference classically can result in actual differences in behavior constructively. Therefore, classically equivalent notions often “bifurcate” (or “trifurcate” or worse) into multiple inequivalent constructive ones. This tends to happen whenever a concept involves negation and, to a lesser degree, disjunction and existential quantification.

I would like to offer for discussion the notion that the ontic states (bases and classical states, states of reality) and the epistemic states (quantum states, wavefunctions, states of knowledge) have been consistently treated to different axiomatizations in physics, perhaps even in the same paper/textbook without the author realizing it. Furthermore, I am offering that the multiple overlaping epistemic states which one can have for the same ontic state in quantum mechanics derives from the bifurcation between the classical mathematics one uses on the ontic states and the constructive mathematics on the epistemic states. It comes from the use of the existential quantification which one passes up from the ontic realm up to the epistemic realm when one talks about the elements of the Hilbert space or the space of quantum states. Further, we should find in the articles listed above that they represent an attempt to eliminate the constructive mathematics at the epistemic level. In particular, we see in Hardy’s paper the central axiom is of “Ontic Indifference” and here it is stated:

“Ontic indiﬀerence. Any quantum transformation on a system which leaves unchanged any given pure state, $| \psi \rangle$, can be performed in such a way that it does not aﬀect the underlying ontic states, $\lambda \in \Lambda_{| \psi \rangle}$ in the ontic support of that pure state.”

What we should note is the presence of the existential quantifier being passed up from the ontic states to the epistemic states. It is here where we can begin with dissecting this recent work in terms of foundations. Below, we the definition of “ontic support”:

“By the ontic support of a given state, $|\psi \rangle$, we mean the set, $\Lambda_{|\psi \rangle}$, of ontic states, $\lambda$ which might be prepared when the given pure state is prepared (i.e. those ontic states that have a non-zero probability of being prepared when the given pure state is prepared)”

]]>- Stanford Encyclopaedia of Philosophy online, contents is free online in the article by article html format (for now, they pledge for support to stay so…) ! Good quality stuff online. I added the link to philosophy, and will later add it to math archives.

Specially good for usage and references in our foundational entries on quantum mechanics is that they have excellent online articles quantum logic and probability theory, quantum mechanics: Kochen-Specker theorem, quantum mechanics and quantum mechanics: von Neumann vs. Dirac.

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