Quantum Information theory, in my opinion, has been responsible for an upsurge in further studies and analysis in several topics pertaining to Linear Algebra and Operator Theory, these are some of the disciplines deeply touched by QIT , I myself work on an abstract mathematical formalism where a side interest is to study the two key objects of QIT, namely the Density matrices and the Unitary operators acting as logic gates and governing the dynamics, in light of my formalism, i believe that QIT will definitely lead to many interesting developments in Mathematics/Mathematical Physics in the years to come.

I will be uploading a paper soon on the topic of DeSitter Space, Conformal Field Theory in Information Theory as the substructure of space-time.

To your question, yes there is one.

If you know a little bit about information theory, you would know that there is Shannon's entropy, which is the measure of uncertainty in information, measured in bits.

It turns out that in quantum cryptography Von Neumann's entropy, which has the same function as that of Shannon, but applied to the quantum medium, is used.

In my article https://www.researchgate.net/publication/327111273_Description_entropique_de_la_gravite_par_la_thermodynamique_de_fluides_relativistes_et_par_la_theorie_de l'information

I explain the link between these two notions. Here is an excerpt.

Shannon's entropy, due to Claude Shannon, is a mathematical function that corresponds to the amount of information contained or delivered by an information source: the more redundant the source, the less information it contains. is thus maximal for a source whose all symbols are equiprobable Shannon's entropy can be seen as measuring the amount of uncertainty related to a random event, or more precisely to its distribution.Generally, the logarithm will be in base 2 ( binary basis).

[...]

Von Neumann's entropy is identical to that of Shannon, except that Shannon uses the variable ρ, a density matrix. As Serge Laroche writes, this equation can be used to calculate the degree of entanglement of two particles: if the two particles are not entangled, the entropy is zero. Conversely, if the entanglement between the two particles is maximum, the entropy is maximum, considering we do not have access to the subsystem. In classical mechanics, zero entropy means that events are certain (only one possibility) whereas in quantum mechanics this means that the density matrix is ​​a pure state of ψ. But in quantum physics measurements are usually unpredictable because the probability distribution depends on the wave function and the observable. "

I invite you to read a scientific article entitled "Entropy and Its Inequalities" which speaks precisely of this "conjugation" between these two forms of entropies. I put you the link:

http://rickbradford.co.uk/QM6Entropy.pdf

Thank you Roman. Verlinde's work, your paper (in French) and the holographic principle are very relevant to this topic.

Dear Michel,

My answer consists of two questions: (a) Is logic (in general sense) grounded upon the 2nd law of thermodynamics (i.e. entropy, in general sense), or any other types of entropy in particular sense (e.g. von Neumann entropy)? (b) Is the term 'qubit' not a grammatical contradiction? Since 'bit' is 'binary' and quantum information theory is engaging in quantum parallelism, i.e. 'many' simultaneously?

Thank you for the right answer! Marc.

Dear Mark,

About quantum logic

https://en.wikipedia.org/wiki/Quantum_logic

and there is research about the paradoxes of quantum mechanics (e.g. quantum contextuality) that contadicts classical logics.

May be mathematics helps to circumvent these paradoxes hence my question. At some stage of my research I coined "geometric contextuality" (see Fig. 11 in the reference below for a popular sketch)

https://arxiv.org/pdf/1605.07118.pdf

Besides, qubits do exist in the real world of course.

Michel

Dear Michel,

Thank you very much for the links. Would you be so kind as to give me the right answer to this very short double question:

-Is wave a set? If yes, can a particle be an element of that set?

Thank you in advance for the answer! Marc.

You may have a look at this link

https://en.wikipedia.org/wiki/Quasi-set_theory

Thank you, Michel Planat :)

Marc Carvallo

The second principle of thermodynamics allows to define the arrow of time. It is thanks to it, and only to it, that we know that we can not go back in time for example, because the physical phenomena are irreversible. In a way, it is more "logical" to rely on the entropic principle, since it follows the temporal evolution of physical reality.

There are certain processes for which the phenomenon is reversible. Take for example an elastic that deforms you. According to the elasticity of it, it can, when you let go, return to its original state, and this without adding energy.

Some phenomena can be made reversible. But this will require a certain amount of energy, since the entropy is only the variation of the energy as a function of the temperature. For the example cited above, there is no need to provide energy. It is a spontaneous reversibility without energy input.

After, Shannon's entropy does not work exactly the same. It is not a thermodynamic entropy, but a purely statistical entropy from the theory of communication. Some phenomena are reserrable. All that is needed is to exchange new information, reducing the entropy. However, disturbances (such as spurious noise) can be a cause of irreversibility of information.

In quantum mechanics, quantum decoherence, Heisenberg uncertainty and quantum laws force, in a way to another, to make all quantum phenomena irreversible. Von Neumann's entropy, if it were to be described for example by a time-dependent wave function (cf. Shrödinger's equation), we could know the previous state, but not return to this state, since the phenomena are probabilistic. .

Qubit simply designates the principle of quantum state superposition. It is very difficult to explain verbally, people tend to say that the quantum superposition explains the fact that two states are simultaneous, "at once" to such state, and "at the same time" to another state. In fact, quantum physics just says that the two states follow each other by some sort of probabilistic law. To be clearer, if you take a phi function defining the wave function, you know that it can be split into the sum of two functions phi, take phi 1 and phi 2. This gives phi = phi 1 + phi 2 Phi describes the probability function of two distinct probabilities functions. Phi thus describes the "superposition" of the two states, each of the two states defining the probability of the quantum system studied.

Qubit wholesale, that's it. You have the probability of having 1, the probability of having 0, the sum of the two states giving the qubit, two distinctive probabilities described by one and the same "state" of probability. Hence the expression "superposition".

I do not know if I've been clear, it's hard to explain verbally. Quantum physics has the gift of being mathematically clear, but to explain it verbally .... it's the big "brothel";)

Thank you, Roman.

I found something for you to think about

Though Landauer famously said "information is physical," it turns out that information is not so physical after all.

here are some details:

https://phys.org/news/2016-07-refutes-famous-physical.html

Michel

Dear Michel,

I just read the wikilink you sent me but I could not find the right answer I am searching after. Would you then be so kind as to give me the short and concise answer upon that double question? Thanks! Marc.

Dear Roman,

From your answer I take only 1 point: so there is a sequence between phi 1 and phi 2. So, (a) even quantum physics is not able to annull the 2nd law of thermodynamics (re: sequence). In other words, our classical logic remains the same; it does'nt change the classical logic and mathematics notwithstanding its own being as non-classical physics; (b) is our logic and mathematics in principle and in fact, at closer look, grounded upon the 2nd law of thermodynamics? (c) if phi 1 and phi 2 are different, then they are according to the logical law of identity not identical, only simultaneous; (d) please explain me more about the logical distinction between identity and simultaneity (or: quantum parallelism). Thanks for your answer. Marc.

Thank you for your reference, Michel Planat

Indeed, I knew, well, 2, 3 years ago, that the Landauer limit had been exceeded. However, this does not mean that there is no limit. Landauer was inspired by Boltzmann's equations to define his equation, but that does not mean that there is no theoretical limit.

I think it is simply impossible, by virtue of the first principle of thermodynamics, not to use energy for a phenomenon to become reversible. Irreversibility will always be a foundation of physics and I think no one can prove that reality is not following this irreversible process. Otherwise, it would come into question "the arrow of time".

Dear Marc Carvallo

Regarding the second principle of thermodynamics.

I can not be more specific in what I just said. However, I will try to rephrase.

If we hold to the entropic principle and to the thermodynamic principle, the evolution of reality is based on the principle of causality: a phenomenon A, which leads to a phenomenon B, leading to the phenomenon C etc ... All that the we know in physics that it is impossible for a phenomenon B to occur before the A, since it is the A that causes the phenomenon B to occur.

The causality of phenomena has in fact two sources, one of which I have just recalled, and which I forgot to mention. I'm sorry.

The first is the irreversibility of physical phenomena, whether of a quantum, relativistic or classical nature. The second law of thermodynamics makes explicit that entropy (hence disorder, or uncertainty of information) can only increase in an isolated system (system in which no exchange with the outside is performed).

If we consider our universe as a closed system, then the entropy of the universe can only increase. This is a fact.

Then, the most interesting area to study and which will probably allow you to better understand what I am telling you is the study of the thermodynamics of black holes. According to the works of Ted Jacobson, Jacob Bekeinstein or Stephen Hawking (and other scientists' goods), they concluded that the information engulfed by a black hole - for example a 3-dimensional object - is proportional to the air from its horizon, a surface that has only two.

From there, they generalized the principle by stating this: "the complete description of any physical system occupying a region of space can be defined on the only border of its horizon".

However, the horizon of a black hole only grows by swallowing a large amount of "energy". Energy, therefore, is itself porportional to the amount of information.

If we follow this logic, it means that energy is the source of information. You can take any physical system, whether quantum or not, the principle remains the same. There is an undeniable link between energy and information.

Therefore, any energy supply can lead to reduce entropy, thus increasing the amount of information. The opposite can also happen. You can bring energy to make a system more chaotic, and thus reduce entropy and thus increase the amount of information.

Where it's complicated is to understand the interpretation of entropy. In the physics of the thermodynamics of black holes, scientists say that entropy is equivalent to the amount of information. This is fundamentally wrong. Entropy is a measure of the uncertainty of information. That's basically the amount of information upon which uncertainty is based, not the amount of information itself. This is another debate, and I am thinking of doing an article on hoographic theory, which I think gives a misinterpretation of information.

Regarding quantum physics.

Thomas Boisson, a science group on Facebook, had written a survival guide in quantum physics. I'm going to translate it into English so that you better understand what it's all about.

I put you in file attachment the original (in French) and the version translated into English.

So I quote this document, concerning the quantum superposition:

"Until a quantum system has not been measured, its status is undefined. This means that even observable can take many different values ​​depending on a set of probabilities (called probability density) defined by the wave function.

For example, as an electron was not measured, it may have 15% probability of being at a position A, 12% probability of being at a position B, 27% probability of being a position C ... Thus, we often say that the particle "is in many places at once." Actually that is not true, it's just that as considered probabilities, the particle can be in a particular location. Likewise with the speed, power, spin etc.

This range of different values ​​that can take the quantum system before measurement is called "superposition state". To return to the fact that all these values ​​are superimposed without none is set. Each value is mathematically formalized by a vector called state vector contained in the state space, i.e. the space including all state vectors (and therefore all possible values).

When the measurement is performed, a set value for each observable is fixed, it is said that there is the projection value on an eigenvector. So every observable sees assigned a specific value, determined from all values ​​as the wave function probabilities. So there is only one value for the speed, value for position, a value for the spin etc. It is said that the measure causes the collapse of the wave function, because with the measure all probability disappear just to leave one."

From the moment when there is a collapse of the wave function, it is impossible to go back, because quantum physics is defined according to the parameters of chance. If you could go back, you would have another result, precisely because of this chance.

Hoping it helped you.

Thank you Roman, below a link that motivates the concept of 'quantum contextuality', more or less what you are writing

https://en.wikipedia.org/wiki/Quantum_contextuality

Of course this is not the last word!

Dear Roman,

Many thanks for your extensive answer and both links. They are very interesting. I am going to read it very carefully.

For (i) regarding the 2nd law of thermodynamics my suggest to you might be this. Please scratch down (in understandable language) only as a try for your planned article on holographic theory which according to you would give misinterpretation of information. Thanks, Marc.

For (ii) regarding quantum mechanics: So, from logical and mathematical point of view ph1 and ph2 are not identical, only simultaneous? Thanks in advance for your answer, Marc.

Thank you Michel for your second link. I 'll read it more carefully. Secondly, I have signed up for your MDP but up to now never answered. Maybe it may regard to that difficult password. Please do send me the email-alerts of your MDP. Thanks, Marc.

Marc, what kind of problem are you experiencing?. Just follow the MDPI link at the top of this question to possibly submit a contribution for the corresponding special issue. Good luck. Michel