Reading the opening post of a software question
https://www.researchgate.net/post/Hasnt_software_engineering_been_relying_on_an_untested_hypothesis_of_computer_science_by_mistakenly_assuming_the_hypothesis_to_be_a_fact
encouraged me to publish the following about Quantism.
FOUR EXAMPLES OF ENTANGLEMENT
1.- SPATIUM ENTANGLEMENT
Consider the standard elementary geometrization of "absolute" three dimensional physical space. In this context instead of "point in physical space" we will say "spatium state".
For a given space coordinate system with XYZ axis, spatium state S corresponds with a point P in geometric XYZ space, that is, S corresponds with a triple of numbers in \R^3. Point P projects over coordinate points Px, Py, Pz, each belonging one of the coordinate axis, hence
P = Px + Py + Pz
If furthermore Px=\alpha Ux, Py=\beta Uy, Pz=\gamma Uz with Ux, Uy, Uz unit vectors we have that P is the following linear combination
P=\alpha Ux + \beta Uy + \gamma Uz
But each of the normalized vectors Ux, Uy, Uz corresponds with some spatium state, say Sx, Sy, Sz. Therefore the Cartesian, traditional, high school decomposition according to coordinate axis becomes the following postmodern phrase:
** Spatium state S is a spatium entanglement of Sx, Sy, Sz **
The spatium physical state S is therefore the spatium physical superposition of the normalized spatium physical states Sx, Sy, Sz.
Can I pretend this is a new, mysterious physical property of space?
2.- SPATIUM-TEMPUS ENTANGLEMENT
Add now a clock and a numerical time dimension T. Consider a space-time physical event S and call it a "spatium-tempus state" with corresponding space-time four-vector P decomposable as
Px=\alpha Ux, Py=\beta Uy, Pz=\gamma Uz, Pt=\delta Ut
where Ux, Uy, Uz, Ut have been normalized. These correspond with spatium-tempus states Sx, Sy, Sz, St. We can now say
** Spatium-tempus physical state S is the spatium-tempus physical entanglement of spatium-tempus physical states Sx, Sy, Sz, St **
Does someone disagrees that this is deep Physics?
3.- MULTIPLEX ENTANGLEMENT
Suppose a physical system consists of physical states S described by means of points of an n-dimensional differentiable manifold M^n. This means that each S corresponds with a P of M^n. In this situation instead of "physical state" we will say "multiplex state". I did check the dictionary. Latin translation of "manifold" is "multiplex".
Let the open subset V of M^n be such that it contains point P corresponding to physical multiplex state S, and assume additionally that V is the domain of a coordinate chart
f:V\to \R^n
By hypothesis, there is a point p in \R^n such that f(P)=p. Let e_1, ... , e_n be the canonical basis of \R^n. Things will now advance towards esoteric virtuality since, generally speaking, f(V) need not be all of \R^n.
Being a vector in \R^n element p can be written as
p=p_1 e_1 + ... + p_n e_n
If there are multiplex physical states S_1, ... , S_n such that f(S_j)=e_j it will be said that e_j determines the "real" multiplex physical state S_j. Otherwise e_j determines a "virtual" multiplex physical state that need not be specified. Never mind what these virtual physical states could really be. They are often invoked by reputable writers. Whether states are real or virtual, it can now be said that
** Multiplex physical state S is the multiplex entanglement of S_1, ... , S_n **
It should be possible to refine this example to include Lagrangian and Hamiltonian systems.
How much physical insight does this provides?
4.- QUANTUM ENTANGLEMENT
Consider quantum states S corresponding with normalized wave functions. Let \psi be a normalized finite linear combination of normalized quantum eigenfunction \psi_j of some self-adjoint qunatum mechanical operator H
\psi = \mu_1 \psi_1 + ... + \mu_n \psi_n
If S_j are the quantum physical states corresponding to the the eigenfunctions \psi_j we can say that
** Quantum state S is an entanglement of quantum stationary states S_1, ... , S_n **
It must be remarked that in practice the operator H does not have to be explicitly given, nor its eigenvalues and eigenfunctions have to be calculated.
Quantum entanglement does not seem much more than the entanglements above.
5.- CONCLUSION
Examples 1, 2, 3 are entanglements formulated in the context of classical Physics. Example 4 is quantum entanglement.
Is "quantum entanglement" a substantial physical concept? Or is just traditional "linear combination"?
Cordially,
Daniel Crespin
Mathematically, the difference between classical and quantum physics is a possible incompatibility of observables of the latter theory. For this reason, if quantum entanglement is a typically quantum mechanical feature, its origin must be related to incompatibility of quantum mechanical observables. It is possible to understand incompatibility of such observables in terms of mutual disturbance in joint non-ideal measurement of incompatible observables. I have discussed this issue extensively in my book W.M. de Muynck, Foundations of quantum mechanics, an empiricist approach, and in articles that can be consulted on my web site
http://www.phys.tue.nl/ktn/Wim/muynck.htm. In particular, violation of the Bell inequalities by quantum mechanics can be understood in this way.
Hello Alexander and Willem.
Thanks very much for your interesting and well focused comments.
Quantum entanglement is nonsensical in a much deeper sense than explained above. Let us see why.
Superposition of complex wave functions is nothing but old fashioned linear combination. Generally speaking linear combinations of complex wave functions make sense because complex wave functions constitute a complex vector space. Enter normalization.
Since Quantism interprets "wave function modulus squared" as a probability density, the wave functions must be normalized. This is usually stated as follows:
"The space of quantum states consists of Hilbert rays"
The problem here is that the space of Hilbert rays is exactly a complex projective space, which is a highly non-linear manifold.
If entanglement of quantum states is to make sense, the linear combination of Hilbert rays should always result in some well defined Hilbert ray. This amounts to stating that the projective space is a linear space.
The complex projective space, however, can never be continuously equivalent (homeomorphic) to a linear space. That is because PE has non zero cohomology on all even positive dimensions while for linear spaces such cohomology is null.
Of course, any set with the same cardinality of a complex vector space can be made into a vector space, but in the context of quantum states this is an unacceptable, globally discontinuous artificial procedure.
If the fantasy of a vector space structure on PE were possible, probability theory would have been dealing with "the vector space of probabilities" since long, long ago.
That linear combinations of Hilbert rays do not result in a well defined Hilbert ray implies that the desired manipulation of Hilbert rays is impossible. Thus, quantum entanglement is forced to rely ---instead of linear combinations--- on some sort of undefined and hard to explain "physical intuition". But irrationality is not acceptable as a proper setting for science.
What about experiments? Well, they are performed, and interpreted. But if the interpretation is in terms of nonsensical mathematics, or in terms of non-existing mathematical entities, that is very, very bad.
With best and most sincere regards,
Daniel Crespin
I think in your post you consider "superposition" (for example of quantum wave functions) actually an entangled quantum state is a quantum state that can not be breaked up as two quantum states forming a tensor product. The way to characetrize entanglement is by showing that the state viloates the Bell Inequality.
Daniel, you might try this approach: The difference between quantum and classical entanglement is not in the math, but in how long it takes the math to complete.
For example, after reading the spin direction (spin state) of an electron in the real world, you can "fix" the probability function of a second spin-entangled electron by sending it a hidden signal at the speed of light. The result is a fully classical model of entanglement. The problem is that if you want this model to predict known physics accurately, you will be forced to shut down your modeling of anything else until the second electron probability function receives that signal and is properly updated. If you don't do that, you will end up with a violation of conservation of angular momentum, and that is what the universe at large is far more concerned about preserving than such mundane issues as the speed of light. (There is always a conservation principle hiding behind every form of entanglement, by the way.)
Making a classical mathematical model "quantum" does not so much change the math as it does its interpretation. In a quantum model the hidden signal propagates "instantly" across the distance that is being modeled. Alas, real computers don't work that way, and that is precisely the issue that Feynman noticed in his famous paper in which he almost accidentally invented the concept of quantum computing. The best that real computers can do is help alleviate the speed-of-light delays by keeping the processes that represent real entangled particles very close to each other within a single computer -- which is the exact opposite approach to the increasingly popular trend of "distributed computing," I should note.
However, even with such massive scale-downs of distances made possible by modern computers, the speed of light constraint eventually begins to win out. As a result, accurate modeling of even small numbers of entangled particles quickly becomes computationally intractable. Conversely, this is also why there is such intense interest these days in using quantum mechanics to increase processing speed. A true quantum computer can correlate probability functions in ways that defy the speed of light constraints of normal computation.
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