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Classical, realist theories just fail to produce unitary gadgets

Nature, quantum mechanics organize gadgets along complex matrices; realism doesn't

In 1925, after 15-25 years of searches in the dark when all the good physicists knew that a radical change of the foundations of physics was urgently needed, physics switched to the quantum mechanical underpinnings. Those are totally and obviously different than classical physics – although they may produce classical physics as a limit – and it's unsurprising that virtually every complex enough experiment is predicted to end up differently by quantum mechanics than by any classical theory.

The difference was nevertheless made explicit in many toy examples starting with Bell's theorem. Bell's theorem studies various types of measurements made on two qubits. It's a straightforward situation – pretty much the simplest situation in which some entanglement exists – and one may calculate the prediction made by a general classical local theory with hidden variables. It produces results that obey some general inequalities and indeed, those inequalities are violated by quantum mechanical predictions as well as observations (the latter two agree).

The choice of this example is arbitrary and the conflict between classical and quantum physics is utterly unsurprising. You may see the difference between classical and quantum physics in many other, sometimes clearer, setups. Yes, all the good founders of quantum mechanics knew that their new theory was totally unequivalent to anything you could get in classical physics.

We may demonstrate the difference between the two frameworks more conceptually, using "harder tasks" that the experimenters should solve. For example: Pick a physical realization of an \(N\)-dimensional Hilbert space and study how the quantum information carried by this object may be transformed by reversible gadgets. These tasks will be harder for the experimenter to perform but they will be more directly linked to fundamental mathematical structures of quantum mechanics (while the form of Bell's inequality is not fundamental at all – but the Bell's experiment is easier for the dumb experimenter).

For \(N=2\), we deal with a qubit. The spin of the electron (up or down) is the simplest example. The general pure state is\[

\ket\psi = a\ket\uparrow + b\ket\downarrow, \quad a,b\in\CC

\] You may do many measurements on this qubit. You are asked to classify all the possible gadgets that may measure "something" about the electron's spin and that have the property that it's possible to prepare the state \(\ket\psi\) repeatedly in such a way that for this state, the gadget always returns the same results. What are the possible gadgets?

It's simple. In quantum mechanics, all the possible nontrivial measurements of a qubit are equivalent to the measurement of the spin along some axis \(\hat n\in S^2\). Choose an axis in 3D, measure the spin up-or-down relatively to this axis, and the gadget answers "up" or "down". It is possible to prepare electrons with the spin up (or down) along a chosen axis and if the spin is measured once again, the gadget returns the same result.

The sphere \(S^2\) of possible gadgets – simply differently oriented spin up-or-down measurements – may be generally, for any \(N\), visualized as the set of all orthonormal bases in the Hilbert space. That space may be written as\[

U(N) / U(1)^N.

\] For every orthonormal basis in the Hilbert space, any \(U(N)\) transformation produces another good solution. But the \(U(1)^N\) transformations inside \(U(N)\) produce the same solution because they only change the phases of the basis vectors. One of the \(U(1)\) groups may always be chosen to be the diagonal one so the quotient above is also\[

SU(N) / U(1)^{N-1}.

\] In the qubit case \(N=2\), \(SU(2)/U(1)\) is geometrically equivalent to \(S^3 / S^1 = S^2\). The gadgets really differ just by the orientation in space. For higher values of \(N\), there are many possibilities and the \(SU(N)\) group manifold is much larger than the \(SO(3)\) manifold of rotations in 3D. For example, pick a spin-1 particle that has a three-dimensional Hilbert space, \(N=3\). Quantum mechanics says that you may still measure a quantity that has possible results \(0,1,2\) and there are many ways to do so. The ways to do so may be arranged along the manifold \(SU(2) / U(1)^2\).

Similarly, you may classify all experimentally possible reversible gadgets that transform the particle with the \(N\)-dimensional Hilbert space. "Reversible" means that there exists a gadget that undoes these changes so that the combination of the two gadgets, \(UV=1\), gives you the identity. The combination of the two gadgets, the original one and the inverse one, is required to return the particle to the same state so that the results of the "guaranteed" measurements, which we always require to exist, is the same as if there were no gadgets at all. And just to be sure, we demand the same (between the \(UV\) gadgets and \(1\)) probabilistic distributions for the measurements applied to all other initial states.

Similarly as in the case of the measurements, the space of possible reversible gadgets is the \(SU(N)\) group manifold! It's not just a speculation. You may just hire a team of clever experimenters to work for many years and actually produce many such reversible gadgets for the \(N=3\) case. They will find out that only an 8-dimensional continuous space of such gadgets is possible and they may be arranged along the \(SU(3)\) group manifold. I can tell you how such gadgets may be constructed. You may obtain them as a sequence of several \(U(1)\) rotations in several planes by general angles (a complex, higher-dimensional generalization of Euler's angles).

These are basic properties of quantum mechanics. Observables are organized as \(N\times N\) Hermitian matrices and the transformations are unitary \(SU(N)\) matrices. An obvious fact – but a fact that is never emphasized in the mostly irrational discussions about "interpretations" of quantum mechanics – is that classical physics doesn't have such matrix-based structures at all. As I stressed in May 2019, the classical i.e. realist theories always replace the unitary transformations by the more dumb "permutations of the points in the phase space". And those just can't be the same. Similarly, classical i.e. realists replace the "Hermitian matrices" set of possible measurements by "functions on the phase space". And those are completely different measurements.

I could discuss both, the measurements or the reversible transformations. The difference between classical and quantum mechanics would be shown in a very similar way. Let me just discuss the "set of reversible transformations" of the object that carries some information. OK, in quantum mechanics, those gadgets may be arranged as points along the \(SU(N)\) group manifold.

That's a continuous space and the permutation group \(S_K\) is a discrete set. It is very obvious that if you want a classical i.e. realist theory to emulate the \(SU(N)\)-like continuum of possible gadgets, the number \(K\) defining the permutation group \(S_K\) has to be infinite, parameterized by an index that takes values in a continuum. This "continuously infinite" increase of the permutation group obviously means that the number of points in the phase space \(K\) is continuously infinite even for objects that are described by a finite-dimensional Hilbert space quantum mechanically. This choice of a classical theory with a continuous phase space means that we really turn the wave function into an objectively real classical wave.

As I have emphasized many times, such a classical theory with infinitely many states immediately leads to general problems that contradict the observations. The number of states is infinite and therefore the entropy should be infinite. You create a new atom and an antiatom, that increases the number of degrees of freedom in your realist entropy by an infinite amount, and therefore the atom and the antiatom should have a huge entropy and therefore heat capacity. But they only have the entropy and heat capacity comparable to "one bit per molecule", experiments clearly show. That's why the whole strategy to "many \(K\) in a classical theory continuously infinite" doesn't really work.

But it's not the only way to show that it doesn't work.

If you choose to fake quantum mechanics by a classical theory where the wave function is a classical, i.e. objectively real and in principle measurable, wave, then your theory makes the prediction that the possible gadgets – both the measurement apparatuses and the reversible gadgets – are far more numerous than implied by quantum mechanics (and confirmed by experiments). Both the set of measurement gadgets; and the set of reversible gadgets would be infinite-dimensional manifolds.

The smallest number of classical degrees of freedom by which the \(S_K\) permutation group may emulate the \(SU(2)\) transformations appears when \(K\) is infinite and the indices take values in \(S^2\). With such a classical or realist theory, you are really saying that the different spin states along the \(S^2\) are in principle perfectly distinguishable from each other. That's equivalent to saying that the squared absolute value of the wave function is a general probability-like function on an \(S^2\).

But such a function may be \(1\) near the North Pole of the sphere – but may be zero at some point that is very close to the North Pole. It follows that your theory predicts that it's possible to create gadgets that perfectly reliably distinguish these two states, spin-up electron states polarized along two different but very close axes. Be my guest to make such a prediction. It's very clear that this prediction disagrees with the known experiments.

To argue in this way, I do rely on the fact that no one has created such a gadget yet. If you have a spin-up electron along the \(z\) axis, the measurement of the spin along some very nearby axis \(z'\) produces the same answer as the measurement along \(z\) with a probability that is almost 100%. \(\cos^2 \theta/2\) is the probability that they agree, as you know from Born's rule very well. According to quantum mechanics, you can't really do anything with the high correlation between the outcomes of these two measurements that exists for a very small \(\theta\) because non-orthogonal states in the Hilbert space simply aren't mutually exclusive.

But in your classical or realist theory, you should be able to break this correlation between the measurements along the related axes. Your theory clearly predicts that such a perfect discrimination between the nearby spin-up states is possible. So why don't you show us how to do it experimentally? It's really fair to say that the burden of proof is on your side. Your theory predicts that something omnipresent is possible – all the measurements and transformations that could exist if the wave function were a classical wave – but experiments show that this thing seems impossible in every single case. While this proof isn't perfectly complete and rigorous, the evidence against your conjecture is extremely strong.

The infinite huge capacity of atoms predicted by your theory is a complete argument that kills your theory.

But even if you didn't really understand these arguments and their strength, you should still have some common sense and basic intuition. The experiments clearly show that the complex matrices (for measurements) and the unitary matrices (for transformations of the qubit-like information) are omnipresent in Nature, in agreement with quantum theory. The complex matrices and unitary transformations are absolutely unnatural from the viewpoint of any classical theory.

Classical theories say that the measurement apparatuses may be identified with functions of the phase space \(f(x_i,p_i)\) which look nothing like "Hermitian matrices". In particular, the set of Hermitian matrices has a natural noncommutative multiplication operation (multiplication of matrices) while the functions on the phase space don't have one. (And when they do, like in the case of the star-product, you get an equivalent formulation of quantum mechanics.)

Similarly, the transformations in classical theories want to look like permutations of the phase space. Again, that's a very different group geometrically from the unitary groups. If you start with a classical or realist theory, the most obvious types of it won't have any unitary structures in it and they will clash in experiments. You may start to fix these lethal bugs and when you do it right (e.g. after you demand that the Wigner quasiprobability distribution on the phase space isn't a general non-negative function, as implied by a fully classical theory), you will arrive to the star-product and similar structures, an equivalent formulation of quantum mechanics. All the "improvements and modifications" that you need to do with your classical theory are unavoidable and forced upon you by rather well-defined and obvious experimental facts.

Quantum mechanics is really forced upon us. It follows from experimental facts. The randomness follows from experimental facts and so does the Born rule that shows the probabilities of agreement between the measurements done on states that can actually be prepared. The Hermitian matrix shape of the set of possible measurement apparatuses is pretty much an experimental fact and so is the manifold of unitary matrices parameterizing the possible reversible gadgets that transform your particles. Even if you didn't want to see some of the unavoidable contradictions between these experimental facts proving the QM structures on one side; and your classical theory on the other side, you should still be able to see that there is at least nothing encouraging about your classical theory that could make you believe that you are on the right track to describe the experiments by a correct theory. And even the absence of good news would be pretty bad.

Nature doesn't smell, walk, and barks like your smelly animal. It is probably because Nature isn't your smelly animal. The efforts to build alternative theories denying quantum mechanics are signs of either complete technical incompetence of the "interpreter" or his dishonesty that is clearly driven by some ideological prejudices, prejudices that are comparable to those believed by fanatical religious believers.

And that's the memo.

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