Friday, March 25, 2016 ... /////

I had to copy the title from Backreaction because it is unusual and refreshing to see that Sabine Hossenfelder reacts totally sensibly to a two-minute dialogue about quantum mechanics. (Chad Orzel and Michael Motherboard Byrne wrote a criticism of Nye's answer, too.)

OK, so Bill Nye was a confused engineer and clown who has starred as a scientist on a TV show for kids (after Nye plagiarized Prof Proton and stole his wallet – and I must warn you, Prof Proton is a fictitious fictitious scientist, not a real fictitious scientist let alone a real scientist). When they grew up, some of these kids still couldn't distinguish a fairy tale character from a real tooth fairy so they actually started to believe that he is a scientist and he can answer questions they have about science.

For example, as the video above shows, two days ago, Nye's fan named Tom from Western Australia asked about the quantum entanglement. If quantum entanglement may transfer the information instantaneously or superluminally (Tom wanted to boast how educated he was, so he also pointed out that another term for "quantum entanglement" is "quantum spookiness", not bad, Tom!), how will it change the world?

First, the "CEO of The Planetary Society" Bill Nye replied on "Big Think" (so much modesty and sanity here) that he incorrectly loved the man. Why is it incorrect to love Tom? Because he's totally confused by similar nonsense about quantum mechanics that the media talking heads such as Nye are speading on a daily basis.

As Einstein figured out in 1905 when he discovered the special theory of relativity, no information can ever propagate faster than the speed of light (in the vacuum) let alone instantaneously. If this were possible, the Lorentz symmetry – the right way to switch to the perspective of a different inertial observer – would allow us to describe the same causation as a sequence of events where the effect has preceded the cause. It's clearly impossible.

The information and influence can only propagate from the past to the future and because this has to be true from all inertial reference frames' perspectives, the information and influence may only propagate by the speed that never exceeds the speed of light (in the vacuum).

It's an absolutely universal and exact law of Nature that admits no exceptions (at least as long as the physics may be described as some "objects added on top of a basically flat Minkowski background"). And quantum mechanics doesn't and can't violate Einstein's principle of locality, either.

Now, the quantum entanglement has been understood since the mid 1920s when quantum mechanics was born – even though the name was only coined by Einstein in 1935 (when he wrote papers that attempted to find problems in quantum mechanics).

Quantum entanglement is basically omnipresent, experimentally proven and easy to demonstrate, completely understood since the 1920s, and even though Bill Nye claims that it only becomes useful when people construct the time machine (which they never will) or when they extract energy from black holes (they may in some distant future but the relationship of this technology with quantum entanglement is very limited), quantum entanglement is actually manifested in pretty much every other phenomenon we observe in Nature.

Quantum entanglement is nothing else than the quantum description of clean, fully quantum mechanically behaving (e.g. small enough) systems (objects) $A,B$ whose properties just happen to be correlated. That's it. Although Einstein wanted to claim otherwise, there is nothing spooky about it. It is the perfect quantum mechanical counterpart of the correlated probabilistic distributions that already existed in classical physics.

Imagine that two objects in classical physics $A,B$ are described by some variables $a_i,b_j$ where the indices take several values and $a$ and $b$ are properties of objects $A$ and $B$, respectively. For every choice of $(a_i,b_j)$, we may ask what's the probability of that choice. If the possible values are continuous, we must talk about "probability densities" instead of probabilities.

But to simplify things a bit, let us assume that the number of options is finite and they are discrete. The generalization to the countable or continuous spectra is straightforward. OK, for every choice of values, we have the probability$P_{AB}(a_i,b_j)$ that the combined system $(A,B)$ finds itself in the state with properties given by the particular numbers $a_i,b_j$. If the systems $A,B$ are uncorrelated, we may write this function of many variables as$P_{AB}(a_i,b_j) = P_A(a_i) P_B(b_j).$ It means that the complicated function of many variables on the left hand side may be written in terms of functions of many fewer variables on the right hand side which seemingly contain much less information. Now, almost no functions of many variables can be "factorized" in this way. For example, $\sin(a)\cos(b)$ is factorized but $\cos(a+b)$ is not.

Whenever the objects $A,B$ described by the probabilities interact with each other, the probabilities get complicated and mixed up and we produce one of the more general probabilistic distributions that can't be factorized: a correlated distribution.

Similarly, in quantum mechanics, we may replace the probabilistic distributions by the density matrices $\rho$ and the density matrix of the "bi-object" that may be written as the simple (tensor) product of the two distributions$\hat\rho_{AB} = \hat \rho_A \otimes \hat \rho_B$ describes a composite i.e. "bi-object" system where the two subsystems $A,B$ are perfectly independent or uncorrelated. In most cases, this won't be the case. It will almost never be the case after $A,B$ interacted with each other. And that's enough to claim that $A,B$ are entangled.

I must emphasize that the general density matrix $\rho_{AB}$ carries much more information than density matrices $\rho_A$ and $\rho_B$. If the Hilbert spaces for $A$ and $B$ are $N_A$ and $N_B$-dimensional, the Hilbert space for $A+B$ is $N_AN_B$-dimensional, the tensor product, and the general density (square) matrix contains $(N_A N_B)^2$ entries, much more than either $N_A^2$ or $N_B^2$, the number of entries in the density matrices for $A$ or $B$ separately (and more than the sum, too).

If we describe the quantum mechanical objects $A,B$ in a way that maximizes our knowledge so that it's still compatible with the laws of physics (especially the uncertainty principle), quantum mechanics says that the objects are described by a wave function or a pure state or a state vector or a ket – basically synonyms. The non-entangled state of $A,B$ occurs when$\ket{\psi_{AB}} = \ket{\psi_A} \otimes \ket{\psi_B}$ where the tensor product sign $\otimes$ in the middle is usually omitted. The probabilities are computed as combinations of squared absolute values of the wave functions. So the factorization of $\ket\psi$ is totally analogous and has the same physical meaning as the factorization of $\rho$ or $P$ above.

Again, in typical cases, especially after $A,B$ interacted, the wave function for the system $A+B$ won't have the special product form, and that's enough to say that $A,B$ are entangled. They almost always are, especially after an interaction of the two objects. If $A,B$ are entangled, the wave function for $A+B$ has to be (and may be) written as a sum of several (at least two) products of the form $\ket{\psi_{A,m}}\otimes \ket{\psi_{B,n}}$.

Similarly, the entanglement may also "evaporate" if we learn a complete information about one object, e.g. $A$, by a set of measurements done on this single object only. When we do so, we know that $A$ finds itself in a particular pure state $\ket{\psi'_A}$ and the state of $A+B$ has no choice but to get factorized.

An EPR experiment: the story without tooth fairies and nonsense

The description of "EPR entanglement experiments" is simple. Two objects $A,B$ interacted sometime in the past (or they were born together in the past). That's how their properties got correlated. Some "relative properties" (e.g. answers to the question: Does $A$ have the same spin as $B$?) may be perfectly determined due to the high correlation that was created by the interaction of $A$ and $B$. Even without calculating the full exact results, such perfect correlation e.g. concerning the spins may be proven e.g. from the conservation of the angular momentum and by other arguments.

But this doesn't contradict the fact that the individual properties of $A$ (its spin) or similarly the individual properties of $B$ may be completely undetermined (random). This is enough to say that $A,B$ are entangled – quantum correlated. The entanglement survives when $A,B$ are physically separated. And when we measure them, we observe the correlations (and often perfect correlations) between the properties of $A$ and $B$ even though the individual properties of $A$ or $B$ may seem "random". Quantum mechanics gives us verified formulae to calculate the probability of each combination of the measured outcomes (for $A$ or $B$ or $A+B$). There's nothing mysterious and no signals or influences were sent: the correlations are the consequence of the interaction between $A$ and $B$ at the beginning when $A$ and $B$ were basically touching or overlapping each other!

Quantum mechanics is true and conceptually differs from classical physics

Because quantum mechanics calculates predictions in a way that seriously differs from the framework of classical physics in many serious respects, the predictions of quantum mechanics and classical physics will be different.

In particular, two systems may be perfectly entangled in "all possible, mutually non-commuting, properties" (spins with respect to different axes) even though the value of each property is random when measured individually. Quantum mechanics tells us that we can measure various properties associated with mutually non-commuting linear operators (and the non-zero commutator means that two quantities can't have sharp values at the same moment, and the measurement of one property always disturbs the predictions for future predictions of the other, non-commuting property); the set of quantities that may be measured in classical physics is different (commuting functions on the phase space). One can't ever reduce a quantum mechanical theory to a classical one because $\hbar\neq 0$ contradicts $\hbar=0$. On the other hand, most classical theories may be obtained as limits of some quantum mechanical theories because $\hbar=0$ may be approximated by $\hbar\neq 0$ when we send $\hbar\to 0$.

The mechanisms and mathematics of all these things are understood and they have been measured since the 1920s, too.

The time machine won't be constructed because it would produce closed time-like curves that violate the rules of logic. But the quantum entanglement is being applied in tons of contexts, anyway. A very fancy industry where the quantum entanglement is talked about all the time is "quantum computation". But we don't need to look at these fancy things.

Benzene ring: an esoteric but elegant example of entanglement

Quantum entanglement is literally everywhere in physics, chemistry, and beyond.

Look at this chemical formula. It's actually the original copy of something printed in 1865 and contains the internal structure of benzene as understood by Kekulé. Six carbon atoms with six attached hydrogen atoms are organized to a hexagon. Long before people knew quantum mechanics or could see atoms under microscopes (and even decades before atoms were used to explain the Brownian motion and other things), German chemist Friedrich August Kekulé (a descendant of a Czech post-1620 protestant emigrant, so sort of ethnically Czech) was already able to guess the correct arrangement of carbon and hydrogen atoms in benzene, something I find sort of amazing. (Johann Loschmidt of Carlsbad, Bohemia who was ethnically German knew about the circle shape of the benzene molecule with shared electrons already in 1861 but he was confused about details.)

The chemists could already do so many amazing things while being ignorant about much of physics let alone quantum mechanics! ;-) If you have some simple explanation why and how chemists could determine the structure of molecules in those ancient times, let us know.

But we need to study some details. The benzene molecule ${\rm C}_6 {\rm H}_6$ was one of the favorite examples of two-level systems used by Feynman in his lectures.

Look that back in 1865, Kekulé drew two pictures. The six carbon atoms in the ring are labeled 1,2,3,4,5,6 and some of them are connected by single bonds, others by double bonds. Each carbon atom has four "legs" (electrons it's ready to share) because only the $n=2$ electrons in the carbon shells $1s^2 2s^2 2p^2$ are loose enough to play with other atoms, one of those is dedicated to the hydrogen atom associated with a given carbon atom, and three "legs" are ready to attach to other (carbon) atoms. It's $3=1+2$. So one bond going from the carbon to an adjacent carbon has to be single, the other has to be double, and they have to alternate as a consequence.

But there are two ways to do so. The double bonds may be either those between 1-2, 3-4, 5-6 (left picture), or between 2-3, 4-5, 6-1 (right picture). Which of these two is the "real" molecule of benzene? Well, there's really no difference in "style" between them because you may get one of them by a rotation by 60 or 180 degrees from the other (in either direction). The left benzene probably can't smell differently than the right one because you rarely make a stinky aroma pleasant by turning your body (or the dish) by 60 degrees. :-) It reminds me of a new TV commercial where a mother asks a kid to "turn the meat over" and the kid rotates the roasting pan by 180 degrees around a vertical axis. "I said the meat, not the pan".

Nye is making a point about quantum mechanics.

Quantum mechanics adds a great new layer of cleverness. Quantum mechanics actually implies that if you place the nuclei at the given places, both pictures by Kekulé are equally likely. The actual ground state of the molecule of benzene is$\ket{\Psi_+} = \frac{ \ket{12,34,56} + \ket{23,45,61} }{\sqrt{2}}$ where the pairs of numbers in the kets denote the location of the double bonds. That's great, a superposition. In classical physics, you could have 50% of the molecules respecting the left Kekulé picture and 50% behaving as the right picture – but you could always imagine that a particular molecule has a particular choice for the double bonds. But in quantum mechanics, even one single molecule may be and actually demonstrably is intrinsically undecided whether it prefers the 12,34,56 or 23,45,61 choice for the double bonds! 50% chance for both options.

Benzene is an obscure compound but in reality, superpositions are absolutely omnipresent in chemistry, atomic physics, and quantum mechanics in general. Superpositions and the quantum entanglement changed the world as soon as it was created. They were needed and are needed for everything in the world as we know it.

In fact, the truth that the ground state of benzene is a superposition implies that the binding energy is significantly deeper than it would be otherwise. The ability of the double bonds to choose "where will we sit", the superposition form of the wave function, tells us that the energy eigenvalues are split (the corresponding eigenstates have the relative plus or minus sign in the $\ket\Psi$ state) and one of these two energies is actually lower than the average of the two (while the other$\ket{\Psi_-} = \frac{ \ket{12,34,56} - \ket{23,45,61} }{\sqrt{2}}$ is higher, and that state is unstable), so the superposition allows us to lower the energy and make the benzene molecule particularly stable. And if a molecule (or any other bound state) is more stable, it becomes more likely that you will find such molecules in Nature. Nature wants to save energy and these "deeply bound" molecules (or nuclei etc.) are popular final states of processes and less likely to decay or get disrupted. So the relevance of the superposition is what makes the molecule more important in Nature! The superpositions aren't "bizarre exceptions". On the contrary, they are a part of the pedigree for those who are important and omnipresent.

Once the molecule of benzene has enough time to emit the "MASER photon" which makes it almost guaranteed to drop from the originally allowed $\ket{\Psi_-}$ higher state to the lower-energy $\ket{\Psi_+}$, we may be sure that the molecule finds itself in the true ground state $\ket{\Psi_+}$ with the lower energy. In other words, we may be sure that it is in the superposition state!

Like pretty much every general enough superposition, the superposition in $\ket\Psi$ may be interpreted in terms of the quantum entanglement. We may ask "what is the strength of the bound [single/double] between 1-2" and "what is it between 2-3". Because the bonds are alternating – in both states contributing to $\ket\Psi$ – we know that the answers to the questions are exactly opposite.

We know that if the 1-2 bond is single, the 2-3 is double, and vice versa. The bonds are entangled. The correlation between these two Yes/No questions is "perfect" despite the fact that the character of the 1-2 bond itself (or the 2-3 bond itself) is completely uncertain: it has 50% chance to be single and 50% chance to be double. It's guaranteed that the odds are 50-50 and the entanglement exists after the time needed for the molecule to spontaneously emit the "MASER photon", as I said. As long as objects have some time or enough time to interact and/or lower the energy (in a region), the entanglement is absolutely unavoidable.

(We could actually measure other questions than just "is a given bond double or single", measure operators that don't commute with the operator answering this "single/double" question. We could still see a perfect correlation between the adjacent bonds, as implied by the maximum entanglement.)

Feynman (Volume III, chapters 8-12 and others) discusses many other physical examples of these superpositions (and entanglement) – the hydrogen molecule, the ammonia molecule (plus MASER: in 2012, I wrote a longish post on ammonia and MASERs), and many others. The elements and compounds look different and they smell different but the mathematics governing these simple quantum mechanical models is identical in all cases.

The ammonia molecule is a tetrahedron of a sort where one vertex, the nitrogen atom, may be pushed above or below the triangle of the hydrogen atoms. Because of quantum tunneling, if the nitrogen starts "above", it has a probability amplitude to get "below" as well, and it oscillates as a result. Consequently, the actual stationary states (energy eigenstates) are equal superpositions of "above" and "below". This fact creates another two-level system, one that makes MASERs (microwave cousins of LASERs) possible. I chose the benzene to discuss entanglement because it's easier to divide the benzene molecule into "regions" (or "spatially defined subsystems"). We ask whether there is a double bond in the regions between 1-2 and between 2-3 etc. Incidentally, a simple search shows that Feynman hasn't used the word "entangled" or "entanglement" once in his whole lectures! So this reinterpretation of the superpositions using the "entanglement" between the 1-2 and 2-3 regions is my "pedagogic contribution" although I am the first one to agree that the value of this contribution that goes beyond linguistics and pedagogy is nearly zero.

Whenever there is some correlation in composite systems that have to be cleanly described by state vectors and where the interactions are nonzero (and they almost always are, if the systems are close enough), some general superpositions will be relevant and the subsystems will be unavoidably entangled.

The entanglement isn't a fantasy or a tooth fairy. It's something that is found everywhere around us. In fact, even the "nearly maximum entanglement" (and the EPR photon pair or the benzene molecules were examples) is found at very many places. On the contrary, it would be supernatural to find non-entangled states of nearby subsystems because it would mean that for some reasons, the interactions between the parts are non-existent or behave as if they were non-existent – so the objects behave as invisible ghosts. Objects that physics says to "exist" don't want to be this invisible. Quantum entanglement is what makes the world normal!

In 1935, Einstein co-wrote the EPR paper and invented the term "spooky action at a distance" as his loaded synonym for the quantum entanglement; in 1935-1936, Schrödinger coined the equivalent, more serious and less loaded term "entanglement". But if one uses the correct theory to make predictions (and study the actions and decide whether there is an action at all), namely quantum mechanics, he sees that there is no action, there is nothing spooky about the entanglement and the correlations that follow from it, and there is no question about the existence – and, in fact, physicists' correct and complete understanding – of quantum entanglement and many particular entangled physical systems.

By talking about time machines and about the waiting for a revolution concerning the quantum entanglement (the actual revolution in these matters took place almost 100 years ago), Bill Nye shows that he is at most a stupid caricature of a science guy. He just doesn't have a damn clue about modern science and I am pleased to see that even Sabine Hossenfelder was willing to point this obvious fact out.

Hossenfelder said that by the number of "would-be pro-science" followers, Nye is somewhere in between Neil deGrasse Tyson and Brian Cox. I guess that Tyson is the ultimate superstar. One may see that as a quantum variable, the number of followers is entangled (OK, I simply mean correlated) with the scientific illiteracy of the men. They're increasing functions of each other. Sadly, the more incompetent you are, the more celebrated as a science guy you become in this postmodern world.

Incidentally, some two years ago, Neil deGrasse Tyson was answering a similar question about quantum entanglement and he got it wrong, too (although maybe he was less stupidly wrong than Bill Nye). He believes that the superluminal signalling becomes impossible for large objects but for the small ones, it does take place. Well, it doesn't.