Saturday, October 20, 2012

Entanglement is nothing else than correlation described in the quantum language

Quantum entanglement is arguably the most "sexy" feature of quantum mechanics when it comes to the exploitation and abuse of quantum mechanics by popular books and TV programs. When I talk to the laymen and wannabe physicists who are excited about quantum mechanics, they are often excited about the entanglement.

Entanglement is the ultimate justification of the New Age memes that everything is connected with everything else, that quantum mechanics endorses souls that can separate themselves from the matter and that bring physics closer to religion, and that the laws of relativity have been abolished by quantum mechanics.

Don't get me wrong. Quantum entanglement is cool. It probably sounds bizarre to almost everyone who is starting to learn quantum mechanics. However, it's also completely mundane, generic, ordinary, and typical. All the supernatural implications of entanglement in the previous paragraph are bogus. And whoever keeps on thinking that "entanglement can't be true" or "it must be challenged and tested all the time" hasn't yet completed the learning of the basics of quantum mechanics. He or she hasn't reconciled himself or herself with the fact that classical physics died about a century ago.

I was directly led to write this blog entry by the question about an entangled electron-positron pair at the Physics Stack Exchange. Pipsi asked:
Usually when we talk about entanglement, we mean entangled spin states (of electrons) or polarizations (of photons). My questions are:

Does pair production guarantee the product electron and positron entangled?

If there's no observer measuring either particle, can we say the types, or charge, of the particles are also entangled, with a wavefunction like:$\frac{1}{\sqrt{2}} \zav{ \ket{+e} \pm \ket{-e} } ?$
You may see that the spirit of the question is the following: Quantum entanglement is such an unbelievable thing. Could it be possible that even particle physics is capable of achieving something so amazing and supernatural that the wonderful atomic physicists are doing in their labs of scientific witchcraft?

I've already quite a long time ago noticed that in particle physics we usually do stuff that quantum-computing people will call an "entanglement". We just don't phrase it like that, because we are used to it and we aren't much "in awe" about it.

So the "entanglement" you are talking about is long known in particle physics. The earliest reference I know is this:

Pion-Pion Correlations in Antiproton Annihilation Events”, Phys. Rev. Lett. 3 (1959), no. 4, 181–183.

As you see, it is for pions (charged, actually).

The more "modern" review is this:
Bose–Einstein and Fermi–Dirac interferometry in particle physics”, Rep. Prog. Phys 66 (2003) 481.
(Below the answer, you may read some violent exchanges between your humble correspondent and Emilio Pisanty, a man who is in awe about entanglement.)

Exactly, Kostya. Quantum entanglement is completely common and omnipresent in particle physics – and any other discipline of science that routhnely observes quantum processes – and if particle physicists were in constant awe about it, they couldn't really focus on their work because quantum entanglement belongs among the basics.

In particular, whenever we create or recoil particles whose information is undetermined, they are always entangled. For example, if the angular momentum of the initial state is zero, the final state of spinning particles has to be proportional to$\eq{ \ket\psi &= \frac{ \ket{{\rm part.}\, 1\uparrow,{\rm part.}\, 2 \downarrow} - \ket{{\rm part.}\, 1\uparrow,{\rm part.}\, 2 \downarrow} }{\sqrt{2}}=\\ &=\frac{\ket{\uparrow\downarrow}-\ket{\downarrow\uparrow}}{\sqrt{2}} }$ If you re-express the spin to the basis "up" and "down" relatively to any other axis, you get the same form of the state (up to an overall normalization constant which is unmeasurable).

It's the standard Einstein-Podolsky-Rosen entangled state (EPR). Instead of these complicated terms suggesting a link with the supernatural world, particle physicists just call it "a singlet". It's a damn ordinary singlet, a rotationally invariant $$j=0$$ state constructed out of two $$j=1/2$$ degrees of freedom. Similar entangled states appear if you have two $$j=1$$ particles and if you start with more complicated states or end up with more complicated states, the detailed form of the final state will differ but you may be sure that it will be an entangled state in almost all cases. Quantum entanglement is pretty much unavoidable once you get to the level of elementary particles (and even atoms).

What is so new about quantum entanglement?

In classical physics, especially when the laws of the classical theory are local (as required by special relativity), objects possess "objective and independent properties". For example, if you have two 4 GB Flash cards, each of them is classified by 30+ billion classical bits of information at each moment. The content of the other card must be independent from the first one. So you have 60+ billion classical bits.

However, if you're not sure what the state of all the bits is, but you still assume that classical physics is applicable, you have to describe your knowledge about the state of the two Flash cards by a probability distribution. There are 60+ billion bits whose values $$0,1$$ may be combined in $$2^{60,000,000,000}$$ ways or classical configurations (these configurations get generalized to points in the phase space if you consider continuous variables and not just bits). For each of these configurations, you must decide about the probability that the configuration is realized, so you end up with a rather large number of real numbers, the probabilities of each configuration:$p_i, \quad i\in \{1,2,3,\dots, 2^{60,000,000,000} \}.$ The most general description of what you think about the memory chips involves this exponentially insanely high number of real parameters $$p_i$$. Now, these numbers $$p_i$$ may remember many patterns. For example, you may be sure that one of the Flash cards contains $$000\dots 000$$ and the other one contains $$111\dots 111$$ and you just don't know which one is which. Chances may be 64% and 36% that the Flash card with $$111\dots 111$$ is the red one or the blue one, respectively. So the Flash cards are correlated. All these patterns, and many more complicated patterns, may be encoded to the probability distribution, in this case given by the probabilities $$\{p_i\}$$.

Going quantum

The key fact one must understand when she is switching to quantum mechanics – the right fundamental theoretical framework to describe our Universe – is that quantum mechanics generalizes the latter, the probability distributions, and not the former, the objective state of bits.

In classical physics, the "objective independent state of the bits" could have been unknown but such ignorance could have been considered to be "just a personal disadvantage". You could always believe that there exists a "real state of the bits" – something that God, or at least the Big Brother, may know or does know. The fundamental description of Nature would be given by the knowledge of the Big Brother himself, so the usage of probabilities was a sign that someone was using a non-fundamental description of the fallible humans who are ignorant about certain things which is why they have to use the sinful concept of probabilities.

However, quantum mechanics tells you that there is no Big Brother. There is no objective state of the bits that someone could in principle know with certainty. The probabilities are completely essential. So according to quantum mechanics, even if our knowledge about the chips is "maximized", the state of the two Flash cards is given by the pure state$\ket\psi = \sum_{i=1}^{2^{60,000,000,000}} c_i \ket{i},\quad c_i\in\CC.$ In the classical probabilistic description, we used to have the super-insane exponentially large number of probabilities $$p_i$$. In quantum mechanics, we have the same number of probability amplitudes $$c_i$$. The only difference here is that $$c_i$$ are complex, not real. The squared absolute values of these complex numbers play the very same role as the probabilities,$p_i = |c_i|^2,$ however, the complex phase of each $$c_i$$ is actually as important as the absolute value and it will show up when you try to calculate the probabilities of different measurements, measurements of "whether the state is found in a linear superposition", questions that are not just trying to emulate some classical questions about the bits.

And be sure that these questions are not contrived or unnecessary; they are exactly as natural as the "classical ones". For example, if you talk about the spin, the measurement of $$j_z$$ may look like the measurement of the "classical bits" but in this basis, the measurement of $$j_x$$ and $$j_y$$ or any combination of the components (which are exactly as important as $$j_z$$, due to the rotational symmetry) will look like a measurement "whether the state is in a particular complex superposition". This boils down to the fact that $$j_x,j_y,j_z$$ don't commute with each other – the nonzero commutators (and not any non-locality or something like that!) are the real primary fact that logically distinguishes quantum mechanics from classical physics and that is behind the ability of quantum mechanics to circumvent Bell's inequalities and similar restrictions of classical physics.

I mentioned that $$\{p_i\}$$ could remember various correlations between the two Flash cards. Because $$\{c_i\}$$ produces its own $$\{p_i\}$$ as well, it also remembers correlations. Again, if you decide to measure the values of individual bits and you never measure the value of observables that "don't commute with these bits" – and most observables don't commute with these bits, so it's a big sacrifice – the pure quantum state will behave exactly in the same way as the classical probabilistic distribution.

The only difference is that in quantum mechanics, there is no Big Brother. The probabilistic description optimized for the knowledge of an observer is the most fundamental description one may ever get. Even in classical physics, the belief in the Big Brother was useless, to say the least, because you were not the Big Brother so you never knew "everything". The Big Brother was a philosophical dogma, a tooth fairy, that you rarely used in strict predictions of messy systems. Quantum mechanics is telling you that this tooth fairy doesn't exist at all.

If you think about it rationally, it can't cause any paradox because the tooth fairy was just an unsubstantiated belief of your childhood. What you were actually doing since you became a "self-sufficient child" was to use the laws of physics and your probabilistic knowledge whether or not certain propositions are right to deduce whether or not other propositions are true. Quantum mechanics allows you to do exactly the same thing. It is just no longer compatible with the Big Brother or a tooth fairy.

Anti-quantum zealots are like children

The more I interact with them, the more I feel that the people who keep on talking about hidden variables, many worlds, pilot waves, mechanisms of collapses etc. – people who refuse to understand that the most fundamental description of the Universe is intrinsically probabilistic in character – resemble children who just can't live with the insight that the tooth fairy was just an illusion, a trick, an approximation in which playful parents are equated with the most divine layer of the Cosmos.

For example, take this Emilio Pisanty guy, a Mexican currently in London who studies some "quantum dynamics". He was feeling uncomfortable with Kostya's answer that entanglement has been mundane for more than half a century. So he wrote:
It's important to note, though, that there's an extra step to go from correlations to entanglement. From a brief look at the references it looks like they measure correlations that are well described by an entangled state, but they do not make efforts to prove they do not admit classical-correlations models. This is fine, but it is spooky how many "quantum" effects admit hidden-variable explanations.
You see he is an immature child who just needs to listen to stories about "spooky" things (much like the climate alarmists?) – they're a part of his neverending childhood – and about hidden-variable models that try to create the illusion of a tooth fairy after he has clearly seen that there can't be any tooth fairy but he refused to believe his own eyes.

I told him that the desire to make things "spooky" was exactly what Kostya described by the words that "these people are in awe about entanglement". One may be "in awe" about "spooky" things but one doesn't have to be. More materially, I told him that almost all states in quantum mechanics are entangled. Recall that by definition, the only non-entangled states are the states of the form$\ket\psi = \ket{\psi_{A,i}} \otimes \ket{\psi_{B,j}}.$ They're tensor products of particular states describing the subsystems. But you see that this state only depends on $$\dim\HH_A$$ complex numbers describing the Hilbert space of the system $$A$$ and $$\dim\HH_B$$ complex numbers describing the Hilbert space of the system $$B$$. Well, subtract one because if you scale the first factor $$A$$ up by a factor and reduce the second factor $$B$$ by the same factor, a complex number, you get the same $$\ket\psi$$. So allowing an arbitrary normalization and unphysical phase, we need a certain number of complex numbers (the sum of the dimensions minus one).

On the other hand, the full Hilbert space for the $$AB$$ system has the dimension that is the product of the dimensions. However,$\dim \HH_A + \dim \HH_B -1 \ll \dim \HH_A \cdot \dim \HH_B.$ Well, for small dimensions, I should have written $$\lt$$ and not $$\ll$$ but at least the modest inequality is satisfied already for $$A,B$$ describing two qubits (two-level systems). The non-entangled state represent a small-dimension submanifold in a greater manifold, so they're a measure-zero subset of the Hilbert space. Almost all states in the Hilbert space are entangled; they are nontrivial complex linear superpositions of different tensor product states. If the two subsystems $$A,B$$ have interacted sometime in the past, it's pretty much guaranteed that they emerged in an entangled state.

Also, if the final particles are in any entangled state which may be as simple as the singlet, it's trivial to experimentally verify that the correlation will affect the measurements of the spins relatively to any axis – exactly like in the EPR measurements of "the" entanglement that is presented almost supernaturally. It's so well-known, trivial, and rudimentary – both experimentally and theoretically – that particle physicists just don't even talk abott it.

(Note that once you "fully" measure $$A$$ or $$B$$, i.e. acquire a complete set of commuting observables that describe it, the entanglement disappears because after the measurement, you know that e.g. $$A$$ has to be in a particular known state $$\ket{\psi_A}$$ and therefore the full system $$AB$$ must be in $$\ket{\psi_A}\otimes \ket{\psi_{B}}$$ where $$\ket{\psi_B}$$ is unknown before you measure $$B$$ as well. So the entanglement only exists prior to the full measurement, and as you measure things, it goes away. Quantum entanglement is the "charge of correlation" that the subsystems acquired by their mutual contact in the past and that they "discharge" as they live and expose their independent lives later.)

So if someone has a problem with the entangled states, he has a problem with 99.99999...% of the Hilbert space. The Hilbert space describes what the world may look like according to quantum mechanics. So if you have a problem with entanglement, you have a problem with 99.99999...% of what quantum mechanics may be saying about the world!

It's completely irrational – a form of a self-confusing propaganda – to try to hype non-entangled states or other states that are more compatible with a classical description. Most states (almost all states, both in the informal sense and in the strict sense of measures) in quantum mechanics are entangled. Most experiments (in both senses) are incompatible with a hidden-variable or otherwise "intrinsically classical" description, too. What's the point of using the completely invalid classical description (in the context of atomic experiments) as a zeroth approximation, of discussing the violation of this zeroth approximation? When you get to the level of individual spins or individual qubits or other pieces of quantum information, our Universe violates the basic assumptions of classical physics almost entirely. Any agreement with the classical intuition is a coincidence, an artifact of very special and uninteresting assumptions about the state, and someone's desire to focus on these special cases means that he wants to use an arbitrarily weak noise or infinitesimal glimpse of classicality to resuscitate his old belief in a tooth fairy.

But there's clearly no tooth fairy.

Those old children don't want to hear anything of the sort. So Emilio Pisanty wrote another reply:
Yes. Nevertheless, one must keep in mind that there is an extra step from correlations to entanglement. This is not crucial here, but the distinction can be quite important: for a strong example, in quantum cryptography, any improvement over classical protocols must come from nonclassical correlations. I wouldn't call this a lack of will to accept QM is right, but rather a cautiousness in scientific adventuring as well as a precise, quantitative understanding of how quantum an actual experiment is.
You see that it's still the same rubbish, just using slightly different words so that the refutation must use different words, too. However, the beef in these wrong claims is still the same.

It is not true that there is an "extra step from correlations to entanglement". Entanglement is just the generic, typical, and universal relationship between two (or more) physical systems that is, according to quantum mechanics, synonymous with correlation. There is no extra step from correlations to entanglement. Within the framework of quantum mechanics, they are the same thing. Every time you construct a pure state of a composite physical system in which the subsystems are correlated, this pure state will be inevitably "entangled". And vice versa. Entangled states predict correlated probabilities of measurements of $$A,B$$. In quantum mechanics, entanglement and correlation is the very same thing.

When we deal with "classical information", the only reason why the quantum aspects of the correlation may be unimportant is that the value of each qubit is "copied many times", i.e. to many qubits (the number of copies moreover grows with time, due to decoherence), and the classical system (or computer) is constructed to preserve the agreement between all the copies. This assumption – implied by keeping the voltage at each place at 0 V or 5 V or something like that – effectively bans all the observables that don't commute with the "classical bit" operators. It bans superpositions, allows one to reinterpret the quantum probabilities as classical observables, and when the computer is healthy and digital, it even keeps these observables in a discrete set in the course of the evolution.

A way to make Emilio's sentence about the "extra step" valid is to say that there is an "extra step" from classical correlations to quantum correlations i.e. entanglement. But this shouldn't be shocking. After all, there is an "extra step" from classical physics to quantum physics which affects pretty much everything. It affects how they describe Nature. They describe correlations differently, too. After all, as mentioned in the "inequality for dimensions" above, almost all the information carried by the state vector in quantum mechanics is about correlations. And yes, most of what quantum mechanics says is "new" so it follows that most of what quantum mechanics says about correlations must be different than in classical physics, too.

There is no room in 2012 for "cautiousness in scientific adventuring" of accepting quantum mechanics as the right language. Such discussions may have been OK 80 years ago but they are not OK today. People who try to stick to the classical intuition in 2012 are on par with the people who keep on believing in geocentrism. Classical physics doesn't work for anything in the world of spins and elementary particles, so of course it doesn't work for correlations, either.

Entanglement is being particularly assaulted and challenged and retested etc. because it's a particular indisputable way to show the incompatibility of our quantum world (and its quantum mechanical theoretical description) with the classical framework. But even if you don't talk about entanglement, this incompatibility may be easily demonstrated. In the Physics Stack Exchange discussion, people talked about entangled spins – because spins are the most convenient discrete quantum numbers where such a discussion makes sense.

But another thing that these overaged children believing tooth fairies overlook is that the spin itself is incompatible with classical physics, too. Even if you only consider one spin-1/2 particle which can't be entangled with anything else because you assume there is nothing else, it's still a system that is totally incompatible with classical physics. First of all, the spin may be carried by point-like particles but classically, such particles would have to spin at an infinite angular speed to have a finite angular momentum which would violate the laws of relativity.

Of course, it's not a real problem for the classical limit because in the classical limit, the whole spin disappears. Its magnitude is a small multiple of $$\hbar$$, the reduced Planck's constant, and the classical limit is defined by $$\hbar\to 0$$. There's just no spin left in the classical limit. That's a good thing because if such a discrete quantum number survived in classical physics, it would be a catastrophe. The electron spin may be$s_z\in\{ +\frac \hbar 2, -\frac\hbar 2 \}$ and if a particular component of a vector were constrained to belong to a discrete set, the symmetry under continuous rotations would be manifestly violated: discrete numbers such as $$\pm 1/2$$ cannot be transformed to each other by continuous rotations from $$SO(3)$$. Quantum mechanics avoids this conclusion because it fundamentally rejects the existence of any objective $$s_z=\pm \hbar /2$$. Instead, it says that the most fundamental description is given by complex probability amplitudes $$c_\uparrow$$ and $$c_\downarrow$$ and these complex probability amplitudes are continuous which means that they nicely transform under $$SU(2)\approx SO(3)$$ (the isomorphism holds locally on the group manifold).

Another crazy thing is that the wave function doesn't return to itself after a rotation by 360 degrees (it changes the sign whenever the spin is half-integer). Everything we know in the classical world should return to itself. (Well, there could also be classical spinor fields in classical physics but no one has thought about them before the birth of quantum mechanics and they couldn't describe the measurements of individual particles' spins, anyway.)

This point is much more general. Nature crucially depends on "basic descriptive numbers" that are continuous. Classical physics needs observables such as $$\vec x, \vec p, \vec E(\vec x)$$ etc. that are continuous because it says how they evolve as functions of a continuous time $$t$$ and if the degrees of freedom were discrete, they couldn't really ever controllably change. Quantum mechanics changes it: it allows objects (e.g. the spin of an electron) to "discontinuously change" but that's because it doesn't describe the "objective state of objects" itself; it describes the probabilities (more precisely probability amplitudes) and those change continuously with time. In classical physics, there was only one explanation of a "weak influence": the observable properties changed by a small amount due to this influence. In quantum mechanics, there's a new type of a "weak influence": the observable properties may change by a finite (or even large) amount but the probability of this happening may be tiny.

So the consistency requires that all real or complex numbers describing possible outcomes of measurements that constitute a "discrete set" are inevitably just probability amplitudes. If they were an "objective reality", the rotational symmetry, other continuous symmetries, and the ability of the system to evolve in (continuous) time would be invalidated.

The overaged children must suffer when they read such rudimentary comments. They will look for another contrived setup that will re-energize their belief that there must exist a classical or hidden-variable tooth fairy. And they will use any new, perhaps more detailed or more convoluted, proof that there can't be any tooth fairy and the world is intrinsically quantum and probabilistic as a reason to believe in the Big Brother anyway – and to believe that the Big Brother must be even bigger than previously thought because He is able to imitate all these blasphemous quantum, non-realist laws. That will always strengthen, and not weaken, their search for the Big Brother.

These folks are analogous to hopeless religious bigots.

And that's the memo.