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Tuesday 22 November 2022

Oh, What An Entangled Web...

Allan Adams
Quantum Mechanics is not a difficult subject to understand, but it does require being prepared to shed our naïve intuitions. The intuition we most have to be ready to shed is the notion of 'particles' being little chunky things. It's an intuition regulars will recognise as one we've looked at from a few different angles, from the perspective of 'duality', which is not a real thing out there in the real world but the mashing together of two analogies, neither of which is very good. 

Analogies, metaphors, similes... these are the tools we work best with in education. In other words, we learn to understand things best by comparison to other things we already understand. The problem being, of course, the things we think of as particles aren't really like anything in our experience, so we have nothing to compare them to. This leads to a situation in which we end up making crude comparisons between individual behaviours of very different things, and this is what duality is. We've learned that there is no wave, and no particle, and no duality. There is something else with behaviours we associate with those things. From here on, I'm going to shed it completely and replace it with 'system'.

Today, I want to look at yet another source of enormous confusion in Quantum Mechanics stemming from thinking of systems as chunky, yet it's really not an inherently quantum principle, but something trivially occurring in our everyday experience; entanglement.

Entanglement is yet again one of those principles in QM which absolutely ties people in knots, yet it shouldn't, because it's really straightforward once you have the right intuition. The hope here is the reader will get to the end of this and wonder what all the fuss was about because, in my humble, there shouldn't be any. So what is entanglement, then, and why does it cause such problems? To answer this, we really need to start at the beginning and talk about conservation. This is another topic we've approached from various angles, but one thing we haven't covered in any detail is how conservation appears in Quantum Mechanics. One might think it needn't, but that's a mistake, because conservation is fundamentally what QM is all about*. Here's where we begin to learn about conservation: 
Energy may be neither created nor destroyed.

There you go, the classical first law of thermodynamics. Of course, this particular formulation is an updated version of a very naïve construct - naïve in this case both in the sense of being informal (in natural language, as opposed to formal language, i.e., mathematics) and in the sense of being rather incognisant of its flaws. The best way to express these flaws is to say the statement is highly problematic without enormous qualification. It's not a statement robustly applicable to ongoing processes in any kind of thermodynamic system, because the laws of thermodynamics are inherently about comparisons between equilibrium states, and the wheels fall off hard if you try to conduct detailed calculations of the energy content of a system during an energetic process. All else aside, Heisenberg's Uncertainty Principle always looms large, and can trivially change the energy content of any system from moment to moment, especially during things like system expansion or contraction because, as we've learned, no wave can exist between two boundaries unless those boundaries are an exact multiple of π radians of the wavelength, which is precisely how energy is quantised and why microwave ovens are a thing\(^†\).

It is worth noting that the classical first law has been entirely replaced in modern physics, and has the status of 'lie to children', which is not to say actually dishonest, just something that we know to be untrue but serves as a good basis for the understanding of conservation. In modern parlance, the equivalent of the first law of thermodynamics is Noether's Theorem, and the best vernacular statement of the underlying principle is that the laws of physics don't change over time. No, really!

Anyhoo, let's start with our classical expression, and see where it takes us.  

Picture a perfectly still pond, and drop a ball bearing in it. You can already picture in your mind's eye the concentric rings radiating out from the point of impact. Because the surface of the pond is two-dimensional, the energy of the rings will spread out evenly directly as a function of distance, for exactly the same reason that light and gravity follow an inverse-square falloff due to being in three dimensions. Let's add another slight idealisation and make our pond one-dimensional, or at least close to it, by making our pond long and narrow. This means that there will be no falloff other than the initial spread. We'll also only focus on a single wave for simplicity. It might look a bit like this. 

Note the distribution of the energy, always exactly equal and opposite. It should be quite suggestive when I say they have 'symmetry' which, as we learned in our look at Noether's Theorem, tells us that something is being conserved. When they meet the end of the pond, they'll reflect back off the ends in a manner that is exactly equal and opposite and meet in the middle. There is no time, absent other influences, when the behaviour of these two waves is anything other than exactly equal and opposite. Note also there is no transfer of information between these waves at any time. When they meet, they'll essentially pass straight through each other, with a brief flurry of destructive and constructive interference, but their distribution will always be exactly equal and opposite as long as no other influences are introduced.

That, right there. That's entanglement. Not an analogy, not a simile, and not a metaphor, but full entanglement exactly as it appears in QM. This is what entanglement actually is. This is the intuitional foundation that should be taken into any discussion of quantum entanglement. Because of this conservation, we can model this as a single system. No, it IS a single system, and we can model it using a single function; a wavefunction. As an aside, this is also an extremely good intuition for how we create entangled systems in the first place, because all processes that emit energy pretty much produce entangled systems.

Where QM differs is in a few small but critical areas and, again, they have to do with symmetry and conservation, but also with uncertainty. Experiments in entanglement in QM are usually centred on a very particular conserved quantity, namely spin. Spin is a perfect quantity for this purpose, because it has both uncertainty and simple straightforward binary properties simultaneously.

Spin is a conserved quantity in QM. It should be, because it's a kind of angular momentum, and angular momentum is also a conserved quantity. The conservation of angular momentum as it appears in Noether's Theorem is a consequence of symmetry under a particular kind of translation, a rotation. It tells us, in its basic vernacular form, that the laws of physics don't change based on which direction you're facing. The way spin actually appears in the systems of the standard model is not identical to angular momentum in the classical world, but it isn't enormously different either. The major difference is that, in systems, spin is an intrinsic quality rather than arising from a behaviour. A simple analogy should suffice to show the distinctions. 

Picture a Harlem Globetrotter doing tricks with a basketball, and ending with the classic of having the ball spinning on a fingertip. That force on the ball as it rotates is angular momentum. Now imagine stopping the ball from rotating but, oddly, the force still being there even though the ball isn't moving. This is what it means for spin to be intrinsic. It's a basic property of the entities in question, and not anything to do with their behaviour, as it is in the case of the basketball.

While we have this analogy on the table, let's use it to talk about some other characteristics of system spin. Picture the basketball again, but take the finger away and imagine it freely rotating in space. Now imagine that you choose a direction from which to observe the ball. In the classical world, any but the axis of rotation that it was set spinning about will show the surface of the ball passing your eye radially, with the rotation about the same axis at all times. Not so with 'systems'. In particular, because we're dealing with sets of conjugate variables subject to superposition and uncertainty, whichever axis you choose to observe, 100% of the system's spin will be about the axis of your choosing, either clockwise or counter-clockwise. This is because the spin about any given axis of observation is a superposition of spin about all other axes, in direct comparison to the Uncertainty Principle for position and momentum, wherein any well-defined position is a superposition of momenta, and where any well-defined momentum is a superposition of positions. 

System spin has one other oddity, and it's to do with orientation. Pick a spot on your basketball and draw a dot on it. Now look at the dot and rotate the ball. If it's a standard ball in 4D spacetime, it will go through one full revolution before the dot arrives back at its original location. Not so with systems, for which this is a quantity based on the exact spin number of the system. For systems with integer (whole-number value) spin, the behaviour is just like that of the basketball, and one full rotation reorients the spin to its original position. For systems with half-integer spin (½, 1½, etc.), they have to go through two full rotations to reorient in this way\(^‡\). 

It's really important to note that superposition and uncertainty are again not inherently quantum. They are, in fact, pretty basic behaviours of waves. Superposition is an example of linearity, just like linearity in mathematics. If you take any two waves and add them together, you get a wave. That's linearity. Any wave can be constructed as a sum of other waves (or, indeed, as a sum of sines and cosines). That's superposition. In Quantum Mechanics, take any two wavefunctions \(\Psi_1\) and \(\Psi_2\) and add them together, you get a wavefunction \(\Psi_{1+2}\). Take any two solutions to the Schrödinger Equation and add them together, you get a solution to the Schrödinger Equation. 

So, let's get back to entanglement, and in particular the notion of 'action at a distance', because this is both where the wheels fall of a bit and where all of the above will start to come into focus, and I'm going to resurrect an analogy I tried before, but it wasn't well-developed, and let's look at why the notion of transfer of information in entanglement experiments doesn't stack up, and a simple explanation of why it can't be used for faster-than-light communications.

Suppose I take two shoes and put them in separate boxes, handing one of them to you and the other to your twin, and send you off in opposite directions. Neither of you knows whether you have a left shoe or a right shoe until you make an observation. When you open your box revealing the left shoe, you know your twin has the right shoe.

Now do the same, only for socks.

The difference here is that, of course, a sock doesn’t have chirality (handedness) until an observation is made, which in physics means an interaction (no consciousness required). We have specific ways of interacting with socks, and it depends entirely on the nature of the thing that it interacts with (what kind of observation we make; the observer effect). As soon as you make an observation by putting on a sock, the chirality is nailed down. Of course, where socks differ is that, even in a pair, chirality is a choice we make. Again, not so with systems. In exactly the same way that our entangled waves above are always symmetrical, spins statistics among entangled systems are always symmetrical. This is analogous to to your socks each being in a superposition of chirality, of being both left and right simultaneously (like Schrödinger's Cat being alive and dead), until an observation is made on one of them and the other instantly having opposite chirality, even if your twin has buggered off to the other side of the universe. 

There's also the small matter of the broader implications of uncertainty, namely that this parameter has no definite value until observation, and that includes the chirality. Heisenberg's Uncertainty Principle tells us that, until an observation is made, all parameters are in superposition. This means that the superposition of spin axes is always and exactly counter-correlated in the constituents of an entangled system. 

The upshot of this uncertainty relation is this; spin states are in a constant state of flux, whether in entangled systems or not. When we bring entanglement into the mix, we’re talking about multiple ‘systems’ as a single, unified system, with a single wavefunction describing the system. All the fluctuations and superposition states on the one are exactly counter-correlated at all times, because their spin statistics are a conserved quantity. There is never any time before, or during an observation at which these quantities are anything other than directly counter-correlated. No information is ever transferred between them at any point. The observation tells you which direction the intrinsic angular momentum of the one system is, which is correlated at the instant of observation in exactly the same way as at all other times. No information has been transferred, they were always exactly equal and opposite, in the same way our waves in the pond are.

It's also important to note that, when we make an observation - remember that this means an interaction - we've done what we said we couldn't do and maintain symmetry in our pond example; we introduced another influence, destroying entanglement and meaning they evolve as separate entities from the point of interaction onwards. This is the so-called 'observer effect' writ large, about which you can read more in the links below.

I hope I've shown, at bottom, entanglement is nothing more nor less than a way of talking about the distribution of conserved quantities and, because they’re conserved, they must always be counter-correlated. That is, in the same way that, for example, virtual particle pairs are always the same mass but opposite in sign, and for exactly the same reasons, when quantum numbers are distributed among multiple systems, they will always be exactly equal and opposite in sign, because the sum total of their conserved quantities are in fact a single conserved quantity, but it's distributed. This is just conservation in action, and is entirely as a rudimentary understanding of conservation would bring us to.

Where the wheels come off, then, is entirely as thinking of them as ‘particles’, which is a natural language term with a lot of baggage. The particles in the standard model are really not particles (modern thought has it they're excitations in fields), and almost all of the intuitional pitfalls arise because the notion of little chunky things is very, very tempting to our middle-world intuitions, but we really need better intuitions.

Further reading:

Paradox! A Game For All The Family More on spin, superposition and the EPR Paradox.

Give Us A Wave! More on waves and QM.

Well, Blow Me Down! Conservation laws and intuition.

On The Shoulders Of... More on Noether's Theorem.

* The Schrödinger Equation is, at bottom, a way of talking about quantities that have no definite value until observation in a way that ensures that all symmetries are maintained; that all conserved quantities are conserved. It is Noether's Theorem writ large.

\(^†\) Without this quantisation, the energy inside an oven would be infinite. This is known as the 'ultraviolet catastrophe', and it was solved by Planck. The exact solution comes in the form of a constant, the dimensional integration of energy over time, which distributes the energy of a wave along its wavelength. This is the famous Planck Constant.

\(^‡\) This number is very, very important to us, because systems with half-integer spin are fermions, and are thus subject to the Pauli Exclusion Principle, which means they can't occupy the same space. An interesting consequence of the way spin statistics work is that they're also additive. For some atoms, such as Helium 4, the spins statistics add up in such a way that the overall spin number is an integer, and this means they aren't subject to Pauli Exclusion, and this results in some very odd behaviour, particularly when it's cooled to extreme temperatures. The atoms can get arbitrarily close together, and this is the basis for the fourth phase of matter, the so-called Bose-Einstein condensate, which does some really weird things indeed.

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