Do quantum correlations violate special relativity?

It turns out that quantum information and foundations researchers tend to disagree on the answer to that question.  I’m at the APS March Meeting in Boston and have had a number of stimulating discussions with people, this being one of them.  I was surprised by some of the reactions I got when I said quantum mechanics violates axioms of special relativity and/or was non-local.

Bell’s inequalities include locality in their derivation.  There are a few different ways one can express this, but, for example, Sakurai says that locality is represented in the fact that Alice’s “result is predetermined independently of [Bob's] choice as to what to measure” [J.J. Sakurai, Modern Quantum Mechanics, (Addison-Wesley-Longman: Reading, 1994), p. 228].  Bell himself noted this in several places.  Violations of Bell inequalities would, therefore, seem to imply that quantum mechanics is non-local.  Outside of the more fringe interpretations, I did not think this point was under dispute.  It seems I was wrong.  It also seems that there might be a difference between something being non-local and something truly violating special relativity.

The more I thought about it today, the more I think the debate is really over how to interpret special relativity and not how to interpret quantum mechanics (I think that, in itself, is interesting).  Consider the two independent measurement events.  Those two events, each represented by some action on one-half of an entangled particle pair, must be correlated in some way.  As such, their past lightcones must overlap (technically, the past lightcones of everything overlaps at some point, but what we could say is that they share a common event in their past).  However, the crucial point is that the events that represent the choices that Alice and Bob make regarding which basis to measure, are completely independent in that they do not share some common “origin” event (or, since the Big Bang, is a common “origin” event for everything, we could say instead “the chain of events leading to a common origin is too long for them to be correlated”).  As such, they are (or can be) spacelike separated and yet a correlation will be found.

One (dare I say incorrect?) response to this could be that the freedom to choose a measurement basis doesn’t actually do anything actively to the particles.  For example, if Alice chooses to measure spin along the z-axis, her result will be correlated with Bob’s only if he chooses to measure along the same axis.  If he measures, say, along the y-axis he gets a random result but crucially doesn’t know it is random until he talks to Alice and compares notes.

But we know that choosing a measurement basis physically alters the state of the particle being measured.  This is the heart of quantum mechanics – you must interact with the particle in order to measure it and it is very difficult to do so without altering the particle’s state!  In special relativistic terms, changing some object’s state involves the action of a force (defined relativistically to be dP/dt where P is the relativistic momentum).  Thus, from the standpoint of special relativity, a measurement by Alice should produce a force on the particle she is measuring while another force produced by Bob’s measurement (assuming their measurements are roughly at the same time) must produce something else instantaneously.  These forces cannot be causally linked and yet they are somehow correlated.  This is a fundamental violation of special relativity!

One counter-argument that has been put forward this week by a couple of people is that there’s no violation of special relativity because nothing propagates between the two events.  The two events are quantum correlated, so it’s still a non-classical phenomenon, but it doesn’t violate relativity.  But what does it mean to be quantum correlated versus classically correlated?  It is my contention – and that of others – that the very definition of quantum correlations are that they’re non-local and thus in violation of special relativity!  From an empirical/operational viewpoint, I really see no other way to explain this.  As I have noted before, citing a mathematical result (e.g. a non-factorable state) as evidence for something physical, has serious interpretational problems.

So, my conclusion is (and always has been) that quantum correlations do, indeed, violate special relativity.

What exactly is information, anyway?

[Note: this post is somewhat related to an ongoing discussion about "interestingness" and "complexidynamics" that was initially inspired by Sean Carroll's FQXi presentation and has been subsequently discussed by Scott Aaronson (see here and here) and Charlie Bennett.]

We know how to measure information (both classical and quantum), we know how to encode information, and we know how to transmit information.  But do we really know what it is?

Consider the following example.  Imagine a painter who paints an exquisite painting.  Is the painting itself information?  Certainly we can encode it in bits and qubits, but is the encoding of the painting the same thing as the original painting?  In other words, is information the same thing as that which it represents?  I can get plenty of copies of the Mona Lisa, but they aren’t the Mona Lisa.

Now suppose our painter never shows anyone else and the painting is eventually lost in a fire.  The painter forgets about it and eventually dies.  No record of this painting ever existed (after the fire) outside of the painter’s own head and he/she eventually died.  To use a turn of phrase from a novel I’m currently reading (Murakami’s 1Q84), quite clearly something – who knows what – was “irretrievably lost” once that painter died.  Now this raises some interesting questions.

If we take Wheeler’s “it from bit” seriously, then information is the core component of the universe.  While the physical atoms of the painting and the painter are, of course, not lost, the specific “macrostate” (“message”), if you will, that they represented, is clearly gone.  The same can be said if we merely say that information represents reality rather than constructs it and even if we say it merely encodes reality.

Given recent results concerning information causality and our penchant for referring to entropy as a measure of information, this should raise some intriguing questions.  Now, even if you postulate an infinite universe or a multiverse to compensate for the lost information (if that is indeed what is lost), there’s still a problem: the fact remains that that painting in that particular universe (or part of an infinite universe) was destroyed (I suppose there are some subtle distinguishability issues that this raises).

The ongoing discussion about “interestingness” and complexidynamics aside, I think this raises a very thorny issue: just what exactly is information anyway?

Information and the nature of mass

I’ve had an idea bouncing around in my head since last fall when I had an interesting discussion on the nature of mass on LinkedIn.  It actually follows up on some ideas that I outlined in Physics Today about seven years ago.  It all centers on the concept of mass: what is it?  There seem to be two schools of thought on this issue (aptly represented by myself and Frank Wilczek in the aforementioned article).  The first school of thought assumes that mass is merely energy.  The second school of thought says that, although it can be transformed into energy, it isn’t quite the same thing.  One of the arguments against the former is that it (perhaps unintentionally) hides the significance of the spacetime metric, something Lev Okun has pointed out.  To fully understand this, however, we need to understand how it is mathematically defined (something everyone agrees on).  Specifically, mass is defined (mathematically) to be the magnitude of the four-momentum vector.

What’s tricky here is that the magnitude of four-vectors aren’t quite like that of other vectors – the signature plays a role.  Specifically, the square of the temporal component of the four-momentum (called the relativistic energy) has the opposite sign to the square of the spatial components (called the relativistic momentum).  So, the magnitude of the vector above works out to

Crucially, this quantity is frame-independent.  As such it is often referred to as the “invariant” mass (also “rest” mass).  Okun’s argument (one I happen to agree with) is that by merely saying mass is the same as energy, implies that momentum is the same as energy (given the above equation), and thus the difference in the sign of the signature is “brushed under a rug.”  This difference is very important since it points out the fact that time and space, though similar, are not quite the same exact thing (that’s a discussion for another time).  Note that the four-momentum is conserved which means that it is conserved by component.  We’ll see what that means in a moment.

To illustrate some of these concepts – and raise some interesting questions – let’s consider the following thought experiment.  Imagine a very thin, spherical optical cavity the size of, say, a star, i.e. a very thin, hollow sphere with perfectly (or near perfectly) reflective material coating the inside, that is at “rest” (roughly inertial).  Now suppose that the cavity is filled with neutral pions.  Neutral pions decay to photons.  Now at some point prior to any pion decay, the total four-momentum of the system will be

where we assume there are n pions.  The system has a total mass equal to the magnitude of the system’s four-momentum vector (large enough for stuff to orbit it in this example).

For the sake of simplicity, let us assume that any pion decays to two photons (this is the primary decay mode with probability 0.98798) that, in order to conserve four-momentum, must move off in opposite directions.  After some time, t, all the pions have decayed and the total four-momentum is

where we now have 2n photons rattling around inside the cavity.  Note that, in order to conserve four-momentum, each decay produces two photons that move off in opposite directions.  Given the large, randomly oriented quantity of pions, on average we expect the total relativistic momentum of the photons to equal zero.  This is true even though the photons from a single decay may hit the insides of the cavity at different times because we assume that there is a comparable decay on the other side the balances it out (note that it might be possible for the cavity to “jiggle” a bit if these decays are slightly uneven temporally, but in the end it’s center-of-mass should not have budged an inch).

Naively, we might be tempted to say that the system has lost a great deal of mass.  But it clearly hasn’t since, assuming four-momentum is conserved, it must be that

where the system’s mass is the magnitude of each of these vectors.  So, while the pions’ mass has seemingly disappeared – turning into a bunch of massless photons – the system’s mass remains unchanged.  In fact, if we suppose that a planet orbits our contrived “star” (cavity), assuming the cavity is internally perfectly reflecting, the planet should experience no change in its orbit.  Thus, in this broad sense, mass is conserved (invariant).  It’s crucial for the definition, however, that we’re dealing with four-vectors here.  If it were an ordinary vector, none of this would work.  That’s one reason the metric is so important.

Now, if we view this from the standpoint of information, is information created, destroyed, or conserved in this process?  Traditionally, despite being spin-1 particles (which can serve as a representation of a qutrit), photons only have two degrees of freedom and so they are typically treated as a representation of qubits.  Either way, it’s fairly easy to see how we can encode information in the photons.  Pions, on the other hand, are spin-0.  How could we possibly encode information in plain, old neutral pions?  I suppose there might be a way, but it still begs the question of whether information gets created in this process or whether it’s conserved.

So now let’s ask ourselves what happens if the inside of the cavity is not a perfect reflector.  In this case, let us assume that a certain fraction of all photon collisions with the cavity result in a photon being absorbed by the cavity.  Given enough absorptions, the cavity will begin to heat up, re-radiating thermal photons, some outward and some inward.  Eventually, the cavity should radiate enough heat outward that the total mass of the cavity plus all the photons inside it is now lower.  In other words, in a process that’s somewhat analogous to Hawking radiation and the evaporation of black holes, our “cavity” radiates away its mass.  Does this have any effect informationally?

The next question that should be asked, of course, is what would be different for this system if we treated it quantum mechanically? In that case, all the photons (and pions, for that matter) can be assigned wavefunctions and we can do a modified “particle in a box” model where the box is now a sphere.  I haven’t yet given this enough thought.  We’ll have to wait and see.  If anyone out there has thoughts on this, please do share.

What is a particle?

Is the answer obvious?  It is perhaps not as obvious as you might think.  I was in a discussion with a colleague of mine yesterday about the “reality” of fields.  I’m not convinced that they’re anything more than a mathematical convenience championed by people wishing to sweep the “action at a distance” problem under a rug.  I can’t think of an experiment that can only be interpreted in terms of fields (most interference experiments can be interpreted statistically – let’s ignore the recent PBR paper for the moment).  He countered that the same could be said for particles.  I suppose I could concede that they could exist, but that we might not be able to prove it since it would be impossible to make a measurement of anything without involving a “particle” interpretation somewhere in the chain of interpretations leading from the measurement process to our conscious brains.  In any case, he countered my assertion in the most logical way by asking me: what is a particle?

As a foundationalist, I probably am guilty of over thinking this, but the only thing I could come up with that could differentiate a particle from a field was that a particle is a local concept whereas a field is not.  Fields have spatial extension (of course this requires the existence of space and quite possibly time as well, but that’s an argument for another day).  Particles – fundamental particles, anyway – are mere points.  Note that in my above assertion about the requirement of a particle somewhere in the interpretation chain, I make no distinction between the classical and quantum cases because, like many other quantum foundations and information folks, I take the view that the classical world is a special case of the quantum world.  As such, “spooky action at a distance” can be explained by virtual particles (for interactions) or mere correlations (for odder things like entanglement).

At any rate, I am still at a loss in terms of trying to find a better differentiation between a particle and a field that is not, in some way, self-referential (in other words, saying that a particle is the quantization of a field is, in my mind, self-referential to some extent).  However, stay tuned for a commentary on the nature of mass that might add a little twist to this whole thing.