Alice Entangled Land

My goal in this post is to discuss some potential objections to the possibility of finding a deterministic, discrete model of physics, such as the kind Alex has been talking about on this blog, that would support both Special Relativity and Quantum Mechanics. In particular, I hope to address the implications of analogues to Quantum Entanglement for such models.  I’m not proposing any particular model; rather, I’m making the argument that a certain class of models, discrete and deterministic in the classical computation sense, could exhibit both Special Relativity and Bell’s Inequality without paradox.  Perhaps I am tilting at a straw man here; it’s not clear to me that this is uniformly believed to be a problem among mainstream physicists (at least not lately). However, I’ve found that many people with a passing knowledge of the issues immediately conclude that there is a deep contradiction, which could make the kind of models we would like to design non-starters.  I hope to present a rough picture of how such models might work.

The Game

Scott Aaronson here proposes a game where two players attempt to correlate their actions:

We’ve got two players, Alice and Bob, and they’re playing the following game. Alice flips a fair coin; then, based on the result, she can either raise her hand or not. Bob flips another fair coin; then, based on the result, he can either raise his hand or not. What both players want is that exactly one of them should raise their hand, if and only if both coins landed heads. If that condition is satisfied then they win the game; if it isn’t then they lose. (This is a cooperative rather than competitive game.)

Clarification: Bob and Alice can both raise their hands or both keep them down in the case where they don’t both throw heads. Also, the time interval within which they must raise their hands is too quick for them to share any information at light speed (ie the events form a space-like interval).

In a classical world, with no quantum-ish non-local correlations (what Einstein referred to as “spooky action at a  distance”), Bob and Alice can win at most 75% of the time, by keeping their hands down (or up) on every turn.  However, using particles with quantum entangled spin states, they can do better.  Aaronson refers to this, which allows Alice and Bob to win the game more than 80% of the time, by cleverly choosing which angles to measure these entangled particles’ spin.  I’d like to discuss a made-up world, where Bob and Alice can win this game 100% of the time, by virtue of the properties of physics in this imaginary world.

Bob and Alice are experimental scientists living in a virtual world we have programmed on a classical computer (ie one that works according to the principles of a Turing machine).  According to their observations, this world exhibits Special Relativity — there is a maximum speed at which anything can move, which also acts as a limit on the speed of transfer of information.  Measurements conform to the familiar Lorentz transforms of Special Relativity, making all non-accelerating, non-rotating inertial frames equivalent, as far as any observations from within the world show. However, this world also exhibits something similar to the Quantum Entanglement we see in our world — except it’s even more pronounced.

In this world, there exists a particle with some interesting properties, which we will call Q. This particle has an orientation, and can be set to point in a certain direction.  When it is excited to a certain energy level, the Q particle breaks up into two subparticles.  Each subparticle travels at the speed of light — one in the direction the particle was pointing, and the other in precisely the opposite direction.  One of two measurements can be taken on each subparticle, which we will call up and down.  The result is always either yes or no.  There is no obvious pattern to the results; they appear completely random, with yes and no each occurring with equal probability.

However, the results of Alice and Bob’s measurements are surprisingly correlated.  If the measurements of both particles are ‘up’, the results always disagree (Bob measures yes and Alice measures no, or vice versa).  Otherwise, the measurements are always the same — they both measure yes, or both measure no.  Stated differently, Bob and Alice’s measurements always agree unless they both choose to measure ‘up’, in which case they always disagree.

The Q particle allows Bob and Alice to win the game with 100% success: they measure up or down depending on whether their coin comes up heads or tails, and raise their hands if the measurement result is ‘yes’.

Behind The Scenes

It’s worthwhile to imagine how this virtual world might be implemented.  As Alex has discussed on this blog, there are plausible computational models of physics for which Special Relativity is the subjective experience of entities doing experiments within the model.  However, all such models we are aware of have a ‘real’ frame of reference within which the world is actually computed.  The question we would like to pose is: does the addition of the Q particle allow Alice and Bob, our intrepid virtual scientists, to ‘pierce the veil’ of Relativity, and deduce the ‘real’ frame of reference underlying the simulation?

In Special Relativity, the notion of simultaneous events is not a meaningful one.  Instead, any two events are considered to form a space-time interval, which can be either space-like, time-like, or light-like.  Time-like intervals are those where each event is within the ‘light-cone’ of the other event.  Such events have a strict ordering in time — one comes before the other.  An implication of this is that the earlier event may cause, or at least have an effect on, the later event.  Space-like intervals are those where the two events that make up the interval originate outside each other’s light cones, and therefore can have (according to Relativity) no direct causal influence on each other.  The events in our Bob/Alice game are assumed to form space-like intervals; they cannot have a causal relationship.  Is it possible for this to be true, and yet for the Q subparticles to be correlated in the way that we have described?

As discussed, from a computational perspective, the world must be simulated using some — possibly arbitrary — preferred reference frame.  In this frame of reference, we treat the world as Euclidean in nature: every point in space has 3 coordinates (x, y, and z), and every event occurs at an absolute point in space and time, ie a 4-vector (x, y, z, t).  From this perspective, events are strictly ordered in time — an event with a smaller t precedes an event with a larger one.

Let’s assume that in this ‘real’ reference frame, in which we actually compute the world, Alice’s measurement happens first.  In this instance Alice decides (by randomly flipping a coin) to measure in the ‘up’ position.  We (the wizards behind the curtain, who run this Matrix-like world) need to provide her with a measurement immediately (what matters is that the measurement must be provided before Bob flips his coin).  Let’s say we give her a ‘yes’ answer.  (How we decide what answers to give is an important issue, but we’ll deal with that later — for now, assume we flip our own coin).  Now it’s Bob’s turn, and he also flips heads, so measures ‘up’.  We need to give him a ‘no’, to maintain the properties of Q as defined (if both scientists measure ‘up’, their measurements must disagree).   If Bob had flipped tails, we would have given him a ‘yes’ answer instead.

Impact on Relativity

Naively, it might be assumed that the addition of the Q particle breaks down the subjective experience of Special Relativity experienced by the inhabitants of our virtual world.  After all, it’s clear that, behind the scenes, we are making decisions that affect events in a space-like interval. Specifically, Alice’s decision about her measurement (up or down), along with our response, is used in a faster-than-light fashion to craft the response to Bob’s measurement.  Can’t Bob and Alice deduce somehow that the ‘real’ reference frame is the one where Alice goes first?

No, they can’t.  From Bob and Alice’s perspective, the (apparently) random nature of the outcomes makes that experiment impossible.  To make this concrete, let’s replay the events, this time computing them from a reference frame in which Bob’s measurement precedes Alice’s.  In this version of reality, Bob threw heads, and we gave him a ‘no’.  Later, Alice threw heads too, so we gave her a ‘yes’.  If she had thrown tails, we would have given her a ‘no’.   Given the information available to Bob and Alice from observations made inside their world, either perspective makes perfect sense.

For any series of such measurements, including any decision Bob and Alice might make, and any arbitrary frame of reference, we can construct a narrative that is consistent with all the facts.  From our God’s eye view, we can say that there is one true reality — the ontological truth of the matter. However, the fact remains that Bob and Alice, from an epistemological point of view, simply do not have enough information to deduce that reality.  The best theory they can devise, as responsible scientists, is a probablistic one.  Intriguingly, this theory includes analogues of both Special Relativity and Quantum Mechanics, and is as good a theory as they can expect to discover, given that they are not privy to the ‘coin flips’ we wizards use to provide responses to their measurements.

Determinisim

How does this model hold up when we stipulate that it must in fact be completely deterministic?  It is sometimes assumed that the randomness inherent in Quantum Mechanics is fundamental, and perhaps it’s meaningless to look for a ‘deeper’ theory to explain what we observe.  Certainly, in Bob and Alice’s world (as outlined so far), that would be a reasonable position for them to take.  Within their Universe, Bob and Alice do experiments that appear to produce random results.  This randomness in turn isolates them from discovering the ‘real’ reference frame we use to compute the evolution of their Universe, thereby ensuring that all reference frames are on an equal footing — and that they cannot use the Quantum Entanglement of the Z particle to perform faster-than-light communication.

Let’s assume we use a sophisticated Pseudo-Random Number Generator to decide how to respond to measurements of the Q subparticles. We have seeded this PRNG with some ‘random’ data from our own Universe.  Wait you say — there’s randomness! You promised us a deterministic model! Well perhaps; one might say there’s a wee bit of spam in the pudding… however, this could be a small, finite number of bits put into the system at the beginning, and so constitute initial conditions. The point here is that there is no need to continually pump new random data into the system for every fundamental interaction. As Wolfram and others have shown, you don’t need that many bits for an algorithm to produce a huge amount of apparent complexity and randomness.

In this picture of our virtual Universe, Bob and Alice might end up having a collegial disagreement about the ultimate nature of their reality.  Bob might say that, since the result of every measurement on subparticles ever made has appeared to be totally random, that randomness is at the heart of their Universe, and they should accept this fact, and be happy with a stochastic model that best explains what the likely results of measurements will be.  Alice, on the other hand, may insist that a deeper, deterministic theory is still possible.  All she needs to do to prove this, it seems, is to somehow ‘break’ the PRNG that we use to compute their world.  If, through some amazing coincidence, she were able to guess the seed of the PRNG at some point in time, she might be able to deduce the ‘real’ reference frame, and from then on, predict the exact results of every future experiment.  Among other things, this would allow faster-than-light communication, since the decision on how to measure a subparticle can have an instantaneous impact on a distant measurement.  Sounds to me like Alice is on the right track.

Conclusion

The class of models we’ve outlined here show that there is no inherent contradiction between Special Relativity and Quantum Entanglement, as long as we postulate that the results of certain measurements appear to observers within the system to be random in nature.  Afficionados of Cellular Automata and chaos theory will not need to be convinced that this restricted type of randomness can be produced within a discrete, deterministic theory, relying only on classical computation methods to evolve its state. Rather than wizards flipping heavenly coins, or even running standard pseudo-random number generators to decide the results of measurements, it seems plausible that the source of apparently random behavior in our Universe could come from the easy chaos that even extremely simple algorithms are capable of producing.  We can choose to stay with stochastic theories, like Bob, or — like Alice — we can push ourselves to imagine what kind of underlying fabric might reasonably produce the kind of results we see with the experiments we have so far been able to conceive and reduce to practice.  This sort of imagining might one day result in an experiment that could decide between these two worldviews.