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.

Reviews and Relativity

In 2002, Stephen Wolfram published his book, A New Kind of Science. About a month later, Scott Aaronson published a review of it which included a proof intended to demonstrate that the kind of discrete, deterministic universe Wolfram described was a scientific impossibility. I only just read this review, which makes me rather late to the party.

I like Aaronson’s review a lot, not because of what it has to say about NKS, but because the proof it contains. This proof, in my opinion, is one of those rare, wonderful moments in which a scientist with relatively mainstream views takes the time to refute a position in digital physics in a precise fashion. Out of such moments, stronger theories are made.

For those who’re interested, the review can be found here. I encourage all those who’re interested in this topic to take a look–particularly at Section 3.2.

For those who aren’t inspired to take a look, the gist of the proof is this: Any model that incorporates both quantum entanglement and special relativity is going to run into situations in which a measurement B in one reference frame precedes the event A that appears to precipitate it. The same situation must be viewable in other reference frames in which the events appear the other way around. The proof points out that a completely discrete model like the one Wolfram proposes lacks the quantum mechanical tools that usually help us resolve such scenarios. In the discrete case, either event A causes event B, or vice versa.

The proof is important because it’s not specifically directed at Wolfram’s ideas, but rather all fully discrete models of physics. What the proof proposes, in essence, is that complete discrete models are fundamentally incompatible with what we see in experimental physics.

I think I know what’s wrong with this proof and I’ll try to make my thinking on the topic clear here. If anyone out there disagrees with what I have to say, I’d be delighted to hear about it. To be honest, my idea of what’s wrong is so simple that I can’t quite believe that nobody else has said it. Quite possibly, there’s something massively obvious that I’m missing. If that’s the case, I can’t wait to learn what it is.

I believe that Aaronson’s proof fails because of the literal requirement of Assertion 2, which states:

R satisfies the relativity postulate. That is, assuming the causal network approximates a flat Minkowski spacetime at a large enough scale, there are no preferred inertial frames.

I would argue that while the proof may work so long as Assertion 2 is true, there’s no requirement that it hold. This is because we don’t know that spacetime actually conforms Minkowski space. We only know that whenever we observe objects traveling through space at less than the speed of light, their behavior is consistent with that model.

It’s true that every observation we’ve ever made has been rigorously, perfectly consistent with the Minkowski-space model, but we also know that we can never actually prove that spacetime conforms to Minkowski-space from basic philosophy of science. Notably, the work of Karl Popper.

To quote the mighty Wikipedia on Popper’s work:

Logically, no number of positive outcomes at the level of experimental testing can confirm a scientific theory, but a single counterexample is logically decisive: it shows the theory, from which the implication is derived, to be false.

In other words, we can never prove that something is true–only that it’s false. This concept is important here because spacetime is a bit like dark matter–we can never measure it directly. We can only ever measure the motion of particles traveling through it. I would argue that this changes the requirements for a working model of digital physics. Namely, the requirement becomes that particles within our model must always travel in a Lorentz-invariant fashion.

This distinction is key because if we can create other models of spacetime for which Lorentz-invariant motion always holds, but for which discretization works properly, then Aaronson’s proof fails for that case.

Are there such models? Doesn’t Special Relativity require Minkowski space? So far as I understand the topic, yes there are such models, and no, Relativity doesn’t need it. For an alternative model that I can’t find a problem with, all we need to do is a little algebra.

Here is the expression that defines the properties of Minkowski space, in units where the speed of light is 1:

s^2 = t^2 – x^2 – y^2 – z^2

To get something a little nicer, let’s just get rid of those pesky minus signs by moving our spatial axes to the other side of the equation. Then we get this:

t^2 = s^2 + x^2 + y^2 + z^2

Suddenly we have something that’s flat and local. But what does it mean in practice? It means that we need a simulation with an extra compact dimension, in addition to the three we’re used to looking at, that codes for the spacetime interval s. Motion in this compact dimension operates as a measure of the ‘subjective time’ that a particle experiences. With each iteration, particles travel at fixed velocity in some direction that combines motion in s, x, y and z. Simulation steps are then ordered along the axis t, which we might think of as ‘objective time’. I have a video of particles traveling this way on the web, and which I’ve mentioned in a previous post. You can find it here.

“But,” I hear you say, “that doesn’t look like Special Relativity, for a start, there’s a preferred frame of reference–namely the one through which we’re viewing the simulation”. Yes, it’s true that from outside the simulation, there’s a preferred frame, but there isn’t one when viewed from inside. Different reference frames are manifested as different angles with respect to the compact dimension, and motion in each direction is exactly the same. From within the simulation, measurements are completely consistent with the Minkowski-space model because the math governing them is identical.

“But what about Lorentz boosts?” you may ask. “What about Lorentz contraction? How come just one extra dimension is necessary? Don’t you need three?” Only one dimension is necessary because we know that to all extents and purposes, particles are point-like. Particles without extent don’t experience Lorentz-contraction. All of the physical properties that we observe of them emerge from their subjective experience of time.

Using this model starts making a difference when we get to the line in Aaronson’s proof at the bottom of page 10.

Then for all Z we require the following, based on what observers in different inertial frames could perceive:

This line and those that follow presuppose that in our discrete model, what an observer perceives as simultaneous is actually simultaneous. In other words, there is some discrete link directly connecting cause and effect. This is true in the Minkowski-space approximation, but in our compact-dimension model, it’s not. An observer perceives two events as simultaneous simply because the light from those events reaches him at the same time with respect to the objective time axis t.

What this means for examples such as the one that Aaronson raises, is that from outside our discrete simulation, we always know exactly when a particle interaction occurs, even if observers within the simulation may never be able to agree. It doesn’t matter that in some reference frames, effect B appears to precede cause A, because the perceived ordering of events no longer implies that the controlling simulation treats them the same way.

One of my current projects is a simulation that will hopefully make this point absolutely clear. I intend to track the subjective experiences of a large number of pseudo-particles traveling across a discrete space approximation that uses an extra compact dimension of the sort I describe. It is my belief that by constructing a secondary graph from the set of their subjective-time paths, it should be possible to obtain a causal set graph that approximates Minkowski space. Tools to measure the properties of such graphs have been developed by theorists working in Causal Set theory. By applying those tools, it should be possible to confirm that the experience of Special Relativity in a discrete simulation doesn’t require that the supporting graph mimic Minkowski-space directly.

This still leaves us with the topic of how exactly to encode quantum entanglement in a fully discrete system, as Aaronson’s proof relates as much to this topic as it does to relativity. This topic, though, is perhaps one for another post. However, it is worth stating that modeling entanglement in its most basic form appears to be extremely straightforward. The models I’ve built so far use something rather like the ‘long-range thread’ approach that Wolfram describes in his book, and it appears to work fine. Encouraging a particle to collapse into one of two spatially disjoint positions is easy in discrete models–the Jellyfish algorithm I’ve described in previous posts revealed this behavior on its own without any coaxing from me.

Ironically, the trickiest problem I’ve encountered in this area isn’t entanglement, but the encoding of information in geometric form. In order to create a working Bell Inequality simulation, we have to be able to simulate particle orientation and have two particles that retain their orientation in a coordinated way that is linked to the shared particle state we wish to collapse. This turns out to be tricky–particularly at the tiny scales at which my simulations run. It may be that there are better ways to manage Bell’s Inequality than the tools I’m currently using. Dan Miller, who also posts on this blog, has some interesting ideas in this arena which he will hopefully share in a later post.

To conclude, let me say that there is one old saying with which I ferociously disagree, and it is this: better to keep your mouth shut and have others think you’re a fool and to open it and remove all doubt. This sentiment negates learning. If you think I’ve illustrated ignorance or folly in this posting, call me on it. If you believe in science, this is your opportunity to share what you know to a willing audience. From me, you will hear only thanks.