Tag Archives: Probability

The Zen of the Quantum Omlette

[Quantum mechanics] is not purely epistemological; it is a peculiar mixture describing in part realities of Nature, in part incomplete human information about Nature, all scrambled up by Heisenberg and Bohr into an omelette that nobody has seen how to unscramble. Yet we think that the unscrambling is a prerequisite for any further advance in basic physical theory. For, if we cannot separate the subjective and objective aspects of the formalism, we cannot know what we are talking about; it is just that simple.” [1]

— E. T. Jaynes

Note: this post is about foundational issues in quantum mechanics, which means it is rather long and may be boring to non-experts (not to mention a number of experts). I’ve tried to use simple language so that the adventurous layman can nevertheless still get the gist of it, if he or she is willing (hey, fortune favours the brave).

As I’ve said before, I think research on the foundations of quantum mechanics is important. One of the main goals of work on foundations (perhaps the main goal) is to find a set of physical principles that can be stated in common language, but can also be implemented mathematically to obtain the model that we call `quantum mechanics’.

Einstein was a big fan of starting with simple intuitive principles on which a more rigorous theory is based. The special and general theories of relativity are excellent examples. Both are based on the `Principle of Relativity’, which states (roughly) that motion between two systems is purely relative. We cannot say whether a given system is truly in motion or not; the only meaningful question is whether the system is moving relative to some other system. There is no absolute background space and time in which objects move or stand still, like actors on a stage. In fact there is no stage at all, only the mutual distances between the actors, as experienced by the actors themselves.

The way I have stated the principle is somewhat vague, but it has a clear philosophical intention which can be taken as inspiration for a more rigorous theory. Of particular interest is the identification of a concept that is argued to be meaningless or illusory — in this case the concept of an object having a well-defined motion independent of other objects. One could arrive at the Principle of Relativity by noticing an apparent conspiracy in the laws of nature, and then invoking the principle as a means of avoiding the conspiracy. If we believe that motion is absolute, then we should find it mighty strange that we can play a game of ping-pong on a speeding train, without getting stuck to the wall. Indeed, if it weren’t for the scenery flying past, how would we know we were traveling at all? And even then, as the phrasing suggests, could we not easily imagine that it is the scenery moving past us while we remain still? Why, then, should Nature take such pains to hide from us the fact that we are in motion? The answer is the Zen of relativity — Nature does not conceal our true motion from us, instead, there is no absolute motion to speak of.

A similar leap is made from the special to the general theory of relativity. If we think of gravity as being a field, just like the electromagnetic field, then we notice a very strange coincidence: the charge of an object in the gravitational field is exactly equal to its inertial mass. By contrast, a particle can have an electric charge completely unrelated to its inertia. Why this peculiar conspiracy between gravitational charge and inertial mass? Because, quoth Einstein, they are the same thing. This is essentially the `Principle of Equivalence’ on which Einstein’s theory of gravity is based.

Einstein

These considerations tell us that to find the deep principles in quantum mechanics, we have to look for seemingly inexplicable coincidences that cry out for explanation. In this post, I’ll discuss one such possibility: the apparent equivalence of two conceptually distinct types of probabilistic behaviour, that due to ignorance and that due to objective uncertainty. The argument runs as follows. Loosely speaking, in classical physics, one does not seem to require any notion of objective randomness or inherent uncertainty. In particular, it is always possible to explain observations using a physical model that is ontologically within the bounds of classical theory and such that all observable properties of a system are determined with certainty. In this sense, any uncertainty arising in classical experiments can always be regarded as our ignorance of the true underlying state of affairs, and we can perfectly well conceive of a hypothetical perfect experiment in which there is no uncertainty about the outcomes.

This is not so easy to maintain in quantum mechanics: any attempt to conceive of an underlying reality without uncertainty seems to result in models of the world that violate dearly-held principles, like the idea that signals cannot propagate faster than light, and experimenters have free will. This has prompted many of us to allow some amount of `objective’ uncertainty into our picture of the world, where even the best conceivable experiments must have some uncertain outcomes. These outcomes are unknowable, even in principle, until the moment that we choose to measure them (and the very act of measurement renders certain other properties unknowable). The presence of these two kinds of randomness in physics — the subjective randomness, which can always be removed by some hypothetical improved experiment, and the objective kind of randomness, which cannot be so removed — leads us into another dilemma, namely, where is the boundary that separates these two kinds of uncertainty?

E.T. Jaynes
“Are you talkin’ to me?”

Now at last we come to the `omelette’ that badass statistician and physicist E.T. Jaynes describes in the opening quote. Since quantum systems are inherently uncertain objects, how do we know how much of that uncertainty is due to our own ignorance, and how much of it is really `inside’ the system itself? Views range from the extreme subjective Bayesian (all uncertainty is ignorance) to various other extremes like the many-worlds interpretation (in which, arguably, the opposite holds: all uncertainty is objective). But a number of researchers, particularly those in the quantum information community, opt for a more Zen-like answer: the reason we can’t tell the difference between objective and subjective probability is that there is no difference. Asking whether the quantum state describes my personal ignorance about something, or whether the state “really is” uncertain, is a meaningless question. But can we take this Zen principle and turn it into something concrete, like the Relativity principle, or are we just by semantics avoiding the problem?

I think there might be something to be gained from taking this idea seriously and seeing where it leads. One way of doing this is to show that the predictions of quantum mechanics can be derived by taking this principle as an axiom. In this paper by Chiribella et. al., the authors use the “Purification postulate”, plus some other axioms, to derive quantum theory. What is the Purification postulate? It states that “the ignorance about a part is always compatible with a maximal knowledge of the whole”. Or, in my own words, the subjective ignorance of one system about another system can always be regarded as the objective uncertainty inherent in the state that encompasses both.

There is an important side comment to make before examining this idea further. You’ll notice that I have not restricted my usage of the word `ignorance’ to human experimenters, but that I take it to apply to any physical system. This idea also appears in relativity, where an “observer in motion” can refer to any object in motion, not necessarily a human. Similarly, I am adopting here the viewpoint of the information theorists, which says that two correlated or interacting systems can be thought of as having information about each other, and the quantification of this knowledge entails that systems — not just people — can be ignorant of each other in some sense. This is important because I think that an overly subjective view of probabilities runs the risk of concealing important physics behind the definition of the `rational agent’, which to me is a rather nebulous concept. I prefer to take the route of Rovelli and make no distinction between agents and generic physical systems. I think this view fits quite naturally with the Purification postulate.

In the paper by Chiribella et. al., the postulate is given a rigorous form and used to derive quantum theory. This alone is not quite enough, but it is, I think, very compelling. To establish the postulate as a physical principle, more work needs to be done on the philosophical side. I will continue to use Rovelli’s relational interpretation of quantum mechanics as an integral part of this philosophy (for a very readable primer, I suggest his FQXi essay).

In the context of this interpretation, the Purification postulate makes more sense. Conceptually, the quantum state does not represent information about a system in isolation, but rather it represents information about a system relative to another system. It is as meaningless to talk about the quantum state of an isolated system as it is to talk about space-time without matter (i.e. Mach’s principle [2]). The only meaningful quantities are relational quantities, and in this spirit we consider the separation of uncertainty into subjective and objective parts to be relational and not fundamental. Can we make this idea more precise? Perhaps we can, by associating subjective and objective uncertainty with some more concrete physical concepts. I’ll probably do that in a follow up post.

I conclude by noting that there are other aspects of quantum theory that cry out for explanation. If hidden variable accounts of quantum mechanics imply elements of reality that move faster than light, why does Nature conspire to prevent us using them for sending signals faster than light? And since the requirement of no faster-than-light signalling still allows correlations that are stronger than entanglement, why does entanglement stop short of that limit? I think there is still a lot that could be done in trying to turn these curious observations into physical principles, and then trying to build models based on them.

Advertisements

The Adam and Eve Paradox

One of my favourite mind-bending topics is probability theory. It turns out that, for some reason, human beings are very bad at grasping how probability works. This is evident in many phenomena: why do we think the roulette wheel is more likely to come up black after a long string of reds? Why do people buy lottery tickets? Why is it so freakin’ hard to convince people to switch doors in the famous Monty Hall Dilemma?

Part of the problem is that we seem to think we understand probability much better than we actually do. This is why card sharks and dice players continue to make a living by swindling people who fall into common traps. Studying probability is one of the most humbling things a person can do. One area that has particular relevance to physics is the concept of anthropic reasoning. We base our decisions on prior knowledge that we possess. But it is not always obvious which prior knowledge is relevant to a given problem. There may be some cases where the mere knowledge that you exist — in this time, as yourself – might conceivably tell you something useful.

The anthropic argument in cosmology and physics is the proposal that some observed facts about the universe can be explained simply by the fact that we exist. For example, we might wonder why the cosmological constant is so small. In 1987, Steven Weinberg argued that if it were any bigger, it would not have been possible for life to evolve in the universe —  hence, the mere fact that we exist implies that the value of the constant is below a certain limit. However, one has to be extremely careful about invoking such principles, as we will see.

This blog post is likely to be the first among many, in which I meditate on the subtleties of probability. Today, I’d like to look at an old chestnut that goes by many names, but often appears in the form of the `Adam and Eve’ paradox.

(Kunsthistoriches Wien)
Spranger – Adam and Eve

Adam finds himself to be the first human being. While he is waiting around for Eve to turn up, he is naturally very bored. He fishes around in his pocket for a coin. Just for a laugh, he decides that if the coin comes up heads, he will refuse to procreate with Eve, thereby dooming the rest of the human race to non-existence (Adam has a sick sense of humour). However, if the coin comes up tails, he will conceive with Eve as planned and start the chain of events leading to the rest of humanity.

Now Adam reasons as follows: `Either the future holds a large number of my future progeny, or it holds nobody else besides myself and Eve. If indeed it holds many humans, then it is vastly more likely that I should have been born as one of them, instead of finding myself rather co-incidentally in the body of the first human. On the other hand, if there are only ever going to be two people, then it is quite reasonable that I should find myself to be the first one of them. Therefore, given that I already find myself in the body of the first human being, the coin is overwhelmingly likely to come up heads when I flip it.’ Is Adam’s reasoning correct? What is probability of the coin coming up heads?

As with many problems of a similar ilk, this one creates confusion by leaving out certain crucial details that are needed in order to calculate the probability. Because of the sneaky phrasing of the problem, however, people often don’t notice that anything is missing – they bring along their own assumptions about what these details ought to be, and are then surprised when someone with different assumptions ends up with a different probability, using just as good a logical argument.

Any well-posed problem has an unambiguous answer. For example, suppose I tell you that there is a bag of 35 marbles, 15 of which are red and the rest blue. This information is now sufficient to state the probability that a marble taken from the bag is red. But suppose I told you the same problem, without specifying the total number of marbles in the bag. So you know that 15 are red, but there could be any number of additional blue marbles. In order to figure out the probability of getting a red marble, you first have to guess how many blue marbles there are, and in this case (assuming the bag can be infinitely large) a guess of 20 is as good as a guess of 20000, but the probability of drawing a red marble is quite different in each case. Basically, two different rational people might come up with completely different answers to the question because they made different guesses, but neither would be any more or less correct than the other person: without additional information, the answer is ambiguous.

In the case of Adam’s coin, the answer depends on things like: how do souls get assigned to bodies? Do you start with one soul for every human who will ever live and then distribute them randomly? If so, then doesn’t this imply that certain facts about the future are pre-determined, such as Adam’s decision whether or not to procreate? We will now see how it is possible to choose two different contexts such that in one case, Adam is correct, and in the other case he is wrong. But just to avoid questions of theological preference, we will rephrase the problem in terms of a more real-world scenario: actors auditioning for a play.

Imagine a large number of actors auditioning for the parts in the Play of Life. Their roles have not yet been assigned. The problem is that the director has not yet decided which version of the play he wishes to run. In one version, he only needs two actors, while in the other version there is a role for every applicant.

In the first version of the play, the lead actor flips a coin and it comes up heads (the coin is a specially designed stage-prop that is weighted to always come up heads). The lead actress then joins the lead actor onstage, and no more characters are required. In the second version of the play, the coin is rigged to come up tails, and immediately afterwards a whole ensemble of characters comes onto the scene, one for every available actor.

The director wishes to make his decision without potentially angering the vast number of actors who might not get a part. Therefore he decides to use an unconventional (and probably illegal) method of auditioning. First, he puts all of the prospective actors to sleep; then he decides by whatever means he pleases which version of the play to run. If it is the first version, he randomly assigns the roles of the two lead characters and has them dressed up in the appropriate costumes. As for all the other actors who didn’t get a part, he has them loaded into taxis and sent home with an apologetic letter. If he decides on the second version of the play, then he assigns all of the roles randomly and has the actors dressed up in the costumes of their characters, ready to go onstage when they wake up.

Now imagine that you are one of the actors, and you are fully aware of the director’s plan, but you do not know which version of the play he is going to run. After being put to sleep, you wake up some time later dressed in the clothing of the lead role, Adam. You stumble on stage for the opening act, involving you flipping a coin. Of course, you know coin is rigged to either land heads or tails depending on which version of the play the director has chosen to run. Now you can ask yourself what the probability is that the coin will land heads, given that you have been assigned the role of Adam. In this case, hopefully you can convince yourself with a bit of thought that your being chosen as Adam does not give you any information about the director’s choice. So guessing that the coin will come up heads is equally justified as guessing that it will come up tails.

Let us now imagine a slight variation in the process. Suppose that, just before putting everyone to sleep, the director takes you aside and confides in you that he thinks you would make an excellent Adam. He likes you so much, in fact, that he has specially pre-assigned you the role of Adam in the case that he runs the two-person version of the play. However, he feels that in the many-character version of the play it would be too unfair not to give one of the other actors a chance at the lead, so in that case he intends to cast the role randomly as usual.

Given this extra information, you should now be much less surprised at waking up to find yourself in Adam’s costume. Indeed, your lack of surprise is due to the fact that your waking up in this role is a strong indication that the director went with his first choice – to run the two-person version of the play. You can therefore predict with confidence that your coin is rigged to land heads, and that the other actors are most probably safely on their way home with apologetic notes in their jacket pockets.

What is the moral of this story? Be suspicious of any hypothetical scenario whose answer depends on mysterious unstated assumptions about how souls are assigned to bodies, whether the universe is deterministic, etc. Different choices of the process by which you find yourself in one situation or another will affect the extent to which your own existence informs your assignation of probabilities. Specifying these details means asking the question: what process determines the state of existence in which I find myself? If you want to reason about counterfactual scenarios in which you might have been someone else, or not existed at all, then you must first specify a clear model of how such states of existence come about. Without that information, you cannot reliably invoke your own existence as an aid to calculating probabilities.