Buhjillions

Jeff Atwood, over at Coding Horror posed an interesting little puzzle about probability:

Let’s say, hypothetically speaking, you met someone who told you they had two children, and one of them is a girl. What are the odds that person has a boy and a girl?

To put it in more precise language so we can concentrate on probability and not nuance of word choice, the person means that at least one child is a girl.  At first I was tempted to say what a lot of people came up with in the comments: 50%.  If one is a girl, my thought process goes, then we’re just looking at the probabilities for the other child, and surely those are not affected by the child we know about.

This is, of course, wrong.  In that argument we fail to take into account that having two children are distinguishable events, and we don’t know which child they were talking about when they said one was a girl.  When I actually wrote it down, then the solution became more clear.  Having two children gives 4 possibilities in terms of their gender (B for boy, G for girl):

BB, BG, GB, and GG

In learning that at least one is a girl, we can eliminate BB.  We cannot eliminate BG or GB because we’re not told which child was being referred to when we were told one is a girl.  Of the 3 remaining, 2 have one boy and one girl, so the solution is 2/3 or about 67%.

But wait!  Why should order matter?  As expressed by one commenter:

All the children learned probability theory and forgot how to think normally! Why would you care if the first one is a boy or a girl..they didn’t tell that their first child was a Girl, now did they? So, you have three choices: [BB, GB, GG]

To a certain extent, one is entirely justified in formulating the solution in terms that don’t include the ordering.  It wasn’t asked for in the solution or mentioned it in the problem.  However, if you formulate the problem in this way you are forced to abandon an implicit assumption we made in the previous reasoning: that all possibilities are equally likely.  If we leave out order, we can simplify our notation and just count the number of boys, and know that the rest are girls (leaving aside the relatively rare occurrence of gender ambiguity).  So our possible cases are [2, 1, 0].  However, the respective probabilities for these cases are [25%, 50%, 25%].  That is to say, having one boy and one girl is twice as likely as having two girls.  With this in mind, it’s easy to see that the solution should be 2/3.

But why are the probabilities equal when you include order, and not equal when you don’t?  Maybe you don’t even believe me.  The answer has a very deep connection to physics, and so my advice to any doubter is to try it out with a pair of coins!  Get two coins, flip them, and record the number of heads.  Repeat this 20 or 30 times and you’ll handily see that exactly 1 head comes up roughly twice as often as either 2 heads or 0 heads.  It doesn’t even matter whether you flip them at the same time or whether the coins are easily distinguishable!  Even seemingly identical coins are distinguishable in principle.  No two coins are exactly alike at the molecular level, and even if they were, it would be possible to track them individually through the air during a flip.  By only recording the number of heads we are throwing out some information which is, in principle, available to us.

Any time we don’t include information which, in principle, exists, then we don’t get equal probabilities.  However, we can still work out the probabilities of our incomplete description.  In thermodynamics, our incomplete description (in this case, the number of heads) is called the macrostate, and a complete description that uses all the information available in principle is called a microstate.  To find the probabilities of the macrostates, we have to weight them by the number of different microstates that give that macrostate.  In the case of exactly 1 head, this has two microstates (HT and TH).  The other macrostates each have only one microstate, thus exactly 1 head is twice as likely as either 2 or 0.

The number of different microstates that correspond to a particular macrostate is a measure of our lack of information.  When we get a macrostate of 2 heads, we know exactly which microstate we’re in—we have complete knowledge.  But imagine that we had 100 coins instead of 2.  There is only one microstate that has 0 heads, but there are 100,891,344,545,564,193,334,812,497,256 different microstates for 50 heads.  50 heads is astronomically more likely than 0 heads.  But just knowing that there are 50 heads leaves us without much knowledge of the microstate: there are over 100 thousand trillion trillion of them to choose from!  The measure of this is called entropy (technically, the logarithm of the number of microstates).  In our boy-girl example, having one boy and one girl has a higher entropy because we don’t know the order.  Entropy is sometimes called a measure of disorder.

In thermodynamics the macrostate of a system is given by things like overall temperature, volume and pressure, whereas microstates would have to be given in terms of the positions and velocities of each molecule.  That information is present, in principle (at least up to a quantum-mechanical limit), so it has a real effect on the probability.  Just like in coin-tossing the probabilities of the macrostates are weighted by the number of microstates that correspond.  The more likely macrostates must have higher entropy.  This is the origin of the famous 2nd Law of Thermodynamics.  Since macrostates of high entropy are so much more likely, random processes always end up there; the more elements in the system, the more this probability becomes like a simple fact.

Why won’t physicists leave these poor cats alone?! Is it dead? Alive? Both?

As if the ideas about where quantum mechanics melds into classical physics weren’t already confusing enough, new experimental work seems to confirm the theory that the transition from multiple possibilities to a single observed outcome is far from instantaneous and is, in some cases, reversible. In my previous post I described the old standard quantum mechanics view of the instantaneous collapse of possibilities at the moment of observation—but the modern view is that this is far from the whole story. It turns out that these observations, or measurements in physics parlance, can vary in “strength” depending on how much information they give us about the state of the system.

To illustrate this point, I’ll appeal to a famous thought experiment put forth by a skeptical Erwin Schrödinger, now known as Schrödinger’s Cat: imagine an experiment where an ordinary housecat is placed in a large steel box, along with what Schrödinger called a “diabolical” device consisting of a single atom of a radioactive substance, a detector, and a phial of poison gas. When the atom decays radioactively the detector goes off and smashes the phial, killing the cat. The box is sealed, and the experimenters/cat detractors wait until the probability that the atom has decayed is one half.

N.B. depending on the species of atom, they could be waiting anywhere from nanoseconds to several billion years.

They then ask the question, how do we describe the state of the cat? Classically, we’d say that the cat is either alive or dead, with 50% chance each. Quantum mechanics says that since we haven’t opened the box and observed the cat yet, it is in an equal “superposition” of alive and dead, which is to say, some weird quantum state of both at the same time. Since when opening the box, the state “collapses” to either alive or dead, you might ask what possible difference could it make how you describe the cat?, half the time it will be alive, half the time dead.

This is where things get tricksey. Imagine that, instead of opening the box, I put my ear to the side and listen. Cats meow occasionally, when they’re alive, and don’t when they’re dead. So if I hear a meow, then I can safely assume that the cat is alive. But, what if I hear nothing? I don’t know for sure that the cat is dead, since it might have simply chosen not to meow while I was listening. However, the fact that I didn’t hear anything still gives me some information about what’s happening in the box. In quantum mechanics, this is called a weak measurement. I now expect to be more likely to find a dead cat in the box by virtue of my measurement of hearing nothing. If I had a well characterized cat, which meows randomly at a certain average rate, then I could calculate new probabilities of the cat being alive or dead.

This is, in effect, what experimenters at UC Santa Barbara have done. Their “cat” is a loop of superconductor whose electrical properties can be in one of two different states, labeled 0 and 1 for this discussion. They prepare the loop in a superposition of 0 and 1 using a pulse of radio waves. They then perform a weak measurement of the system, where if the state is 1, they have some probability to detect it, like the cat meowing if it is alive. If the state is 0, then they will detect nothing. They then perform a full “strong” measurement, which detects either 0 or 1 (equivalent to opening the box). For the runs where they detected nothing during the weak measurement, they were more likely to get a 0 during the final measurement.

So far, so good, but this is where it gets interesting. They then added a second radio pulse to their procedure after the weak measurement. The effect of this second pulse is to swap states 0 and 1, so if the cat was dead, it would get reincarnated, but if alive, it would be killed (and here the prospects for doing this with actual cats probably end). They then performed the same weak measurement on the loop after the swap. For the runs where both weak measurements detected nothing, the final state was exactly the original, equal superposition of 0 and 1!

The weak measurements only partially collapsed the quantum state, leaving it in a state which was not an equal superposition, but which could be reversibly “uncollapsed” by performing the same weak measurement on the other state. We used to think, according to the “standard” interpretation of quantum mechanics that the measurement induced collapse was both instantaneous and irreversible, but new experiments in this realm are forcing us to reconsider!

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The original paper, published to the arXiv. Read this if you’re a physicist, or are hardcore.

Nature News article about this research. This is behind a paywall. Oxford people should read it fine if you’re on campus or the VPN. Others, ask your respective institutions/employers. Read this if you’re a scientist.

Science Daily article about this research. Read this if you’re interested, but don’t have a science background.

While Einstein is best remembered by the general public for his theory of relativity, this was largely complete by 1916. Quantum mechanics, by that time, was only just beginning to be developed into a full, general theory. Einstein made important contributions to quantum mechanics early on, in 1905 using the hypothesis that light is quantized to explain the photoelectric effect (Nobel prize!), and spent much of the later part of his career thinking and writing about quantum mechanics. He was a vocal critic of the way quantum mechanics describes the universe—notably the fact that quantum mechanics is inherently non-deterministic in the sense that it can, in general, only predict the probability that a certain outcome will happen, regardless of how much is known about the system and its initial state. This is the origin of the now famous (and often paraphrased) remark

I, at any rate, am convinced that He [God] does not throw dice.

Einstein argued that quantum mechanics must be an incomplete description of reality, and that there must be additional information, termed “hidden variables,” which determine the course of events, and which are as of yet, inaccessible to our scrutiny. It’s hard to argue with this view. How can one make the case that in calculating a quantum mechanical probability we then know all that we ever can know about the future of our system? How can we say that there isn’t just a little bit more beneath the surface which is not yet within reach of our instruments?

Quantum mechanics introduces an unsettling moment of truth: a nanoscale Seldon Crisis in which the system goes from having many possible outcomes to just one. This moment of actually throwing the proverbial dice occurs (according to the standard interpretation of quantum mechanics during Einstein’s lifetime) at the instant in which the system is observed. At that instant the system collapses from many possibilities, with one being chosen at random to be the outcome actually observed. The illusion of determinism comes from things which are more or less constantly observed. They are simply not left unperturbed long enough for the hallmarks of quantum mechanical behavior to disturb that illusion.

With this observation effect, it seems as if the results of measurements do not exist independently—they don’t come into being, they are not “real” until the results are actually observed. This is a hard thing to get one’s mind around. Quantum mechanics seems to say that there is no independent, objective reality without observation (there is a great deal of philosophical work these days to try to resolve this without invoking Einstein’s “hidden variables,” but most of it is beyond my current understanding).

This is exactly the old “if a tree falls in the forest” question, but with an actual physical theory at the center of the (ostensibly) philosophical debate. Einstein said he believed that the moon continued to exist even when he wasn’t looking at it. If quantum mechanics is God throwing dice, then hidden variables are a bit like playing cards. You don’t know what card will come up when you say, “hit me,” but you can be reasonably assured that the outcome does actually exist before the card is turned over.

Photo by Steven Nickells, CC Licensed.

On the first day of undergraduate physics at Olin College, Mark Somerville, my professor drew the following diagram on the board:

In this course, he said, we’d be dealing with “big, slow stuff.”  The dividing line between slow and fast is pretty well defined, with fast being any sizable fraction of the speed of light (what fraction depends on how much precision you require).  However, the dividing line between small and big is a little muddier.  Traditionally, quantum mechanics is thought to be important mostly for sizes on the order of a single atom (less than 1 billionth of a meter).  However, the microscopic quantum effects play a huge role in large systems, such as the sun.  Without quantum mechanics, we could not explain nuclear fusion, or the series of dark bands (called absorption lines) observed when you split sunlight very carefully with a prism.  Or, more prosaically, the orange glow given off by sodium-vapor street lamps.

Still, the large scale motion of convection currents, solar flares and sunspots can be understood using classical physics.  The same goes for pretty much everything moving on scales larger than atomic dimensions—most of the time.  The tiniest speck of dust which can be seen under a microscope still behaves according to the laws of motion set out by Newton over 300 years ago.  In practice, the dividing line between “small” and “big” was somewhere on the order of 10 to 100 nanometers (billionths of a meter).

Or so we thought.

What the above diagram doesn’t do a good job demonstrating is that the theories are not really that separate.  Even though some of its early founders were skeptical of taking the puzzling and bizzare consequences of quantum mechanics too seriously, the modern view seems to be that quantum mechanics is the theory which accurately describes the world—on all length scales!  Quantum mechanics makes the correct predictions about the flight of a bumblebee or a diesel engine, at least in principle.  But it’s easy to demonstrate that quantum mechanics reduces, with incredible accuracy, to the old classical mechanics in most situations on large scales: physical laws which are much easier to apply.

In fact, quantum mechanics isn’t really a physical theory in the same way that relativity is—it’s more like a framework for creating physical theories.  Within this framework there have been several extremely successful theories, like quantum electrodynamics (the famous QED of Richard Feynman) which describes how light and matter interact to form stable atoms, as well as solids, liquids and gasses which we know and love.  And so, while we can show how QED is simply a more general theory which reproduces the earlier classical theories, we would also like to build a quantum theory of relativity, also known as quantum gravity, which would extent quantum mechanics’ reach into the upper right quadrant of the diagram as well.  Experimentalists in this field are building things like gravity-wave dectectors, and theorists are trying to sort through the tricky mathematics of extra dimensions and sets of space transformations called symmety groups.

Thinking about the world in a quantum mechanical way is not easy—there are plenty of apparent paradoxes to try and wrap your mind around.  I’m sure you’ve heard of at least some of them: the uncertainty principle, Schrödinger’s Cat, the wave-particle duality, quantum interference, and the Multiverse or Many Worlds to name a few.  I don’t, however, imagine that getting used to the old physics was very easy when it was new either.  How do you contemplate a solar system held together without crystalline spheres or string or the grace of God?  Some unseen force moving through the cosmos to arc the planets gracefully around.  How mundane that it’s the same force that makes toast land butter-side down.

David Deutsch ended one of his excellent lectures on quantum mechanics by asking a question, which I will now paraphrase: What would it feel like to live according to the laws of quantum mechanics?  To live with fundamental uncertainty and wave-particle duality in a tiny corner of the one of an uncountable number of parallel universes exploring every possibility?

Why, the same way it feels now.  Quantum mechanics seems strange and spooky, but it’s the same physics that governs the air we breathe and the light we see by.