Saturday, November 03, 2012

Quantum casino: less than zero chance

Guest blog by Johannes Koelman, the Hammock Physicist

Human thought has led to a variety of remarkable and profound insights. Many of these insights are well established and have been embraced by a significant portion of the global population.

The earth being round, the atomistic nature of matter, our unremarkable place in the universe, and us being a product of evolution, all being examples of such insights. Other insights, although unanimously embraced by experts, have a long way to go for a larger population to accept them. More than for any other subject, this holds for quantum physics. No other product of human thought is as profoundly mysterious as quantum theory.

Unfortunately, the subject is not easy to digest and, to make things worse, it is often misrepresented in the pop-science literature. Those eager to understand the quantum are often fed with confusing slogans and misleading analogies.

A year ago I dedicated two blogposts (here and here) to the mysteries of the quantum. Judging from the reactions, these posts must have been useful to at least a few of the readers. The simple thought experiment presented (dealing with Albert's socks) allows the reader to explore the weird world of quantum physics, an experience that likely will challenge the reader's view on reality. A view that - for all of us - is heavily biased by sensory perceptions limited to classical (non-quantum) physics.

Yet, I feel one question remained insufficiently discussed in my earlier posts. And that is the question how physicists deal with quantum reality. Let's see if I can fill this gap by further exploring Albert's quantum socks.


Albert's chest of quantum drawers, although most spooky in its behaviors, appears rather unremarkable and featureless from the outside. Three rows, each consisting of three drawers, make up this piece of furniture. Each night, when Albert is sound asleep, his housekeeper fills this cabinet with fresh socks. The next morning, whenever Albert pulls open a drawer, he is presented either with a single sock or an empty drawer. Unfortunately for Albert, he can not open just any number of drawers. Each morning, his search for fresh socks is limited to the opening of three drawers: any three drawers forming a horizontal row or any three drawers forming a vertical column.

Initially, out of habit, each morning Albert opens a horizontal row of drawers. He does this without giving much thought to which of the three rows to open. After a few days, he starts noticing a pattern. Whichever row he opens, either all three drawers are empty, or two drawers each contain a sock leaving one empty drawer. In other words, a row of drawers always contains a total even number of socks. Many days pass by, and Albert never encounters a row not containing an even number. That makes sense: each row containing an even number of socks tells Albert that each morning the chest must be filled with an even number of socks. An observation comfortably compatible with the notion that socks come in pairs.

One morning, Albert decides to deviate from his fixed ritual, and he opens a vertical column of drawers: the three leftmost drawers. Interestingly, this time he observes an odd number of socks distributed over the three drawers opened. The next day, he opens the same leftmost column of drawers, and again observed an odd number of socks.

Albert gets curious about the other columns. What number of socks will they reveal? The next morning he opens the middle column. Again an odd number of socks. The next day he once more checks the middle column. Once again he observes an odd number of socks.

Albert is a clever guy, and he now realizes he can predict with certainty that the rightmost column of drawers must behave differently from the two columns already inspected. The rightmost drawers for sure must contain a total number of socks that is even. This is obvious as an odd number of socks in the rightmost drawers, added to the number of socks in the middle column (observed to be odd) and the leftmost column (also observed to be odd) would result in an odd total number of socks in the chest. A contradiction, as he had already observed, based on opening horizontal rows, that the total number of socks in the chest is always even.

The next morning he eagerly opens the three rightmost drawers. To his astonishment he observes an odd number of socks.

The next few mornings he randomly checks the various columns. Each attempt reveals an odd number of socks. Albert realizes something must have gone wrong. Maybe his housekeeper coincidently changed from filling the chest with an even number of socks to filling it with an odd number of socks, just at the time he switched over from opening horizontal rows to vertical columns? The next morning Albert again opens a horizontal row. An even number of socks stares him in the face. He starts randomly switching between horizontal rows and vertical columns. Horizontal rows always deliver even numbers, vertical columns odd numbers.

This drives Albert crazy. The results he is obtaining are logically impossible. "At any given morning if I would open three rows", Albert reasons, "I would end up with an even number of socks. However, would I open three columns, I would end up with an odd number of socks. Yet in both cases I would have opened the same nine drawers. How can this be?"


"Nice story" you might react, "but nothing more than a fantasy. Obviously chests of drawers like these logically can't exist in our world".

Well, as a matter of fact, they do. I can put that even more strongly: all evidence points at our world consisting solely of devices like Albert's drawers. It just happens to be that these drawers tend to be very, very small and that we generally observe the aggregate behavior of many drawers without being able to distinguish critical details such as the distinction between even and odd counts.

Many physicists deal with quantum reality day in, day out. How have they come to terms with a physical reality that presents us with devices like Albert's chest of drawers? Physicists don't seem to lose any sleep over the many counter-intuitive notions and apparent paradoxes surrounding quantum theory. Are they oblivious to the utterly strange world view that the quantum represents?

Not at all. The point is that the vast majority of physicists simply have stopped worrying and have embraced a practical approach described by the catch-phrase:
"Shut up and calculate!"
This phrase was coined by Cornell physicist and science communicator David Mermin, in an attempt to mock the widely-accepted Copenhagen interpretation of quantum physics. Mermin was not at ease with the Copenhagen position put forward by Niels Bohr in the famous Bohr-Einstein debates. Ironically, two things happened following the publication of Mermin's phrase. Firstly, Mermin's own ideas on how to interpret quantum physics started to converge towards the Copenhagen interpretation. Secondly, the "shut up and calculate" phrase achieved the status of a popular instrumentalist approach to quantum physics equated with eschewing all interpretation.

Let me try to offer you the "shut up and calculate" approach as a choice for coming to terms with the quantum. To do that, I need to explain how to calculate sock counts for Albert's drawers.


The mathematical machinery behind quantum devices like Albert's chest is compelling and carries a great beauty. It is based on a generalized probabilistics that adds likelihoods not as a linear sum, but rather as a Pythagorean sum. I would love to present this quantum math to you. But alas, the math is too involved to explain in a blog like this.

Fortunately, the complex quantum math built on Pythagorean sums can be mapped on the much simpler classical math based on straightforward linear sums. Physicists apply such mappings all the time when studying what is known as "the classical limit" of quantum systems: the behavior of large quantum systems that can be accurately represented with classical (non-quantum) laws of physics.

Eugene Wigner, one of the quantum pioneers, discovered in the early 1930's that something weird happens when approximating quantum physics with classical probabilities. He discovered that one can describe quantum systems with probabilities that add linearly, but these probabilities are no longer guaranteed to be non-negative.

Paul Dirac, Wigner's brother in law, later wrote a paper that discusses the use of concepts like negative energies and negative probabilities in quantum physics:
"Negative energies and probabilities should not be considered as nonsense. They are well-defined concepts mathematically, like a negative of money."
Years later the idea of negative probabilities received increased attention when Richard Feynman started to popularize the idea:
Trying to think of negative probabilities gave me a cultural shock at first, but when I finally got easy with the concept I wrote myself a note so I wouldn’t forget my thoughts..."
In describing Albert's chest of drawers, how far do we get with negative probabilities? Before you read my negative probabilities description for Albert's chest, I urge you to try it yourself first. It's a straightforward and instructive exercise.

Ok, here is a probabilistic description of Albert's drawers. A total of 10 configurations are relevant:
  1. Nine configurations consist of two drawers in a single row being filled, and all other drawers being empty. These nine configurations each carry a 1/6 probability.
  2. One additional configuration features an empty chest of drawers. This configuration carries a minus 1/2 probability.

It is easy to see that all probabilities add up to unity, as they should. The negative probability for the 'all empty' state is a strange beast, but a beast that doesn't show up in the end result when calculating observable probabilities.

For instance, let's calculate the probabilities for observing the various numbers of socks in the bottom row of Albert's chest. In the above table you can read off that we have seven realizations with an empty bottom row, six with +1/6 probability, and one with –1/2 probability. The total probability for an empty bottom row is therefore +1/2. Furthermore, we have three realizations with a bottom row filled with two socks, each with a probability of +1/6. The total probability for finding two socks in the bottom row is therefore +1/2. So we have two equal likelihoods of finding zero or two socks, each with a positive (+1/2) value, and we are guaranteed to retrieve an even number of socks from the bottom row.

Repeating the same calculation for a column leads to the result that a column renders one sock with unit probability, hence a guarantee for an outcome of an odd number of socks.

What happened in both these cases is that the negative probability gets cancelled by positive probabilities describing the same outcome for the row or column under consideration. As a result, rows lead to even numbers and columns to odd numbers, and never does a negative probability turn up its ugly head.

This obviously doesn't hold when determining the number of socks in the entire chest of drawers. A negative probability does show up prominently, and it does not get cancelled as there are no positive probabilities describing the same configuration. However, we don't need to worry about this, as the configuration with negative probability does not correspond to a possible observation. Only three drawers (one row or one column) can be opened, not the whole chest.

What this means is that one can extract only a limited amount of information out of Albert's chest. Getting more information out of it (opening more drawers) is impossible as it would lead to negative probabilities to become evident. This is a manifestation of Heisenberg's uncertainty principle for Albert's chest.

Heisenberg uncertainty, destructive interference, spooky action at a distance, violation of Bell's inequalities, all of quantum physics can be mapped on negative probability models. So why don't physicists use such negative probability models? The answer is simple: negative probability models are rather clumsy representations of what is better described using "Pythagorean probabilities".

But as a model to get some understanding of the fundamentals of quantum physics, negative probability models do carry a value largely unexplored by pop-science writers.

All of this places some of Einstein's quotes in a different light.

God does play dice. And malicious he is. Or how would you characterize anyone who throws dice loaded to render negative chances, and who manages to keep this so well-hidden from us?

Disclaimer by LM: The views and opinions expressed in this article are those of the author but they do reflect the position of TRF. By the way, if you want the relevant operators etc., the closest discussion on TRF is this treatment of the GHZM experiment. For those who want the realization of this particular "drawer", it's caller the Mermin-Peres magic square (thanks, Johannes) and it acts on just two qubits (two spins). The \(3\times 3\) operators are\[

\,1\otimes \sigma_z& \sigma_z\otimes 1& \sigma_z\otimes\sigma_z\\
\sigma_x\otimes 1\,& \,1\otimes \sigma_x& \sigma_x\otimes \sigma_x\\
-\sigma_x\otimes \sigma_z&- \sigma_z\otimes \sigma_x &\sigma_y\otimes \sigma_y\\

\] Each pair of operators in the same row commutes; each pair in the same column commutes. Check it. The third entry in each row is a product of the previous two so the "even number of socks in each row" is guaranteed. I added minus signs on the third row so that the product in each column is \(-1\), too. Note that for the individual drawer measurements, \((+1)=\)"yes sock" and \((-1)=\)"no sock". So the correlations are guaranteed regardless of the state vector.

It's left as an exercise to the reader to determine the state vector for which the probabilities are as indicated by Johannes; and to find out how the Wigner distribution which has 9 times +1/6 and 1 times –1/2 should be constructed.

This magic square has been used as a tool in quantum pseudo-telepathy and exaggerations of this sexy term are guaranteed. ;-)


  1. nice to see a superposition of you two in the same (blog)spot on the reference frame, where "spot on the reference frame" has to be understood in the cooperian sense, as in:

    "That is my spot, in an every changing world, it is a single point of consistency. If my life were expressed as a function on a four dimensional Cartesian coordinate system, that spot, from the moment I first sat on it, would be 0-0-0-0."

  2. Many thanks to Dr. Koelman for this instructive and entertaining guest blog! It is this sort of accessible treatment that I had in mind when some time ago I asked that an introduction to "q

  3. I willingly and expectantly am prepared to travel to Wherever (and forge by birth certificate) to start over from grade one IF the speculative 'quantum theory for toddlers curriculum' become realized 'there'! ;-)

  4. Dear Dr. Koelman,

    thanks for the nice and fun article. I didn't know that negative probabilities can work as model for quantum mechnical states.

    However it seems there is no formalism using negative probabilites to calculate the time evolution of a physical system or at least a way to decide which states including negative probabilities are allowed quantum mechanical states. If this is true I would regard this as an educational toy model and not something deep. I would rather say the negative probabilities come from the fact that one insists that there is a physical realtive before measurement.

  5. The no-realism of quantum mechanics can be seen by considering only one particle. The 2 operators (Pauli Matrices) Sx, Sz have a square equal to identity. However the product (Sx Sz Sx Sz) equals minus Identity (and this is true whatever the state is). This is obviously different from a realistic theory, where a analog of this product must be equal to Identity. This deep reason is that there is a non-trivial structure describing states (here, concerning spin, a 2 dimensional real or complex structure), and this implies a non-trivial structure for operators too, and so generally no commutation between operators, and so no realism, because operators change the physical state of the particle. However, there is a difference between spin and position/impulsion. With position/impulsion, you always find non-trivial structures, but with spin, it depends on the dimension of the space-time : With 4 space-time dimensions, there are 2 possible polarizations for the photon, so it is a non-trivial structure. But this will not be case wih 3 space-time dimensions : in this case, there is only one polarization for the photon, and so it becomes a trivial structure, with trivial operators, etc..

  6. Hi Trimok.
    with just one particle you could always argue that there is an objective reality but our measurements disturb it, The strong arguments against reality come from the space like separated correlated states because you need to give up special relativity to save reality. This part is in fact also missing in the current blog post. With a single chest only somebody (nature) could watch which drawers you open and add or remove socks to the other drawers.

  7. 30, 2013, 9:01:00 AM Reading this post reminds me of my old room mate! He always kept talking about this. I will forward this article to him. Pretty sure he will have a good read. Thanks for sharing!