A promising proof of the so-called Kadison-Singer conjecture was announced yesterday. Mathematicians are excited, because this conjecture is equivalent to a remarkable number of open problems in other fields. And mathematicians just love to make connections between wildly disparate fields.
Here is another reason to be excited: the Kadison-Singer conjecture has important consequences for the foundations of physics! Settling this conjecture shows an important way in which our experiments are enough to provide a complete description of a quantum system. Here’s why.
Quantum Mechanics: The Simple Picture
The Kadison-Singer question comes out of thinking about how to describe a quantum system. Here is a simplified picture of how that works. When we describe matter in quantum mechanics, there are two main things we are concerned with:
- Observables: Each observable represents a property whose values we can measure, and
- States: Each state is a “probability measure,” which determines the probability of measuring each value of each observable, given our experimental setup.
This is a lot like the situation with a coin: a property that we can measure is the side that is facing up; it can have value “heads” or “tails.” When the coin is in a state of flipping in the air under normal circumstances, the probability of measuring each value is 1/2. So, the state “flipping normally” determines a probability measure. And that measure assigns probability 1/2 to each value that the observable “which side of the coin is facing up” can take.
Quantum mechanics is a lot like that, except we’re measuring the fundamental properties of matter. Our observable might be the spin of an electron, which can take the values “up” or “down.” And our starting state might be an electron flying out of an oven and through a magnetic field. But the principle is the same, and in this case, the state happens to assign probability 1/2 to each value.
Now, here is a crucial distinction for understanding the Kadison-Singer conjecture. One thing that is special about quantum observables is that there are some that can be measured simultaneously, and some others that can’t. Two observables that can be simultaneously measured are called “compatible,” and are otherwise called “incompatible.”
For example, you can simultaneously measure position and spin. But you cannot simultaneously measure position and momentum; the latter is known as Heisenberg’s uncertainty principle. Position and spin are compatible observables, but position and momentum are not. This is a simple fact of quantum life. Indeed, according to many, it is the crucial distinction between the quantum and the classical descriptions of the world.
Dirac suggested a simple formula for taking experiments and using them to describe a quantum system. The following is a slight improvisation on what Dirac actually wrote; the historically-precise may prefer to think of it as a “Dirac-inspired” picture.
We want to know the properties of a quantum system, meaning the probabilities (the states) that get assigned to all the measurable quantities (the observables). How can we use experiments to learn about this? Here is one way.
- Write down what you can measure simultaneously. Collect together a set of compatible observables, and keep adding to it until it is as large as it can be. For example, you could take a finite set of observables (say, position and spin ), and then just add in all the functions of those observables (like , , etc.). The resulting set is called a “maximal abelian algebra.” This set represents the most we can learn from a single experiment.
- Determine the basic probabilities (i.e., pure states). Arrange your lab in some way, and then repeat an experiment that determines the values of those observables. Then do it again. After enough repetitions, you’ll be able to choose a probability distribution for the experiment. A structurally “basic” probability distribution is called a pure state.
- Generalize this information to the entire system. We now want to extend this probability distribution to all the other observables, those that are not compatible with our set. That is, we want to choose a probability distribution that applies to all observables, and which is the same as our probability distribution when restricted to our set of compatible observables. And, if we may ask so much, we’d like this extension to be unique, so that we can be sure we’ve chosen the right one.
The last step is what’s interesting. Dirac’s procedure requires us to extend what we learned about the basic quantum probabilities of compatible observables to all the other observables as well, and to ensure that there is only one way to do so. Is this always possible? That is the Kadison-Singer conjecture.
The Kadison-Singer Conjecture
The Kadison-Singer conjecture says we have at least one way to completely characterize a quantum system on the basis of what we can learn from experiments. We can take what we know about the quantum probabilities of simultaneously measurable quantities, and uniquely extend this to all the other measurable quantities as well.
In particular, if we have determined a “basic” probability distribution on a set of compatible observables that is as large as it can be, then this probability distribution extends uniquely to all the other observables.
Here is the precise statement of this claim.
Kadison-Singer Conjecture. Let be a discrete maximal abelian subalgebra of , the algebra of bounded linear operators on a separable Hilbert space. Let be a pure state on that subalgebra. Then there exists a pure extension of to all of , and that extension is unique.
Proof of this statement provides a very nice assurance, that our experiments really are enough to describe quantum systems as we understand them. (Note for those familiar with quantum theory: I have included some notes on the terminology of this statement in the last section of this post.)
Unfortunately, the result is not without caveats, and an important one is the “discreteness” caveat. Although a great many physical descriptions satisfy discreteness, there are also some continuous ones (some of which are relevant to quantum field theory) that are not discrete, and for which the conjecture does not apply. Indeed, Kadison and Singer showed in the second theorem of their original paper that the conjecture fails for such continuous geometries!
History of the conjecture
Kadison and Singer wrote a number of wonderful papers on the mathematical foundations of quantum theory in the 1950’s. The conjecture that is named after them comes from the “Related Questions” section of their 1959 paper on extensions of pure states. On the question of uniqueness, they say that “we incline to the view that such an extension is non-unique.” Kadison and Singer seem to have thought that their conjecture is probably false! You might have expected this too, if you had just proven that uniqueness fails in the continuous case. This makes it all the more surprising that the conjecture turns out to be true.
In settling the Kadison-Singer conjecture, a number of other equivalent propositions are have also been settled. Those familiar with the mathematics of quantum theory may enjoy reading this paper by Casazza and Tremain on some equivalent formulations, which is very accessible from that perspective.
Some mathematical notes
For the statement of the Kadison-Singer conjecture above, here are a few definitions.
We say that an algebra is discrete if it contains “atoms,” i.e. minimal projections such that only if . This is the familiar situation for most. For example, a 1-dimensional projections onto a ray in a Hilbert space is a minimal projection for .
A state is a probability distribution that is -additive on families of mutually orthogonal projections. (Often, one takes a state to be just a positive normalized functional, and defines a state to be normal if it is -additive.) One of the cornerstones of quantum theory, called Gleason’s Theorem, says that a (normal) state can always be implemented by a density matrix, in that there always exists a density matrix such that for all .
A state is pure if it cannot be expressed as a non-trivial convex combination of other states, i.e., for only if . In this sense, pure states are “basic.” The pure states on can always be implemented by Hilbert space vectors, in that there exists some such that for all .
11 March 2014. Danny points out an interesting example in the comments, which clarifies the importance of requiring that all states be “normal” in the last sentence of this post.
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