May 5, 2007

This Week's Finds in Mathematical Physics (Week 251)

John Baez

Last week I mentioned the conference on "Philosophical and Formal Foundations of Modern Physics" in Les Treilles, an estate near Nice. On our last night there, the chef showed us his telescope. We saw the phase of Venus, mountains on the Moon, and - best of all - the rings of Saturn! They were beautiful. I was reminded of Galileo, who had to make do with a much cruder telescope.

Here's an even better view - a photo taken by the Cassini probe on March 1st, from a distance of 1.2 million kilometers:

1) Cassini-Huyghens, Tourniquet shadows,

I learned some fun stuff about the foundations of quantum mechanics at Les Treilles, so I want to mention that before I forget! I'll take a little break from the Tale of Groupoidification... though if you've been following carefully, you may see it lurking beneath the surface.

Lately people have been developing "foils for quantum mechanics": theories of physics that aren't classical, but aren't ordinary quantum theory, either. These theories can lack some of the weird features of quantum theory... or, they may have "supra-quantum" features, like the Popescu-Rohrlich box I mentioned last week.

The idea is not to take these theories seriously as models of our universe - though one can always dream. Instead, it's to explore the logical possibilities, so we can see quantum mechanics and classical mechanics as just two examples from a larger field of options, and better understand what's special about them.

Rob Spekkens is a young guy who used to be at the Perimeter Institute; now he's at DAMTP in Cambridge. At Les Treilles he gave a cool talk about a simple theory that mimics some of features of quantum mechanics:

2) Evidence for the epistemic view of quantum states: a toy theory, Phys. Rev. A 75, 032110 (2007). Also available as quant-ph/0401052.

The idea is to see how far you get using a very simple principle, namely: even when you know as much as you can, there's an equal amount you don't know.

In this setup, the complete description of a physical system involves N bits of information, but you can only know N/2 of them. When you do an experiment to learn more information than that, the system's state changes in a random way, so something you knew become obsolete.

The fraction "1/2" here is chosen for simplicity: it's just a toy theory. But, it leads to some charming mathematical structures that I'd like to understand better.

In this theory, the simplest nontrivial system is one whose state takes two bits to describe - but you can know at most one. Two bits of information is enough to describe four states, say states 1, 2, 3, and 4. But, since you can only know one bit of information, you can't pin down the system's state completely. At most you can halve the possibilities, and know something like "the system is in state 1 or 3". You can also be completely ignorant - meaning you only know "the system is in state 1, 2, 3 or 4".

Since there are 3 ways to chop a 4-element set in half, there are 3 "axes of knowledge", namely

Is the system's state in {1,2} or {3,4}?
Is the system's state in {1,3} or {2,4}?
Is the system's state in {1,4} or {2,3}?

You can only answer one of these questions.

This has a cute resemblance to how you can measure the angular momentum of a spin-1/2 particle along the x, y, or z axis, in each case getting two choices. Spekkens has a nice picture in his paper:

                    |  {2,4}
                    | /
                  / |
             {1,3}  |
This octahedron is a discrete version of the "Bloch ball" describing mixed states of a spin-1/2 particle in honest quantum mechanics. If you don't know about that, I should remind you:

A "pure state" of a spin-1/2 particle is a unit vector in C2, modulo phase. The set of these is just the Riemann sphere!

In a pure state, we know as much as we can know. In a "mixed state", we know less. Mathematically, a mixed state of a spin-1/2 particle is a 2×2 "density matrix" - a self-adjoint matrix with nonnegative eigenvalues and trace 1. These form a 3-dimensional ball, the "Bloch ball", whose boundary is the Riemann sphere.

The x, y, and z coordinates of a point in the Bloch ball are the expected values of the three components of angular momentum for a spin-1/2 particle in the given mixed state. The center of the Bloch ball is the state of complete ignorance.

In honest quantum mechanics, the rotation group SO(3) acts as symmetries of the Bloch ball. In Spekken's toy version, this symmetry group is reduced to the 24 permutations of the set {1,2,3,4}. You can think of these permutations as acting on a tetrahedron whose corners are the 4 states of our system. The 6 corners of the octahedron above are the midpoints of the edges of this tetrahedron!

Since Spekkens' toy system resembles a qubit, he calls it a "toy bit". He goes on to study systems of several toy bits - and the charming combinatorial geometry I just described gets even more interesting. Alas, I don't really understand it well: I feel there must be some mathematically elegant way to describe it all, but I don't know what it is.

Just as you can't duplicate a qubit in honest quantum mechanics - the famous no-cloning theorem - it turns out you can't duplicate a toy bit. However, Bell's theorem on nonlocality and the Kochen-Specker theorem on contextuality don't apply to toy bits. Spekkens also lists other similarities and differences.

All this is fascinating. It would be nice to find the mathematical structure that underlies this toy theory, much as the category of Hilbert spaces underlies honest quantum mechanics.

In my talk at Les Treilles, I explained how the seeming weirdness of quantum mechanics arises from how the category of Hilbert spaces resembles not the category of sets and functions, but a category with "spaces" as objects and "spacetimes" as morphism. This is good, because we're trying to unify quantum mechanics with our best theory of spacetime, namely general relativity. In fact, I think quantum mechanics will make more sense when it's part of a theory of quantum gravity! To see why, try this:

3) John Baez, Quantum quandaries: a category-theoretic perspective, talk at Les Treilles, April 24, 2007,

For more details, see my paper with the same title (see "week247").

This fun paper by Bob Coecke gives another view of categories and quantum mechanics, coming from work on quantum information theory:

4) Bob Coecke, Kindergarten quantum mechanics, available as quant-ph/0510032.

To dig deeper, try these:

5) Samson Abramsky and Bob Coecke, A categorical semantics of quantum protocols, quant-ph/0402130.

6) Peter Selinger, Dagger compact closed categories and completely positive maps, available at

Since the category-theoretic viewpoint sheds new light on the no-cloning theorem, Bell's theorem, quantum teleportation, and the like, maybe we can use it to classify "foils for quantum mechanics". Where would Spekkens' theory fit into this classification? I want to know!

Another mathematically interesting talk was by Howard Barnum, who works at Los Alamos National Laboratory. Barnum works on a general approach to physical theories using convex sets. The idea is that in any reasonable theory, we can form a mixture or "convex linear combination"

px + (1-p)y

of mixed states x and y, by putting the system in state x with probability p and state y with probability 1-p. So, mixed states should form a "convex set".

The Bloch sphere is a great example of such a convex set. Another example is the octahedron in Spekken's theory. Another example is the tetrahedron that describes the mixed states of a classical system with 4 pure states. Spekken's octahedron is a subset of this tetrahedron, reflecting the limitations on knowledge in his setup.

To learn about the convex set approach, try these papers:

7) Howard Barnum, Quantum information processing, operational quantum logic, convexity, and the foundations of physics, available as quant-ph/0304159.

8) Jonathan Barrett, Information processing in generalized probabilistic theories, available as quant-ph/0508211.

9) Howard Barnum, Jonathan Barrett, Matthew Leifer and Alexander Wilce, Cloning and broadcasting in generic probabilistic theories, available as quant-ph/0611295.

Actually I've been lying slightly: these papers also allow mixtures of states

px + qy

where p+q is less than or equal to 1. For example, if you prepare an electron in the "up" spin state with probability p and the "down" state with probability q, but there's also a chance that you drop it on the floor and lose it, you might want p+q < 1.

I'm making it sound silly, but it's technically nice and maybe even conceptually justified. Mathematically it means that instead of a convex set of states, you have a vector space equipped with a convex cone and a linear functional P such that the cone is spanned by the "normalized" states: those with P(x) = 1. This is very natural in both classical and quantum probability theory.

Quite generally, the normalized states form a convex set. Conversely, starting from a convex set, you can create a vector space equipped with a convex cone and a linear functional with the above properties.

So, I was only lying slightly. In fact, a complicated web of related formalisms have been explored; you can learn about them from Barnum's paper.

For example, the convex cone formalism seems related to the Jordan algebra approach described in "week162". Barnum cites a paper by Araki that shows how to get Jordan algebras from sufficiently nice convex cones:

10) H. Araki, On a characterization of the state space of quantum mechanics, Commun. Math. Phys. 75 (1980), 1-24.

It's a very interesting paper but a wee bit too technical for me to feel like summarizing here.

Some nice examples of Jordan algebras are the 2×2 self-adjoint matrices with real, complex, quaternionic or octonionic entries. Each of these algebras has a cone consisting of the nonnegative matrices, and the trace gives a linear functional P. The nonnegative matrices with trace = 1 are the mixed states of a spin-1/2 particle in 3, 4, 6, and 10-dimensional spacetime, respectively! In each case these mixed states form a convex set: a round ball generalizing the Bloch ball. Similarly, the pure states form a sphere generalizing the Riemann sphere.

Back in "week162" I explained how these examples are related to special relativity and spinors in different dimensions. It's so cool I can't resist reminding you.

Our universe seems to like complex quantum mechanics. And, the space of 2×2 self-adjoint complex matrices - let's call it h2(C) - is isomorphic to 4-dimensional Minkowski spacetime! The cone of positive matrices is isomorphic to the future lightcone. The set of pure states of a spin-1/2 particle is the Riemann sphere CP1, and this is isomorphic to the "heavenly sphere": the set of light rays through a point in Minkowski spacetime.

This whole wonderful scenario works just as well in other dimensions if we replace the complex numbers (C) by the real numbers (R), the quaternions (H) or the octonions (O):

So, it's all very nice - but a bit mysterious. Why did our universe choose the complex numbers? We're told that scientists shouldn't ask "why" questions, but that's not really true - the main thing is to do it only to the extent that it leads to progress. But, sometimes you just can't help it.

String theorists occasionally think about 10d physics using the octonions, but not much. The strange thing about the octonions is that the self-adjoint nxn octonionic matrices hn(O) only form a Jordan algebra when n = 1, 2, or 3. So, it seems we can only describe very small systems in octonionic quantum mechanics! Nobody knows what this means.

People working on the foundations of quantum mechanics have also thought about real and quaternionic quantum mechanics. hn(R), hn(C) and hn(H) are Jordan algebras for all n, so the strange limitation afflicting the octonions doesn't affect these cases. But, I wound up sharing a little cottage with Lucien Hardy at Les Treilles, and he turns out to have thought about this issue. He pointed out that something interesting happens when we try to combine two quantum systems by tensoring them. The dimensions of hn(C) behave quite nicely:

dim(hnm(C)) = dim(hn(C)) dim(hm(C))

But, for the real numbers we usually have

dim(hnm(R)) > dim(hn(R)) dim(hm(R))

and for the quaternions we usually have

dim(hnm(H)) < dim(hn(H)) dim(hm(H))

So, it seems that when we combine two systems in real quantum mechanics, they sprout mysterious new degrees of freedom! More precisely, we can't get all density matrices for the combined system as linear combinations of tensor products of density matrices for the two systems we combined. For the quaternions the opposite effect happens: the combined system has fewer mixed states than we'd expect.

This observation lurks behind axiom 4 in this paper:

11) Lucien Hardy, Quantum theory from five reasonable axioms, available as quant-ph/0101012.

Another special way in which C is better than H or R is that only for a complex Hilbert space is there a correspondence between continuous 1-parameter groups of unitary operators and self-adjoint operators. We always get a skew-adjoint operator, but only in the complex case can we convert this into a self-adjoint operator by dividing by i.

Here are some more references, kindly provided by Rob Spekkens. The pioneering quantum field theorist Stückelberg wrote a bunch of papers on real quantum mechanics. Spekkens recommends this one:

12) E. C. G. Stückelberg, Quantum theory in real Hilbert space, Helv. Phys. Acta 33, 727 (1960).

This is a modern review:

13) Jan Myrheim, Quantum mechanics on a real Hilbert space, available quant-ph/9905037.

What I find most fascinating is the connection between real quantum mechanics and time reversal symmetry. In ordinary complex quantum mechanics, time reversal symmetry is sometimes described by a conjugate-linear (indeed "antiunitary") operator T with T2 = 1. Such an operator is precisely a "real structure" on our complex Hilbert space: it picks out a real Hilbert subspace of which our complex Hilbert space is the complexification.

It's worth adding that in the physics of fermions, another possibility occurs: an antiunitary time reversal operator with T2 = -1. This is precisely a "quaternionic structure" on our complex Hilbert space: it makes it into a quaternionic Hilbert space!

For more on these ideas try:

14) Freeman J. Dyson, The threefold way: algebraic structure of symmetry groups and ensembles in quantum mechanics, Jour. Math. Phys. 3 (1962), 1199-1215.

15) John Baez, Symplectic, quaternionic, fermionic,

From all this one can't help but think that complex, real, and quaternionic quantum mechanics fit together in a unified structure, with the complex numbers being the most important, but other two showing up naturally in systems with time reversal symmetry.

Stephen Adler - famous for the Adler-Bell-Jackiw anomaly - spent a long time at the Institute for Advanced Studies working on quaternionic quantum mechanics:

16) S. L. Adler, Quaternionic Quantum Mechanics and Quantum Fields, Oxford U. Press, Oxford, 1995.

A problem with this book is that it defines a quaternionic vector space to be a left module of the quaternions, instead of a bimodule. This means you can't naturally tensor two quaternionic vector spaces and get a quaternionic vector space! Adler "solves" this problem by noting that any left module of the quaternions becomes a right module, and in fact a bimodule, via

xq = q*x

But, when you're working with a noncommutative ring, you really need to think about left modules, right modules, and bimodules to understand the theory of tensor products. And, the quaternions have more bimodules than you might expect: for example, for any automorphism of the quaternions:

f: H → H

there's a way to make H into an H-bimodule with the obvious left action and a "twisted" right action, where q acts on x to give

x f(q)

Since the automorphism group of the quaternions is SO(3), there turn out to be SO(3)'s worth of nonisomorphic ways to make H into an H-bimodule!

For an attempt to tackle this issue, see:

17) John Baez and Toby Bartels, Functional analysis with quaternions, available at

However, it's possible we'll only see what real and quaternionic quantum mechanics are good for when we work in the 3-category Alg(R) mentioned in "week209", taking R to be the real numbers. Here:

This could let us treat real, complex and quaternionic quantum mechanics as part of a single structure.

Dreams, dreams....

Addenda: In email, Scott Aaronson pointed out this nice webpage:

18) Scott Aaronson, Lecture 9: Quantum,

He wrote:

I talk all about the known differences between QM over the complex numbers and QM over the reals and quaternions (including the parameter-counting difference you mentioned, but also a couple you didn't), and why the universe might've gone with complex numbers.
His lecture also cites this paper:

19) Carlton M. Caves, Christopher A. Fuchs, and Ruediger Schack, Unknown quantum states: the quantum de Finetti representation, available as quant-ph/0104088.

which Rob Spekkens also pointed out to me.

The quantum de Finetti theorem is a generalization of the classical de Finetti theorem. Both classical quantum de Finetti theorems are about n copies of a system sitting side by side in an "exchangeable" state: a state that's not only invariant under permutations of the copies, but lacking correlations between the different copies!

Here's the quantum de Finetti theorem. Suppose you have an "exchangeable" density operator ρn on H⊗ n - that is, one such that for each N ≥ n, there's a density operator ρN on H⊗ N which 1) is invariant under permutations in SN and 2) has ρ as its marginal, meaning that

Tr (ρN) = ρn

where Tr is the partial trace map sending operators on H⊗ N to operators on H⊗ n. Then, ρn is a mixture of density matrices of the form ρ ⊗ ... ⊗ ρ: a tensor product of n copies of a density matrix on H.

This is completely plausible if you know what all this jargon means.

And now for the punch line: This theorem would fail if we did quantum mechanics using the real numbers!

Of course, this is related to the fact I mentioned this Week, namely that for real quantum mechanics, "the whole is more than the product of its parts" in a more severe way than for complex quantum mechanics.

Bob Coecke wrote:

The standard references on quaternionic QM are:

20) D. Finkelstein, J.M. Jauch, S. Schiminovich and D. Speiser, Foundations of quaternion quantum mechanics, Journal of Mathematical Physics 3, 207 (1962).

21) D. Finkelstein, J.M. Jauch, S. Schiminovich and D. Speiser, Some physical consequences of general Q-covariance, Helvetica Physica Acta 35, 328-329 (1962).

22) D. Finkelstein, J.M. Jauch, S. Schiminovich and D. Speiser, Principle of general Q-covariance, Journal of Mathematical Physics 4, 788-796 (1963).

A standard structural result in the order-theoretic vein which separates Reals, Complex Numbers and Quaternions from ''non-classical fields'' is:

23) M. P. Soler (1995) Characterization of Hilbert spaces with orthomodular spaces, Comm. Algebra 23, pp. 219-243.

It does this relative to the order-theoretic characterization of Hilbert spaces:

24) C. Piron (1964, French) Axiomatique Quantique, Helv. Phys. Acta 37, pp. 439-468.

25) I. Amemiya and H. Araki (1966) A Remark on Piron's Paper, Publ. Res. Inst. Math. Sci. Ser. A 2, pp. 423–427.

24) C. Piron (1976) Foundations of Quantum Physics, W. A. Benjamin, Inc., Reading.

A nicely written recent survey on this stuff is:

26) Isar Stubbe and B. van Steirteghem (2007) Propositional systems, Hilbert lattices and generalized Hilbert spaces, chapter in: Handbook Quantum Logic (edited by D. Gabbay, D. Lehmann and K. Engesser), Elsevier, to appear. Available at

It is not clear to me how exactly this order-theoretic stuff relates to the thick categorical axiomatics for QM John mentioned above. One key difference is that in the order-theoretic axiomatics one failed to find an abstract counterpart to the Hilbert space tensor product. (ie without having to say that we are working in the lattice of closed subspaces of a Hilbert space) On the other hand, the categorical approach starting from symmetric monoidal categories takes that description of compound systems as an a priori. Singling out the complex numbers is done in terms of two involutions on morphisms, one covariant and one contravariant, where the covariant one capture complex conjugation ie the unique non-trivial automorphism characteristic of complex numbers. The contravariant one captures transposition and together they make up the adjoint.

Here "thick" refers to working with categories which nice big hom-sets, instead of mere posets or preorders, which are categories with at most one morphism from one object to another.

Rob Spekkens also gives some references on quantum computation in real quantum mechanics. He writes:

See also:

27) C. M. Caves, C. A. Fuchs, P. Rungta, Entanglement of formation of an arbitrary state of two rebits, available as quant-ph/0009063.

It's also worth noting that quantum computation and quantum cryptography do not require the complex field. Have a look at:

28) T. Rudolph and L. Grover, A 2 rebit gate universal for quantum computing, quant-ph/0210187.

I actually know of no information-theoretic task whose possibility is contingent on the nature of the number field.

More discussion (and pictures!) can be found at the n-Category Café.

© 2007 John Baez