Skip to the Main Content

Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

May 6, 2007

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

Posted by John Baez

In week251 of This Week’s Finds, hear some of what I learned at Les Treilles. First, Spekkens’ theory of "toy bits" — one of several "foils for quantum mechanics". Then, Howard Barnum on the convex set approach to general physical systems, and Lucien Hardy on the mysteries of real and quaternionic quantum mechanics.

Here’s a picture of Howard Barnum, taken at Les Treilles by Marc Lachièze-Rey:

Here’s a picture of Rob Spekkens (left) talking to Lucien Hardy (right):

And here I am, pondering the mysteries of the universe — or more likely, the mysterious power of coffee to combat jet-lag:

You can see more pictures of Les Treilles here.

Posted at May 6, 2007 1:29 AM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/1263

21 Comments & 1 Trackback

Re: This Week’s Finds in Mathematical Physics (Week 251)

Scott Aaronson pointed out this nice webpage:

As he says:

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.

I’ll add this link to the body of week251, and also this paper:

Carlton M. Caves, Christopher A. Fuchs, and Ruediger Schack, Unknown quantum states: the quantum de Finetti representation.

The quantum de Finetti theorem is a generalization of the classical de Finetti theorem. Both classical and quantum de Finetti theorems are about nn 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 ρ n\rho_n on H nH^{\otimes n} — that is, one such that for each NnN \ge n, there’s a density operator ρ N\rho_N on H NH^{\otimes N} which 1) is invariant under permutations in S NS_N and 2) has ρ n\rho_n as its marginal, meaning that

Tr(ρ N)=ρ nTr (\rho_N) = \rho_n

where TrTr is the partial trace map sending operators on H NH^{\otimes N} to operators on H nH^{\otimes n}. Then, ρ n\rho_n is a mixture of density matrices of the form ρρ\rho \otimes \cdots \otimes \rho: a tensor product of nn copies of a density matrix on HH.

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 in week250, 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.

Posted by: John Baez on May 7, 2007 2:33 AM | Permalink | Reply to this

Measure how BADLY they fail; Re: This Week’s Finds in Mathematical Physics (Week 251)

This may be a foolish qubit of intuition, but:

“This theorem would fail if we did quantum mechanics using the real numbers!”

leads me to wonder about quantifying the extent of the failure, for some class of deformations of the reals, the complex numbers, the quaternions, the octonions…

Similarly, the “toy bit” examples are individual points in a superspace of toy QM. Might it be useful to ask about them within *-algebras, as von Neumann had set up a playground for generalizations of QM?

I’ve played with finite *-algebras, and I find interesting combinatorial properties.

My hunches might be only a millionth as useful as your hunches, or Terry Tao’s hunches, but in the spirit of openly practicing mathematical speculation, thanks for allowing me these obvious comments.

Posted by: Jonathan Vos Post on May 7, 2007 3:03 AM | Permalink | Reply to this

Re: Measure how BADLY they fail; Re: This Week’s Finds in Mathematical Physics (Week 251)

Jonathan Vos Post wrote:

Similarly, the toy bit examples are individual points in a superspace of toy QM. Might it be useful to ask about them within *-algebras, as von Neumann had set up a playground for generalizations of QM?

It’s not a *-algebraic theory, or even in the set of convex theories investigated by Barnum and Co. It’s crucial to Spekkens’ constructions, and particularly to the analog of superposition, that the state-space is discrete. Finding a good mathematical formalism for his theory (I suspect finite fields may be the way to go) and placing it within a comprehensive framework for generalized theories would be very interesting.

If category theory has anything to say about this, particularly if it can significantly unify the different approaches to toy theories and/or generalizations of quantum theory, then I would definitely become a lot more interested in it.

Posted by: Matt Leifer on May 12, 2007 12:26 AM | Permalink | Reply to this

Re: Measure how BADLY they fail; Re: This Week’s Finds in Mathematical Physics (Week 251)

I may be misunderstanding your comment, but there are finite, finite-dimensional, and discrete *-algebras.

[PDF]
Quantum and classical pseudogroups. Part I. Union pseudogroups and …
File Format: PDF/Adobe Acrobat
A finite-dimensional star algebra d is a C*-algebra if and only if the. following positivity condition is satisfied:. if A9. AO then A*AeO. (cf. I-5]). …
www.springerlink.com/index/W75568068568345X.pdf - Similar pages
[PDF]
Star-algebra for Reasoning with Angular Directions in 2D-Space
File Format: PDF/Adobe Acrobat - View as HTML
from a finite discrete set of values between 0 and 360 degrees, e.g., {x=0°, …. inverse, set union and intersection operations, forming the Star-algebra( …
rutcor.rutgers.edu/~amai/aimath04/AcceptedPapers/Mitra-aimath04.pdf -

Posted by: Jonathan Vos Post on September 2, 2007 10:26 PM | Permalink | Reply to this

Re: Measure how BADLY they fail; Re: This Week’s Finds in Mathematical Physics (Week 251)

In that case it’s time to learn some category theory Matt. Check out these toy quantum categories and in particular the joint paper with Spekkens the group theoretic origin of non-locality for qubits which provides a category-theoretic unification of the toy theory and quantum theory, pinpoints their key differences and establishes why one is non-local while the other one is local.

Posted by: bob on July 2, 2009 7:30 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 251)

Spekkens and I spent an afternoon trying to think about his theory as quantum mechanics over some finite field, but failed — we almost came close to proving it couldnt’ work.

I feel sure category theory would have interesting things to say about this theory, but only after I know what category Spekkens’ theory is talking about. I don’t yet know precisely what the state spaces are in his approach, and (more importantly) what the allowed maps between them are. He’s worked out the answers in some special cases, but I haven’t seen the general answer, and I’m not having much luck so far figuring it out. This is not a shortcoming of category theory: it’s shortcoming in my understanding of what his theory really says!

I should have tried hard to get him to explain this when I was in Les Treilles, but we were having too much talking about other things.

Posted by: John Baez on May 12, 2007 6:25 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 251)

On Spekkens’ toy system and finite geometry

Background–

  • In “Week 251” (May 5, 2007), John wrote:
    “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…. 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 the n-Category Cafe ( May 12, 2007, 12:26 AM, ) Matt Leifer wrote:
    “It’s crucial to Spekkens’ constructions, and particularly to the analog of superposition, that the state-space is discrete. Finding a good mathematical formalism for his theory (I suspect finite fields may be the way to go) and placing it within a comprehensive framework for generalized theories would be very interesting.”
  • In the n-category Cafe ( May 12, 2007, 6:25 AM) John Baez wrote:
    “Spekkens and I spent an afternoon trying to think about his theory as quantum mechanics over some finite field, but failed — we almost came close to proving it couldnt’ work.”

On finite geometry:

The actions of permutations on a 4 × 4 square in Spekkens’ paper (quant-ph/0401052), and Leifer’s suggestion of the need for a “generalized framework,” suggest that finite geometry might supply such a framework. The geometry in the webpage John cited is that of the affine 4-space over the two-element field.

Related material:

Posted by: Steven H. Cullinane on September 2, 2007 10:03 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 251)

Both classical and quantum de Finetti theorems are about nn 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!

That’s an odd way of describing what de Finetti was doing with the concept of ‘exchangeability’. Remember he’s the one famous for ‘Probabilities do not exist’.

What his theorem does is to show that if you are totally indifferent between the members of certain collections of sequences of events, then your beliefs may be modelled as though the events were produced by a Bernoulli process, and you had a prior distribution over the parameter pp of that process.

So it’s all about representing your own subjective state of belief, and not about the way the world is. Of course, what happens in the world may lead you to revise your accessment of exchangeability. You ought to save yourself some of your meta-prior, so that if you see a sequence of coin tosses which begins with 70000 heads and is followed by 30000 tails, you don’t just update the Bernoulli-esque part of your belief to arrive at a posterior centred around 0.7, but rather come to expect long sequences of the same side of the coin to occur in the future.

Similarly, Fuchs and Caves are interested in exploring how much of the quantum formalism is describing one’s state of information rather than being about the world as it is in itself.

Posted by: David Corfield on May 7, 2007 1:04 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 251)

David wrote:

That’s an odd way of describing what de Finetti was doing with the concept of ‘exchangeability’. Remember he’s the one famous for ‘Probabilities do not exist’.

Interesting! But I can’t remember that, since I never knew it. I’d never heard of the de Finetti theorem until folks started telling me the quantum de Finetti theorem fails in quantum mechanics over the real numbers!

So, I must be foisting a wholly different interpretation on his theorem than he did. But hopefully I got the math of it right — though I only stated it for the quantum case.

In both the classical and quantum cases, it’s a characterization of mixed states that are mixtures of ‘independent identically distributed’ states.

Similarly, Fuchs and Caves are interested in exploring how much of the quantum formalism is describing one’s state of information rather than being about the world as it is in itself.

Yeah, I noticed that stuff when looking through their paper, but I tried to ignore it and dig out the mathematical ‘meat’: the theorem I stated here. I guess I was just in that kind of mood.

Posted by: John Baez on May 7, 2007 7:01 PM | Permalink | Reply to this

Baez on complex number advantage; Re: This Week’s Finds in Mathematical Physics (Week 251)

As John Baez wrote (and he can supply the URLs better than I):

“… the complex numbers have a distinct advantage …

… Only in this case can we turn any
self-adjoint complex matrix into a skew-adjoint one, and vice versa, by multiplying by i.
… We don’t just want a Jordan algebra
… we don’t just want a Lie algebra
… we want something that’s both …”.

“… the complex numbers have a distinct advantage …” and here I will take that remark out of context to mention a point, made by Stephen Adler in his (Oxford 1995) book Quaternionic Quantum Mechanics and Quantum Fields (pp.10-11):

“…
standard quantum mechanics [is formulated] in a complex Hilbert space
… The special Jordan algebras are equivalent … to the Dirac formulation in … real, complex, or quaternionic Hilbert space

the … exceptional Jordan algebra … of the 27-dimensional non-associative algebra of 3x3 octonionic Hermitian matrices … corresponding to a quantum mechanical system over a two- (and no higher) dimensional projective geometry that cannot be given a Hilbert Space formulation …

Zel’manov (1983) … proved that in the infinite-dimensional case one finds no new simple exceptional Jordan algebras …”.

Posted by: Jonathan Vos Post on May 7, 2007 6:55 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 251)

Well if you’re that way inclined then the complex numbers are just the even subalgebra of 2d real clifford algebra and quaternions are just the even subalgebra of 3d real clifford algebra. After that it gets harder to understand, possibly because there are bivectors which are not the product of two vectors [which might mean that bivectors part company from 2-forms, if I knew what 2-forms were, but I don’t, even though I passed Prof Wall’s course 37 years ago].

Anyway I was just reading Doran and Lasenby’s book on “Geometric Algebra for Physicists” and came across the following quote, and I wondered if anyone would like to comment on the claim by the Clifford Algebra fans that you can do all that quantum stuff without complex numbers:

(top of page 283) The full Dirac equation is now

(1)ψIσ 3eAψ=mψγ 0 \nabla\psi I \mathbf{\sigma}_3 - e A\psi = m\psi\gamma_0

A remarkable feature of this formulation is that the equation and all of its observables have been captured in the real algebra of spacetime, with no need for a unit imaginary. This suggests that interpretations of quantum mechanics that place great significance in the need for complex numbers are wide of the mark.

Posted by: Robert Smart on May 17, 2007 12:39 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 251)

In fact, I am inclined to like that attitude.

While it is slightly bothersome that the Hestenes-Doran-Lasenby-et al. school keeps emphasizing how everything appears in a suggestive different light from their point of view, without (so far, as I am aware, but you will now immediately correct me) actually doing something with that which gets us to a point where we haven’t been before – while this is slightly bothersome, it doesn’t mean that they are not right!

It just means that somebody still may have to make that next step.

Something is going on, and nobody understands it yet. And it involves understanding what Clifford algebra really is and what supersymmetry really is. I claim. And I also claim that we know what anything really is once we know how it categorifies.

We have talked about that recently.

Posted by: urs on May 17, 2007 1:53 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 251)

Let’s not forget that we are a part of this universe, not an external observer. It is not necessarily the case that we can find out exactly how it works. There might be multiple different processes which produce this universe as we experience it. From a science point of view, explanations that give the same predictions are the same. For example Doran-Lasenby-Gull have a gauge theory of gravity on a flat spacetime which they claim to prove is identical, prediction-wise, to Einstein’s curved spacetime. If there are indistinguishable choices of explanation then it is often helpful to move between them to see a particular problem from different viewpoints. Will categorifications be like that?

Posted by: Robert Smart on May 18, 2007 12:06 AM | Permalink | Reply to this

Quaternions et al - some references

The standard references on quaternionic QM are:

* D. Finkelstein, J.M. Jauch, S. Schiminovich and D. Speiser, Foundations of quaternion quantum mechanics, Journal of Mathematical Physics 3, 207 (1962)
* D. Finkelstein, J.M. Jauch, S. Schiminovich and D. Speiser, Some physical consequences of general Q-covariance, Helvetica Physica Acta 35, 328-329 (1962)
* 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:

* 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:

* C. Piron (1964, French) Axiomatique Quantique, Helv. Phys. Acta 37, pp. 439-468.
* I. Amemiya and H. Araki (1966) A Remark on Piron’s Paper, Publ. Res. Inst. Math. Sci. Ser. A 2, pp. 423–427.
* C. Piron (1976) Foundations of Quantum Physics, W. A. Benjamin, Inc., Reading.

A nicely written recent survey on this stuff is:

* 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. Download-able from http://www.win.ua.ac.be/~istubbe/

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.

Posted by: bob on May 7, 2007 12:12 PM | Permalink | Reply to this

Re: Quaternions et al - some references

relates to the *thick* categorical axiomatics

Sorry, what does “thick” mean in this context?

Posted by: urs on May 7, 2007 1:19 PM | Permalink | Reply to this

Re: Quaternions et al - some references

“Thin” refers to the fact that preorders are categories with hom-sets either singletons or empty. The order-structure, ie some special kind of orthomodular lattice, then captures the Hilbert space structure via Piron’s Thm. The “thick” varient has all linear maps in the Hom-set. Hilbert space is now captured by compact closure, which internalizes the Hom-sets as spaces (eg Deligne’s thm). Both approaches have a clear operational (or instrumentalist) connotation (eg Piron’s book for the case of lattices and my my notes for SMCs) but it’s not clear to me how they precisely relate, neither physically not mathematically. The obvious thing to do would be just to look at the category of those lattices but no notion of morphism seems to give the desired symmetric monoidal structure that allows to derive protocols such as quantum teleportation, provides trace structure etc.

Posted by: bob on May 7, 2007 2:21 PM | Permalink | Reply to this

Re: Quaternions et al - some references

Thanks for all the quaternionic QM references, Bob!

Bob wrote:

It is not clear to me how exactly this order-theoretic stuff relates to the *thick* categorical axiomatics for QM John mentioned above.

This is a fascinating puzzle! I ran a seminar on it once, but nobody took notes, and we didn’t get very far… Mike Stay got pulled over to working on the quantum lambda-calculus instead.

The basic idea went like this. Say we start from the ‘thick’ description in terms of a category CC of ‘state spaces’ and ‘processes’ — for example, the category Set\mathrm{Set} of sets and linear operators, or Hilb\mathrm{Hilb} of finite-dimensional Hilbert spaces and operators, or nCobn\mathrm{Cob} of compact oriented (n1)(n-1)-manifolds and cobordisms between them.

Next, say we want to get back the good old ‘thin’ description of logic in terms of posets, where the partial order is ‘implication’. We eventually want to get something like a lattice of propositions about the state for each ‘state space’ — that is, each object of CC. But, we want all these lattices to fit together nicely!

Clearly, each object has a poset of subobject, and we can think of this partial order as ‘implication’. In some nice cases (Set\mathrm{Set} or Hilb\mathrm{Hilb}) this poset will be a lattice. In some very nice cases of the ‘non-quantum’ or ‘cartesian’ flavor (e.g. Set)\mathrm{Set}), it will be a Heyting algebra or even a Boolean algebra. In some very nice cases of the ‘quantum’ or ‘symmetric monoidal category with duals’ flavor (e.g. Hilb\mathrm{Hilb}), it will be an orthocomplemented lattice. And, in a bunch of these nice cases (e.g. both Set\mathrm{Set} and Hilb\mathrm{Hilb}), we get a functor

F:C[lattices]F : C \to [lattices]

sending each object to its lattice of subobjects. Here the trick is making sure the morphisms in CC give lattice homomorphisms!

In nice cases, FF will be lax monoidal, so there will be a lattice homomorphism

F x,y:F(x)F(y)F(xy)F_{x,y}: F(x) \otimes F(y) \to F(x \otimes y)

This sends a pair (P,Q)(P,Q) consisting of a proposition about the system xx and a proposition about the system yy to a proposition ‘PP and QQ’ about the joint system xyx \otimes y. I guess a new thing about quantum systems is that F x,yF_{x,y} need not be an isomorphism? That is, there are propositions about a joint quantum system that can’t be built from propositions of the form ‘PP (about the first system) and QQ (about the second)’ using the lattice operations?

One could have a lot of fun figuring out the precise conditions under which all these nice things happen. The ‘cartesian’ cases have been pretty much worked out already — a good place to start is Johnstone’s Elephant, with its discussion of regular categories and the like. So, the fun lies in a similar investigation of the ‘symmetric monoidal category with duals’ cases. Here one important trick is to focus, not on arbitrary subobjects, but on ‘isometric’ ones, i.e. monomorphisms f:xyf: x \to y such that f *f=1 xf^* f = 1_x. Clearly one needs to do something like this to get an orthocomplemented lattice.

If I could clone myself, I’d set one copy working on this project: the reconstruction of ‘thin’ quantum logic from the ‘thick’ categorical semantics of quantum systems. I do my best by having lots of grad students. But, I can’t clone my quantum state!

Maybe Isar Stubbe would be interested in this project. He’s one of the few people who knows all the relevant stuff.

Posted by: John Baez on May 7, 2007 7:55 PM | Permalink | Reply to this

Re: Quaternions et al - some references

In a computer science jargon this means extracting a “theory of properties” from a “type theory”. The first one turns out to be important when one cares about verification of hardware, protocols or programs.

In this context it’s probably also worth to mentioning a paper by John Harding:

in which he gives a categorical criterion on when objects decompose as orthomodular posets. This generalises his previous results in:

  • J. Harding (1996) Decompositions in quantum logic. Trans. Amer. Math. Soc. 348, 1839–1862.

although again, it’s not clear to me what the connections are with the above mentioned thick axiomatics, … he will be talking about this work at:

Posted by: bob on May 8, 2007 2:47 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 251)

Howdy. :-)

Don’t you know what’s the business with that p-adic quantum mechanics? Please, enlighten. It seems this subject is omitted by John Baez. (There are some fearful, but arousing interest, motivic integrals they are somehow connected to p-adic integrals, eh?)

Thank you very much. :-)

Posted by: nosy snoopy on May 8, 2007 4:42 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 251)

I don’t understand pp-adic quantum mechanics very well, but the main reason I didn’t include it here is that it doesn’t seem to be in the same family of generalizations of quantum mechanics that I was mainly focused on here: generalizations where observables take values in \mathbb{R}!

As far as I can tell, the best thing about pp-adic quantum mechanics is that it fits with complex quantum mechanics inside of adelic quantum mechanics. This lets us reduce certain (special) computations in complex quantum mechanics to computations in pp-adic quantum mechanics for all pp, or vice versa.

But, this is about the extent of my knowledge.

Posted by: John Baez on May 8, 2007 8:26 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 251)

Howard Barnum wrote (in part):

Hi John —

I noticed one thing that may be an accidental slip—you wrote:

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 convex 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.

I suspect you meant

“we can’t get all density matrices for the combined system as linear combinations of tensor products of density matrices…”

since with “convex combinations” this just says we have entangled states, true even in complex quantum mechanics, whereas the point I think you want to make for real quantum mechanics is that (if the combination rule is to take the tensor product of the underlying real Hilbert spaces, and let the state space be the positive semidefinite selfadjoint operators on that tensor product) we actually have more dimensions than available in the linear span of the product states.

Incidentally, in Barrett’s paper, and in the four-author paper we wrote on broadcasting, it’s an assumption (although we’ve discussed what happens if it’s relaxed) that the combined system’s state space lives in the span of the product states. But I guess you noticed this.

Another thing you might find interesting, since you noticed Carl Caves, Ruediger Schack, and Chris Fuchs’ work on de Finetti theorems and their observation about real QM is that Jon Barrett and Matt Leifer have a generalized de Finetti theorem in the convex sets framework (in particular, in Jon’s version of it). Apparently, the condition that joint states lie in the span of individual states is crucial. I’m not sure exactly how they model multiple-system combination, though, as the work isn’t published and I believe they are intensely working on writing it up. (Actually it could be on quant-ph by now… I don’t scan it very frequently.)

Come to think of it, Leonid Gurvits and I have another result that relates real and complex QM, to some extent: the real unentangled “unnormalized density matrices” form a radius-1 ball around the identity in the restriction of the cone of positive semidefinite real symmetric matrices to the span of the products, even for an arbitrary number of systems, whereas the radius must decrease exponentially for complex QM. It’s in

I don’t have any enlightening thoughts about its significance.

Anyway, I know you have other things to do but thought I’d share the thoughts that sprang to mind on reading your piece.

I’ll have more to say sometime on monoidal categories of convex sets, but I too haven’t had time to think deeply about them this week (presentation at a project review, paperwork….).

Cheers,

Howard

Indeed, writing “convex combinations” when I meant “linear combinations” was a slip of the fingers, probably caused by the fact I’ve been wondering how cogent this argument against real QM really is. Convex combinations of density matrices have a pretty clear operational meaning. The meaning of the linear combinations here is more mysterious to me. I guess it’s related to the ‘quantum tomography’ stuff.

Posted by: John Baez on May 12, 2007 6:32 AM | Permalink | Reply to this
Read the post The n-Café Quantum Conjecture
Weblog: The n-Category Café
Excerpt: Why it seems that quantum mechanics ought to be the de-refinement of a refined theory which lives in one categorical degree higher than usual.
Tracked: June 8, 2007 6:27 PM

Post a New Comment