Anthony Aguirre is looking for postdoc at Santa Cruz in Physics of the Observer

Anthony Aguirre points out that UC Santa Cruz is advertising for postdocs in the "Physics of the Observer" program; and although review of applications began in December with a Dec. 15 deadline "for earliest consideration", if you apply fast you will still be considered.  He explicitly states they are looking for strong applicants from the quantum foundations community, among other things.

My take on this: The interaction of quantum and spacetime/gravitational physics is an area of great interest these days, and people doing rigorous work in quantum foundations, quantum information, general probabilistic theories have much to contribute.  It's natural to think about links with cosmology in this context.  I think this is a great opportunity, foundations postdocs and students, and Anthony and Max are good people to be connected with, very proactive in seeking out sources of funding for cutting-edge research and very supportive of interdisciplinary interaction.  The California coast around Santa Cruz is beautiful, SC is a nice funky town on the ocean, and you're within striking distance of the academic and venture capital powerhouses of the Bay Area.  So do it!

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Martin Idel: the fixed-point sets of positive trace-preserving maps on quantum systems are Jordan algebras!

Kasia Macieszczak is visiting the ITP at Leibniz Universität Hannover (where I arrived last month, and where I'll be based for the next 7 months or so), and gave a talk on metastable manifolds of states in open quantum systems.  She told me about a remarkable result in the Master's thesis of Martin Idel at Munich: the fixed point set of any trace-preserving, positive (not necessarily completely positive) map on the space of Hermitian operators of a finite-dimensional quantum system, is a Euclidean Jordan algebra.  It's not necessarily a Jordan subalgebra of the usual Jordan algebra associated with the quantum system (whose Jordan product is the antisymmetrized matrix multiplication, $A \bullet B = (AB +BA)/2$).  We use the usual characterization of the projector $T_\infty$ onto the fixed-point space of a linear map $T$$T_\infty = \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{n=1}^N T^n$.  The maximum-rank fixed point is $T_\infty(I)$ (where $I$ is the identity matrix), which we'll call $F$, and the Jordan product on the fixed-point space is the original one "twisted" to have $F$ as its unit:  for $A,B$ fixed-points, this Jordan product, which I'll denote by $\bullet_\infty$, is:

which we could also write in terms of the original Jordan product as $A \bullet_\infty B = \phi(A) \bullet \phi(B)$, where $\phi$ is the map defined by $\phi(X) := F^{-1/2} X F^{-1/2}$.

Idel's result, Theorem 6.1 in his thesis, is stated in terms of the map on all $d \times d$ complex matrices, not just the  Hermitian ones; the fixed-point space is then the complexification of the Euclidean Jordan algebra.  In the case of completely positive maps, this complexification is "roughly a $C^*$ algebra" according to Idel.  (I suspect, but don't recall offhand, that it is a direct sum of full matrix $C^*$ algebras, i.e. isomorphic to a quantum system composed of several "superselection sectors" (the full matrix algebras in the sum), but as in the Euclidean case, not necessarily a $C^*$-subalgebra of the ambient matrix algebra.)

I find this a remarkable result because I'm interested in places where Euclidean Jordan algebras appear in nature, or in mathematics.  One reason for this is that the finite-dimensional ones are in one-to-one correspondence with homogeneous, self-dual cones; perhaps I'll discuss this beautiful fact another time.  Alex Wilce, Phillip Gaebeler and I related the property of homogeneity to "steering" (which Schrödinger considered a fundamental weirdness of the newly developed quantum theory) in this paper.  I don't think I've blogged about this before, but Matthew Graydon, Alex Wilce, and I have developed ways of constructing composite systems of the general probabilistic systems based on reversible Jordan algebras, along with some results that I interpret as no-go theorems for such composites when one of the factors is not universally reversible.  The composites are still based on Jordan algebras, but are necessarily (if we wish them to still be Jordan-algebraic) not locally tomographic unless both systems are quantum.  Perhaps I'll post more on this later, too.  For now I just wanted to describe this cool result of Martin Idel's that I'm happy to have learned about today from Kasia.

ITFP, Perimeter: selective guide to talks. #1: Brukner on quantum theory with indefinite causal order

Excellent conference the week before last at Perimeter Institute: Information Theoretic Foundations for Physics.  The talks are online; herewith a selection of some of my favorites, heavily biased towards ideas new and particularly interesting to me (so some excellent ones that might be of more interest to you may be left off the list!).  Some of what would have been possibly of most interest and most novel to me happened on Weds., when the topic was spacetime physics and information, and I had to skip the day to work on a grant proposal.  I'll have to watch those online sometime.  This was going to be one post with thumbnail sketches/reviews of each talk, but as usual I can't help running on, so it may be one post per talk.

All talks available here, so you can pick and choose. Here's #1 (order is roughly temporal, not any kind of ranking...):

Caslav Brukner kicked off with some interesting work on physical theories in with indefinite causal structure.  Normally in formulating theories in an "operational" setting (in which we care primarily about the probabilities of physical processes that occur as part of a complete compatible set of possible processes) we assume a definite causal (partial) ordering, so that one process may happen "before" or "after" another, or "neither before nor after".  The formulation is "operational" in that an experimenter or other agent may decide upon, or at least influence, which set of processes, out of possible compatible sets, the actual process will be drawn, and then nature decides (but with certain probabilities for each possible process, that form part of our theory), which one actually happens.  So for instance, the experimenter decides to perform a Stern-Gerlach experiment with a particular orientation X of the magnets; then the possible processes are, roughly, "the atom was deflected in the X direction by an angle theta," for various angles theta.  Choose a different orientation, Y, for your apparatus, you choose a different set of possible compatible processes.  ("The atom was deflected in the Y direction by an angle theta.")  Then we assume that if one set of compatible processes happens after another, an agent's choice of which complete set of processes is realized later, can't influence the probabilities of processes occuring in an earlier set.  "No signalling from the future", I like to call this; in formalized operational theories it is sometimes called the "Pavia causality axiom".   Signaling from the past to the future is fine, of course.  If two complete  sets of processes are incomparable with respect to causal order ("spacelike-separated"), the no-signalling constraint operates both ways:  neither Alice's choice of which compatible set is realized, nor Bob's, can influence the probabilities of processes occuring at the other agent's site.   (If it could, that would allow nearly-instantaneous signaling between spatially separated sites---a highly implausible phenomenon only possible in preposterous theories such as the Bohmian version of quantum theory with "quantum disequilibrium", and Newtonian gravity. ) Anyway, Brukner looks at theories that are close to quantum, but in which this assumption doesn't necessarily apply: the probabilities exhibit "indeterminate causal structure".  Since the theories are close to quantum, they can be interpreted as allowing "superpositions of different causal structures", which is just the sort of thing you might think you'd run into in, say, theories combining features of quantum physics with features of general relativistic spacetime physics.  As Caslav points out, since in general relativity the causal structure is influenced by the distribution of mass and energy, you might hope to realize such indefinite causal structure by creating a quantum superposition of states in which a mass is in one place, versus being in another.  (There are people who think that at some point---some combinations of spatial scales (separation of the areas in which the mass is located) and mass scales (amount of mass to be separated in "coherent" superposition)) the possibility of such superpositions breaks down.  Experimentalists at Vienna (where Caslav---a theorist, but one who likes to work with experimenters to suggest experiments---is on the faculty) have created what are probably the most significant such superpositions.)

Situations with a superposition of causal orders seem to be exhibit some computational advantages over standard causally-ordered quantum computation, like being able to tell in fewer queries (one?) whether a pair of unitaries commutes or anticommutes.  Not sure whose result that was (Giulio Chiribella and others?), but Caslav presents some more recent results on query complexity in this model, extending the initial results.  I am generally wary about results on computation in theories with causal anomalies.  The stuff on query complexity with closed timelike curves, e.g. by Dave Bacon and by  Scott Aaronson and John Watrous has seemed uncompelling---not the correctness of the mathematical results, but rather the physical relevance of the definition of computation---to me for reasons similar to those given by Bennett, Leung, Smith and Smolin.  But I tend to suspect that Caslav and the others who have done these query results, use a more physically compelling framework because they are well versed in the convex operational or "general probabilistic theories" framework which aims to make the probabilistic behavior of processes consistent under convex combination ("mixture", i.e. roughly speaking letting somebody flip coins to decide which input to present your device with).  Inconsistency with respect to such mixing is part of the Bennett/Leung/Smolin/Smith objection to the CTC complexity classes as originally defined.

[Update:  This article at Physics.org quotes an interview with Scott Aaronson responding to the Bennett et. al. objections.  Reasonably enough, he doesn't think the question of what a physically relevant definition of CTC computing is has been settled.  When I try to think about this issue sometimes I wonder if the thorny philosophical question of whether we court inconsistency by trying to combine intervention ("free choice of inputs") in a physical theory is rearing its head.  As often with posts here, I'm reminding myself to revisit the issue at some point... and think harder.]

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How does one prove that domains of positivity of symmetric nondegenerate bilinear forms are self-dual cones?

On mathoverflow, I've asked how one proves that domains of positivity of symmetric nondegenerate bilinear forms on real vector spaces are self-dual cones.  A bilinear form on a real vector space $X$ is just a map $B: X \times X \rightarrow \mathbb{R}$ that is linear in each argument.  (In other words, if you fix $y$, the function $B(x,y)$ that takes $x$ to $B(x,y)$ is linear, and similarly if you fix the other argument.)  It's called nondegenerate if the only $x \in X$ such that $B(x,y)=0$ for all $y$, is $x=0$.  And, of course, it's symmetric if for all $x,y$, $B(x,y)=B(y,x)$.

A closed domain of positivity of such a form is a maximal set $Y$ such that $B(Y,Y) \ge 0$.  Maximal means maximal in the ordering of sets by containment, i.e. $Y$ is not contained in any other set $Z$ satisfying $B(Z,Z) \ge 0$.  This notion was introduced, or at least used, by Max Koecher in the late 1950s, in work that led to the celebrated result, published in 1958 in "The Geodesics of Domains of Positivity"  (well, actually, "Die Geodatische von Positivitatsbereichen" (Mathematische Annalen 135 (1958), pp. 192--202)), that homogeneous self-dual cones in finite dimension are precisely the cones of squares in finite-dimensional formally real (aka Euclidean) Jordan algebras.  Indeed, probably the very interpretation of the main result of that paper as concerning homogeneous self-dual cones relies on the identification of domains of positivity with self-dual cones that I'm looking for a proof of.

If the form were positive semidefinite, i.e. $B(x,x) \ge 0$ for all $x$ (which implies symmetry) then a domain of positivity for would clearly  be a self-dual cone.  This is practically the definition of a self-dual cone.  The dual of a cone in a real inner product space is the set of all vectors $y$ whose inner product with everything in the cone is nonnegative---and the definition of an inner product on a real vector space is that it's a nondegenerate positive semidefinite bilinear form.  A self-dual cone is one that's equal to its dual cone.

For our definition of domain of positivity, the form was required only to be symmetric, not necessarily also positive semidefinite.  Nevertheless, according to things I've read, its domains of positivity are self-dual cones.   These domains are not necessarily unique, of course, although they are maximal, i.e. no one of them contains another).  Although I have a vague recollection of having seen a proof that they are self-dual, I haven't been able to find the paper or come up with a proof.

It's easy to prove that such a domain is a pointed, convex, closed cone.  A natural way to prove that it is a self-dual cone would be to exhibit a positive semidefinite form $B'$, depending on $B$ and possibly also on $Y$, such that $Y$ is a domain of positivity of $B'$.   An idea for how to do this involves the fact that such a form can be diagonalized: we can find a basis $x_i$ for the vector space such that the matrix with elements $M_{ij} := B(x_i, x_j)$ is diagonal, with diagonal elements $\epsilon_i \pm 1$.  The number of $-$ signs on the diagonal is the signature of the form.  A natural candidate for $B'$ is the Euclidean inner product $\langle u, v \rangle := \sum_i u_i \cdot v_i$ in the basis $x_i$ (i.e $u_i, v_i$ are the components of $u, v$ in this basis).  That is, we just change the $-1$'s to $+1$'s in the diagonal form of $B$.

Nondegenerate symmetric bilinear forms are interesting for a variety of reasons.  One of them is that they are closely related to the metric structure on a pseudo-Riemannian manifold.  Something like the following is true: you specify such a form at each point of the manifold, in such a way that the forms at the various points are nicely related to each other, and you've specified the geometric structure of a pseudo-Riemannian manifold.  (One restriction here, I believe, is that the signature of the forms has to be the same everywhere; the forms also need to vary somewhat smoothly, in a manner I should look up and summarize, but not now.)  For example, in the general relativistic description of spacetime, the spacetime manifold has signature $-1, 1, 1, 1$.  Or $1, -1, -1, -1$; people use different conventions.  I'm attracted to $1, -1, -1, -1$, because the odd-one-out corresponds in relativity theory to time, and this way, the forward and backward light cones are the (only!) domains of positivity for the form.  I.e. the form is $B(v, v') = tt'- (xx' + yy' + zz')$; we have $B(v, v) = t^2 - (x^2 + y^2 + z^2)$ (here $v = (t, x, y, z)$, etc...).  Interestingly, with the other choice of signature, the domains of positivity consist of spacelike vectors, and there is a continuum of them.  To get a picture of what's going on, consider one time and two space dimensions, with signature $1, -1, -1$.  You can visualize this in $\mathbb{R}^3$, with the vertical $t$ axis as time (associated with the $-1$ diagonal element of the form) and the horizontal $xy$ planes for constant time $t$ as a spacelike plane.  If you rotate the 45 degree line between $t$ and say the $x$ axis, around the $t$ axis, you get the boundary of a double cone, the forward and backward light cones.  But similar cones pointing along any ray in the $xy$ plane are clearly domains of positivity for the form.  I suspect lots of other cones---basically, any self-dual cone you can fit into the "conic doughnut" that is the closed complement of the double light-cone, i.e. into the spacelike (and null) vectors, are also domains of positivity for this form.

My main reason for interest in the problem isn't pseudo-Riemannian geometry, however.  More on the main reason later.  (It has to do with the Koecher result cited above).

If you found this problem first on mathoverflow, and you have the answer, please post your answer there, and link to it here if you feel like it; if you encountered it first here, please post the answer here indicating you encountered the problem here, and it would be nice if you'd also post it on mathoverflow indicating you found it on my blog.  We can have a little race between we happy few who read this blog, and the overflowing mathematicians.  I know who I'm betting on---your mission, readers, should you choose to accept it, and should any of you actually exist, is to prove me wrong!

(Thanks to Will Jagy, of MSRI, for noticing that I defined nondegeneracy wrong here at first: as requiring that the only $x$ for which $B(x,x)=0$ is $x=0$. This wrong definition, corrected above, of course says that the form has no nontrivial "isotropic" or "null" vectors (ones for which $B(x,x)=0$). And we certainly don't want to assume that! Sorry about the slip-up, which I dont think affected anything else in the post.)

Gross, Mueller, Colbeck, and Dahlsten: "All reversible dynamics in maximally non-local theories are trivial"

David Gross, Markus Mueller, Roger Colbeck, and Oscar Dahlsten have considered the "maximal non-signaling tensor product" of "boxlets", and shown that the reversible dynamics of this state space consists just of permutations of the systems (the boxlets) followed by reversible local transformations (i.e., ones on the individual boxlets).

What the heck does that mean, you ask?  Well, "boxlets" were introduced in several contexts.  In the "operational quantum logic" literature they're sometimes called "semiclassical test spaces".  In quantum foundations and informations, they were introduced as a generalization of a notion of Popescu and Rohrlich, who introduced the two-measurement, two-outcome-per-measurement boxlet in order to describe correlations between measurement results on distinct systems that can be stronger than quantum correlations, but still don't allow someone ("Alice") in possession of one of the systems to signal to the other ("Bob") just by making measurements on her system.

A "boxlet" is a system on which there are M distinct alternative measurements one can make, each with K outcomes.  (More complex versions allow different measurements to have different numbers of outcomes.)  The allowable states of a boxlet are given by specifying M probability distributions, each one over K alternatives:  for each measurement, the probabilities of each of its K outcomes.  All possible lists of M such distributions are allowed; this is a convex, compact subset of an MK dimensional vector space (one dimension for each probability).  The M normalization constraints mean that this set lies in an MK-M (i.e., M(K-1)) dimensional affine subspace  (higher dimensional generalization of the line, plane, etc... of high-school geometry).  The possible states of a pair of such systems are given by the "maximal tensor product" of a pair of these compact convex state spaces.  The technical definition of maximal tensor product of state spaces can be found here.  Another way of defining this is that it's the state space of the  Foulis-Randall tensor product  of the "test spaces" (definitions reviewed in Sections II and IV of this paper) describing each of the boxlets.  A test space is just a collection of subsets of some set; the elements of the set interpreted as measurement outcomes, and the subsets, called "tests", as measurements.  The semiclassical test space of a boxlet like the ones I described above just consists of a set of MK elements, partitioned into M sets of K elements.  A state on a (finite, like the ones in question) test space is a function from the set to the real numbers between zero and 1, i.e. to probabilities, such that for each test, the probabilities of the elements of the test add up to one.  The Foulis-Randall tensor product of two test spaces just takes their Cartesian product, and allows any probability assignments such that the "marginal states" obtained by fixing a measurement on one side and marginalizing all the joint distributions of this fixed measurement with measurements on the other side, is independent of which measurement is marginalized over.  That is, Alice can't signal to Bob (by affecting the probabilities of the outcomes of one or more of his measurements) just by her choice of measurement.

Now, a transformation of the state space is an affine map from the state space to itself (i.e. one that preserves convex combination, which seems only reasonable), and a reversible one is one that has an inverse that is also an affine map of the state space.  So what GMCD are saying is that, if you combine boxlets this way, there are no very interesting reversible dynamics:  just combinations of local reversible dynamics on the individual boxlets, and permuting the boxes amongst themselves.

An interesting question is, can one extend this result to maximal tensor products of *arbitrary* systems with convex state space (locally equipped, let's say, with the maximal set of possible effects)?

See the comments on the Information Causality thread at Dave Bacon's blog for a bit more discussion (and related interesting matters).