# 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.

# Quantum Information PhD studentships, University of Warwick

Animesh Datta informs me that there are three quantum information PhD studentships available at Warwick.  One looks theoretical, on "sources and signatures of quantum enhanced information processing", another involves "Processing and Controlling Quantum Information in NV Centre Architectures", and a third is on integrated photonics and intended for students from Singapore.

# 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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# Duke Ellington Sacred Concerts---Oxford University Jazz Orchestra and Schola Cantorum Oxford

Just came from an extraordinary concert at the Sheldonian Theatre in which the Oxford University Jazz Orchestra and the Schola Cantorum of Oxford performed a version of Duke Ellington's Sacred Concerts, with two pieces from composer and baritone Roderick Williams' Oxford Blues Service inserted in the Sacred Concert running order.  This constituted the second half of the program; I'll perhaps write in another post about the first half, which featured many good things but a sound balance that was slightly problematic at times, with the band occasionally drowning out the excellent guest soloist, alto saxophonist Nigel Hitchcock.  (I can't allude to the first half, though, without mentioning the really superb singing of first-year Olivia Williams in "Lookin' Back" and "Feelin' Good".)  In the second half, the balance was suddenly almost perfect, the bass acoustic throughout, the swing consistent and unforced, and immediately with the meditative baritone saxophone solo, originally performed by Harry Carney, that introduces "In the Beginning God" we were immersed in Duke Ellington's world of sound and his personal take on religion and spirituality.  Besides the excellence of the band, choir, and soloists, the conducting and preparation of the musicians by Schola conductor James Burton was clearly crucial to the success of this performance.  Nigel Hitchcock's beautiful alto playing was another crucial ingredient, but the regular band members who played key solos, like the baritone sax in "In the Beginning", the clarinet in "Freedom", the plunger-muted trumpet in "The Shepherd" did themselves and the Duke proud as well.  The Roderick Williams pieces "Gray Skies Passing Over" and "The Lord's Prayer"  fit in perfectly, being in a somewhat harmonically lush jazz-to-mid-twentieth-century pop vocal style very similar to parts of the Ellington vocal score, but more contrapuntal, with, I think, an echo of English, and even perhaps Renaissance, church music.

Besides getting real swing from the ensemble, Burton kept things relaxed but accurate, with a real dynamic range, the band in balance with the soloists (Ellington's writing presumably helps here too), expressive phrasing and control over the pace and development of each piece.  "Freedom" was another standout, done with intense feeling and great energy, drawing roars of approval from the audience.  But all the movements were executed superbly, and there were many such moments.  The tap-dancing of Annette Walker, in "David Danced Before The Lord" was another highlight.

This was an utterly professional-sounding performance that felt infused with the passion of people who are together reaching a level they may or may not have reached before, in the zone, giving the audience a musical experience not to be forgotten.  The Sacred Concerts may be a work best experienced live---it was certainly immensely effective, enjoyable, powerful, and moving in this performance.  Bass player and alto Lila Chrisp who is in both groups apparently had the idea that they should join forces in this piece.  I'm very grateful to everyone involved for making this happen and really filling the Sheldonian with the spirit---especially the spirit of Duke Ellington and his band.

# Oxford Kitchen

I've eaten at least twice---once tonight, and once or twice last June or so... at The Oxford Kitchen on Banbury Road in the heart of Summertown in North Oxford.  Extraordinarily good food.  Nice ambience and décor, stylish but relatively casual.  Exposed brick walls, large silk-screen Campbell's soup can print a nice humorous touch.  Set price menu is extraordinary value.  Hake with Jerusalem artichokes, parsley risotto and some kind of thin (like, microthin) veggie crisps just superb.  Nougatine with quince sorbet, thin (like, microthin) wafers tasting of burnt sugar, toasted almonds, and some crumbly stuff mind-blowingly good.  Pumpkin velouté starter good, not mind-blowing, but set off the wine I had very nicely.  At £18.50 for 2 courses, £15.50 for 2, it's one of the best foodie values I've encountered anywhere.   Wines are carefully chosen and if you have wine in restaurants you know how important that is.  2013 Boschendal Chenin Blanc, coastal region South Africa, £5/175ml is kind of the Chablis of Chenin Blancs---pretty dry, a bit of stereotypical apple flavor but mostly just tastes like really solid white wine a bit flinty-seeming maybe but that's probably just a bit of tannin, really long, flavorful finish no doubt stuck to the palate with a bit of that unobtrusive tannin, a smooth, just the slightest bit unctuous, and not at all hot (overalcoholic) mouthfeel.  To get a chardonnay of this quality, you'd be looking at good village-level white burgundy from the Côtes de Beaune or Nuits, like maybe a St.-Aubin at least (or maybe some particular thing, known to the cognoscenti but not to you unless from a restaurant like this...from a lesser appellation like Mercurey) at least, and 3-4x the price.  Ditto on the red side if you wanted a pinot noir as good as the Claro Reserva 2012 Pinot Noir from Chile ( also £5/175ml)... lush, a bit spicy and earthy, with generous berry flavors too but not a fruit or alcohol bomb, balanced.  These folks have done the work of tasting through dozens of ho-hum reasonably-priced wines to find the ones that deliver an experience that is usually (well, I don't usually spend that kind of money) much more expensive.  Talisker 10 year old, neat in a nice wide rocks glass with a bit of water to splash in, was a perfect finish.  Lively balance between peat and brighter more floral notes, filling the nose with perfume and crackling like fire on your tongue.

I had the tasting menu once last year, it too was superb.  Don't miss Oxford Kitchen if you're in town and can get up to Summertown for a meal.

# Des Américains à Paris

Via Ned Rorem, a really nice photo and audio montage promoting a program of a cappella choral music by some of Nadia Boulanger's American pupils: Ives, Copland, Bernstein, Barber, Stravinsky, Copland, Reich, Glass.   If I were in Luxembourg or deep Southwest France in March, I'd definitely go out of my way to hear this. High-resolution photos; it's worth making the video full-screen.

# Viet Cuong, Moth

For at least a few more days you can stream Moth, by classical composer Viet Cuong, performed, at the Midwest Band Clinic, by the Brooklyn Wind Symphony conducted by Jeff Ball, on Performance Today.  It is also available, probably more permanently, at his website and on his Soundcloud page.  I like the piece a lot.  The performance is excellent, really remarkable for an all-volunteer ensemble.  The style is fairly modern for PT, which is to say it is, roughly, in the idiom of tonal Western classical music from the 1920s and 1930s, with perhaps a smidgin of minimalism.  At first listen I thought it made clear use of the language of Stravinksy, especially Petrouchka and Le Sacre du Printemps, as well as of something resembling the post-Stravinsky and neoclassical phase of the 1930s, say, Milhaud, Poulenc, Constant Lambert, but without descending into pastiche.  On my second listen, with better sound, I was a bit taken aback by what I perceive as strong influence from Le Sacre, both in form and in content.   I am less startled by that after further listens.  Form-wise, it intersperses sections with ostinato, theme repetition (certainly key ingredients of Sacre), and other tension building devices (like modulation, especially stepwise upward modulation, which I don't think are found much in Sacre), with more pensive interludes, often tinged with a minor feel.  Just that kind of alternation is a main structural principal of Sacre.  As my references to neoclassicism and modulation above might suggest, there's somewhat more standard tonal content in Cuong's piece, thought it also has very strong Stravinsky-like "modal" or scalar elements, and occasional vaguely Iberian-sounding moments.  (As an aside, just thinking about harmony in Le Sacre makes me wonder if there is any standard dominant-to-tonic resolution at all in the piece---I think not, or not much.)

Cuong knows how to recombine and play with motives, scales, harmonic tropes and other elements to create interest, unify the piece and move things along in a satisfying way.  He shows this from the outset, with a clever motive consisting of a rising and descending scalar figure, played against a similar but inverted figure (or perhaps they are both fragments of the same extended figure that they evolve into, running up and down on flute, changing direction at different pitches), then relaxing into some Iberian-ish sounds.  At 1:30 we get melodic material very reminiscent of Le Sacre, and around 2:10, I think, the first hint of a four-note figure, which one might notate 3 4 2 1 in minor, also very reminiscent of Sacre (indeed it is very close---and would be identical if the last two notes were interchanged---to the initial four notes of a motive, 3 4 1 2 3 1 in minor with the last two notes twice as long as the preceding four, found in Sacre) that will become increasingly important.  Much of this material is developed and cleverly  combined through what sound to me like various key changes.  Around 3:30 things get more urgent, drums, with ostinato and repetition, especially of the four-note theme, and rising modulation.  (I wonder if there is some influence of John Adams' Harmonielehre here; I am reminded of it, but haven't listened to the Adams piece recently enough to tell.  Or maybe I should just can the speculation about influence.)  Around 4:10, quickly peak tension gives way to a mellow contrapuntal woodwind interlude, and there follows a long stretch with some alternation of faster and more complex passages, building a bit more each time, with pullbacks to this sort of mellowness.  Around 6:30 things seem to get more organized for a final buildup.  The ending, with an upward brass gliss emerging out of the ensemble to a momentarily held note, and then a sudden drop to tympani-punctuated chord, reminds me a little bit of Le Sacre too.

The program for this piece seems to be the gyrating flight of a moth before, and eventual immolation in, a flame, which is also in obvious parallel with Le Sacre's program, of a virgin obliged to dance herself to death in a pagan rite.  So I suspect the structural and idiomatic parallels to Le Sacre are no accident, although the overall tone is much lighter, and at 8'38 in this performance, the piece is of course much shorter.  I interpret these parallels, especially as dextrously integrated with harmonic movement at times quite uncharacteristic of Le Sacre, as a bit of a cheeky and light-hearted tour-de-force of compositional virtuosity.  The thematic material does have interest, but might be a little more on the generic side than ideal in places.  That is not really a problem in this piece.   I enjoyed some of the other pieces on his site but did find some of them a bit lacking in gripping melody.  Sound and Smoke I and II sound tailor-made for something like a fantasy movie soundtrack, and are extremely well done.  Part I sounds just as you might think from the subtitle "feudal castle lights", while Part II I found more distinctive.  I have a feeling that with some even stronger melodic material, perhaps some passages with some longer more sustained lines, Cuong could be really dangerous.  Hopefully Cuong will come up with more gripping melodic material in whatever way is necessary, whether from moments of personal inspiration or by ripping it off with exquisite taste à la Stravinsky if necessary.  (I exaggerate, Stravinsky fans... peace, I am one of you.)  Some of Cuong's other pieces show ability in more contemporary idioms.  He is only 24, a graduate student in composition at Princeton.  He is clearly getting a lot of recognition, as the list of awards, commissions, and performances on his webpage shows.  So he probably has a good career assured.  I hope he has his sights fixed on greatness; I'll be very interested to see what comes next.

Bonus:  On that December 12 PT stream, available for a few more days, the Brahms serenade (end of the 2nd hour) performed by the Sinfonia da Camera, if played on a good stereo, is magic.  (On first hearing through a cheap radio I was unimpressed.  Maybe it is all about the bass, although I think an undistorted treble helps too.)

# Isole e Olena 2005 Chianti Classico

A half-bottle of the 2005 Isole e Olena Chianti Classico, consumed a few days ago, was superb. Medium-bodied, with a fair bit of fine but fairly grippy tannins, this was elegant, and for a somewhat tannic wine, somewhat velvety and a pleasure to drink. It didn't seem at all tired or oxidized. Flavors predominantly dried cherry or other red fruits and a hint of pine, at least to my nose.  Super tasty.  Nice long finish. Easily my favorite of the Chiantis I've tasted. I don't remember how much I paid but recent vintages seem to go for $13-15 a half bottle,$20-25 a bottle, which although not cheap, is a bargain if they turn out this well. To judge by how youthful and tannic it still was at 9 years old, I'd guess this one needs to be aged to be at its best---at 9 years it was clearly getting there, but could probably go another 5 or more years and possibly get even more.

Isole e Olena don't appear to have a website; there is more information about them at Giuliana Imports, the Boulder-based importer of this bottle.  Since I have been encountering a lot of claims to the effect that a lot of writing about wine is basically just noise and fashion-following and strongly influenced by things other than the pure olfactory sensation of the wine, I'll point out that their description of the 2011 is very close to my description of the 2005, despite my not having read it (as befits a truly serious wine there are no olfactory notes on the label, either) which suggests to me anyway that Isole e Olena make this wine in a consistent style that can be identified by taste.  Of course this is just one observation, and there is definitely a lot of noise and influence from nonolfactory things like price and reputation and label appearance that enters into people's writing about wine.  (Mention of "red fruits"  or "dark berries"  could easily be influenced by the wine's color, for example, although in my opinion there is usually more to it than that.)  My point is that I think there is a genuine olfactory basis for some of this stuff too.

Based on perusing people's notes on the web (after writing mine), it seems that a lot of people liked this wine young, opinions diverged at about 3-7 years after vintage, and the consensus is more clearly positive over the last few years, suggesting it might have gone through a "dumb"  phase as many wines do during aging.  Also, some people seem to object to the relatively lighter-bodied style, which I happen to love when it is combined, as here, with intensity.  This is a serious producer that has been around at least as long as I've been tasting wine, and based on this sample, their Chianti is indeed a classic.  My sense is that if you have the ability to age it to 9-15 years after vintage you can't go wrong buying multiple bottles of this wine in any decent vintage.

What I just wrote is more meaningful than trying to assign some arbitrary number, but I guess on a Parkeresque 100 point scale, I'd give it something like a 92... and not in the inflated sense where anything you like gets 90---to get 90 or above in my book, a wine has to be at least a bit extraordinary.