This Week's Finds in Mathematical Physics (Week 299)
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From: John Baez on 14 Jun 2010 18:45 Also available at http://math.ucr.edu/home/baez/week299.html June 12, 2010 This Week's Finds in Mathematical Physics (Week 299) John Baez Two weeks ago I went to Oxford to attend a school on Quantum Information and Computer Science, and then a workshop on Quantum Physics and Logic. So, I'll start by telling you about those. I'll show you where you can see videos of the talks! You can learn about string diagrams and Frobenius algebras, and see how people use these in quantum physics. You can even see a program called "Quantomatic", which automatically carries out string diagram computations. And then I'll tell you about Aaron Fenyes' no-cloning theorem for classical mechanics. There was a lot more to the conference than this, but that's all I have the energy for. Why? Well, after I came home, my friend the combinatorist Bill Schmitt paid me a visit. He told me a lot of interesting stuff about about "pre-Lie algebras". These are algebraic gadgets with deep connections to trees, operads, and the work of Connes and Kreimer on renormalization in quantum field theory. So, I want to tell you about that stuff, too. Let's get started. You can see all the talks here: 1) Oxford Quantum Talks Archive, http://www.comlab.ox.ac.uk/quantum/content/ I've been using pictures called "string diagrams" for a long time here on This Week's Finds, and I've tried to explain them, but these talks give a nice systematic treatment: 2) Introduction to monoidal categories and graphical calculus. Lecture 1 by Chris Heunen, available at http://www.comlab.ox.ac.uk/quantum/content/1005005/ Lecture 2 by Jamie Vicary, available at http://www.comlab.ox.ac.uk/quantum/content/1005010/ With the help of these diagrams we can think about many, many things. In particular, we can use them to describe processes in quantum mechanics. Feynman diagrams are an example. But now people in quantum information theory are using them in very different ways. For example, they're using them to study "classical structures" in quantum mechanics. What's a classical structure? The basic idea is simple. You can't "clone a quantum". In other words, you can't build a "quantum copying machine" where you feed in a quantum system in an arbitrary state and have two identical copies pop out, both in the same state. Why not? Well, you could try to measure everything about your system and make a copy where all those measurements have the same values. But you can never succeed! Measuring one thing will change the values of other things you already measured, in uncontrollable random ways. So you can never know everything about your system... not all at once! So, it's impossible to make an exact copy. However, sometimes measuring one thing does *not* mess up the value of something else you measured. Quantities that get along this way are called "commuting observables". A "classical structure" is a set of commuting observables that's as big as possible. And for each classical structure, we can build a copying machine that works for *these* observables. For example, there's no way to take an electron, stick it in a copying machine, and duplicate everything about it. You can measure the spin of an electron along any axis. If you measure the spin along the x axis you get one of two results: "up" or "down". Similarly for the spin along the y axis. However, measuring the spin along the x axis messes up its spin along the y axis. So, you can build a machine that takes an electron, measures its spin along the x axis, and spits out two electrons in that same state: either "spin-up" or "spin-down" along the x axis. But if you put an electron with spin up along the y axis in this machine, it will not be correctly duplicated. In this example, there's a classical structure consisting of the spin along the x axis, and every function of this observable... but this classical structure does *not* include the spin along the y axis. Here's a quick sketch of how the math works. If you're a mathematician, this should be far less confusing than the prose you just suffered through. In fact, you may be left wondering why I turned such simple math into such murky prose. But that's typical of quantum mechanics: the math is crystal clear, but when you try to explain how it describes the real world, it starts sounding mysterious. Suppose H is a Hilbert space. If we were trying to build a quantum copying machine, it would be nice to have a linear operator like this: H -> H tensor H psi |-> psi tensor psi But this operator is not linear, because doubling psi would quadruple psi tensor psi. Life gets easier when we have a classical structure. An "observable" in quantum mechanics is a self-adjoint operator. So, a "classical structure" is a maximal set of commuting self-adjoint operators. If H is finite-dimensional, we can get any classical structure from an orthonormal basis. How? Just take all the self-adjoint operators that are diagonal in that basis. If you pick an orthonormal basis for H, say e_i, there's a unique linear operator that duplicates states in that basis: H -> H tensor H e_i |-> e_i tensor e_i So, this is how a classical structure gives a "duplication operator". Here's where the string diagrams come in. We can draw our duplication operator in a funny symbolic way like this: _______ / \ | | | | | | |\_______/| | | | | | | | | / \ / \ / \ / \ / _ \ / / \ \ /..... / \ .....\ / ` / \ ' \ / `/ \' \ | | | | | | | | \ / \ / \______/ \______/ This is a picture of a 2d surface where one circle comes in and two go out - a kind of metaphor for duplication. We'll see why it's a good metaphor in a minute. There's another thing we can do after we've picked an orthonormal basis for our Hilbert space H: we can define a "deletion operator" H -> C e_i |-> 1 which sends each state in the basis to the number 1. Here C is the complex numbers, a 1-dimensional Hilbert space. We can think of C as a kind of "garbage bin", and think of our operator as "throwing out" states in our basis. We can draw it as a cup-shaped thing, like this: _______ / \ | | | | | | |\_______/| | | | | | | | | \ / \_____/ This is not really a picture of a garbage bin, though it looks like that too. It's a picture of a 2d surface where one circle comes in and *none* go out! Now, the cool part is that our duplication and deletion operators satisfy rules that look very intuitive in terms of these pictures. For example, if we duplicate a state and then delete one copy, it's the same as not having done anything at all: _______ ________ / \ / \ | | | | | | | | | | | | |\_______/| |\________/| | | | | | | | | | | | | / \ | | / \ | | / \ | | / \ = | | / _ \ | | / / \ \ | | / / \ \ | | / / \ \ | | / / \ \ | | | | | | | | | | | | | | \ / | | | | \ / | ..... | | ..... | \____/ | ' ` | | ' ` | |' `| |' `| | | | | \ / \ / \______/ \______/ This rule is "topologically true": we can take the first picture and wiggle it and warp it until it looks like the second one. There are a bunch of other rules. Almost all of them are topologically true - exactly what you'd dream up by playing around with pictures of 2d surfaces. But to write down these rules, you need to notice that any operator between Hilbert spaces has an adjoint going the other way. So, besides duplication H -> H tensor H we have its adjoint H tensor H -> H which we draw just like duplication, except upside-down: ______ ______ / \ / \ | | | | | | | | | | | | |\______/| |\______/| | | | | | | | | \ \ / / \ \ / / \ \_/ / \ / \ / \ / \ / | | | | | ....... | |' `| | | | | \ / \_______/ and besides deletion H -> C we have its adjoint C -> H which we draw just like deletion, except upside-down: _____ / \ / \ | | | | | | | | | ....... | |' `| | | | | \ / \_______/ In math we call these four operators the "multiplication": m: H tensor H -> H the "unit": i: C -> H the "comultiplication": Delta: H -> H tensor H and the "counit": e: H -> C And we can summarize all the rules these operators obey by saying that H is a "special commutative dagger-Frobenius algebra". That's a mouthful, but as I said, almost all these rules come from topologically allowed manipulations on 2d surfaces. For example, here's why multiplication is commutative: __ __ __ __ / \ / \ / \ / \ | | | | | | | | |\__/| |\__/| |\__/| |\__/| | | | | | | | | \ \/ / | | | | \ /` / | | | | \ / ` / \ \ / / / /\ \ \ / / / ` / \ \ \_/ / / `/ \ \ / / /\ \ \ / | / \ | \ / | \__/ | = | | \ / | | \ / | | \ / | | | | | | | | | | | ... | | ... | |' `| |' `| | | | | \___/ \___/ What are all these rules? Well, back in "week268" I defined special commutative Frobenius algebras. So, if you go back and read that, the only extra thing you need to learn today is what's a "dagger-Frobenius" algebra. And that's just a Frobenius algebra that's also a Hilbert space, where the multiplication and unit are adjoint to the comultiplication and counit. For more, try these papers: 3) Bob Coecke and Dusko Pavlovic, Quantum measurements without sums, in The Mathematics of Quantum Computation and Technology, eds. Chen, Kauffman and Lomonaco, Chapman and Hall/CRC, New York, pp. 559-596. Also available as arXiv:quant-ph/0608035. 4) Jamie Vicary, Categorical formulation of quantum algebras, available as arXiv:0805.0432. 5) Bob Coecke, Dusko Pavlovic and Jamie Vicary, A new description of orthogonal bases, available as arXiv:0810.0812 But what's really going on with all these pictures of 2d surfaces? You see them a lot in topological quantum field theory. Indeed, in "week268" I explained that a commutative Frobenius algebra is exactly what we need to get a 2d topological quantum field theory, or TQFT for short. This is a way of making precise the idea that any of these pictures gives a well-defined operator - and warping or wiggling the picture doesn't change the operator. Even better, a commutative dagger-Frobenius algebra is exactly what we need to get a "unitary" 2d TQFT. This means that the upside-down version of a given picture gives the adjoint operator. So, we're seeing a curious fact: A CLASSICAL STRUCTURE ON A FINITE-DIMENSIONAL HILBERT SPACE GIVES A UNITARY 2D TQFT This sounds like it should be important, because it links two subjects with very different flavors: the foundations of quantum mechanics, and topological quantum field theory. But I have no idea what it really means. Maybe you can help me! But before you try your hand at this problem, I should warn you that we don't get *all* unitary 2d TQFTs from classical structures - because we don't get *all* commutative dagger-Frobenius algebras. We only get the "special" ones, which obey this extra rule: ___ ___ / \ / \ | | | | |\___/| |\___/| | | | | | | | | / \ | | / \ | | / __ \ | | | / \ | | | | \__/ | = | | \ / | | \ / | | \ / | | | | | | | | | | | ... | | ... | |' `| |' `| | | | | \___/ \___/ This rule is *not* topologically true. Indeed when it holds, our 2d TQFT is completely insensitive to how many handles our surface has. That makes it sort of boring. For a closed surface, the number of handles is called the "genus". So, the real puzzle is to understand this more mysterious slogan: A CLASSICAL STRUCTURE ON A FINITE-DIMENSIONAL HILBERT SPACE IS THE SAME AS A *GENUS-INDEPENDENT* UNITARY 2D TQFT I've been mulling this over for about a year now, with no great insights. Mathematically it's almost trivial. But physically, I can't tell if it's the tip of an interesting iceberg, or just a coincidence. These talks offered some extra clues: 6) Ross Duncan, Convexity, categorical semantics and the foundations of physics, video available at http://www.comlab.ox.ac.uk/quantum/content/1005102/ 7) Chris Heunen, Complementarity in categorical quantum mechanics, video and slides available at http://www.comlab.ox.ac.uk/quantum/content/1005115/ 8) Simon Perdrix, Classical-quantum graphical calculus, video available at http://www.comlab.ox.ac.uk/quantum/content/1005015/ For more details, try these papers: 9) Bob Coecke, Eric Oliver Paquette and Dusko Pavlovic, Classical and quantum structuralism, available as arXiv:0904.1997. 10) Bob Coecke and Ross Duncan, Interacting quantum observables: categorical algebra and diagrammatics, available as arXiv:0906.4725. 11) Bob Coecke, Quantum picturalism, available as arXiv:0908.1787. 12) Bob Coecke and Simon Perdrix, Environment and classical channels in categorical quantum mechanics, arXiv/1004.1598. Among other things, these papers say what you can *do* with a classical structure. You can do a bunch of things - and you can do them all very generally, because you can do them using just pictures. So, you don't need to be working in the category of finite-dimensional Hilbert spaces: any "dagger-compact category" will do. You can define a special commutative dagger-Frobenius algebra in any such category, and this gives a very general concept of classical structure. This is the sort of thing that makes category theorists drool. But I will restrain myself! I won't work in such generality. I'll just work with finite-dimensional Hilbert spaces, and just sketch a few things we can do with classical structures. I won't even explain how we can do them just using pictures... even though that's the really cool part. For starters, every classical structure determines a "phase group". If the classical structure comes from an orthonormal basis in the way I've described, its phase group consists of all unitary operators that are diagonal in this basis. But the cool part is that we can define this group just using pictures, and prove it's abelian, and so on. We can also use pictures to define "complementarity". In physics we say position and momentum are complementary observables because if you know everything about one, you know nothing about the other. But we can also say what it means for two classical structures to be complementary. For a finite-dimensional Hilbert space, any classical structure comes from an orthonormal basis - and two of them are "complementary" if they come from "mutually unbiased" bases, meaning bases e_i and f_j such that |<e_i,f_j>| is independent of i and j. This means that if you know precisely which state e_i your system is in, it has equal chances of being found in any of the states f_j. So, for example, the spin-up and spin-down states of an electron as measured along the x axis form one orthonormal basis. The spin-up and spin-down states as measured along the y axis form another. And these bases are mutually unbiased. So, knowing everything about the spin along the x axis tells you nothing about its spin along the y axis. And vice versa! When we have two complementary classical structures, we get two ways of making our Hilbert space into a Frobenius algebra. And these are related in a cool way: if we use the multiplication of the first, and the comultiplication of the second, we almost get a "bialgebra"! If we had a bialgebra, this relation would hold: | | \ / |\ /| \ / | \ / | \ / | \ / | \ / | \ | = | | / \ | / \ | / \ | / \ |/ \| / \ | | / \ where I'm writing the multiplication of our first Frobenius algebra like this: \ / \ / | | and the comultiplication of the second like this: | | / \ / \ But in fact this equation holds only up to a constant factor, so we get a "scaled bialgebra". I think this is fascinating because there's a constant interplay between Frobenius algebras and bialgebras in mathematics, and here's yet another example. Now, if you like these pictures, you've got to see "Quantomatic" in action: 13) Lucas Dixon, Quantomatic demo, video available at http://www.comlab.ox.ac.uk/quantum/content/1005019 14) Lucas Dixon, Ross Duncan, Aleks Kissinger and Alex Merry, Quantomatic, http://dream.inf.ed.ac.uk/projects/quantomatic/ This is a program that automatically carries out calculations involving these pictures that I've been drawing! It's a lot of fun. If you know quantum computation, you'll see that we can describe a lot of quantum logic gates, like controlled not gates and Hadamard gates, using these pictures. And if you know 2-categories, you'll realize that the *processes of rewriting diagrams* are actually 2-morphisms in a 2-category! So higher category is sneaking in to the subject here. I bet we'll see a lot more of it in years to come. There were many more interesting talks, but I'm running out of energy, so I just want to say one more thing about "cloning": 15) Aaron Fenyes, There's no cloning in symplectic mechanics, available at http://math.ucr.edu/home/baez/dual/no-cloning.pdf This paper argues that the laws of classical mechanics make it impossible to build a "cloning machine". If we had such a machine, we could put two boxes with balls in them into slots on top of the machine. The machine would copy the position and velocity of the first ball over to the second one. When we were done, the second ball would be a "clone" of the first: a perfect copy. Let me be a bit more precise. The boxes and the balls inside them are identically made. The first ball, the one we want to clone, has an arbitrary position and velocity. Well, of course its position is somewhere in the box! And if you like, we can say it's going no faster than 10 miles per hour. The second ball starts out in some fixed state. Let's say it has zero velocity and it's sitting right in a little dent in the middle of the box. We pop the boxes into the machine. The machine can open the tops of the boxes and insert sensors. When we press a big red button, the machine measures the position and velocity of the first ball. It then does whatever it wants, but after a while two boxes come out of the bottom of the machines... and then a bell rings. And when the bell rings, both balls have the same position and velocity that the first ball had when you pressed the button! A "no-cloning theorem" says you can't build a machine like this, given some assumptions on the laws of physics. The original no-cloning theorem was due to Wooters and Zurek: 16) W. K. Wootters and H. D. Zurek, A single quantum cannot be cloned, Nature, 299 (1982), 802-803. You can see a statement and proof here: 17) Wikipedia, No-cloning theorem, http://en.wikipedia.org/wiki/No-cloning_theorem This was a *quantum* no-cloning theorem. In fact it was very general: it wasn't about balls in boxes, it was about *any* quantum system where states are described by unit vectors in a Hilbert space and processes are described by unitary operators. There have been many no-cloning theorems since then, but what's new about Aaron Fenyes' result is that it applies to *classical* mechanics. Again, it's very general: it's about any classical system where states are described by points in a symplectic manifold, and processes are described by symplectomorphisms. If that sounds scary, well, be reassured that most classical mechanics problems fit into this framework. There are some that can't, but I bet Fenyes' result can be generalized to cover a lot of those, too. So, the big question is: how does this result square with the widely shared intuition that we *can* copy classical information: that we *can* exactly measure the position and momentum of a ball, say, and get two balls into that state? I don't know the answer, but I'd like to - so let me know if you figure it out. Part of the problem is that when we use phrases like "classical structure" in our study of quantum mechanics, we are *not* talking about full-fledged classical mechanics, in which symplectic structures are important. But even after we work through this semantic issue, there's a physics issue left to ponder. After I got back to Riverside, my friend Bill Schmitt visited me. As usual, we spent long evenings listening to music and talking about math. I told you about our last get-together in "week265". Back then, he told me about an amazing 588-page paper on Hopf algebras, combinatorics and category theory. By now that paper has grown into an 836-page book: 18) Marcelo Aguiar and Swapneel Mahajan, Monoidal functors, species and Hopf algebras, available at http://www.math.tamu.edu/~maguiar/a.pdf In his preface, Andre Joyal calls it "a quantum leap towards the mathematics of the future". Check it out! Now Bill is working with Aguiar on developing this theory further. There are some marvelous ideas here... but I'd rather tell you about something else: pre-Lie algebras. The name "pre-Lie algebra" suggests that we're about to do some "centipede mathematics". That's the cruel sport where you take a mathematical concept and see how many legs you can pull off and have it still walk. For example: you take the concept of group, remove the associative law and the identity element, obtaining the concept of "quasigroup"... and then see if there are still any theorems left. A "pre-Lie algebra" sounds like a Lie algebra with some legs pulled off. But actually it's an *associative* algebra with some legs pulled off! Any associative algebra gives a Lie algebra - but you don't need the full force of the associative law to play this game. It's enough to have pre-Lie algebra. That doesn't sound too interesting. But Bill convinced me that pre-Lie algebras are important. They were first named by Gerstenhaber: 19) M. Gerstenhaber, The cohomology structure of an associative ring, Ann. Math. 78 (1963), 267-288. who showed that the Hochschild chain complex of any ring, with grading shifted down by one, is a graded pre-Lie algebra. Later it was noticed that pre-Lie algebras show up in the combinatorics of trees, and are implicit in this old paper by Cayley: 20) Arthur Cayley, On the theory of the analytical forms called trees, Phil. Mag. 13 (1857), 172-176. The fun really starts when we relate these ideas to quantum field theory and operads... ....but first things first! The definition is simple enough. A pre-Lie algebra is a vector space A equipped with a bilinear product such that [L(a), L(b)] = L([a,b]) for every a, b in A. Huh? Here L(a) stands for left multiplication by a: L(a) b = ab and the brackets denote commutators, so [L(a), L(b)] = L(a) L(b) - L(b) L(a) and [a,b] = ab - ba Putting together these formulas, we see the a pre-Lie algebra is vector space equipped with a bilinear product satisfying this scary equation: a(bc) - b(ac) = (ab)c - (ba)c Now, it's obvious that every associative algebra is a pre-Lie algebra: just take this scary equation and erase the parentheses, and you'll get something true. But not every pre-Lie algebra is associative. I'll give you some examples in a minute. When we have an associative algebra, we get a Lie algebra with this bracket: [a,b] = ab - ba But we *also* get a Lie algebra this way from any pre-Lie algebra! That's why they're called "pre-Lie". To check this, take the scary equation above and use it to derive the Jacobi identity: [[a,b],c] + [[b,c],a] + [[c,a,],b] = 0 Try it! It's fun. Honest! Okay - so pre-Lie algebras are a cute generalization of associative algebras which are still good enough to give Lie algebras. But now you probably want me to explain how pre-Lie algebras show up in nature. I'll give three examples. The first is from geometry. The second is from quantum field theory. The third involves operads. First, given a manifold with a flat torsion-free connection D on its tangent bundle, we can make the space of tangent vector fields into a pre-Lie algebra by defining vw = D_v w Second, Connes and Kreimer noticed a certain amazing group that plays an important role in the renormalization of quantum field theories: 21) Alain Connes and Dirk Kreimer, Hopf algebras, renormalization and noncommutative geometry, Commun. Math. Phys. 199 (1998), 203-242. Also available as arXiv:hep-th/9808042. They built this group from Feynman diagrams. How? As you probably know, it's good to build groups starting from Lie algebras. And that's basically what they did. But what Lie algebra did they use? The answer is easy if you know about pre-Lie algebras. But first I want to sketch the usual story. This is much more lengthy and technical... but I want to run you through it, so you can fully appreciate the elegance of the slick approach. In the usual approach, you need to know that any Lie algebra gives rise to something called a "universal enveloping algebra". This is a cocommutative Hopf algebra, so its dual - defined in a careful way - is a commutative Hopf algebra. Connes and Kreimer started by getting their hands on this commutative Hopf algebra. Then they worked back to the Lie algebra. To get their commutative Hopf algebra, they began with vector space whose basis consists of Feynman diagrams. But they also found it helpful to consider a simpler problem, where you start with a vector space whose basis consists of rooted forests. A "rooted tree" looks like this: o o o o \|/ / o o \ / o | o The vertex at the bottom is called the "root". A finite collection of rooted trees is called a "rooted forest" o o o o \| / | o o o o o \ / \ / | o o o o o | | \|/ o o o o Let me show you how to take the vector space whose basis consists of all rooted forests, and make that into a commutative Hopf algebra. To do this, we need to give our vector space a multiplication and a comultiplication. And it's enough to say how these work on basis vectors, which are rooted forests. To multiply two rooted forests, we simply set them side by side to get a new rooted forest. This multiplication is obviously associative. It's also commutative, since we don't care about any "ordering" or "planar structure" on our rooted forests. In short, multiplication is boring. The fun part is comultiplication! To comultiply a rooted forest, we go through all ways of slicing it in a roughly horizontal way. Each slice gives two rooted forests: one below the slice, and one above. Then we form a big sum, where each slice contributes a term (below) tensor (above) where the first factor is the forest below the slice, while the second is the forest above the slice. You can see pictures of how this works in Connes and Kreimer's paper. Starting from this Hopf algebra, you can get a Lie algebra. But there's a vastly quicker way to get this Lie algebra... if you know about pre-Lie algebras, that is. It's: THE LIE ALGEBRA COMING FROM THE FREE PRE-LIE ALGEBRA ON ONE GENERATOR! That's what I call slick. Instead of paragraphs of theorems and pictures, a single devastatingly efficient phrase. But of course we need to see what's lurking in this phrase. Where did the rooted forests go? To answer this, you need to check that the free pre-Lie algebra on one generator has a basis given by rooted trees. Then its universal enveloping algebra will have a basis given by rooted forests! So, the key question is: why does the free pre-Lie algebra on one generator have a basis given by rooted trees? Let me quickly sketch the answer Bill gave me. This may sound a bit cryptic, but I want to write it down before I forget. Suppose you have rooted trees a and b and you attach b to a. More precisely: suppose you connect the root of b to some vertex of a using a new edge, forming a new rooted tree. You can do this in lots of ways, so you'll get a linear combination of trees, say ab. And this how multiplication in the free pre-Lie algebra on one generator works! We can summarize this as follows: ab = b | a Here the picture stands for *any* way of attaching b to a. We should really sum over all of them. When you form a product like (ab)c, different things can happen. We can summarize the possibilities like this: c | (ab)c = b + b c | \ / a a The point is that we can either attach the root of c to a vertex in b, or a vertex in a. There are fewer possibilities when we form a(bc): c | a(bc) = b | a so (ab)c - a(bc) = b c \ / a Now switch b and c in this equation! We get (ac)b - a(cb) = c b \ / a Our rooted trees are not planar, so the answer is really the same: b c c b \ / = \ / a a So, we have (ab)c - a(bc) = (ac)b - a(cb) and this is the definition of a pre-Lie algebra! This calculation reveals the secret meaning of pre-Lie algebras. The secret is that pre-Lie algebras are all about attaching two things by connecting a special point of the first to an arbitary point of the second! Rooted trees are the universal example, so they give the free pre-Lie algebra on one generator. This calculation also reveals that a pre-Lie algebra is really a vector space with a bilinear product whose "associator" {a,b,c} = (ab)c - a(bc) is symmetric in the last two variables. Finally, let me tell you the third way to get pre-Lie algebras: from operads. I'll assume you know about linear operads, which I explained in "week282". Suppose O is any linear operad. Let A be the free O-algebra on one generator. Then A becomes a pre-Lie algebra in a god-given way! You should have seen this coming, since operads are related to trees. The details are explained here: 22) Fr?d?ric Chapoton and Muriel Livernet, Pre-Lie algebras and the rooted trees operad, Int. Math. Res. Not. 2001 (2001), 395-408. Also available as arXiv:math/0002069. The idea is that A is a lot like O, since we get elements of the free O-algebra on one generator by hitting that generator with operations in O. More precisely, we have A = sum O_n / S_n Here O_n is the space of n-ary operations in O, which is acted on by the permutation group S_n. So, we can draw an element of A like this: o o o \ | / ----- | a | ----- | o where a is an n-ary operation in O, but we don't care how the branches of this little tree are permuted. We can multiply two guys like this by summing over all ways of attaching the output of one to an input of the other and composing them using our operad. And, thanks to the "secret meaning of pre-Lie algebras", this makes A into a pre-Lie algebra! This is nice. But over dinner, James Dolan, Bill Schmitt and I came up with an even slicker construction which seems to give the same multiplication on A. A is the free O-algebra on one generator, say x. So, for any element a in A, there's a unique O-algebra endomorphism f(a): A -> A sending x to a. Note that f_x is the identity. By the general philosophy that "an infinitesimal endomorphism is a derivation", the operator (d/dt) f(x + ta)|_{t = 0} is a derivation of A. (For certain familiar sorts of algebras, you may aready know what a derivation is. These are just special cases of a general concept of derivation for O-algebras. I leave it as an exercise to reinvent this general concept.) Now, we can define a multiplication on A by ab = (d/dt) f(x + ta)(b)|_{t = 0} And this is the same as the multiplication I just described. Can we use this slick description to more efficiently prove that A is a pre-Lie algebra? I don't know. One last thing: Any linear operad gives a pre-Lie algebra. But pre-Lie algebras are themselves algebras of a linear operad! This leads to curious self-referential situation, and a nice puzzle. There's a linear operad whose algebras are pre-Lie algebras. As we have seen, for any linear operad O, the free O-algebra with one generator becomes a pre-Lie algebra. So: the free pre-Lie algebra on one generator becomes a pre-Lie algebra in this way. But of course it already *is* a pre-Lie algebra! Do these pre-Lie structures agree? If you give up, you can find the answer here: 23) Dominique Manchon, A short survey on pre-Lie algebras, available at http://math.univ-bpclermont.fr/~manchon/biblio/ESI-prelie2009.pdf But if you want to solve this puzzle on your own, it helps to think about what the operad for pre-Lie algebras looks like. Let's call it PL. It's not hard to guess what it looks like, given everythign I've told you so far. I've told you that the free pre-Lie algebra on one generator has a basis given by rooted trees. And I've told you a general fact: the free O-algebra on one generator is sum O_n / S_n So, taking O = PL, it should come as no surprise that PL_n, the space of n-ary operations in PL, has a basis given by rooted trees with n vertices *labelled by numbers 1 through n*. Modding out by S_n just gets rid of those labels! But how do you compose operations in PL? ----------------------------------------------------------------------- Quote of the Week: I can learn only by teaching. - John Wheeler ----------------------------------------------------------------------- Addenda: Here are some examples that illustrate the comultiplication in the Hopf algebra of rooted forests, taken from Connes and Kreimer's paper: http://math.ucr.edu/home/baez/connes_kreimer_coproduct.jpg Here Delta stands for the comultiplication. BUT BEWARE: these trees have their root on top - I hear that's how they grow in Europe. So, these pictures are upside-down compared to the description I gave earlier! Now to comultiply a rooted forest we slice it in lots of ways and form a sum of terms (above) tensor (below) For example, the terms in the last line of the picture come from these ways of slicing the given tree: http://math.ucr.edu/home/baez/connes_kreimer_admissible_cuts.jpg The first way here is really two ways: we can have the whole tree be above the slice and empty set below, or the empty set above and the whole tree below. So, we get 5 ways to slice the tree, and comultiplying it gives a sum of 5 terms, two of which are equal. Here's another nice introduction to pre-Lie algebras and the Hopf algebra of rooted forests: 24) Frederic Chapoton, Operadic point of view on the Hopf algebra of rooted trees, available at http://www-math.unice.fr/~patras/CargeseConference/ACQFT09_FredericCHAPOTON.pdf BUT BEWARE: he is using right pre-Lie algebras, where I was using left ones. So, his product obeys the law [R(a), R(b)] = R([a,b]) where R(a) stands for right multiplication by a. Right pre-Lie algebras work just as well as left ones. For more discussion visit the n-Category Cafe at: http://golem.ph.utexas.edu/category/2010/06/this_weeks_finds_in_mathematic_60.html ----------------------------------------------------------------------- Previous issues of "This Week's Finds" and other expository articles on mathematics and physics, as well as some of my research papers, can be obtained at http://math.ucr.edu/home/baez/ For a table of contents of all the issues of This Week's Finds, try http://math.ucr.edu/home/baez/twfcontents.html A simple jumping-off point to the old issues is available at http://math.ucr.edu/home/baez/twfshort.html If you just want the latest issue, go to http://math.ucr.edu/home/baez/this.week.html
From: Gerry Quinn on 15 Jun 2010 16:52 In article <hv3o7s$o99$1(a)glue.ucr.edu>, baez(a)math.removethis.ucr.andthis.edu says... > Let me be a bit more precise. The boxes and the balls inside them > are identically made. The first ball, the one we want to clone, has > an arbitrary position and velocity. Well, of course its position is > somewhere in the box! And if you like, we can say it's going no faster > than 10 miles per hour. The second ball starts out in some fixed > state. Let's say it has zero velocity and it's sitting right in a > little dent in the middle of the box. > > We pop the boxes into the machine. The machine can open the tops of > the boxes and insert sensors. When we press a big red button, the > machine measures the position and velocity of the first ball. It > then does whatever it wants, but after a while two boxes come out > of the bottom of the machines... and then a bell rings. > > And when the bell rings, both balls have the same position and > velocity that the first ball had when you pressed the button! [--] > So, the big question is: how does this result square with the widely > shared intuition that we *can* copy classical information: that we > *can* exactly measure the position and momentum of a ball, say, and > get two balls into that state? > > I don't know the answer, but I'd like to - so let me know if you > figure it out. Part of the problem is that when we use phrases like > "classical structure" in our study of quantum mechanics, we are *not* > talking about full-fledged classical mechanics, in which symplectic > structures are important. But even after we work through this > semantic issue, there's a physics issue left to ponder. My intuition is that classical systems *have* a position and momentum. It is not so obvious that they can be measured exactly, although they can clearly in principle be measured to an arbitrary degree of precision. Wouldn't it require infinite energy to specify an exact location? [We are talking about copying rather than measurement here, but the same principles should apply.] - Gerry Quinn
From: Douglas Eagleson on 16 Jun 2010 12:48 On Jun 14, 10:45 pm, b...(a)math.removethis.ucr.andthis.edu (John Baez) wrote: > Also available athttp://math.ucr.edu/home/baez/week299.html > > June 12, 2010 > This Week's Finds in Mathematical Physics (Week 299) > John Baez > > Two weeks ago I went to Oxford to attend a school on Quantum > Information and Computer Science, and then a workshop on Quantum > Physics and Logic. So, I'll start by telling you about those. > > I'll show you where you can see videos of the talks! You can learn > about string diagrams and Frobenius algebras, and see how people use > these in quantum physics. You can even see a program called > "Quantomatic", which automatically carries out string diagram > computations. And then I'll tell you about Aaron Fenyes' no-cloning > theorem for classical mechanics. There was a lot more to the > conference than this, but that's all I have the energy for. > > Why? Well, after I came home, my friend the combinatorist Bill > Schmitt paid me a visit. He told me a lot of interesting stuff about > about "pre-Lie algebras". These are algebraic gadgets with deep > connections to trees, operads, and the work of Connes and Kreimer on > renormalization in quantum field theory. So, I want to tell you about > that stuff, too. > > Let's get started. You can see all the talks here: > > 1) Oxford Quantum Talks Archive,http://www.comlab.ox.ac.uk/quantum/content/ > > I've been using pictures called "string diagrams" for a long time here > on This Week's Finds, and I've tried to explain them, but these talks > give a nice systematic treatment: > > 2) Introduction to monoidal categories and graphical calculus. > Lecture 1 by Chris Heunen, available athttp://www.comlab.ox.ac.uk/quantum/content/1005005/ > Lecture 2 by Jamie Vicary, available athttp://www.comlab.ox.ac.uk/quantum/content/1005010/ > > With the help of these diagrams we can think about many, many things. > In particular, we can use them to describe processes in quantum > mechanics. Feynman diagrams are an example. But now people in > quantum information theory are using them in very different ways. For > example, they're using them to study "classical structures" in quantum > mechanics. > > What's a classical structure? The basic idea is simple. You can't > "clone a quantum". In other words, you can't build a "quantum copying > machine" where you feed in a quantum system in an arbitrary state and > have two identical copies pop out, both in the same state. > > Why not? Well, you could try to measure everything about your system > and make a copy where all those measurements have the same values. > But you can never succeed! Measuring one thing will change the values > of other things you already measured, in uncontrollable random ways. > So you can never know everything about your system... not all at once! > So, it's impossible to make an exact copy. > > However, sometimes measuring one thing does *not* mess up the value > of something else you measured. Quantities that get along this way are > called "commuting observables". A "classical structure" is a set of > commuting observables that's as big as possible. And for each > classical structure, we can build a copying machine that works for > *these* observables. > > For example, there's no way to take an electron, stick it in a copying > machine, and duplicate everything about it. You can measure the spin > of an electron along any axis. If you measure the spin along the x > axis you get one of two results: "up" or "down". Similarly for the > spin along the y axis. However, measuring the spin along the x > axis messes up its spin along the y axis. > > So, you can build a machine that takes an electron, measures its spin > along the x axis, and spits out two electrons in that same state: > either "spin-up" or "spin-down" along the x axis. But if you put an > electron with spin up along the y axis in this machine, it will not be > correctly duplicated. > > In this example, there's a classical structure consisting of the spin > along the x axis, and every function of this observable... but this > classical structure does *not* include the spin along the y axis. > > Here's a quick sketch of how the math works. If you're a > mathematician, this should be far less confusing than the prose you > just suffered through. In fact, you may be left wondering why I > turned such simple math into such murky prose. But that's typical > of quantum mechanics: the math is crystal clear, but when you try > to explain how it describes the real world, it starts sounding mysterious.. > > Suppose H is a Hilbert space. If we were trying to build a quantum > copying machine, it would be nice to have a linear operator like this: > > H -> H tensor H > > psi |-> psi tensor psi > > But this operator is not linear, because doubling psi would quadruple > psi tensor psi. > > Life gets easier when we have a classical structure. An "observable" > in quantum mechanics is a self-adjoint operator. So, a "classical > structure" is a maximal set of commuting self-adjoint operators. If > H is finite-dimensional, we can get any classical structure from an > orthonormal basis. How? Just take all the self-adjoint operators > that are diagonal in that basis. > > If you pick an orthonormal basis for H, say e_i, there's a unique > linear operator that duplicates states in that basis: > > H -> H tensor H > > e_i |-> e_i tensor e_i > > So, this is how a classical structure gives a "duplication operator". > > Here's where the string diagrams come in. We can draw our duplication > operator in a funny symbolic way like this: > > _______ > / \ > | | > | | > | | > |\_______/| > | | > | | > | | > | | > / \ > / \ > / \ > / \ > / _ \ > / / \ \ > /..... / \ .....\ > / ` / \ ' \ > / `/ \' \ > | | | | > | | | | > \ / \ / > \______/ \______/ > > This is a picture of a 2d surface where one circle comes in and two > go out - a kind of metaphor for duplication. We'll see why it's a > good metaphor in a minute. > > There's another thing we can do after we've picked an orthonormal > basis for our Hilbert space H: we can define a "deletion operator" > > H -> C > > e_i |-> 1 > > which sends each state in the basis to the number 1. Here C is the > complex numbers, a 1-dimensional Hilbert space. We can think of C > as a kind of "garbage bin", and think of our operator as "throwing out" > states in our basis. We can draw it as a cup-shaped thing, like this: > > _______ > / \ > | | > | | > | | > |\_______/| > | | > | | > | | > | | > \ / > \_____/ > > This is not really a picture of a garbage bin, though it looks like > that too. It's a picture of a 2d surface where one circle comes in > and *none* go out! > > Now, the cool part is that our duplication and deletion operators > satisfy rules that look very intuitive in terms of these pictures. > For example, if we duplicate a state and then delete one copy, it's > the same as not having done anything at all: > > _______ ________ > / \ / \ > | | | | > | | | | > | | | | > |\_______/| |\________/| > | | | | > | | | | > | | | | > / \ | | > / \ | | > / \ | | > / \ = | | > / _ \ | | > / / \ \ | | > / / \ \ | | > / / \ \ | | > / / \ \ | | > | | | | | | > | | | | | | > \ / | | | | > \ / | ..... | | ..... | > \____/ | ' ` | | ' ` | > |' `| |' `| > | | | | > \ / \ / > \______/ \______/ > > This rule is "topologically true": we can take the first picture > and wiggle it and warp it until it looks like the second one. > > There are a bunch of other rules. Almost all of them are topologically > true - exactly what you'd dream up by playing around with pictures of > 2d surfaces. But to write down these rules, you need to notice that > any operator between Hilbert spaces has an adjoint going the other > way. So, besides duplication > > H -> H tensor H > > we have its adjoint > > H tensor H -> H > > which we draw just like duplication, except upside-down: > > ______ ______ > / \ / \ > | | | | > | | | | > | | | | > |\______/| |\______/| > | > ... > > read more » Beaz goes to school finally. If you read the stuff now, at least he is out of only first predicate. Somebody at the school taught him transform logic finally. Re: "Measuring one thing will change the values of other things you already measured, in uncontrollable random ways. So you can never know everything about your system... not all at once!" This is also said: my quote: "A measure causes observation." It is the first third transform encountered in quantum theory. And should never be read in first theory logic. Sending PhDs to logic school was supposed to be all figured out as part of the school program. Douglas Eagleson eaglesondouglaas123(a)yahoo.com
From: Nicolaas Vroom on 18 Jun 2010 05:23 "John Baez" <baez(a)math.removethis.ucr.andthis.edu> schreef in bericht news:hv3o7s$o99$1(a)glue.ucr.edu... > > What's a classical structure? The basic idea is simple. You can't > "clone a quantum". In other words, you can't build a "quantum copying > machine" where you feed in a quantum system in an arbitrary state and > have two identical copies pop out, both in the same state. > > Why not? Just try it. Try to copy an electron, specific its spin. You will see you cannot. (I expect) The simplest copy process is DNA copying. What you get is RNA. If you copy RNA you get DNA. Compared to such a machine an electron copy machine will switch polarity of the copy. Up will be copied as Down. Anyway that is what you want. The reality is more complex. In reality wat you have are more or less two gyroscopes. What you want is that the direction of one stays fixed, while the direction of other one should be steered in a particular direction. This is a measurement and control problem. (Lisa in space) Related to electrons IMO the biggest problem lies in the control part because of the interactions involved. > Well, you could try to measure everything about your system > and make a copy where all those measurements have the same values. > But you can never succeed! Measuring one thing will change the values > of other things you already measured, in uncontrollable random ways. > So you can never know everything about your system... not all at once! > So, it's impossible to make an exact copy. As I said above: this is a measurement and control problem of both the source/model and its offspring with the emphasis on control. Even if you could measure both directions it does not mean that you can control the direction of the offspring without changing the direction of the source/model. > However, sometimes measuring one thing does *not* mess up the value > of something else you measured. Quantities that get along this way are > called "commuting observables". A "classical structure" is a set of > commuting observables that's as big as possible. And for each > classical structure, we can build a copying machine that works for > *these* observables. I agree at the level of DNA's, PC logic and Lisa. > For example, there's no way to take an electron, stick it in a copying > machine, and duplicate everything about it. You can measure the spin > of an electron along any axis. If you measure the spin along the x > axis you get one of two results: "up" or "down". Similarly for the > spin along the y axis. However, measuring the spin along the x > axis messes up its spin along the y axis. See below. > So, you can build a machine that takes an electron, measures its spin > along the x axis, and spits out two electrons in that same state: > either "spin-up" or "spin-down" along the x axis. But if you put an > electron with spin up along the y axis in this machine, it will not be > correctly duplicated. I doubt if you can even build a machine which only copies in one direction without effecting the direction of the source/model. As I said: this is more than a measurement problem. It is also more than a simple Yes No problem in 3 dimensions, with only 8 discrete states: (1,1,1) (1,1,-1).....(-1,-1,-1) In reality the direction of the spin along each axis is a continuous variable with the value 0 as a possiblity. I fully agree with the topic underdiscussion: You cannot "clone a quantum". IMO this makes the following statement more acceptable: The future of quantum computers is bleak. Anyway this whole issue is much more a physical than a mathematical exercise. Nicolaas Vroom http://users.pandora.be/nicvroom/
From: Nicolaas Vroom on 18 Jun 2010 07:16 "John Baez" <baez(a)math.removethis.ucr.andthis.edu> schreef in bericht news:hv3o7s$o99$1(a)glue.ucr.edu... > What's a classical structure? The basic idea is simple. You can't > "clone a quantum". In other words, you can't build a "quantum copying > machine" where you feed in a quantum system in an arbitrary state and > have two identical copies pop out, both in the same state. > > Why not? Well, you could try to measure everything about your system > and make a copy where all those measurements have the same values. > But you can never succeed! Measuring one thing will change the values > of other things you already measured, in uncontrollable random ways. > So you can never know everything about your system... not all at once! > So, it's impossible to make an exact copy. > > However, sometimes measuring one thing does *not* mess up the value > of something else you measured. Quantities that get along this way are > called "commuting observables". A "classical structure" is a set of > commuting observables that's as big as possible. And for each > classical structure, we can build a copying machine that works for > *these* observables. > > For example, there's no way to take an electron, stick it in a copying > machine, and duplicate everything about it. You can measure the spin > of an electron along any axis. If you measure the spin along the x > axis you get one of two results: "up" or "down". Similarly for the > spin along the y axis. However, measuring the spin along the x > axis messes up its spin along the y axis. > > So, you can build a machine that takes an electron, measures its spin > along the x axis, and spits out two electrons in that same state: > either "spin-up" or "spin-down" along the x axis. But if you put an > electron with spin up along the y axis in this machine, it will not be > correctly duplicated. In the above text the impression is given that the reason why you cannot "clone a quantum" is mainly a measuring problem. I do not agree with that. IMO it is combination between measuring and control with the emphasis on control. If you want to copy an electron (specific its spin) than in fact what you want is to change the direction of the spin of a second electron such that runs parallel with the direction of the first without changing the direction of the spin of the first. The first issue is that if "you" want to copy the spin than "you" must know the initial direction of both spins. The problem that during the measurement process "you" will change the spin of the electron. (i.e. both) For the second electron this is even more critical because you must now its initial direction in order to steer the spin towards the initial position of the first. IMO, I expect, this control problem is the most difficult, partly because this requires a continuous monitoring of the spin of the second electron. For a machine which copies only in one direction the issues are identical. The article also gives the impression that measuring the spin is a "Yes" "No" problem i.e. a discreet problem and that the spin only has 8 different states i.e. (1,1,1), (1,1,-1), .... (-1,-1,-1). This is not the case In fact the direction along the x axis is a continuous variable with zero being the most difficult value. IMO all what is involved is much more a physical than a mathematical issue. Nicolaas Vroom http://users.pandora.be/nicvroom/
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