Are you looking for an interesting idea?

June 7, 2013

A common complaint I’ve heard from my colleagues is that their favourite and best work is their least appreciated and least cited work. (I certainly feel this way.) It is not hard to imagine why this is: probably their best work is the one that contains the most unfamiliar and original ideas, the most difficult calculations, and is probably the least clear to anyone except the author because it is very difficult to explain something truly new.

(This all rather puts me in mind of the quote:

Don’t worry about people stealing an idea. If it’s original, you will have to ram it down their throats.

Howard H. Aiken, as quoted in Portraits in Silicon (1987) by Robert Slater

)

So, if you are looking for something truly interesting to work on why not pick an author you respect and find their oldest least cited paper. (I.e., don’t choose a new one that simply hasn’t been read yet.) Read it forwards and backwards until you completely understand it.

There will surely be some treasure buried in there.

And hey, you can be pretty sure that absolutely noone else will be working on the same thing.

(My personal pick is the intriguing paper of Bill Wootters on entanglement and parallel transport.)

 


Topological order for mixed states

May 29, 2013

In this post I’d like to take the opportunity to talk about the problem of defining what is meant by topological order at finite temperature. I hope to convince you that there is a natural, operationally well motivated, definition for what this means.

But first let’s recall what is meant by topological order.

There appear to be quite a few related yet not quite obviously equivalent definitions floating around: some define it with reference to edge modes, some define it in terms of the topological entanglement entropy, and yet another definition concerns ground state degeneracy on a surface of genus {g}.

The definition I like the most, and which I’ll exploit for the rest of this post is due variously to Xiao-Gang Wen and coauthors and is in terms of local deformation operations. (I’m using the terminology of “local deformation operations” (LDO) instead of Wen’s “LU” to avoid some confusion later on.) To explain this definition we’ll need to work with infinite quantum lattice systems.

Read the rest of this entry »


Guest post on Bohmian Mechanics, by Reinhard F. Werner

May 13, 2013

I went to a conference at the end of April in Bielefeld, entitled “Quantum Mechanics without observers III”, which was the meeting of a European network devoted to the Foundations of Quantum Mechanics. The network has a high percentage of members of the Neo-Bohmian school, so I was a bit of an outside observer, coming mainly to see what that school had been up to in the last couple of years. As the odd operational quantum mechanics guy at a convention of hard-core realists I was therefore part of a small minority. The following impressions were written up mainly as a feedback for the participants of the workshop, presenting an outside angle. I feel encouraged to send this out by a remark of Jürg Fröhlich at the workshop that we should not, out of misplaced politeness, refrain from criticising each other’s scientific positions. I am grateful to Tobias Osborne for posting this text on his blog. This will make it easier for me to collect the comments.

On one level the workshop felt to me like a fundamentalist congregation. To someone not sharing the belief in that particular brand of realism it was especially striking how this belief was enforced time and again by the usual forms of discourse in a one-faith community. Several speakers enjoyed drawing laughter by exposing supposedly absurd quotes from famous physicists, and the hagiography of John Bell kept an amazingly large number of speakers busy. Of course, this saint was claimed whole for the Bohmian camp, and any subtlety left in his writings thoroughly flattened. We heard a sermon on the question whether upon splitting a box with a quantum particle the particle is truly in one or the other box, and a chairman who actually asked the experimentalist speakers to declare their faith in this matter.

(I should add here that not all the Bohmian talks were bad. Stefan Teufel did a good job at presenting an argument. That’s how a workshop can become fruitful.)

So what is the Bohmian belief? I am one of those who see in “local realism” a conjunction of two concepts: locality and realism. Bell’s argument shows that this conjunction is not in agreement with the observed facts. The separation between the concepts is not difficult, something that I expect students to understand. Quantum mechanics as I understand it takes the local option, in the sense of not containing spooky signals. Of course, if you insist on a classical “realist” description they are all over the place. It is clear that if you are altogether unwilling to even debate realism (or “classicality”) you can soak your language in it to such a degree that it would seem like an undeniable demand of basic logic. But that is just sloppy thinking, which is not improved by any degree of shouting or religious devotion. “Realism” has a double meaning in this context. On one hand, it is a basic principle of science, the demand to check any claims against reality, to go for empirical content rather than storytelling. On the other hand, it stands for a particular way of constructing a theory, namely assuming that every individual system has an in principle complete description in terms of its properties (“classicality”). The irony of quantum mechanics is that it brings these two into conflict. Those insisting on the second kind of realism, like the Bohmian school, thereby lose sight of the first: Bohmian trajectories have no connection to empirical fact, and even the Bohmian theory itself claims no connection. So they are just a piece of fantasy. You may call the trajectories the reality givers (I even heard “realizors”) of the theory, and base an “ontology” on them. But they are still but a figment of your imagination.

It goes with this status that there is no way to answer questions about the possible structure of this reality, and to make basic theoretical choices except by appeal to lack of imagination. Why take Nelson’s diffusion constant equal to zero and not one (which gives the pleasing balance between forward and backward derivatives), or maybe 7? Why take wave functions as the description of single systems rather than density operators? I could give some arguments for that. You can drive Bohmian trajectories with density operators just as well, and they would tend to be less singular. Why go for position as the only “real” feature of particles, and not include other variables like spin and momentum, or maybe fewer, like one mystery Reality Bit, which nature chooses at random. All this is possible, and equally irrelevant. If you tell me you don’t believe in such arbitrary constructions, I can only say, “Fine, but then you should perhaps go one step further and scrap Bohmian positions and wave functions tagged on individual particles along with the rest.”

Let me try to explain it with a quote from Feynman (Messenger Lecture 1964, a few sentences after, and in elaboration of, his often quoted “I think I can safely say that nobody understands quantum mechanics”),

If you will simply admit that maybe [Nature] does behave like this, you will find her a delightful, entrancing thing. Do not keep saying to yourself, if you can possibly avoid it, `But how can it be like that?’ because you will get ‘down the drain’, into a blind alley from which nobody has escaped. Nobody knows how it can be like that.

(So my paraphrase of what he really says is “Nobody understands quantum mechanics IN CLASSICAL TERMS”.)

I read this as suggesting that one should suspend judgement on things like naive realism. Try to come to grips with what you can see in a lab. Try to build your concepts around that. Then it may be that realism at the quantum level is found to contribute to the enterprise, but on the other hand it might not. I think his warning can be exemplified well by the Bohmian community, going down that very drain 60 years ago, and indeed never coming back, except to declare that they saw Jesus at the bottom of it. To me physics is just too interesting to get seriously caught up in this. If you have heard Harald Weinfurter’s talk, ask yourself what the insistence that “spin isn’t real” could have added to the very competent explanations he gave. In order to make any analysis of a quantum optics experiment you do talk about spins and internal states of atoms and photons and polarization; positions are in no way privileged parts of the explanation. You will not be naive realist about any of these concepts. Certainly, no working physicist I know would think of measurement as uncovering predetermined values (a battle that Bell apparently still found necessary to fight). On the other hand, it seems to me that on philosophical grounds Bohmians (and the philosophers present at the workshop) would deny rationality of this discourse since it gives a shit about ontology. So what do you make of this? Is it really worth saving Physical Reality at the expense of real physics?

Let me give another example, from a conversation with Roderich Tumulka, who was quite patient with me. (The conversations on the side of the workshop were certainly more illuminating for me than most of the talks). I tried to describe in that conversation what kind of theorem I would call a solution of the measurement problem. Effective information loss is clearly part of it, as is the need to show that certain properties of macroscopic systems are stable against the way we interact with them. To me the question whether “the moon is there when nobody looks” is related to its quantum description only in the way that you have to PROVE IT from a better under understanding of macroscopic quantum systems. The trivial solution “Have no fear, its Bohmian particles are going to be somewhere” does not even begin to tackle this problem. Coming back to my description of the measurement problem, any Bohmian will understand that Roderich was probably not interested. To him my description was that of a lot of hard mathematical work, which in the end would fail to solve the MP, because there would still be a superposition; “effective collapse is not collapse”. At this point Nicolas Gisin dropped by and enlightened me: what I was after here (and to me would be all you could hope to rationally argue for) was merely “fapp” collapse. This sums it up nicely. To me the “fapp fixed outcomes” problem is a target on which even partial progress is highly welcome. It would require an increase of our understanding of complex systems and an improvement of our mathematical technique. Assuming that to be solved, there would be virtually nothing left of the measurement problem, except maybe a two line historical comment in a paper. The Bohmian perspective seems to be the opposite. You don’t care about the hard problem, but only about that last, utterly trivial bit.

The proposed Bohmian “solution of the measurement problem” is that the pointer is really somewhere, because the nuclei it is composed of are “really” somewhere, assembled in a pointerlike shape. The same observation is central to the claim of empirical equivalence between Bohmian Mechanics and Quantum Mechanics. At the end of the day everything is supposed to be recorded in a pointer position or ink on paper (There is something cozily old-fashioned about the insistence on position here. Magnetization of tapes, or storage on a USB stick or the colour of pixels on a screen are apparently unsuitable for macroscopic records.) This will be the same in quantum mechanics, and since these theories supposedly make the same predictions about positions, the two are “empirically equivalent”. Note how this argument grants that quantum mechanics had no measurement problem in the first place, since it apparently takes it as unproblematic that there will be agreement. The empirical content of Bohmian Mechanics entirely rests on this bridge. Again, it is left entirely to the quantum physicists to work out how stable pointer positions come about. Bohmian Mechanics will then extend a blessing of Reality. That’s all it does.

There was a time when the claim of empirical equivalence was made in a stronger form, namely that the two theories agree about the positions of quantum particles. This was part of the initial appeal of the theory: In spite of Heisenberg’s criticism of the notion of trajectories, here they are! Great! And if you look at positions at any one time, even the probabilities will come out right! Not any more, though, I am told. Position at the quantum level now shares the fate of spin: it is not real, but has to be indirectly inferred from an experiment (guaranteed as just mentioned to agree with quantum mechanics). Indeed, the agreement is shaky as can be. I produced a little example the other day (arXiv:0912.3740) showing that two-time correlations, which make sense in both theories, come out differently. So Alice and Bob must be forced to do their measurements at the same time, or agreement is lost. That was, of course, known to many Bohmians, although not often clearly stated. The answers I received about this told me to do it right and include the measurement devices, ultimately reducing everything to macroscopic pointers. That would bring back the empirical equivalence. Fair enough, but I am afraid this opens up a gap. If there is no direct connection between observable and Bohmian positions at the microscopic level, how am I justified to assume it at the macroscopic level? Should we invoke prestabilized harmony? Is this not rather like the measurement problem itself?

So far I have talked about Bohmian theory of measurement. Similar things happen on the preparation side. So what is the meaning of the wave function? To me this is almost the same as the question: What kind of justification can I reasonably give for choosing one wave function rather than another? In a controlled lab experiment the wave function (more likely the density operator) is an attribute of the preparation. I may have a theory about the preparation device, from which to justify an expression or maybe at least an ansatz. This could be tested by subsequent statistical measurements. Even outside the lab, under sufficiently clearly defined circumstances, there may be a justifiable ansatz. There would also be a dependence on those circumstances and how they are specified. The test for any ansatz would likewise be statistical experiments. So a measurement on the light of stars of a certain class, or on the microwave background may be a perfectly acceptable preparation procedure. Only tagging the wave function as some attribute on single particles is known to be a daft strategy, because it gets you into trouble if there is any entanglement. In any case, here is the absolutely most boring thing you can say about wave functions (It is anyhow false when applied to subsystems, i.e., anything below the universe level): “Every system really and truly has one, even if you can never find out which”.

Two lines I heard many times now are “Bohmian Mechanics is simple and beautiful, because it just needs 2 equations” and “Bohmian Mechanics explains Quantum Mechanics”. Now for the first, I could offer a further simplification: drop the Q-dot equation. The real reason the theory is simple is because you are very modest in your goals. If you don’t want to go into the details of the physics, it is easy to stay simple. If you just want Physical Reality restored to satisfy your philosophical needs, (“some Q is real, but spare me the details”) you can even drop the first equation, and leave it all to God: He knows what is real, and you can sleep reassured. Now that would be a really simple theory achieving as much in the field of Reality search as Bohmian mechanics. For the second claim, I see a pattern here: “To explain QM, invent P, to get ‘BM=QM and P’. From this you explain QM by forgetting P”. I am not impressed.

So what is the disagreement in the end? Should it be locality or realism? Should it be quantum mechanics in minimal statistical interpretation, with an operational stance, or Bohmian Mechanics, or maybe something else? I guess Bohmians and I agree that the choice is not between equally viable positions. We only disagree about which one it is. To me one is a sound basis for doing physics, including theoretical and mathematical physics with a foundational interest, and the other has turned out to be fairly sterile. In 60 years the number of interesting new physical or even mathematical problems from the Bohmian and Neo-Bohmian community has been rather modest. The workshop certainly didn’t convince me otherwise, although the hope was what made me come. Bohmian Mechanics feels to me like a theologian explaining the origin of the universe. He could say: “With all your physics, which anyhow does not cover the singularity, you cannot explain Why it happens, but theology can”. I can see that many people would go for that sleeping pill. But it is a really lousy contribution to cosmology nonetheless.

One last thing: I am always ready to play also with whacky ideas, like Bohmian trajectories. To me the trajectories used to be the most interesting part of the theory, even though the Bohmian community rarely seemed to bother to find out anything about them. What kind of physics would Bohm’s Demon see, by which I mean that hypothetical entity with direct access to the Reality of Bohmian trajectories, but to nothing else? So on Thursday evening I made a bet with Nicolas Gisin on a purely mathematical statement concerning Bohmian trajectories in the presence of detectors. He was on the anti-Bohm side, so I chose pro-Bohm, partly influenced by a heuristic argument Roderich Tumulka gave earlier that day. The stakes are a good bottle of wine, and I hereby put a second bottle as a prize for the person (most likely a Bohmian) who comes up with a pertinent theorem. I describe that in a separate post.


I’m back

May 13, 2013

This blog and my twitter account have been very quiet for the past year. This time it wasn’t entirely due to laziness or a lack of motivation. Instead, as an experiment, I gave up all forms of social networking and online news: I stopped reading all blogs, I uninstalled the twitter and facebook apps, I deleted all bookmarks to anything resembling an online news service, and I removed all chat programs. Apart from one or two minor lapses, in order to obtain some contact details, the closest I was to anything resembling a social network is arxiv.org and the closest news source has been (occasionally) the radio. (Not internet radio though.) I did, however, read email (alas, this seems to be necessary for daily functioning in a large institution in the modern world…)

Why did I do this? I think it is an understatement to say that the internet is distracting and I increasingly had the suspicion that news and social networks did not make me happier. I also suspected that the internet was a serious drain on productivity. So, what is the result? Disappointingly, my productivity didn’t soar. It turns out that you can be distracted by things that aren’t the internet, e.g., unfortunately, books. This problem must be solved by some other productivity measure (however, visiting a lifehacker site is still strictly forbidden!).

What about happiness? This is more interesting: I think I am, in general, actually a teensy weensy bit happier. It is abundantly clear that “giving up the internet” is not a miracle cure for the malaise of modern life. But one does feel noticeably calmer on a day to day basis when not exposed to people being wrong on the Internet.

But didn’t I miss out on loads of important things? Well, it seems not. It gives you a sense of perspective to stop living in the sometimes hollow echo chamber that is the blogo-twitter-gplus-facebook-sphere. I’m sure I missed out on loads of really important scandals and outrages. I do know that I missed out on a couple of really very good blog articles: but the nice thing is that the good ones I did miss got recommended to me by word of mouth.

I am now returning to a more active internet presence: I strongly believe in open access science, and open science in general. I also believe that social networks are a fantastic medium with which to interact with others in science. I just wish there was a really simple way to filter out all the negative stuff.

To kick things back off on this blog I will be right back with a guest post ;)


Is quantum mechanics an effective theory?

May 16, 2012

What is the ultimate theory governing Nature? All evidence to date strongly suggests that quantum mechanics (QM) is this theory. However, it could still be that QM is simply a really good effective theory which breaks down if we are able to perform experiments with sufficiently high energy and precision. If this is the case, what sort of “post-quantum theory” could QM be replaced with? Assuming only that special relativity is correct, one can postulate “generalised probabilistic theories” (GPTs) as the framework to explore such alternatives.

What is a GPT? To define it one only imposes a handful of axioms, which are introduced in a paper of Barrett, based on previous work by Hardy. They are:

Assumption 1. The state of a single system can be completely specified by listing the probabilities for the outcomes of some subset F of all possible measurements. These are the fiducial measurements. These probabilities can be written arranged in a vector P.

Assumption 2. For each type of system, the set of allowed normalized states is closed and convex. The complete set of states S is the convex hull of the set of allowed normalized states and 0.

Assumption 3. For each type of system, there is a set T of allowed transformations. A set of transformations T mapping S to itself. The set T includes the transformation that maps all states P to 0 and is convex.

Assumption 4 (Local Operations Commute). Consider a joint system composed of systems A and B. Suppose that an operation is performed on system A alone with outcome OA and an operation on system B alone with outcome OB. The final unnormalized state of the joint system does not depend on the order in which the operations were performed. In particular, this implies that the joint probability of getting outcomes OA and OB does not depend on the ordering of the operations.

These aren’t the only sets of axioms one could use, and there has been plenty of work tinkering with them. However, for convenience, we take these as our working definition of a GPT. One of the consequences of these assumptions is that the set of possible states, S, and the set of measurement effects, F, are dual convex bodies. Thus, basically, a GPT is completely specified by the assumptions after specifying either S or F. We choose to work with F.

GPTs have received considerable attention recently, both as a foil to better understand the features of QM, and as a powerful abstract way to reason about correlations and relativity. These investigations have lead to many interesting results, including simplified and improved cryptographic schemes and primitives. An interesting consequence of a GPT beyond QM is that they yield violations of CHSH inequalities beyond those possible in QM, so they are basically ruled out by experiment.

But, let’s suppose for the moment that Nature isn’t described by QM, and rather by some other GPT. A natural question then arises: why is QM such a good effective theory? A natural answer, which we investigate in a recent preprint, is that experimental imperfections prevent us from observing any post-quantum phenomena.

Suppose that Nature is described by a GPT with a high-dimensional state space and corresponding high-dimensional set of all possible measurements. Observational limitations, such as detector resolution, mean that it is impossible to access most of these theoretically possible measurements. If physically implementable measurements are those chosen from some typical subset (a precise definition is given our paper) then we show that the bipartite correlations arising in any experiment can be modelled, to a high degree of precision, by those of QM. Note that the tripartite and multipartite correlations could go beyond those exhibited by QM: a sufficiently refined experiment involving three or more particles could exhibit behaviour going beyond that possible within QM!

It is interesting to contrast our setting with that of decoherence, which models the passage from the microscopic to the macroscopic classical world. The crucial difference here is that decoherence arises from the correlations developed between a given particle and many other inaccessible particles (in the GPT framework it is rather likely that decoherence will always leads to an effective classical theory). By way of contrast, we considered only a few particles in isolation: roughly speaking, we studied the case where only the “local dimensions” are effectively truncated.

Our argument makes use of the the concentration of measure phenomenon, epitomized by Dvoretzky’s theorem which states, roughly, that a random low-dimensional section of a high-dimensional convex body looks approximately spherical (check out the figure in our paper for an illustration). This result then allows us to exploit the observation that spherical state spaces can be simulated by sections of quantum mechanical state spaces. Our approach also owes much to the recent work showing that bipartite correlations may be modelled by QM when the constituents locally obey QM.

Putting all this together we obtained the main result that:

If the local measurements in a GPT are chosen from a typical section of the convex body of all possible measurements then, with a high degree of accuracy, they do not yield any post-quantum prediction for the bipartite scenario.


We’re on GitHub!

May 15, 2012

The open science movement seems to be picking up steam these days thanks, in no small part, to Michael Nielsen’s “Reinventing Discovery“. Complementary to this has been an increasing clamor for scientists to open up and release the computer codes involved in scientific papers. This is an issue I feel strongly about: it is rather hard for a scientific result to be called “reproducible” if the code the result is based on is withheld.

Strongly motivated by these issues I’ve recently been investigating ways to share and open the source code for the numerical quantum many body projects I’m involved in here in the quantum information group, Hannover. The social coding website GitHub provides a great platform to do exactly this. Thus I’d like to take this opportunity to introduce a new open science open source project evoMPS, led by Ash Milsted, and hosted on GitHub.

EvoMPS is an actively developed implementation of some of the new methods to simulate quantum many body systems using the time-dependent variational principle reported on in these papers.

EvoMPS is also partly the result of our frustration with the expensive licenses required by commercial scientific software (e.g., Matlab and Mathematica) and the irritations of maintaining connections to license servers etc. We wanted that anyone could download the code and experiment with it without having to pay exorbitant fees for commercial software. Thus, to implement the project we looked around for open source alternatives to Matlab and eventually settled on Python + Scipy/Numpy, (although I must say, the new Julia language looks extremely interesting). Python seems really well suited to numerical quantum many body calculations as it allows one to swap in highly optimised LAPACK/BLAS libraries and also one can insert optimised c code using cython.

I am very excited by the results, so please do clone/fork evoMPS today and take a look.

If you’d like to contribute please don’t hesitate to contact us.


Four positions in the quantum information group, Hannover

May 1, 2012

We would like to draw your attention to four openings in the Quantum Information Group at Hannover (RF Werner and TJ Osborne).

We are looking for either postdocs or PhD students with appointments for up to three years, and will also consider shorter term flexible appointments. Late applications will be considered until positions are filled.

Senior postdocs might also be interested in an assistant professorship which we hope to be able to advertise shortly.

The official ad can be found here.

We would appreciate it if you would forward this to potential candidates.


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