This morning I was re-reading David Deutsch’s classic paper “Quantum Theory, the Church-Turing Principle and the Universal Quantum Computer”, Proc. of the Roy. Soc. London A, 400, 97-117 (1985) This is the paper where he explicitly shows an example of a quantum speedup over what classical computers can do, the first time an explicit example of this effect had been pointed out. Amusingly his algorithm is not the one most people call Deutsch’s algorithm. But what I found funny was that I had forgotten about the last line of the article:
From what I have said, programs exist that would (in order of increasing difficulty) test the Bell inequality, test the linearity of quantum dynamics, and test the Everett interpretation. I leave it to the reader to write them.
I guess we are still waiting on a program for that last problem?
To my knowledge, the strongest evidence we have that quantum mechanics is dynamically linear are the observations summarized by Richard Thompson in his 1989 Nature review “Is quantum mechanics linear”.
Those observations are from (then) molecular beam experiments and (now) trapped particle experiments. In essence, carefully isolated quantum systems (1) exhibit a relative linewidth of order 10^{-15} or less and (2) exhibit a linear response to external driving fields.
These two attributes (spectral purity and linear response) are compatible with the QM state-space being a flat Hilbert manifold … but they are also compatible with the QM state-space being a product manifold of polynomial Schmidt rank … `cuz heck … even the Bloch equations on a Riemann sphere exhibit spectral purity and linear response.
Thus, the evidence for the widespread belief that QM has been proved to be linear is less rigorous than is generally appreciated. In particular, as far as I know, none of the modern writers on this topic have quantitatively or qualitatively improved upon Thompson’s 1989 analysis … references in this regard would be *very* welcome.
LOL … perhaps the following “Church-Turing lemma” makes the point more crisply: Any experiment whose data can be verifiably simulated with computational resources in P/i> is compatible with a non-Hilbert model of quantum dynamical flow.
As everyone knows, Scott Aaronson and Alex Arkhipov have been thinking hard about quantum experiments that (provably and/or near-provably) cannot be simulated in P … this is work that I admire very much.
I think Deutsch believes that the existing algorithms already vindicate the Everett interpretation.
Setting aside the Everett interpretation comment, even testing the linearity of QM is not trivial. In particular, if you’re doing a case-by-case analysis of the inputs to your experiment or using density matrices to describe what’s going on you’re on very thin ice.
Dave says: Bacon posts are never off topic on this blog
Then here’s a baconesque reading suggestion: Doctor Mirabilis … Jame Blish’s SciFi biography of the medieval scientist Roger Bacon. … who was the 13th century’s version of David Deutsch.
Blish’s wonderfully eccentric Bacon book is out-of-print, but it is available on the used book market … also in the UW library stacks … it gets rave reviews from the cognizenti on Amazon.com …so your wife’s gift-giving needs are met! 🙂
Off topic, but all thigs Bacon
http://www.taobacon.com/
Jeff
Bacon posts are never off topic on this blog 🙂
I am hoping that weblogs like Quantum Pontiff and Shtetl Optimized will become a little bit more lively this week, in the event that Scott live-blogs his lecture at the TEDxCalTech Event, Feynman’s Vision: the Next 50 Years, which takes place today (Friday) at CalTech.
Scott has titled his lecture Physics in the 21st Century: Toiling in Feynman’s Shadow. Hmmm … the prospect of fifty more years of “toil” sounds kinda dark … hopefully at least some of the TEDxCalTech speakers will echo the memorable words of Steve Martin’s character Navin R. Johnson:
Because heck … don’t pretty much *all* of Feynman’s most famous lectures lay out fresh STEM roadmaps?
Seriously, we should all admire the courage of the speakers for even taking the stage at this event … because they’re going to have to share that stage with none other than Kongar-Ol Ondar:
As Feynman might have said … this event is gonna be terrific! 🙂
I think you meant “Mythic Man” — he is only “Mythical” in another universie 🙂
Maybe it’s debatable what light quantum computing sheds on the WMI, but I think the idea it’s compatible with the alternative states being in the same space-time but not violating conservation laws is rubbish. Really, if say you send a photon through a beamsplitter and two detectors a photon can reach, and if you say that somehow the photon is really in both detectors, then you violate conservation of mass-energy etc. because there “really is” twice as much energy as before. It’s not just about information, but mass-energy too.
Neil, an “old-but-good” article that tiptoes up to the issues you mention is Ludvig Faddeev’s Symplectic structure and quantization of the Einstein gravitation theory (1971). This is the article that introduced the ideas, founded in geometric dynamics, that rapidly grew into the formalism that today is known Fadeev-Popov quantization.
Fadeev-Popov quantization methods are at present the dominant formalism for pretty much all gauge field theories, including in particular the nonabelian gauge fields of the Standard Model … it is interesting that these methods were discovered by a straightforward line of reasoning that is geometric rather than algebraic, and that makes no reference to Hilbert space.
To put it another way (and just to maintain a pulse in the quantum physics blogosphere), when we are puzzled by field-theoretic questions like “What is the gravitational field of a single photon traversing an interferometer?”, then we begin to feel that our physical intuition needs strengthening.
But if we are puzzled by field-theoretic questions like “What is the gravitational field of a Fadeev-Popov ghost particle traversing an interferometer?”—which is conceivable as a gedankenexperiment—then we begin to appreciate that our mathematical intuition needs strengthening just as much as our physical intuition … or perhaps more.
On John Baez’ weblog I discuss the above points from a category-theory point of view … on Scott’s weblog from a STEM/QIT Roadmap point of view … can’t help feeling that nowadays, a species even more endangered than stoked quantum bloggers is stoked quantum readers & posters.