Posted, without comment:
quant-ph/0606017 [abs, ps, pdf, other] :
Title: Can measuring entanglement be easy?
Authors: S.J. van Enk
Comments: No
O brave new quantum world!
O brave new quantum world!
Posted, without comment:
quant-ph/0606017 [abs, ps, pdf, other] :
Title: Can measuring entanglement be easy?
Authors: S.J. van Enk
Comments: No
It was tough (for me) to make sense of that article. But maybe, that’s because demonstrating entanglement being NP-hard, all papers are going to be tough, from now on!
There seems to be a convergent focus in mathematics on tough problems. An article today in the Asian Journal of Mathematics claims A Complete Proof of the Poincar̩ and Geometrization Conjectures РApplication of the Hamilton-Perelman theory of the Ricci flow, which is one of the Millenium Prize Problems.
What’s the connection to quantum computation? Well, the set of all (unentangled) qubit product states is a submanifold of the larger Hilbert space, and if we normalize the product states, this submanifold is obviously compact. It also has a natural Kahler form that is associated with a Fubini-Study metric, and it is not hard to show that the Ricci curvature tensor is proportional to this metric, i.e., product states are compact Kahler-Einstein manifolds, and furthermore, are geometric solitons under the induced Ricci flow that is the focus of the above article. Unsurprisingly, the product-state scalar Ricci curvature is a constant integer, and therefore, these manifolds are in some sense as smooth as manifolds of their topology can be.
The point being, that hard problems in quantum computation and quantum entanglement are intimately linked to similarly hard problems in the algebraic geometry of complex manifolds, and to similarly hard problems in quantum model order reduction (MOR). Everyone is climbing the same mountain.
We’ll be discussing this point of view at John Marohn’s MRFM Summer School, at Cornell later this month.
Harvard geometer Shing-Tung Yau asserts that these advances will “heavily influence the development of physics and engineering — our QSE Group wishes we knew what he meant by “engineering” in saying this.
By the way, for physics students who are daunted by the literature of (complex) algebraic geometry, a natural progression is (1) calculate quantum manifold curvature “the hard way” by treating the Hilbert space as a real manifold, using the methods of Weinberg’s Gravitation and Cosmology, and implementing the calculations via your favorite Mathematica package. (2) Using the same Mathematica package, switch to complex coordinates “z” and “zBar”, treating “i” as just another parameter of the transform, such that “z” and “zBar” are independent coordinates. Of course, any scalar quantity you compute will be invariant under this coordinate transform. (3) Now notice that your calculations runs orders of magnitude faster, because huge blocks of the Riemann curvature tensor are zero (assuming that you picked a metric that is analytically “nice”, or technically speaking, that is Kahlerian). (4) Now you are equipped to read Flaherty’s Hermitian and Kahlerian Geometry in Relativity, and also Schutz’ Geometrical Methods of Mathematical Physics; these authors explain why the geometry of complex manifolds is so much simpler than the geometry of real manifolds. (5) Now, finally, you can at least begin to study the masters, e.g., Chern, Yau, etc.
Of course, this is like learning C++ programming by studying BASIC first — some peope will say that it permanently damages your brain.
He truly is the king of comments.
My lengthy posts are a losing battle against …
[b]The Slashdot Conjecture:[/b] The mathematical and physics problems that arise naturally in everyday life are in complexity class NP-hard.
[b]The Slashdot Corollary:[/b] Meaningful discussion of real-world problems requires either oversimplification or humor (preferably both).
I’ll try to do better on the oversimplification and humor!
Scott: I agree with everything you say, all of which is exactly correct and expressed with great logical clarity, IMHO.
To extent what you say, and apply it in the real world, is part of the profession of an engineer; hence the need to interface articles like van Enk’s with urgent technological and social issues. And it is at this global interface that we begin to appreciate that an exponentially large number of articles can be written about simplified instantiations of pretty much any NP-hard problem (which when you think about it is very good news).
It is not too common (yet) for North American complexity theorists to write about these topics, but for authorization and encouragement, we can look to Harvard geometer Shing-Tung Yau, who writes beautifully about Perspectives on Geometric Analysis (an article with 755 references … amazing!). On the other hand, Yau writes with equal beauty and clarity about Personal recommendations for the advancement of Chinese technology in the Harvard Asia Pacific Review.
What has been the concrete benefit of Shing-Tung Yau’s federative point-of-view? Well, Yau’s work has attracted the notive of the renowned Hong Kong businessmen and philanthropists Ronnie and Gerald ChanMorningside Foundation is investing hugely in fundamental mathematical and engineering research.
And when I say “hugely”, I mean Apollo-scale — but focussing on Kahler space and biospace instead of outerspace. With typical Chinese pragmatism, the Apollo-style program that China has launched is IMHO is guaranteed to produce human-scale social and economic benefits. And China’s program is not some future idea — it is already well underway.
Over on Lance Fortnow’s blog, everyone is wondering about how to increase funding for theoretical computer science. Well, IMHO, they don’t have to look any farther than the federative partnership of Shing-Tung Yau and Ronnie Chan to see how this can be achieved.
Hmmmm … the above post scores high on “oversimplification” and low on “humor” — but China’s Apollo-style technology development effort is real and serious, and therefore, so is my post on it.
PS: another great article in the above-referenced Harvard Asia Pacific Review is Amartya Sen’s Globalizing What?
History, economics, equity, and efficiency.
The connection with complexity theory is two-fold. FIrst, Sen quotes the Buddha: There are some questions that can be asked of which there are no answers. Hey, the Buddha knew complexity theory!
Second, a specific question that Sen’s essay suggests, and which is IMHO a central question for (applied) complexity theorists, is this: do open global markets exist whose efficiency can be optimized by algorithms of P-time complexity? The point being, that a global market that is free and open, but NP-hard to optimize, can only be efficiently exploited by entities with large-scale computational resources. Derivative traders?
And if you don’t know who Amartya Sen is, shame on you!
John: Just because testing entanglement of an N*N mixed state is NP-hard, doesn’t mean it’s hard to prove it experimentally for specific states like Bell pairs! Indeed, for the small states that Steven is discussing, computational complexity seems completely irrelevant. Note that (title notwithstanding!) Steven’s criticisms are only of a particular experiment; he’s not arguing for inherent hardness in anything like the sense a computer scientist would mean.