In search of an isomorphism

I’m working on some group-theoretic stuff related to n copies of a unital quantum channel and am in search of some isomorphism between the dihedral group and the unitary group U(n).

Now, what’s interesting is that the unitary group U(1) is isomorphic to the circle group, i.e. the geometric circle, while can be thought of as representing the discrete rotations of an n-gon. So one would think that as , would become isomorphic to U(1) since the n-gon geometrically approaches a circle. But is isomorphic to which geometrically is a line infinite in both directions.

Technically, for what I’m interested in, I suspect U(1) would be too restrictive anyway.  I’m more interested in finding a direct isomorphism between and U(n) or, at the very least, something broader than U(1) and, preferably, broader than SU(2) as well.

So if you’re a regular reader of this blog and you know of any such isomorphism, post it here!

March Meeting Madness: Wrap-up

I’m back from the March Meeting having skipped most of today’s sessions in favor of hours and hours of airports (but only about 100 actual minutes of flying time!).  Here are some additional notes:

• Got to see a great invited talk entitled ‘Quasicrystals in Medieval Islamic Architecture’ by Peter Lu as part of the session ‘Physics and Art.’  It was good enough to inspire me to join the Forum on Physics and Society which sponsored the session.  More on Peter’s work is here: www.peterlu.org.  There were others in that session I wanted to see, but had to run to my own session.
• Frank Tabakin gave an interesting talk on a new Fortran 90 code for running large quantum computing simulations.  Their basic research started out by using a Mathematica 7 add-on called QDensity (something I use sometimes, though I prefer Quantum) and their Fortran 90 code, called QCMPI, ran with several parallel processors (including 4 connected MacBooks!).
• Mark Coffey gave a nice talk on Schmidt decomposition and CNOT complexity.  Unfortunately I didn’t write down the arXiv reference but I did find this reference from the ACM.
• Haitao Quan discussed the work he has been doing with Michael Zwolak and the always interesting Wojciech Zurek (who ‘smuggled’ German beer into a conference last fall) on quantum Darwinism for environments in mixed states.  Again I can’t find an arXiv reference for this (though there are a number on general quantum Darwinism).  I’m particularly interested in this since I think some of my work on static environments may be related to Zurek’s ‘envariance’ (entanglement assisted invariance).
• Winton Brown, one of Lorenza Viola’s grad students, discussed their work on asymptotic convergence rates for statistical moments of pseudorandom circuits.  Their approach to asymptotic convergence was interesting but very different from the approach I’ve been taking lately in my studies of Birkhoff’s theorem (I’m using a group theoretic approach).  The basic idea is summed up here.
• I hate hotels.  I much prefer cabins, cottages, etc.
  
• I hate food on wooden skewers.
• I had a great time visiting my cousins Thursday afternoon and evening!  The Carnegie Natural History museum was really interesting.
  
• Dave Bacon has topped me in the ‘quirky lecture’ department.  He apparently once tied himself up in class to demonstrate some disentanglement method.

Pittsburgh is actually a pretty nice city.  It’s very much in the mold of Buffalo (though larger).  But their airport leaves a bit to be desired (even worse is Philly’s).

March Meeting Musings

I’ve been very busy at the March Meeting and have not had the chance to sit down and blog about it until now.  But here are some highlights that particularly struck me:

• Lorenza Viola gave an interesting talk about quantum gates that actually correct their own errors (see arXiv:0810.0698).  The room was packed to the hilt, I was at the back, and Lorenza talks fast so I missed a point here and there, but I am quite intrigued by such an idea.
• I need to learn more about Cartan involutions.  Both Byron Drury and Peter Love gave talks on the subject and Byron, in fact, was the Leroy Apker Award winner.  Congrats to Byron!  (see: arXiv:0806.4015).
• Bob Griffiths is awesome.
• Bob Griffiths isn’t quite sure what all the fuss is about regarding entanglement.
• Once again, someone from the IOFPR submitted an abstract to the foundations session but did not show up.
• Todd Brun talked about CTCs.  That makes two straight years that we’ve had talks on CTCs. (see: arXiv:0811.1209).
• Unfortunately Partovi withdrew his abstract.  I was really looking forward to it since it challenges our natural assumptions about the quantum/classical contrast. (see: arXiv:0708.2515).
• Kim Fook Lee gave an interesting talk on coherent light fields being used for entanglement-based communication.  It gives me some ideas that I want to take back to the laser guys in my department. (see: arXiv:0810.1998).
• I am not a morning person.
• Picked up a couple of exam copies of interesting looking books from Princeton University Press including on on Compuational Physics that I am thinking of using to develop a full course for since we now have a Computational Physical Sciences Program.
• The Topical Group on Quantum Information NEEDS MORE MEMBERS – and active ones at that!

I’m skipping most of today’s talks and giving my own tomorrow.

An ironic likeness

I couldn’t resist making this when I saw the picture of Bernie Maddoff below.  It was just so creepily reminiscent of Washington that it begged for attention.  God, the irony is just surreal.

Weather realities

Let’s get something straight.  Buffalo is neither that cold nor that snowy.  With the recent signing of Terrell Owens by my beloved Bills, my equally beloved hometown has received quite a bit of press in recent days.  Nearly all the articles mention something about how cold and/or snowy Buffalo is.  But let’s look at some actual weather statistics.  Of all of these top ten lists, Buffalo appears on only one.  It is, oddly enough, the sixth rainiest city in the US, but is not in the top ten snowiest (though Syracuse is, as I often tell people) or coldest (has no one ever been to freakin’ Minnesota before?).

Buffalo isn’t the only place whose weather gets misrepresented in the national media, however.  As it turns out, New York and Miami both have a higher annual rainfall total on average than Seattle.  And check this out: the average annual temperatures of New York, St. Louis, and San Francisco differ by only 1.8ºF.

Oh, and where exactly is the snowiest place in the United States?  It would be in California, to be exact.  Blue Canyon, California, down the road a ways from Lake Tahoe.  Blue Canyon averages 240 inches of snow per year.  The runner-up is Marquette, Michigan with 128 inches.  As for the coldest place, well, it’s no surprise it’s International Falls, Minnesota, a place that is actually colder on average than Fairbanks, Alaska (which doesn’t even make the list).

Update:  Apparently it is debated whether Fairbanks is colder or not.  Anchorage certainly isn’t since it is warmed by the Pacific.  In any case, here are some more stats that include some more remote areas.  Technically, there really isn’t one place that holds the title to any of these, actually.  See this journal article.  Here’s yet another list with considerably more cities from New York State showing up – but not Buffalo.

Liquid water on Mars?

The Phoenix spacecraft that recently landed on Mars may have actually landed in a small puddle of liquid water.  Apparently the economy is so bad this didn’t even make headlines on the major news outlets as far as I can see.  But, anyway, if it truly is liquid water the chances Mars currently has or once had some form of microbial life just increased quite a bit.  Very cool.

Links: bloody battles and dead Daleks

• My colleagues David George and Matt Gonzales of the Saint Anselm Classics Department have a new eight-week mini-series coming out on the History Channel called Battles, B.C. Matt has guest lectured in a History of Physics class I taught last fall and David actually has an undergraduate degree in physics.  I asked him once why he became an archeologist and he said something about a “buxom redhead.”
• A friend forwarded me this link to an article about some folks cleaning up a swamp in Hertfordshire who dredged up a Dalek head.  Apparently BBC did film Dr. Who in Hertfordshire once in the early ’80′s.  Apparently.  Hmm…

Cayley tables & a “new” math blog

OK, so it’s not new, per sé, it’s just new to me.  That would be the great – and very informative – blog Division by Zero.  I ended up there as I was making a review sheet for an exam I’m giving next week in my Modern Algebra class and stumbled on an interesting post regarding Cayley tables.  I find Cayley tables very interesting since they’re related to things like “magic squares.”  In class, I also liken them to Sudoku puzzles though the latter are a tad more complex.  But they both share the fact that an entry can only appear once per row and once per column.  I found this analogy worked pretty well with students familiar with Sudoku.