I was more than a little disappointed when Apple ran their marketing campaign a few years ago using pictures of many famous civil rights people and other people like Einstien. The exploitation of good people for making money seemed awfully sickening to me.
That's right, because we all know that MSFT didn't amass any of their wealth by abusing their monopoly position to push smaller more-innovative companies out of business. No, MSFT never exploited the efforts of "good people for making money".
I mean, if you're comparing these two companies, perhaps Apple is more likely to exploit someone in an advertisement, while MSFT is more likely to drive them out of business and steal their marketshare. Which is worse? You seem to be more disappointed with Apple, but I disagree.
Asimov just might like it, as long as it complied with the lesser-known Zeroth Law of Robotics - "A robot may not injure humanity or, through inaction, allow humanity to come to harm."
Okay, I think we're in near agreement with the theory at this point, and just debating semantics.
BTW, I re-read my original response to your first response, and it's bloody awful, no wonder you thought I was a fool. Like I said, I accidentally killed the tab, so I tried to whip up something quickly before missing too much of the Daily Show, and wound up mangling terms in my haste, eg vector space vs Hilbert space, etc.
Basically I didn't agree with how you said combining two qubits together is simple vector addition, because as we've beaten to death by this point it's really tensorial, like a linear combination of two linear combinations, yada yada yada.
I still disagree with you about quantum mechanics being more enigmatic than classical mechanics, but as you've said, most quantum computing routines don't require knowledge of group theory and only 'baby' tensor stuff. (Although if you want a good understanding of the actual quantum mechanics underneath, especially in regards to real physical systems such as the hydrogen atom, than those concepts become more important).
That applet I linked to basically lets you choose any input qubit, operate on it w/ a number of gates (the Pauli gates, Hadamard, and some others), and see the resulting qubit, in real time. Displays both the spinor representation, as well as graphically on the Bloch Sphere, for both qubits. So you can see in real-time the effects of the various single-qubit gates on any qubit value. Granted there's not much useful stuff to do with only single qubits, but it's a start. I've got the two-qubit version in the works, haven't touched it in about a year.
I've also debated such a 'dummy' quantum computing guide, becuase like you said, qubits aren't very difficult conceptually wise. In fact, that was part of my motivation in making these Java applets. If you're serious about working on this and would like to colloborate on something, send me an email at the address on the applet page.
My whole point is that the O'Keefe style of management, namely cutting important science missions in the name of 'safety', is changing. Especially since the Columbia accident happened on O'Keefe's watch, while Griffin comes in with a clean record.
My friend, back in 1997, claimed that sometime a few years earlier he and his friends were trying to hack some game on his Mac, so they were browsing the various files with a hex editor. Apparently one of the files, somewhere in the middle, had alot of text saying "blahblahblahblahblahblahblahblahblah" for many 'pages', and at some point in the middle said "why are you reading this?"
Hell, maybe this example is even common knowledge amonst the slashdot crowd.
Well let's see, NASA administrator Sean O'Keefe says it will never be visited by a space shuttle repair mission again
Um dude, O'Keefe has been gone from NASA for nine months now, your article link is almost a year old. One of the first things that the new administrator Michael Griffin did when he took over the reins was to try to figure out ways to keep Hubble alive. Griffin's an actual scientist, unlike O'Keefe who's a career-track manager. And thus sees the important of Hubble, which has been indispensible for astronomical research.
"At his April, 2005 confirmation hearing before the U.S. Senate, as well as in subsequent statements, new NASA Administrator Michael D. Griffin testified to the extraordinary scientific value of Hubble. He indicated his desire to take the robotic servicing mission "off the plate" on the basis of mission complexity, and reconsider an SM4 Shuttle-astronaut mission to Hubble. His rationale is that after the Shuttle's Return to Flight ("RTF", currently scheduled for July of 2005), and in particular after all the Shuttle improvements that precede RTF, NASA will essentially have a "new" Shuttle vehicle and system in the context of astronaut and mission safety. After successes in RTF and the following flight, if analysis shows that the risk levels associated with a Hubble mission are sufficiently low and manageable, SM4 could be reinstated by the Administrator."
No, I understand the quantum mechanics quite well but you don't need the full power of quantum mechanics to do quantum computing.
Okay, we're in total agreement here. The whole point of my reply was because in your original post you said there's nothing enigmatic about quantum mechanics, and that adding qubits is trivial. I disagree with both those statements. Granted, the quote you replied to in your original quote says "quantum mechanics" but the rest of your post deals primarily with quantum computing.
However, I still disagree with you that it's trivial to add qubits, which is mathematically equivalent to adding angular momentum. Like I said, a two-qubit case is relatively simple, but things get much more complicated in higher-order systems. Maybe such higher-order systems aren't as prevalent in quantum computing, so perhaps such topic doesn't need to be known in depth for most quantum computing algorithms.
when you form the tensor product your're writing it as the sum of the spin-0 and spin-1 irreps (shorthand: 2*2 = 1+3). You need to do this to understand the total spin of the combined system. But this has nothing to do with quantum computing.
Okay, you basically proved my point, with your tensorial shorthand of 2x2=1+3. That is not simple vector addition, at least not in my book.
Perhaps we differ in outlook in that you regard such a tensor product as 'baby stuff', as per your original post, whereas I think it's not as trivial. In that regards we can just leave it that you're much smarter than me if you consider tensor products as simplistic math. But again, this simple 2x2 case is fairly easy, and if this is as difficult as things get for most quantum computing routines, than you're right this isn't that much of a big deal.
But things are more complicated in general, such as in physical angular momentum rotations, which are important in rotational and atomic systems, where IIRC O(3) operations can lead to 3x3=5+3+1 irreducible forms, and can yield nifty tricks like the version of Wigner-Eckart for a closed surface integral containing the product of three spherical harmonics. But again, I guess such techniques aren't useful in quantum computing.
You must be a particle physicist or something.
Condensed matter experimentalist, currently working with mesoscopic transport and superconducting nanowires. What is your field of research?
But we're not looking at multiple-spin systems. You're going off at a tangent. None of the basic quantum computing algorithms (Shor's algorithm, Grover's algorithm etc.) make any reference to combining spin.
But I disagree, and you did demonstrate this earlier with your 2x2=1+3. That IS a multiple-spin system, and it DOES deal with Clebsch-Gordon coefficients, just the most simplistic ones there are. AFAIK, the controlled-not is one of the most fundamental gates in quantum computing, and that does deal with two-qubit systems, in which case the level of entanglement is built into the 4-element complex state.
And the reason I keep bringing up Clebsch-Gordon is merely to demonstrate, which apparently you DO understand, the concept that adding angular momentum isn't quite trivial as it would appear at first glance. At least not to me. And since qubits ARE representations of SU(2) algebrea, and are synonymous with a spin-1/2 particle, it is valid to talk about them from this point of view.
But granted I'm not as familiar with that many theorems of quantum computing, especially any error-correcting. But I know that quantum teleportation requires the spin-singlet state, in which case entanglement IS quite important. Perhaps this routine isn't really considered within the scope of quantum computation? Additionally, I also thought some quantum computing routines are possible because they can effectively circumvent the no-cloning limitation by using entanglement as a means to build in some kind of redundancy. But that's just a hand-wavy concept, maybe
I won't respond to all of this post. You clearly are unfamiliar with quantum mechanics and are just throwing around keywords to sound impressive.
At first I thought you were trolling, and you gave me a good laugh. After responding to this, my conclusion is that you know some quantum computing stuff, but you really don't have a decent understanding of the underlying quantum mechanics.
An isolated qubit is represented by a 2D (complex) vector space. Where does the group theory come in? Read Shor's paper on factoring numbers.
Okay, don't just take my word for it. Take a look at this book . Notice in the description it says "A group theoretic abstraction of Shor's algorithms completes the discussion of algorithms." Want another example? This course specifically teaches some group theory before getting to the quantum algorithms.
You suggested in your first post that any system of more than one qubit merely involves 'adding' their vector spaces. Ie, you said this
"What is enigmatic about adding two vectors in a vector space? I can't stand the way popular science press insist on making bizarre statements about the most trivial mathematics and science in an attempt to make it more interesting."
Okay, please demonstrate how to form the spin-singlet state by _adding_ the vectors of the two individual spinors. Now explain how to form the m=0 spin-triplet state by again adding the vectors of the two spinors. You cannot do this with 'baby stuff' arithmetic as you insinuated in your original post.
Ie, you're increasing the size of your Hilbert space, so it's not basic addition at all.
You would be right if you said both could be formed as a linear combination of two individual up/down and down/up states. But each of the individual states themselves is a spinor, so by 'combining' these two states. you're not doing simple 'vector addition'. You are actually dealing with tensors, and group theory, only you're not realizing it. And while this might be relatively simple for two qubits, once you add many qubits to the system the complexity increases quickly.
In fact, you need to know next to nothing about group theory to understand what a qubit is. An isolated qubit is represented by a 2D (complex) vector space. Where does the group theory come in?
You're right that an isolated qubit can be represented by a two-element spinor. The group theory comes in for systems with multiple qubits. Here's a quick explanation for a two-qubit system. For reasons I explained in the original post, there are only two free parameters, hence an easy Bloch Sphere representation, for one qubit. Two qubits together, however, have SIX free parameters (the overall system state has four complex elements, but it's normalized and also has a meaningless phase). How do you take two qubits with two degrees of freedom, and combine them to get a system with six degrees of freedom? The combination itself is NOT simple vector addition, because you cannot merely _add_ two qubits together. The description of where the extra degrees of freedom come from when combining two qubits is described by the group theory. Or you can talk about from a tensor point of view by including irreducible tensors, etc.
As far as I can see you're just part of the grand conspiracy to make Quantum Mechanics, and especially Quantum Computing, seem far more mysterious than it is. Shame on you.
Coupled systems ARE enigmatic, and I'm sure you've heard of EPR and Bell's Inequality. If you think it's "baby stuff", then you seem to be one of those scientists that just follow through calculations 'plug-and-chug', without any comprehension of the underlying concepts.
So as I said previously, at first I thought you were trolling, but I now think I understand where you're coming from. I guess in your experience
Dammit, I had a huge in-depth reply with lots of wiki links, and I accidentally killed the tab! Well, here's a quick re-do, but I'm not going to spend as much time.
What is enigmatic about adding two vectors in a vector space?
Nothing about adding vectors. However - qubits are NOT vectors, they're representations of SU(2) algebra.
States in a quantum computer are elements of a vector space. You learn what vector spaces are in the first year of an undergraduate course in mathematics. This is baby stuff.
No, this is very wrong, it's not simple vector space, it involves group theory, specifically that of Lie Groups. Additionally, spin states aren't in a vector space, but in a Hilbert Space.
basically - a qubit is a two-element spinor, each element is a complex number, so that leaves four independent terms. However, the whole spinor must be normalized, and additionally there's an overall phase that's meaningless. So that leaves only two free parameters which completely describe the qubit. And these parameters can be transformed such that they lie nicely on the surface of a sphere, known as the Bloch Sphere. I assume that's what you mean when you are talking about qubits as vectors. but that only works for individual vectors, it completely breaks down when dealing with systems of vectors because it ignores the entanglement between multiple qubits!
However, any elementary student of quantum mechanics learns quickly that spin, which is an embodiment of angular momentum, has individual components that DO NOT COMMUTE. Basically - a quantum vector would allow you to simultaneously know the x,y, and z components. Not true with angular momentum, you can only know something about the overall length of the vector and the component of the vector on only ONE dimension. Therefore, any angular momentum in quantum mechanics is really pointing somewhere on a ring of uncertaintly along a sphere.
Now, when you look at a system of two spins, and qubits can be represented as spin-1/2 system, you must use the Clebsch-Gordon coefficients to add them. Angular momentum addition is actually an esoteric topic. It is straightforward once you learn it, but it's definitely not as straightforward as adding two vectors.
So, two qubits, or two spin-1/2 particles (eg electrons) can be put in a combined system in many ways. The eigenstates of an individual qubit would be up and down, but it turns out you cannot simultaneously know the up/down of the two qubits in the total system, OR you cannot know the total spin. So - you have four eigenstates of TOTAL SPIN that are a spin-zero system (the l=0 spin singlet ) or the triply-degenerate l=1 spin-one system (the spin triplet). In the singlet state, there is only m=0, but the triplet can have m=+1, 0, or -1. Remember these quantum numbers from high-school chemistry? However, if you know you have a triplet, you cannot know the directions of the individual spins (unless you're in the state of both UP or both DOWN).
So basically, this is a bit more enigmatic than common-sense will dictate, and it's certainly much more complex (in the mathematical sense too) than simple vector mathematics.
The thing I find amazing about research with black holes is that effectivly we will never be able to prove this by direct observation.
And also without direct observation of the poster by the scientists in question, since they already presented their poster the day before yesterday at the poster session at AAS. When I first read in TFA that they'd be presenting their results at the AAS Winter Meeting , I thought it might be cool to see the talk or poster, but alas, it already happened.
Although I'm actually a condensed-matter physicist, as per the other posts on the phase-transitions thread I just typed up. But my girlfriend will be at AAS, so that's always a good excuse to anonymously crash the festivities:-)
In short... this does nothing for our "understanding" of phase changes.
Wrong, I highly suggest you take a Phase Transitions and Critical Phenomena class if you want to see the utility of methods such as this. As I noted in another post, though, this isn't the first method to allow computer simulations of points arbitrarily close to criticality, there have been other algorithms (eg Cluster algorithm) to allow this too. But every new algorithm to get past critical slowdown is very useful.
What we've learned in the past several decades in critical phenomena is how parameters change close to the critical point. For example, look up Critical Exponents and Scaling. What is very interesting is that critical exponents are unique to a universality class. So if you are able to take a new system and show that it boils down (no pun intended) to a previously-studied universality class, you can know instantly how various parameters will scale and change as a function of temperature, magnetic field, etc, close to the critical point.
And to give you an example of this, look at Superconductivity. It was originally discovered by Onnes in 1911, but it took 46 years until the BCS Theory was adequately able to explain how Cooper Pairs form and how resistanceless supercurrent can flow quantum-mechanically. Such a theory is referred to as 'microscopic', meaning it deals with the fundamental physics involved, specifically the electron-phonon-electron interaction and how the Fermi sea is unstable to Pair condensation.
However - alot of work was done prior to BCS dealing with 'macroscopic' theory, whereby certain laws were able to be formed (eg London equations for classical electrodynamics of a superconductor), we just didn't understand how or why they were valid.
One such important example is the Ginzburg-Landau theory (Landau won the Nobel Prize decades ago, Ginzburg just got it a couple of years ago), which extends Landau's Theory of 2nd-Order Phase Transitions to use a complex order parameter, which can vary in space. This yields the Ginzburg-Landau equations, which describe VERY WELL the behavior of a superconductor close to the transition point. It was using these equations that Josephson was able to come up with the concept of the Josephson Effect (earning him a Nobel Prize). And Abrikosov was able to come up wit the idea of Type II superconductors and vortices (he also won a Nobel Prize for this work). And after the BCS theory was understood, Gor'kov was able to show that the Ginzburg-Landau equations are a limiting case of the BCS theory close to the critical point.
However, the point of all this is that it was shown, before the microscopic BCS interactions were understood, scientists were able to do ALOT of things with the Ginzburg-Landau equations. What makes these so great is that they are able to approximate quantum mechanics decently, which the London equations were unable to do. And the best part is that scientists today (myself included) still use Ginzburg-Landau equations to model superconductors. It's just that much easier to use these equations for many interactions than the lower-level BCS theory. But amazingly, these equations were known BEFORE the BCS theory!
So back to your comment, such study of critical phenomena teaches us a great deal about systems in criticality, even if the methods involved are decoupled from the microscopic physics. Especially if one can determine the universality class of an unknown system. And for very complicated systems, critical exponents will be difficult to determine analytically and must be solved numerically. Hence the importance of simulations and algorithms such as this.
"This is the first time a computer program could simulate a phase transition because the computers would always bog down at what's known as the 'critical slowdown.' We figured out a way to perform a kind of end-run around that critical point slowdown and the results allow us to calculate certain critical point properties for the first time."
There have been previous methods to look at systems arbitrarily close to the critical point in phase transitions, and the article is misleading when it says this area has been off-limits to computers. I haven't read the actual Physical Review Letters article, but it appears the authors have come up with a novel algorithm, perhaps more ideally suited for fluids, to overcome critical slowdown. But this is not the first such algorithm, and there has been loads of prior computer simulation of phase transitions and critical phenomena to boot.
For example, in the Ising Model or the Potts Model, one can examine system parameters arbitrarily close to the critical point, in finite time, using a Cluster Algorithm. This page gives some information on how the cluster algorithm. The page has a java applet graphically depicting the system for a variety of algorithms.
Just for completeness, here's an Ising model applet that I wrote, which doesn't just have a system animation, but allows you to calculate and plot data (specific heat, magnetization, etc) as the system passes through the critical point. This applet uses the Metropolis algorithm for time advancement, hence it is subject to critical slowdown. In that respect, the applet is flawed because close to the critical point I don't generate enough Metropolis iterations to ensure the subsequent frame is sufficiently thermally indepdent from the previous state. However, the cluster algorithm would remove these limitations. This applet has actually been used in graduate physics classes at Johns Hopkins to demonstrate magnetic phase transitions.
And also for completeness, here's a Potts model applet, but it doesn't acquire data for plotting like the Ising model. The Potts applet actually uses the Microcanonical ensemble, whereby the energy of the system is conserved, but the Ising applet uses the Canonical ensemble, where the system is in contact with a heat bath at some settable temperature.
And in case anyone's curious, these applets (except for the first one) are part of the Java Virtual Physics Lab , which contains a few different physics java simulations I wrote to help with conceptual understanding.
Sadly, much of the history of physics seems to be directly, or indirectly due to man's need to make war.
I'm not really as nihilistic as that. IMHO, true physics itself is really the study of the music of the spheres. However, it's applied physics and engineering where the dollars come in for war.
But that's my point, Hawking's major work seems to be primarily black holes (eg Hawking radiation, and entropy). Einstein's work spanned a larger set of subfields of physics. Hawking also isn't the only person working on these topics, and hasn't opened up entirely new fields of research.
Additionally, Einstein used experimental data (eg his 'miraculous year' article on Brownian motion, which used IIRC sugar water). And he also proposed experiments to test the validity of his work (eg, measure the deviation of a star during solar eclipse). AFAIK Hawking hasn't done this either. Although in Hawking's defense, in Cosmology it's much harder to do experiments.
After the invention of the atomic bomb, governments realized that physicists could actually do something useful. Funding poured in and physics became a business.
That's right, it's not like governments ever funded people to use physics to predict the trajectory of a bullet from a large gun before the atomic bomb. And moreso, it's not like they ever decided that for large distances the Coriolis force and air resistance need to be properly accounted for and thus they never needed to fund the development of electric computers.
Nor did they ever need to understand physics to figure out the design of aerofoil wings, or the best shape to make ship hulls, before the atom bomb.
Hawking has contributed to the fields of GR and cosmology, but can you tell me the major discoveries and research he's conducted? Just because he writes a pop-science book and you've heard of him doesn't make him a 'great' in physics. Of course it doesn't mean he's not 'great' either.
So he's done some novel things within cosmology, along with Penrose, Rees, and others, but how does that compare with Einstein? Which of Hawking's discoveries do you think is worthy of a Nobel Prize, specifically why should Hawking get one over other cosmologists? Einstein should have had at least a few more Nobel prizes (special relativity itself is worthy, not to mention GR, and his study of Brownian Motion is pretty good too).
While Hawking is well-known (he'd probably be less famous if he wasn't in a wheelchair), Einstein's research ran a much wider gamut, including opening up entirely new areas of physics.
Einstein did amazing research across the whole gamut of physics, that's something that is much harder to do these days. For example, his miraculous year, he posited the theory of special relativity, came up with the photoelectric effect (which was a major leap for the study of quantum mechanics), and documented Brownian Motion (which was a major proof for accepting statistical mechanics of particles, especially in fluids).
But that was just one year, he made brilliant subsequent contributions to quantum mechanics and of course the theory of general relativity as well.
Einstein put Relativity on the table, which was previously unknown except to a few as something funky going on with Maxwell's equations under a Galilean transformation. This was an entirely new field. And, when extended into General Relativity, is a huge deal. Not many people get to discover a whole new field of physics like that. Newton did with mechanics. But with E&M, it was several people making discoveries, such as Ampere and Faraday and some others. And the full theory wasn't really collected nicely until Maxwell, who also corrected Ampere's Law. And that's only the classical theory, Quantum Electro-Dynamics is another huge thing. But within classical E&M, you can say Maxwell fully documented it, but it was already an explored field (no pun intended, seriously).
Thermodynamics and Statistical Mechanics had many people make major contributions, specifically when thermodynamics was found to be described entirely within statistical mechanical formalism. Boltzmann made major contributions, eg coming up with entropy and statistical ensembles, but his work wasn't accepted by the community and he ultimately wound up killing himself.
One physicist that may have come close to Einstein in breadth is my favorite, Lev Davidovic Landau. Any graduate student of physics should be familiar with at least some of his ten-volume "Course of Theoretical Physics", otherwise known as Landau-Lifshitz. Landau's grad students were known to be confused during meetings where he would shift topics from superconductivity to hadron interactions, etc. Landau made many amazing contributions, and also won a Nobel Prize, but he wasn't able to open up any entirely new fields of study like Einstein was able to. He made contributions to other fields, such as 2nd-order phase transitions, superconductivity and superfluidity, etc, but no entirely new fields.
Finally, Einstein was also rather active politically and socially, he didn't confine his efforts to the laboratory (well, really his desk since he was a theorist). He also had quite a unique physical appearance, which also contributed to his popularity. But I think, from a popular point of view, his contribution of relativity, which is probably one of the biggest scientific blowbacks to something that was previously accepted as scientifically true and complete, was the dominant factor. Of course scientifically he made many other major contributions, but for the newspapers, trumping over Newton is a rather 'hot' story.
Good God this article, posted at Duke university, is at the intellectual level of a 5 year old. In print and in person, seems like people are getting really dumb.
However, it's not as 'dumb' as someone mistaking a press release for the actual scholarly scientific article.
I didn't find a link to the article in the press release, and I'm too lazy to bother searching through the journal's Table of Contents to find the authors to get the appropriate link to the article itself, so instead I'll cut and paste the relevent part from the press release.
"The findings, published in the January 2006 issue of The Journal of Experimental Biology, challenge the notion that fundamental differences between apparently unrelated forms of locomotion exist."
They fucked us already, we got a package for $35/month, the installers said we can get local channels for an extra $5 monthly, so we said okay. Turns out the installers couldn't install the local antenna due to the structure of our house, although the satellite dish itself is fine. Okay, we thought, no big deal. They tell us we won't have to pay for the local channels.
Anyway, we wind up getting charged the local channels, when we call Dish they blame us. When we demanded to talk to one of the supervisors, he told us that the installer lied to us, that the rate really is $40/month, and that we are responsible. We told him they're charging us for goods and services they didn't provide, the guy doesn't care. We ask to talk to his manager, he absolutely refuses to pass on the call. I've never encountered that before. He seemed highly amused as we got upset over their ripoff shenanigans. His name is Jesse, employee ID # JOV.
After this we called back the installer again, they told us $35 is a valid amount and that the problem is Dish.
Regardless, they are such dickhead nitpickers that they're gaining an extra $60 for the yearly contract with us for local channels, but losing our business permanently when our 1-year contract is up. They demonstrated they'd rather nickle and dime us over lies and misleading statements instead of actually making us happy.
Wiretaps without the oversight of FISA are bad. Bush ignored FISA, the Clinton and Carter and Bush I wiretaps explicitly demanded that authorities comply within the FISA constraints.
There is an entire system of checks and balances to prevent the administration in the White House from gaining too much executive power. FISA is one of these checks and balances, Bush ignored it.
Not to mention that Bush adamantly denied to the press that he wiretaps Americans unless he has a court-approved warrant. Dammit, I can't find that quote now, but it's been kicking around the internets lately.
Hahaha, you just marked me as your 'foe' within the past hour or so. Come on, if you can't engage in reasonable discourse you mark someone as your foe? Congratulations, you're #3 of my 'freaks'. I guess you wouldn't want to waste any time responding to my posts, though. Others have, and we haven't resorted to call names.
I call you a Liar because you tell Lies. Stop saying what Clinton and Carter did is the same thing, or even worse as you claimed, as what Bush has done.
Can you name a SINGLE reason why anti-terror security would be enhanced if Bush did NOT comply with FISA? He can immediately listen to phone calls and emails, he just needs to report these to FISA within 72 hours to get a warrant after the fact.
Seriously, name ONE REASON why he needed to go beyond FISA for this. Him and Cheney say they could have caught some al queda guys in San Diego if they had this power then. WTF??? They could have spied on them just the same, this ruling is nothing but a power grab to remove accountability from Bush.
FISA was enacted to prevent the spying on political opponents Nixon did. The parties involved are kept secret if they are sensitive. Why did Bush go beyond this? It's been revealed Bush spied on vegans and other purely domestic emails of liberal-related groups with absolutely no ties to terror groups.
A republican administration won't be in the White House forever. Honestly ask yourself if you wouldn't have trepidations if Hillary Clinton monitored NRA members, pro-lifers, and pro-intelligent-design groups under the guise of protection from terror. Even if you tell me otherwise, I can't imagine any Republican letting Bush do this while not being repulsed at Hillary Clinton doing the same thing.
Slashdot Mirror
That's right, because we all know that MSFT didn't amass any of their wealth by abusing their monopoly position to push smaller more-innovative companies out of business. No, MSFT never exploited the efforts of "good people for making money".
I mean, if you're comparing these two companies, perhaps Apple is more likely to exploit someone in an advertisement, while MSFT is more likely to drive them out of business and steal their marketshare. Which is worse? You seem to be more disappointed with Apple, but I disagree.
Asimov just might like it, as long as it complied with the lesser-known Zeroth Law of Robotics - "A robot may not injure humanity or, through inaction, allow humanity to come to harm."
Did you read that statistic in a peer-reviewed science journal?
BTW, I re-read my original response to your first response, and it's bloody awful, no wonder you thought I was a fool. Like I said, I accidentally killed the tab, so I tried to whip up something quickly before missing too much of the Daily Show, and wound up mangling terms in my haste, eg vector space vs Hilbert space, etc.
Basically I didn't agree with how you said combining two qubits together is simple vector addition, because as we've beaten to death by this point it's really tensorial, like a linear combination of two linear combinations, yada yada yada.
I still disagree with you about quantum mechanics being more enigmatic than classical mechanics, but as you've said, most quantum computing routines don't require knowledge of group theory and only 'baby' tensor stuff. (Although if you want a good understanding of the actual quantum mechanics underneath, especially in regards to real physical systems such as the hydrogen atom, than those concepts become more important).
That applet I linked to basically lets you choose any input qubit, operate on it w/ a number of gates (the Pauli gates, Hadamard, and some others), and see the resulting qubit, in real time. Displays both the spinor representation, as well as graphically on the Bloch Sphere, for both qubits. So you can see in real-time the effects of the various single-qubit gates on any qubit value. Granted there's not much useful stuff to do with only single qubits, but it's a start. I've got the two-qubit version in the works, haven't touched it in about a year.
I've also debated such a 'dummy' quantum computing guide, becuase like you said, qubits aren't very difficult conceptually wise. In fact, that was part of my motivation in making these Java applets. If you're serious about working on this and would like to colloborate on something, send me an email at the address on the applet page.
My whole point is that the O'Keefe style of management, namely cutting important science missions in the name of 'safety', is changing. Especially since the Columbia accident happened on O'Keefe's watch, while Griffin comes in with a clean record.
Hell, maybe this example is even common knowledge amonst the slashdot crowd.
Um dude, O'Keefe has been gone from NASA for nine months now, your article link is almost a year old. One of the first things that the new administrator Michael Griffin did when he took over the reins was to try to figure out ways to keep Hubble alive. Griffin's an actual scientist, unlike O'Keefe who's a career-track manager. And thus sees the important of Hubble, which has been indispensible for astronomical research.
Direct from NASA's Hubble page , it says
Okay, we're in total agreement here. The whole point of my reply was because in your original post you said there's nothing enigmatic about quantum mechanics, and that adding qubits is trivial. I disagree with both those statements. Granted, the quote you replied to in your original quote says "quantum mechanics" but the rest of your post deals primarily with quantum computing.
However, I still disagree with you that it's trivial to add qubits, which is mathematically equivalent to adding angular momentum. Like I said, a two-qubit case is relatively simple, but things get much more complicated in higher-order systems. Maybe such higher-order systems aren't as prevalent in quantum computing, so perhaps such topic doesn't need to be known in depth for most quantum computing algorithms.
when you form the tensor product your're writing it as the sum of the spin-0 and spin-1 irreps (shorthand: 2*2 = 1+3). You need to do this to understand the total spin of the combined system. But this has nothing to do with quantum computing.
Okay, you basically proved my point, with your tensorial shorthand of 2x2=1+3. That is not simple vector addition, at least not in my book.
Perhaps we differ in outlook in that you regard such a tensor product as 'baby stuff', as per your original post, whereas I think it's not as trivial. In that regards we can just leave it that you're much smarter than me if you consider tensor products as simplistic math. But again, this simple 2x2 case is fairly easy, and if this is as difficult as things get for most quantum computing routines, than you're right this isn't that much of a big deal.
But things are more complicated in general, such as in physical angular momentum rotations, which are important in rotational and atomic systems, where IIRC O(3) operations can lead to 3x3=5+3+1 irreducible forms, and can yield nifty tricks like the version of Wigner-Eckart for a closed surface integral containing the product of three spherical harmonics. But again, I guess such techniques aren't useful in quantum computing.
You must be a particle physicist or something.
Condensed matter experimentalist, currently working with mesoscopic transport and superconducting nanowires. What is your field of research?
But we're not looking at multiple-spin systems. You're going off at a tangent. None of the basic quantum computing algorithms (Shor's algorithm, Grover's algorithm etc.) make any reference to combining spin.
But I disagree, and you did demonstrate this earlier with your 2x2=1+3. That IS a multiple-spin system, and it DOES deal with Clebsch-Gordon coefficients, just the most simplistic ones there are. AFAIK, the controlled-not is one of the most fundamental gates in quantum computing, and that does deal with two-qubit systems, in which case the level of entanglement is built into the 4-element complex state.
And the reason I keep bringing up Clebsch-Gordon is merely to demonstrate, which apparently you DO understand, the concept that adding angular momentum isn't quite trivial as it would appear at first glance. At least not to me. And since qubits ARE representations of SU(2) algebrea, and are synonymous with a spin-1/2 particle, it is valid to talk about them from this point of view.
But granted I'm not as familiar with that many theorems of quantum computing, especially any error-correcting. But I know that quantum teleportation requires the spin-singlet state, in which case entanglement IS quite important. Perhaps this routine isn't really considered within the scope of quantum computation? Additionally, I also thought some quantum computing routines are possible because they can effectively circumvent the no-cloning limitation by using entanglement as a means to build in some kind of redundancy. But that's just a hand-wavy concept, maybe
At first I thought you were trolling, and you gave me a good laugh. After responding to this, my conclusion is that you know some quantum computing stuff, but you really don't have a decent understanding of the underlying quantum mechanics.
An isolated qubit is represented by a 2D (complex) vector space. Where does the group theory come in? Read Shor's paper on factoring numbers.
Okay, don't just take my word for it. Take a look at this book . Notice in the description it says "A group theoretic abstraction of Shor's algorithms completes the discussion of algorithms." Want another example? This course specifically teaches some group theory before getting to the quantum algorithms.
You suggested in your first post that any system of more than one qubit merely involves 'adding' their vector spaces. Ie, you said this
"What is enigmatic about adding two vectors in a vector space? I can't stand the way popular science press insist on making bizarre statements about the most trivial mathematics and science in an attempt to make it more interesting."
Okay, please demonstrate how to form the spin-singlet state by _adding_ the vectors of the two individual spinors. Now explain how to form the m=0 spin-triplet state by again adding the vectors of the two spinors. You cannot do this with 'baby stuff' arithmetic as you insinuated in your original post.
Ie, you're increasing the size of your Hilbert space, so it's not basic addition at all. You would be right if you said both could be formed as a linear combination of two individual up/down and down/up states. But each of the individual states themselves is a spinor, so by 'combining' these two states. you're not doing simple 'vector addition'. You are actually dealing with tensors, and group theory, only you're not realizing it. And while this might be relatively simple for two qubits, once you add many qubits to the system the complexity increases quickly.
In fact, you need to know next to nothing about group theory to understand what a qubit is. An isolated qubit is represented by a 2D (complex) vector space. Where does the group theory come in?
You're right that an isolated qubit can be represented by a two-element spinor. The group theory comes in for systems with multiple qubits. Here's a quick explanation for a two-qubit system. For reasons I explained in the original post, there are only two free parameters, hence an easy Bloch Sphere representation, for one qubit. Two qubits together, however, have SIX free parameters (the overall system state has four complex elements, but it's normalized and also has a meaningless phase). How do you take two qubits with two degrees of freedom, and combine them to get a system with six degrees of freedom? The combination itself is NOT simple vector addition, because you cannot merely _add_ two qubits together. The description of where the extra degrees of freedom come from when combining two qubits is described by the group theory. Or you can talk about from a tensor point of view by including irreducible tensors, etc.
As far as I can see you're just part of the grand conspiracy to make Quantum Mechanics, and especially Quantum Computing, seem far more mysterious than it is. Shame on you.
Coupled systems ARE enigmatic, and I'm sure you've heard of EPR and Bell's Inequality. If you think it's "baby stuff", then you seem to be one of those scientists that just follow through calculations 'plug-and-chug', without any comprehension of the underlying concepts.
So as I said previously, at first I thought you were trolling, but I now think I understand where you're coming from. I guess in your experience
What is enigmatic about adding two vectors in a vector space?
Nothing about adding vectors. However - qubits are NOT vectors, they're representations of SU(2) algebra.
States in a quantum computer are elements of a vector space. You learn what vector spaces are in the first year of an undergraduate course in mathematics. This is baby stuff.
No, this is very wrong, it's not simple vector space, it involves group theory, specifically that of Lie Groups. Additionally, spin states aren't in a vector space, but in a Hilbert Space.
basically - a qubit is a two-element spinor, each element is a complex number, so that leaves four independent terms. However, the whole spinor must be normalized, and additionally there's an overall phase that's meaningless. So that leaves only two free parameters which completely describe the qubit. And these parameters can be transformed such that they lie nicely on the surface of a sphere, known as the Bloch Sphere. I assume that's what you mean when you are talking about qubits as vectors. but that only works for individual vectors, it completely breaks down when dealing with systems of vectors because it ignores the entanglement between multiple qubits!
However, any elementary student of quantum mechanics learns quickly that spin, which is an embodiment of angular momentum, has individual components that DO NOT COMMUTE. Basically - a quantum vector would allow you to simultaneously know the x,y, and z components. Not true with angular momentum, you can only know something about the overall length of the vector and the component of the vector on only ONE dimension. Therefore, any angular momentum in quantum mechanics is really pointing somewhere on a ring of uncertaintly along a sphere.
Now, when you look at a system of two spins, and qubits can be represented as spin-1/2 system, you must use the Clebsch-Gordon coefficients to add them. Angular momentum addition is actually an esoteric topic. It is straightforward once you learn it, but it's definitely not as straightforward as adding two vectors.
So, two qubits, or two spin-1/2 particles (eg electrons) can be put in a combined system in many ways. The eigenstates of an individual qubit would be up and down, but it turns out you cannot simultaneously know the up/down of the two qubits in the total system, OR you cannot know the total spin. So - you have four eigenstates of TOTAL SPIN that are a spin-zero system (the l=0 spin singlet ) or the triply-degenerate l=1 spin-one system (the spin triplet). In the singlet state, there is only m=0, but the triplet can have m=+1, 0, or -1. Remember these quantum numbers from high-school chemistry? However, if you know you have a triplet, you cannot know the directions of the individual spins (unless you're in the state of both UP or both DOWN).
So basically, this is a bit more enigmatic than common-sense will dictate, and it's certainly much more complex (in the mathematical sense too) than simple vector mathematics.
And also without direct observation of the poster by the scientists in question, since they already presented their poster the day before yesterday at the poster session at AAS. When I first read in TFA that they'd be presenting their results at the AAS Winter Meeting , I thought it might be cool to see the talk or poster, but alas, it already happened.
Although I'm actually a condensed-matter physicist, as per the other posts on the phase-transitions thread I just typed up. But my girlfriend will be at AAS, so that's always a good excuse to anonymously crash the festivities :-)
Wrong, I highly suggest you take a Phase Transitions and Critical Phenomena class if you want to see the utility of methods such as this. As I noted in another post, though, this isn't the first method to allow computer simulations of points arbitrarily close to criticality, there have been other algorithms (eg Cluster algorithm) to allow this too. But every new algorithm to get past critical slowdown is very useful.
What we've learned in the past several decades in critical phenomena is how parameters change close to the critical point. For example, look up Critical Exponents and Scaling. What is very interesting is that critical exponents are unique to a universality class. So if you are able to take a new system and show that it boils down (no pun intended) to a previously-studied universality class, you can know instantly how various parameters will scale and change as a function of temperature, magnetic field, etc, close to the critical point.
And to give you an example of this, look at Superconductivity. It was originally discovered by Onnes in 1911, but it took 46 years until the BCS Theory was adequately able to explain how Cooper Pairs form and how resistanceless supercurrent can flow quantum-mechanically. Such a theory is referred to as 'microscopic', meaning it deals with the fundamental physics involved, specifically the electron-phonon-electron interaction and how the Fermi sea is unstable to Pair condensation.
However - alot of work was done prior to BCS dealing with 'macroscopic' theory, whereby certain laws were able to be formed (eg London equations for classical electrodynamics of a superconductor), we just didn't understand how or why they were valid.
One such important example is the Ginzburg-Landau theory (Landau won the Nobel Prize decades ago, Ginzburg just got it a couple of years ago), which extends Landau's Theory of 2nd-Order Phase Transitions to use a complex order parameter, which can vary in space. This yields the Ginzburg-Landau equations, which describe VERY WELL the behavior of a superconductor close to the transition point. It was using these equations that Josephson was able to come up with the concept of the Josephson Effect (earning him a Nobel Prize). And Abrikosov was able to come up wit the idea of Type II superconductors and vortices (he also won a Nobel Prize for this work). And after the BCS theory was understood, Gor'kov was able to show that the Ginzburg-Landau equations are a limiting case of the BCS theory close to the critical point.
However, the point of all this is that it was shown, before the microscopic BCS interactions were understood, scientists were able to do ALOT of things with the Ginzburg-Landau equations. What makes these so great is that they are able to approximate quantum mechanics decently, which the London equations were unable to do. And the best part is that scientists today (myself included) still use Ginzburg-Landau equations to model superconductors. It's just that much easier to use these equations for many interactions than the lower-level BCS theory. But amazingly, these equations were known BEFORE the BCS theory!
So back to your comment, such study of critical phenomena teaches us a great deal about systems in criticality, even if the methods involved are decoupled from the microscopic physics. Especially if one can determine the universality class of an unknown system. And for very complicated systems, critical exponents will be difficult to determine analytically and must be solved numerically. Hence the importance of simulations and algorithms such as this.
For example, in the Ising Model or the Potts Model, one can examine system parameters arbitrarily close to the critical point, in finite time, using a Cluster Algorithm. This page gives some information on how the cluster algorithm. The page has a java applet graphically depicting the system for a variety of algorithms.
Just for completeness, here's an Ising model applet that I wrote, which doesn't just have a system animation, but allows you to calculate and plot data (specific heat, magnetization, etc) as the system passes through the critical point. This applet uses the Metropolis algorithm for time advancement, hence it is subject to critical slowdown. In that respect, the applet is flawed because close to the critical point I don't generate enough Metropolis iterations to ensure the subsequent frame is sufficiently thermally indepdent from the previous state. However, the cluster algorithm would remove these limitations. This applet has actually been used in graduate physics classes at Johns Hopkins to demonstrate magnetic phase transitions.
And also for completeness, here's a Potts model applet, but it doesn't acquire data for plotting like the Ising model. The Potts applet actually uses the Microcanonical ensemble, whereby the energy of the system is conserved, but the Ising applet uses the Canonical ensemble, where the system is in contact with a heat bath at some settable temperature.
And in case anyone's curious, these applets (except for the first one) are part of the Java Virtual Physics Lab , which contains a few different physics java simulations I wrote to help with conceptual understanding.
I'm not really as nihilistic as that. IMHO, true physics itself is really the study of the music of the spheres. However, it's applied physics and engineering where the dollars come in for war.
Additionally, Einstein used experimental data (eg his 'miraculous year' article on Brownian motion, which used IIRC sugar water). And he also proposed experiments to test the validity of his work (eg, measure the deviation of a star during solar eclipse). AFAIK Hawking hasn't done this either. Although in Hawking's defense, in Cosmology it's much harder to do experiments.
That's right, it's not like governments ever funded people to use physics to predict the trajectory of a bullet from a large gun before the atomic bomb. And moreso, it's not like they ever decided that for large distances the Coriolis force and air resistance need to be properly accounted for and thus they never needed to fund the development of electric computers.
Nor did they ever need to understand physics to figure out the design of aerofoil wings, or the best shape to make ship hulls, before the atom bomb.
So he's done some novel things within cosmology, along with Penrose, Rees, and others, but how does that compare with Einstein? Which of Hawking's discoveries do you think is worthy of a Nobel Prize, specifically why should Hawking get one over other cosmologists? Einstein should have had at least a few more Nobel prizes (special relativity itself is worthy, not to mention GR, and his study of Brownian Motion is pretty good too).
While Hawking is well-known (he'd probably be less famous if he wasn't in a wheelchair), Einstein's research ran a much wider gamut, including opening up entirely new areas of physics.
Einstein put Relativity on the table, which was previously unknown except to a few as something funky going on with Maxwell's equations under a Galilean transformation. This was an entirely new field. And, when extended into General Relativity, is a huge deal. Not many people get to discover a whole new field of physics like that. Newton did with mechanics. But with E&M, it was several people making discoveries, such as Ampere and Faraday and some others. And the full theory wasn't really collected nicely until Maxwell, who also corrected Ampere's Law. And that's only the classical theory, Quantum Electro-Dynamics is another huge thing. But within classical E&M, you can say Maxwell fully documented it, but it was already an explored field (no pun intended, seriously).
Thermodynamics and Statistical Mechanics had many people make major contributions, specifically when thermodynamics was found to be described entirely within statistical mechanical formalism. Boltzmann made major contributions, eg coming up with entropy and statistical ensembles, but his work wasn't accepted by the community and he ultimately wound up killing himself.
One physicist that may have come close to Einstein in breadth is my favorite, Lev Davidovic Landau. Any graduate student of physics should be familiar with at least some of his ten-volume "Course of Theoretical Physics", otherwise known as Landau-Lifshitz. Landau's grad students were known to be confused during meetings where he would shift topics from superconductivity to hadron interactions, etc. Landau made many amazing contributions, and also won a Nobel Prize, but he wasn't able to open up any entirely new fields of study like Einstein was able to. He made contributions to other fields, such as 2nd-order phase transitions, superconductivity and superfluidity, etc, but no entirely new fields.
Finally, Einstein was also rather active politically and socially, he didn't confine his efforts to the laboratory (well, really his desk since he was a theorist). He also had quite a unique physical appearance, which also contributed to his popularity. But I think, from a popular point of view, his contribution of relativity, which is probably one of the biggest scientific blowbacks to something that was previously accepted as scientifically true and complete, was the dominant factor. Of course scientifically he made many other major contributions, but for the newspapers, trumping over Newton is a rather 'hot' story.
However, it's not as 'dumb' as someone mistaking a press release for the actual scholarly scientific article.
I didn't find a link to the article in the press release, and I'm too lazy to bother searching through the journal's Table of Contents to find the authors to get the appropriate link to the article itself, so instead I'll cut and paste the relevent part from the press release.
Anyway, my point is that the whole electoral college system is utterly ridiculous.
Anyway, we wind up getting charged the local channels, when we call Dish they blame us. When we demanded to talk to one of the supervisors, he told us that the installer lied to us, that the rate really is $40/month, and that we are responsible. We told him they're charging us for goods and services they didn't provide, the guy doesn't care. We ask to talk to his manager, he absolutely refuses to pass on the call. I've never encountered that before. He seemed highly amused as we got upset over their ripoff shenanigans. His name is Jesse, employee ID # JOV.
After this we called back the installer again, they told us $35 is a valid amount and that the problem is Dish.
Regardless, they are such dickhead nitpickers that they're gaining an extra $60 for the yearly contract with us for local channels, but losing our business permanently when our 1-year contract is up. They demonstrated they'd rather nickle and dime us over lies and misleading statements instead of actually making us happy.
There is an entire system of checks and balances to prevent the administration in the White House from gaining too much executive power. FISA is one of these checks and balances, Bush ignored it.
Not to mention that Bush adamantly denied to the press that he wiretaps Americans unless he has a court-approved warrant. Dammit, I can't find that quote now, but it's been kicking around the internets lately.
Hahaha, you just marked me as your 'foe' within the past hour or so. Come on, if you can't engage in reasonable discourse you mark someone as your foe? Congratulations, you're #3 of my 'freaks'. I guess you wouldn't want to waste any time responding to my posts, though. Others have, and we haven't resorted to call names.
I call you a Liar because you tell Lies. Stop saying what Clinton and Carter did is the same thing, or even worse as you claimed, as what Bush has done.
Seriously, name ONE REASON why he needed to go beyond FISA for this. Him and Cheney say they could have caught some al queda guys in San Diego if they had this power then. WTF??? They could have spied on them just the same, this ruling is nothing but a power grab to remove accountability from Bush.
FISA was enacted to prevent the spying on political opponents Nixon did. The parties involved are kept secret if they are sensitive. Why did Bush go beyond this? It's been revealed Bush spied on vegans and other purely domestic emails of liberal-related groups with absolutely no ties to terror groups.
A republican administration won't be in the White House forever. Honestly ask yourself if you wouldn't have trepidations if Hillary Clinton monitored NRA members, pro-lifers, and pro-intelligent-design groups under the guise of protection from terror. Even if you tell me otherwise, I can't imagine any Republican letting Bush do this while not being repulsed at Hillary Clinton doing the same thing.