It was an attempt to take all the existing theory literature and compose it into something like a "user's guide" or "road to success" for the groups who actually want to build a quantum computer.
Just as a side note, topological states can be created in the lab. BECs with controlled generation of quantum vortices are a topological state (technically a state featuring topological defects), and have been around since the late 90's.
I am also certain that topological insulators have been realised too.
Î have never gotten anyone to give me a better definition of "topological state" than "state that is described by a topological quantum number". When I ask what a "topological quantum number" is, they typically start listing examples.
No, just the quantum vortex states I mentioned are topological defects. Topological state seems like a blanket term for any quantum state featuring topological properties. For me, I know my system has topological defect, but as for other systems I'm afraid I can't help.
Not everything needs real-life application to be interesting. Luckily enough, these things are somewhat. Metrology, information storage, macroscopic quantum effects, superfluid turbulence, etc. In reality though I wouldn't expect to see one in your kitchen any time soon.
Well, the utility of what you work on can only be limited by what you think you can do with it. If you have a great idea, then that may make a great paper, or 3, leading to a great job, a career, etc.
Don't rule things out based on application so far. See what interests you most, then pursue that. Working towards a career in an area you aren't interested in is like a long path to suffering and misery, even if it has lots of applications.
@LeeJ.O'Riordan I understand that there are realizations in the lab, what's not been done yet is to show that topological states can be manipulated in the ways needed for information processing.
@0celo7 Uh...states are rays, and going to the projective space of a Hilbert space means going to the space where all points in a ray are identified, so every point in the projective space is actually a distinct state. Every point on the Bloch sphere corresponds to a ray in $\mathbb{C}^2$.
@DanielSank Because quantum states are vectors in Hilbert space up to a complex number $\lambda$. You get a projective Hilbert space by imposing the equivalence $|\Psi\rangle\sim\lambda|\Psi\rangle$ and taking the quotient.
@DanielSank Well, usually, QM is introduced as being set in a Hilbert space, but that states are rays means that the actual geometric object one should be looking at is the projective space. It's how projective representations, and thus half-integer spins arise, for example.
It's also the setting for Wigner's theorem, proving that any symmetry must be implemented as a unitary or anti-unitary operator on the original Hilbert space
@ACuriousMind I swear to Christ you've explained "It's how projective representations, and thus half-integer spins arise, for example." a dozen times but I still don't get it.
@DanielSank Well, there are other things - the entangled states are not a subspace of the combined Hilbert space, but they are a subvariety of the combined projective spaces. One can also do "geometric quantization" in a way that is about finding a way to translate the classical phase space with its symplectic form into the projective Hilbert space with a Kähler structure.
And it drives the point home that "normalization" doesn't really matter because multiplyin by a complex number is a do-nothing operation on the projective space
Ok I guess that might be what I had in mind, except it appplies only to a narrow band of wavelengths (so e.g. white light after passing through it becomes more intense blue
and I am thinking of stacking an arrray of these together
however my optics still need some refinement to see whether my proposal actually makes sense. I will do this later when I had time
@0celo7 Really not much, I think. They just want you to take the hint and transform the definition of that supremum into the definition of the infinmum.
ok, the middle $\le$ is the same reasoning as the middle $<$ in (a), except now $B$ can have a single element, so equality is possible.
Any bound for $A$ must also be a bound for $B$ by definition of subsets on $\mathbb{R}$.
But since $A$ and $B$ needn't have coinciding "smallest" and "biggest" elements, the infimum and supremum can differ.
But the smallest point of $B$ can never be smaller than the smallest point of $A$, thus establishing the first inequality. Then repeat this on the other end.
@0celo7 I don't know why you are being so complicated. You already said that the infimum/supremum of $A$ are bounds for $B$. Now use the definition of infimum/supremum of $B$ as largest/smallest bound.
@0celo7 For every $\epsilon(n) = \frac{1}{n}$, you have an $a(\epsilon)$ with $\lvert a(\epsilon) - s\rvert < \epsilon$. The sequence $\{a_n := a(\epsilon(n))\}$ is the sought-after sequence.
@0celo7 Ah, well, not in this case (but it's pretty hard to tell whether trying to construct something explicitly is the right approach or not a priori)
Let $\epsilon < \lvert a - b \rvert$. Then, $\exists N : \lvert a_k - a \rvert < \epsilon < \lvert a - b \rvert \forall k > N$. Now, I claim that this means $a_k > b \forall k > N$.
@0celo7 I'm not sure whether the book doesn't want you to use the triangle inequality (after all, it is a basic feature of the absolute value on the real numbers)
