Where do I see the diffusion in the Grovers operator? I would think there should be an analogy to classical diffusion but I am not able to see the correspondence.
122484
In the context of , the diffusion operator is defined as $U_s = 2|s\rangle \langle s| - I$ with $|s\rangle\equiv |+\rangle^{\otimes n}$. What is the significance of the term "diffusion"? Does it refer to some physical phenomena?
106947
Can someone show to me the steps to derive the joint state at the bottom of this image, please?   I tried to follow his explanation but I didn't get the same results… This is taken from the lecture notes of Ronald de Wolf in case it may help
21176
When expressing computations in terms of a quantum circuit, one makes use of gates, that is, (typically) unitary evolutions.  In some sense, these are rather mysterious objects, in that they perform "magic" discrete operations on the states. They are essentially black boxes, whose inner workings are not often dealt with while studying quantum algorithms. However, that is not how quantum mechanics works: states evolve in a continuous fashion following Schrödinger's equation.  In other words, when talking about quantum gates and operations, one neglects the dynamic (that is, the Hamiltonian) realising said evolution, which is how the gates are actually implemented in experimental architectures.  One method is to decompose the gate in terms of elementary (in a given experimental architecture) ones. Is this the only way? What about such "elementary" gates? How are the dynamics implementing those typically found?
109412
I've read , I've read , I've read , but I still just don't get it. How does a Quantum Computer work? It sounds very intriguing, but I just can't wrap my mind around it. I would greatly appreciate it if someone could explain it to me as they would explain the basics of it to their 7 year old nephew (who apparently is probably about equal to me in their ability to understand this concept!)   Typically analogies work best when explaining concepts to the unenlightened, but this is quantum theory, so I'm not sure any analogy could exist that can explain such a bizarre concept ;)
101251
This is a question I've been asked several times by students and I tend to have a hard time phrasing it in terms they can understand. This is a natural question to ask and it is not usually well covered in textbooks, so I would like to know of various perspectives and explanations that I can use when teaching.  The question comes up naturally in what is usually students' second course in quantum physics / quantum mechanics. At that stage one is fairly comfortable with the concept of wavefunctions and with the Schrödinger equation, and has had some limited exposure to operators. One common case, for example, is to explain that some operators commute and that this means the corresponding observables are 'compatible' and that there exists a mutual eigenbasis; the commutation relation is usually expressed as $[A,B]=0$ but no more is said about that object.  This naturally leaves students wondering     what is, exactly, the physical significance of the object $[A,B]$ itself?   and this is not an easy question. I would like answers to address this directly, ideally at a variety of levels of abstraction and required background. Note also that I'm much more interested in the object $[A,B]$ itself than what the consequences and interpretations are when it is zero, as those are far easier and explored in much more depth in most resources.    One reason this is a hard question (and that commutators are such confusing objects for students) is that they serve a variety of purposes, with only thin connecting threads between them (at least as seen from the bottom-up perspective).   Commutation relations are usually expressed in the form $[A,B]=0$ even though, a priori, there appears to be little motivation for the introduction of such terminology. A lot of stock is placed behind the canonical commutation relation $[x,p]=i \hbar$, though it is not always clear what it means.  (In my view, the fundamental principle that this encodes is essentially de Broglie's relation $\lambda=h/p$; this is made rigorous by the  but that's quite a bit to expect a student to grasp at a first go.) From this there is a natural extension to the Heisenberg Uncertainty Principle, which in its general form includes a commutator (and an anticommutator, to make things worse). Canonically-conjugate pairs of observables are often introduced, and this is often aided by observations on commutators. (On the other hand, the energy-time and angle-angular momentum conjugacy relations cannot be expressed in terms of commutators, making things even fuzzier.) Commutators are used very frequently, for example, when studying the angular momentum algebra of quantum mechanics. It is clear they play a big role in encoding symmetries in quantum mechanics but it is hardly made clear how and why, and particularly why the combination $AB-BA$ should be important for symmetry considerations.  This becomes even more important in more rigorous treatments of quantum mechanics, where the specifics of the Hilbert space become less important and the algebra of observable operators takes centre stage. The commutator is the central operation of that algebra, but again it's not very clear why that combination should be special. An analogy is occasionally made to the Poisson brackets of hamiltonian mechanics, but this hardly helps - Poisson brackets are equally mysterious. This also ties the commutator in with time evolution, both on the classical side and via the Heisenberg equation of motion.   I can't think of any more at the moment but they are a huge number of opposing directions which can make everything very confusing, and there is rarely a uniting thread. So: what are commutators, exactly, and why are they so important?
113328
This relates a little bit to my previous question (), but it's different in a way and more historical.  One of the postulates of quantum mechanics is:     To every observable in classical mechanics there corresponds a linear, Hermitian operator in quantum mechanics ().   However, absent in the list of postulates is what these operators are.  So is the mathematical expression for each individual operator also a postulate that's not listed, or are they derivable from other axioms?  Specifically, where did  expressions come from?  Maybe the question can better be answered historically.  If I had to guess, I'd say it began with the wavefunction.  Some scientists thought perhaps the squared modulus of the wavefunction corresponded to charge distribution or something, but Born was the one who got it right.  He essentially made a guess that the squared modulus corresponded to the probability of finding a particle at a certain location.  What's funny is that he initially wrote down the wrong expression in his paper (), and his paper even got rejected by the first journal he submitted it to, but the right expression is in a corrective footnote.    But again, Born simply guessed what the squared modulus meant.  There was no procedure; he just thought of an idea and then experiment verified it to be correct.  Thus:  $$\langle\psi|\hat{X}|\psi\rangle = \langle\psi|x\rangle\langle x|\hat{X}|x\rangle\langle x|\psi\rangle = \int d\vec{x}\ \psi^*(\vec{x})\ \psi(\vec{x})$$  So basically in the position basis, the position operator is simply $\hat{X} = x$, which seems pretty intuitive.  I believe the momentum operator came about next; and it was "derived" by fiddling around with the de Broglie relation and just substituting classical expressions for momentum into the Schrodinger equation (can't remember where I read that).  The fact that plugging these classical ideas into a quantum equation just happens to match experiment for momentum measurements seems like a pure coincident to me (it's along the same lines as trying to make the Schrodinger equation relativistic; one way of shoehorning relativity into it gives you the Dirac equation and a different way gives you the Klein-Gordon equation).  I imagine the spin operator came about by simple observation.  An electron goes up or electron goes down in a magnetic field.  Not hard to guess what the expression for that operator might be.  So anyway, am I on the right track here?  Have all of the operators come about simply by guessing?  And if I'm wrong, how did they actually come about?  How would I come up with a new operator for a new type of measurement?
6938
Is there one central reason why there is no "Time" operator in QM?  I know this question has been asked before, but I thought I would try to stimulate some fresh thinking.
10749
I am a physicist. I always heard physicists used the terminology "", "Hermitian", "", and "essentially self-adjoint" operators interchangeably.  Actually what is the difference between all those operators? Presumably understandable by a physicist.
53620
Can anyone point me to a reference (preferably either something online or something a small liberal arts school would be likely to have in its library) that goes through a derivation of the minimum uncertainty wavefunction in more detail than in the Griffiths?  Edit I've moved the second part of my original question to a separate post:
104380
What's wrong in taking the cat as an observer in Schrodinger's experiment? Plz kindly elaborate! And if possible also describe about possible logics if the question bears the answer No.
30729
Recently I'm curious about the topic of  which is not talked about in common quantum mechanics textbooks, i.e. Master equation, Lindblad equation, reduced density matrix, entanglement entropy, correlation, decoherence, , ,   and so on.   Are there some good references (easy and readable or rigorous and thorough) including papers, literatures or textbooks which can elucidate these topics related to open quantum systems?
79938
I'm a mathematician. I have good knowledge of superior analysis, distribution theory, Hilbert spaces, Sobolev spaces, and applications to PDE theory. I also have good knowledge of differential geometry. I would like to study the Semiclassic Analysis, but perhaps I must first study the foundations of quantum mechanics. So I would like to know what book you recommend me to begin studying quantum mechanics. I'm primarily interested in a mathematical point of view. I have seen some books of this type, but I would like to have some other opinion.  Thanks for every reply
12001
Why is  even a valid operation in the first place? I mean it can give us some nice results, but why can we do the ordering for the operators like that?  Is its definition motivated by the relation between the normal ordering and the time-ordered product, which is basically the content of Wick's theorem?
27861
Hm, this just occurred to me while answering another question:   If I write the Hamiltonian for a harmonic oscillator as $$H = \frac{p^2}{2m} + \frac{1}{2} m \omega^2 x^2$$ then wouldn't one set of possible basis states be the set of $\delta$-functions  $\psi_x = \delta(x)$, and that indicates that the size of my Hilbert space is that of $\mathbb{R}$.  On the other hand, we all know that we can diagonalize $H$ by going to the occupation number states, so the Hilbert space would be $|n\rangle, n \in \mathbb{N}_0$, so now the size of my Hilbert space is that of $\mathbb{N}$ instead.  Clearly they can't both be right, so where is the flaw in my logic?
56548
I need some help with functional analysis / Hilbert space theory. If you have a favorite text to recommend, please let me know~  Here is my question:  Given $v_t$ be the "squeeze operator" on $H=L^2[0, 1]$, where $v_t: L^2[0,1] \to L^2[0, \frac{2-t}{2}]$ acts on $f \in L^2[0,1]$ by squeezing the domain of the function. We have that $\{v_t\}$ for $t \in [0, 1]$ is a family of SOT continuous operators. I am wondering why given any $p \in \mathbb{K}(H)$, we have $\{ v_tpv_t^* \}$ is continuous in norm.  I found the following facts on a reference suggested by Wikipedia (Hilbert Space Operators in Quantum Physics), but I am not sure how to prove them or how I may use them...   $T_n \to^{SOT} T$ implies that for any $p \in \mathbb{K}(H)$, we have $T_np \to Tp$ in norm.   Hints or suggestions would be greatly appreciated!
36135
In the Quantum Operations section in Nielsen and Chuang, (page 358 in the 2002 edition), they have the following equation: $$\mathcal E(\rho) = \mathrm{Tr}_{env} [U(\rho \otimes \rho_{env})U^\dagger]$$ They show an example with $\rho_{env} = |0\rangle \langle0|$ and $U = \mathrm{CNOT}$, and claim that the final solution is: $$P_0\rho P_0 + P_1\rho P_1,$$ where $P_0=|0\rangle \langle0|$ and $P_1=|1\rangle \langle 1|$. These are my steps so far to get this, but I don't know how to trace out the environment after this: Let $\rho$ be $|\psi \rangle \langle \psi |$, so that $\rho \otimes \rho_{env} = |\psi, 0\rangle \langle \psi, 0|$. Applying the unitary $U$, we have $$ |00 \rangle \langle 00| \psi, 0 \rangle \langle \psi, 0 | 00 \rangle \langle 00 |  + |00 \rangle \langle 00| \psi 0 \rangle \langle \psi 0 | 10 \rangle \langle 11 | \\  + |11 \rangle \langle 10| \psi 0 \rangle \langle \psi 0 | 00 \rangle \langle 00 |  +  |11 \rangle \langle 10| \psi 0 \rangle \langle \psi 0 | 10 \rangle \langle 11 |. $$ I don't know how to trace out the environment in the above state. Also, I realize that I have considered only a pure state, if anyone can show it for a general state that would be great.
43609
Say we have some state $$|\psi\rangle=\frac{1}{\sqrt{2}}(|0\rangle+i|1\rangle)$$ it is in a quantum superposition of $|0\rangle$ and $|1\rangle$. Its density matrix is $$\rho=\begin{pmatrix}\frac 1 2 &amp; \frac i 2\\ -\frac i 2 &amp; \frac 1 2\end{pmatrix}$$ If we measure $\rho$ we get some classical density matrix of the form $$\rho=\begin{pmatrix}\frac 1 2 &amp; 0\\ 0 &amp; \frac 1 2\end{pmatrix}$$  Is there some nice intepretation of what these cross terms mean, other than just arrising from the phase of the state?