Mathematically what is the difference between pure separable state and entangled state ?  Can anyone explain with equations?
76094
I am currently trying to establish a clear picture of pure/mixed/entangled/separable/superposed states. In the following I will always assume a basis of $|1\rangle$ and $|0\rangle$ for my quantum systems. This is what I have so far:   superposed: A superposition of two states which a system $A$ can occupy, so $\frac{1}{\sqrt{2}}(|0\rangle_A+|1\rangle_A)$ seperable: $|1\rangle_A|0\rangle_B$ A state is called separable, if its an element of the (tensor)product basis of system $A$ and $B$ (for all possible choices of bases) entangled: $\frac{1}{\sqrt{2}}(|0\rangle_A|1\rangle_B+|1\rangle_A|0\rangle_B)$ is not a state within the product basis (again for all possible bases). mixed state: Is a statistical mixture, so for instance $|1\rangle$ with probability $1/2$ and $|0\rangle$ with probability $1/2$ pure state: Not a mixed state, no statistical mixture   I hope that the above examples and classifications are correct. If not it would be great if you could correct me. Or add further cases, if this list is incomplete.   On wikipedia I read about      Another way to say this is that while the von Neumann entropy of the whole state is zero (as it is for any pure state), the entropy of the subsystems is greater than zero.   which is perfectly fine. However I also read on wikipedia a criterion for :     Another, equivalent, criterion is that the von Neumann entropy is 0 for a pure state, and strictly positive for a mixed state.   So does this imply that if I look at the subsystems of an entangled state, that they are in a mixed state? Sounds strange... What would be the statistical mixture in that case?  Moreover I also wanted to ask, whether you had further illustrative examples for the different states I tried to describe above. Or any dangerous cases, where one might think a state of one kind to be the other?
33514
I am trying to get an intuitive idea of how the  affects Quantum computation. My understanding is that given a qubit $Q$ in superposition $Q_0 \left| 0 \right&gt; + Q_1 \left| 1 \right&gt;$, NCT states another Qubit $S$ cannot be designed such that $S$ is equivalent to the state of $Q$.   Now the catch is, what does Equivalent mean? It could mean either that:   $S = S_0 \left| 0 \right&gt; + S_1 \left| 1 \right&gt;$ such that $S_0 = Q_0, S_1 = Q_1$. Or it could mean that $S = Q$, meaning that if $S$ is observed to be some value ( for example 0) then $Q$ MUST be that same value, and vice versa.    So it seems that point 2, occurs anyways in entangled systems (particularly cat-states), so I can eliminate that option and conclude that that No Cloning states, given a qubit $Q$, it's impossible to make another qubit $S$ such that:  $S = S_0 \left| 0 \right&gt; + S_1 \left| 1 \right&gt;$ such that $S_0 = Q_0, S_1 = Q_1$.  Is this correct?
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?
76290
My confusion about quantum theory is twofold:    I lack an adequate understanding of how the mathematics of quantum theory is supposed to correspond to phenomena in the physical world I still have an incomplete picture in my mind of how cause and effect relationships occur at the quantum level of reality.    This is why phenomena such as "entanglement" make absolutely no sense to me. So, in an attempt to come to some understanding of all of this, I would like to know that if what we conceptualize as a "field" is merely an interaction among particles, and particles themselves are actually fluctuations in "fields", then which comes first, particles or fields?
88021
I have a problem understanding the solution provided for the problem here:    It's from the book "Quantum Mechanics - Concepts and Applications", 2nd ed., by N. Zettili (Wiley, 2009).   My issue is that I don't understand the second equality of eq. (6.223). Why do these two integrals cancel each other out?
59014
So I'm an IT engineer and i just watched a video about Quantum entanglement, that somehow when we split the entangled photons, any change happening to one of them will be reflected to the other one INSTANTLY, and therefore we can use this technology to transfer data from one point to the other instantly, basically even faster than the speed of light because no travel is required so we can even send data without the need of encryption!!   and  China has developed a technology like this too   so since i have 0 knowledge about quantum  physics and this blows my mind, can someone here explain this in a simple manner that how is this possible? and does this mean we can communicate with other planets INSTANTLY no matter how far they are ? doesn't this change everything?  also what is this technology called? where we separate the entangled photons and use them to communicate, basically can you guys also tell where should i start because i want to know everything about this technology, but I'm not sure where to start!
49703
It is usually said that the points on the surface of the  represent the pure states of a single 2-level quantum system. A pure state being of the form: $$ |\psi\rangle = a |0\rangle+b |1\rangle $$ And typically the north and south poles of this sphere correspond to the $|0\rangle$ and $|1\rangle$ states. Image: ("Bloch Sphere" by Glosser.ca - Own work. Licensed under CC BY-SA 3.0 via Commons - )    But isn't this very confusing? If the north and south poles are chosen, then both states are on the same line and not orthogonal anymore, so how can one choose an arbitrary point $p$ on the surface of the sphere and possibly decompose it in terms of $0,1$ states in order to find $a$ and $b$? Does this mean that one shouldn't regard the Bloch sphere as a valid basis for our system and that it's just a visualization aid? I have seen decompositions in terms of the internal angles of the sphere, in the form of: $a=\cos{\theta/2}$ and $b=e^{i\phi}\sin{\theta/2}$ with $\theta$ the polar angle and $\phi$ the azimuthal angle. But I am clueless as to how these are obtained when $0,1$ states are on the same line.
49222
In quantum mechanics, one makes the distinction between mixed states and pure states.  A classic example of a mixed state is a beam of photons in which 50% have spin in the positive $z$-direction and 50% have spin in the positive $x$-direction.  Note that this is not the same as a beam of photons, 100% of which are in the state $$ \frac{1}{\sqrt{2}}[\lvert z,+\rangle + \lvert x,+\rangle]. $$ It seems, however, that at least in principle, we could describe this beam of particles as a single pure state in a 'very large' Hilbert space, namely the Hilbert space that is the tensor product of all the $\sim 10^{23}$ particles (I think that's the proper order of magnitude at least).  So then, is the density operator a mathematical convenience, or are there other aspects of quantum mechanics that truly require the density operator to be a 'fundamental' object of the theory?  (If what I mean by this is at all unclear, please let me know in the comments, and I will do my best to clarify.)
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.
59509
recently i was watching a  on quantum computing where the narrators describes that quantum entanglement information travels faster than light!  Is it really possible for anything to move faster than light? Or are the narrators just wrong?  Regards,
8335
From everything I've read about quantum mechanics and quantum entanglement phenomena, it's not obvious to me why quantum entanglement is considered to be an active link. That is, it's stated every time that measurement of one particle affects the other.  In my head, there is a less magic explanation: the entangling measurement affects both particles in a way which makes their states identical, though unknown. In this case measuring one particle will reveal information about state of the other, but without a magical instant modification of remote entangled particle.  Obviously, I'm not the only one who had this idea. What are the problems associated with this view, and why is the magic view preferred?
6845
Why are states describing composite systems given by tensor products and not Cartesian products? They both belong in spaces of equal dimension, what are the differences between the two?
65297
This question is specific to the Copenhagen interpretation, which states that the wave function collapses on interaction. If we have a beam of light reflected off a mirror, whether you see this light as a quantum field or a stream of photons, either way the reflected light is not the same as the original light. The original photons have been absorbed in the mirror and new photons reemitted in a different direction. The wave function also is different accordingly. So on one hand, it would seem that the original wave function collapses at the mirror.  However, a mirror reflection does not affect the results of interference experiments, such as the double slit experiment. Mirrors are used in lasers producing highly coherent light. And finally, mirrors are used in various quantum entanglement setups without affecting the state of entanglement. If this understanding is correct, then it would seem that a mirror does not collapse the wavefunction.  What is the correct understanding and the physical intuition behind it? If the reflected photon is not the original entangled photon, but is reemitted by the mirror, how can it possibly still be entangled with the counterpart of the original photon absorbed by the mirror?
122916
I've been searching for a quantum algorithm to compute the the inner product between two $n$-qubit quantum states, namely $\langle\phi|\psi\rangle$, which is in general a complex number. One can get $|\langle\phi|\psi\rangle|^2$ through thte SWAP test for multiple qubits, but this is not really the inner product as the information of the real and imaginary parts are lost. I came across this , which claims that the task can be done by simply using CZ gates. I'm not sure if I am mistaken, but it seems to me that such a circuit works for computational basis states, but not for general quantum states. Could somebody help me confirm if this is actually the case? If the above algorithm doesn't work for general quantum states, what algorithm would you suggest?
120880
I have come across the equation which comes out of the nothing  in Zettili's book Quantum mechanics concepts and applications p. 167:  $$\psi(\vec{r},t) ~=~ \langle \vec{r} \,|\, \psi(t) \rangle.$$   How do I know that I get function of position and time if i take an inner product of a state vector with an vector $\vec{r}$? Where is the proof?
101642
Quantum entanglement is known to be the exchange of quantum information between two particles at a distance, while quantum superposition is known to be the uncertainty of a particle (or particles) being in several states at once (which could also involve the exchange of quantum information for a particle that is known to be in several locations simultaneously).  I was wondering if all of this was nothing more but the exchange of quantum information between different masses, and if this could clear up all the confusion in terms of how quantum systems connect in this field of science.  A clear explanation for how both of these quantum phenomena work, and if they really are connected (the exchange of quantum information?) would be much appreciated.
103526
I have trouble interpreting the result of a problem. If we have a function $$\psi ( \theta , \phi) = e^{-3i\phi}cos \theta $$ and two operators $$A=\frac{\partial}{\partial \phi} $$ $$B=\frac{\partial} {\partial \theta} $$ It is clear that $\psi(\theta, \phi)$ is an eigenfunction of A and not B. But, on the other hand, I tried to evaluate the operator $[A, B]$ on a general function $\psi(\theta, \phi)$ that is separable, such that $$\psi(\theta, \phi)=\chi(\theta)\xi(\phi)$$, and therefore $$[A, B]\psi(\theta, \phi)=[A, B]\chi(\theta)\xi(\phi)=$$$$=\frac{\partial}{\partial \phi}\frac{\partial} {\partial \theta}(\chi(\theta)\xi(\phi)) - \frac{\partial} {\partial \theta}\frac{\partial}{\partial \phi}(\chi(\theta)\xi(\phi)) =$$$$=\chi'(\theta)\xi'(\phi)-\chi'(\theta)\xi'(\phi)=$$$$=\psi'(\theta, \phi)-\psi'(\theta, \phi)=0$$, which means that the commutator is 0 if applied to a separable function of $\phi$ and $\theta$. I know that if two operators commute, that means that they have a common set of eigenfunctions. Does "common set" mean that every eigenfunction of A is also an eigenfunction of B, or that some of the eigenfunctions of A are also eigenfunctions of B? I previously thought that it was the first choice, but here, $$\psi ( \theta , \phi) = e^{-3i\phi}cos \theta $$ is an eigenfunction of A only. Is there a way to find a general solution for functions that are eigenfunctions for both A and B?