Questions tagged [bloch-sphere]
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48
questions
-1
votes
1answer
34 views
How to compute the axis of a rotation expressed as a combination of Pauli matrices? [closed]
An exercise asked to compute the axis and the angle of the rotation $THTH$.
$(S.1)$ is easy to understand by using the identity $exp(i\theta A) = \cos(\theta) \mathbb1 + i \sin(\theta) A$ for $A$ s.t....
3
votes
1answer
54 views
Constraints on higher-dimensional Bloch vectors
I'm interested in the constraints on the $(4^n-1)$-dimensional generalized Bloch vector (the Bloch vector for $n$ qubits). To the best of my knowledge, these are not analytically characterized for ...
0
votes
0answers
35 views
How to transform coordinates of two-level system's eigenstates from spherical to momentum space?
In almost all references for some two-level quantum system (for example, this Wikipedia article), arbitrary eigenstates are defined in spherical coordinates, and can be written separately as follows (...
1
vote
1answer
67 views
Can all expectation values be constructed using only tensor products of local Pauli operators?
I understand that the Pauli operators and the identity matrix span the space of complex 2x2 matrices.
Let's say you have two qubits and you can perform projective meausurements.
So you can measure ...
0
votes
1answer
154 views
Bloch sphere representation of an eigenvector
I'm trying to work through a problem that wants me to determine the Bloch sphere representation of the eigenvectors of $\sigma_{z}$.
I'm working in bra-ket notation so these would be $\ v_{+} = |0\...
-2
votes
1answer
67 views
How does this term $e^{i\Phi_0}$ get removed in bloch sphere equation?
A qubit can be represented in the form of
$$|\psi\rangle=\alpha|0⟩+\beta|1\rangle$$
where $\alpha$ and $\beta$ are complex numbers. Or a complex number can be expressed by $R e^{i\Phi_0}$.
so the ...
4
votes
1answer
135 views
How to obtain $Y$ rotation with only $X$ and $Z$ rotation gates on the Bloch sphere?
Let's say you have a system with which you can perform arbitrary rotations around the $X$ and $Z$ axis. How would you then be able to use these rotations to obtain an arbitrary rotation around the $Y$ ...
2
votes
3answers
104 views
Quantum Mechanics Notation
I'm studying the Bloch Sphere and just wanted to ask what this notation means:
$|\psi\rangle = \alpha|1\rangle$ for example
I'm just not familiar with the notation in this context if anyone could ...
8
votes
1answer
230 views
Characterisation of the generalised Bloch space in spherical coordinates
I'm so confused by the following definition in the "Quantum Error Correction" by Lidar and Brun that not even sure how to formulate the question properly.
Let $\mathbf n$ denote a unit vector, i.e.,...
0
votes
1answer
45 views
Interaction between joint qubit quantum system [closed]
Consider the following interaction Hamiltonian
$$H = \hbar \mu \sigma_{x} \otimes \sigma_x = \hbar \mu ( |01 \rangle \langle 1 0 | + |10\rangle\langle 01|)$$ acting on the joint states of qubits $\...
1
vote
1answer
156 views
Alternatives to Bloch Sphere? [closed]
The Bloch Sphere is a geometrical representation of the state space of a qubit system. I'm wondering if there are other natural geometrical representations one could use as alternatives to the Bloch ...
1
vote
1answer
127 views
Why are the bloch vector elements u and v also the inphase and quadrature components of the transition dipole moment?
The bloch vectors are given by
$$u=\frac{1}{2}(\rho_{ge}+\rho_{eg})$$
$$v=\frac{1}{2i}(\rho_{eg}-\rho_{ge})$$
$$w=\frac{1}{2}(\rho_{gg}-\rho_{ee})$$
Where $\rho$ is the density matrix for a two ...
0
votes
0answers
23 views
How does pulse duration and shape affect the rotation of a quantum state?
If I have some quantum state $|\psi>$ and I apply a pulse to this to rotate it by 90 degrees, the model would normally assume a rectangular pulse. However, realistically, the pulse will probably ...
1
vote
0answers
37 views
A shaped pulse as a sum of rectangular pulses
I have a pulse with lineshape
$L(ω)=\frac{1}{π}\frac{\frac{1}{2}Γ}{((ω−ω_0)^2+(\frac{1}{2}Γ)^2)}$
in the frequency domain where $\Gamma$ is the pulse width and $\omega_0$ is the resonant frequency ...
1
vote
1answer
148 views
Visual representation of quantum state/phase
Is there a known good way to visualize a quantum state, composed of the sum of eigenstates, with a phase rotating on each state. I am looking for a way to keep up with the state and the phase.
In a ...
1
vote
0answers
49 views
Virtual energy level in the Bloch sphere representation
How the Bloch vector would evolve, if the electron makes a transition to a virtual energy level (placed below the excited state of our two-level atom), as it does in all sorts of phenomena in ...
2
votes
1answer
2k views
How to calculate the bloch vector of a mixed state qubit
As I understand there are pure state and mixed state qubits. Pure states can be computed by
$$|\psi\rangle = \cos(\theta/2)|0\rangle + \exp(i \phi) \sin(\theta/2)|1\rangle . $$
Simple enough. You need ...
1
vote
0answers
92 views
Does an stereographical projection of a physical plane have any physical meaning?
Mathematically, an arbitrary 2D plane can be mapped onto a sphere by stereographical projection. Each line on the plane is equivalent to each line on the sphere. If the sphere rotates under the $SO(3)$...
0
votes
1answer
524 views
Spin-1 representation in the Bloch sphere
I am working now with spin-1 where I have instead of states $|{+1}\rangle$ and $|{-1}\rangle$, as for in spin-1/2 case, I have $|{+1}\rangle$, $|0\rangle$ and $|{-1}\rangle$. How can I check the ...
1
vote
1answer
32 views
Minimal decoherence for a 3 levels system
In the case of a two level system driven by a monochromatic excitation, a minimal amount of decoherence results from the finite lifetime of the excited state. In the optical bloch equations, the ...
0
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0answers
41 views
Probabilities of sequential measurments of spin 1/2 particle
I'm interested in checking why my working for the following problem does not agree with the expected result. In a section describing Bell's Inequality, the following sequence of measurments is ...
2
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0answers
60 views
Open orbits on the Bloch sphere of two level atom
I am studying the Bloch sphere representation of a two level atom in a classical electric field, starting in the ground state at t=0. The Bloch equations for the time evolution are
$$
\begin{align}
...
1
vote
3answers
1k views
If global phase doesn't matter why is relative phase important?
As I understand the state of any $1/2$ spin particle can be expressed as:
$$\chi = \dbinom{\cos(\beta/2)e^{-i \alpha/2}}{\sin(\beta/2)e^{i \alpha/2}} \, .$$
Why is it stated that "a phase common ...
-1
votes
1answer
73 views
spin eigenstates representation in QM
In an exercise in a Quantum Mechanics text (Sakurai Modern Quantum Mechanics) I completed, I showed that the eigenstates $ | \mathbf{S} \cdot \hat{n}; + \rangle$ of $$\mathbf{S} \cdot \hat{n} | \...
0
votes
1answer
126 views
Rotations of spin eigenstates in QM
If you have a state $| \psi \rangle = | \uparrow \rangle$ which is the spin eignstate of the spin operator $\hat{S}_z = \frac{\hbar}{2} \hat{\sigma}_{z}$ then if you view this state as a vector in the ...
3
votes
1answer
192 views
Bloch sphere representation of uncertainty
If we consider the Bloch sphere in quantum mechanics, which is a two level representation of a quantum mechanical system, then any state can be represented as $$| \psi \rangle = \cos\left(\theta/2\...
5
votes
4answers
3k views
Visual interpretation, on the Bloch sphere, when Hadamard gate is applied twice
It's known that the Hadamard operation is just a rotation of the sphere about the $\hat{y}$ axis by 90 degrees, followed by a rotation about the $\hat{x}$ axis by 180 degrees.
On the other hand, $H^{...
1
vote
3answers
1k views
How are linear combinations of qubit states represented in the Bloch sphere?
According to the Wikipedia article on the Bloch sphere, a pure state of a qubit can always be represented as $$| \psi \rangle = \cos \left( \frac{\theta}{2} \right)| 0 \rangle + e^{i \phi} \sin\left(\...
0
votes
1answer
210 views
Is there a way to represent a 3 qubit system using 3 Bloch Spheres?
I am relatively new to the Quantum Computing world and was wondering if representing a 3 qubit system using 3 Bloch Spheres feasible and if so what would the correct way to do it?
I understand a ...
5
votes
1answer
436 views
How to understand Bloch sphere representation?
I'm really new to quantum computation. Now, I'm going through a tutorial article Quantum Computation: a Tutorial (NB: PDF).
I was confused by certain points over there.
So, on page 5, when the ...
-1
votes
1answer
121 views
How to get a desired qubit trought Bloch Sphere rotations? [closed]
Starting from the zero pure qubit, how can I get the normalized qubit $$ \alpha\left|0\right\gt + \beta\left|1\right\gt $$ such that $$ |\alpha|^2 + |\beta|^2 = 1 $$ using the spherical coordinates $(\...
1
vote
0answers
150 views
Cavity quantum electrodynamics: difference between Rabi oscillations and vacuum Rabi oscillations?
I believe the title is pretty self-explanatory: I understand both roughly, but am unable to make a clear distinction.
We use a circuit formulation for which an LC circuit, being a resonator, is the ...
0
votes
1answer
213 views
Rotate quantum state by a given angle
Is there any gates that allow to rotate quantum state by a given angle $\theta$ in XZ plane? I'd like to move from state
$$|\psi\rangle = |0\rangle$$ to state $$|\psi\rangle = {1 \over{\sqrt2}} |0\...
0
votes
1answer
103 views
Bloch sphere rotation axis determined phase of light?
From what I have derived and read (see e.g. pg 13 of the pdf linked to here) the phase $\varphi$ of the radiation incident on a two-level system determines the axis of rotation. In such a case the ...
asked Dec 28 '16 at 6:29
Quantum spaghettification
1,2427 gold badges31 silver badges94 bronze badges
0
votes
2answers
546 views
Understanding the density matrix of a qubit
A density operator $\rho$ for the pure or mixed state of a qubit can be written in the following general form:
$$\rho = \begin{pmatrix}
a+b & c-id \\
c+id & a-b
\end{pmatrix}$$
I know $\rho$ ...
1
vote
1answer
83 views
Two coupled system of which one has dephasing
I'm reading Environment-assisted quantum transport by Rebentrost at all, where they deal with a system of 2 sites that hosts 1 excitation. They describe this in terms of two states: $\vert 1 \rangle$ ...
1
vote
2answers
54 views
Density operators in 2 dimension
Consider any density operator in two dimension. Call it $A$. Let $I$ be the identity matrix, and $\sigma_i, i=x,y,z$ be the Pauli Matrices.
Then we have to show that
$$A=\frac{1}{2}(I+n\cdot \...
0
votes
3answers
463 views
Definition of points on Bloch sphere
In the definition of the Bloch sphere, one demands that $\theta \in [0, \pi]$ ans $\phi \in [0, 2\pi)$ so that any state on the Bloch sphere can be represented by
$$|\phi \rangle= \cos(\theta/2)|0 \...
0
votes
1answer
65 views
Population Decay Model and the Rotating Wave Approximation
The paper which I'm reading mentions the following:
The Hamiltonian,
$$ \frac{\varepsilon}{2}\sigma_z + \sum_k \omega_k b_k^\dagger b_k + \sum_k (\sigma^+ g_k^* b_k + \sigma^- g_k b_k^\dagger), $$...
4
votes
1answer
759 views
Finding rotation axis of an operator on Bloch sphere
I would like to find the action of an operator $V$ on the Bloch sphere. The matrix representation of $V$ is
$V=\sqrt{\frac{i}{2}}\begin{pmatrix} i &-1 \\ -1 &i\end{pmatrix}$.
$\rightarrow |\...
1
vote
1answer
386 views
Find unitary for given rotations on Bloch sphere
I want to characterize a unitary by given rotations on the Bloch sphere.
I know, that when I send in the State $|\Psi\rangle =\begin{pmatrix}1\\0 \end{pmatrix}$, I get the state $U|\Psi\rangle=\...
2
votes
1answer
246 views
2x2 Matrices that are not valid quantum states
Given a 2-dimensional Hilbert space, quantum states can be expressed as $2\times 2$ density matrices. In terms of the Pauli matrices, or Bloch representation, they can be written as
\begin{equation}
\...
1
vote
1answer
3k views
Exact meaning of “pi/2 pulse”
In studying Mach-Zehnder and Ramsey interferometers, I came across the expression "$\pi/2$ pulse". What does it mean exactly? I am working with a Bloch vector representation $(u,v,w)$ of a 2 state ...
0
votes
1answer
208 views
Do the eigenstates of the Pauli operators correspond to the six directions of the 3D world?
I understand that the six eigenstates of the three Pauli operators $X, Y, Z$ correspond to the six poles of the Bloch sphere. By fixing an orthonormal basis of our physical word, does "measuring Pauli ...
2
votes
3answers
1k views
How do I operate on a spin state with a sigma operator?
For any arbitrary spin state $|s\rangle$. How do I operate on it with the Pauli spin matrix, $\hat{\sigma_z}$? Does this have something to do with a Bloch sphere?
0
votes
1answer
1k views
How do I find an expectation value for an electron's magnetic moment?
Given a spin state:
$|s\rangle$ = some linear combination of $|\uparrow\rangle + |\downarrow\rangle$ possibly with an imaginary component. How do you get from the definition of a magnetic momentum ...
1
vote
1answer
310 views
Bloch representation. Why Pauli operators?
Why do I know that a general qubit state can be written as
$$
\rho = \frac 1 2 \big(\mathbb 1 +\vec r \vec \sigma\big)\;\text ?
$$
It is clear that the factor of $1/2$ comes from $\text{tr}\rho=1$. ...
20
votes
6answers
3k views
Why is $\theta \over 2$ used for a Bloch sphere instead of $\theta$?
I'm a beginner in studying quantum info, and I'm a little confused about the representation of a qubit with a Bloch Sphere. Wikipedia says that we can use $$\lvert\Psi\rangle=\cos\frac{\theta}{2} \...
