Questions tagged [bloch-sphere]

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Three dimensional visualization of a qutrit

My question is in reference to the paper "Three dimensional visualization of a qutrit"(https://arxiv.org/abs/1601.07361). The author's start with a symmetric two qubit density matrix written in the ...
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11 views

Bloch sphere hyperspatial multiqubit superpostion

What potentials can extend for a hyper-spatial diagram from a Bloch Sphere in terms of delineating multi-qubit superpositions extending from a quasi-periodic phase index upon reduction of the wave ...
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52 views

Maximum number of “almost orthogonal” vectors one can embed in Hilbert space [closed]

In a Hilbert space of dimension $d$, how do I calculate the largest number $N(\epsilon, d)$ of vectors $\{V_i\}$ which satisfies the following properties. Here $\epsilon$ is small but finite compared ...
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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....
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1answer
58 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 ...
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42 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 (...
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1answer
92 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 ...
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1answer
221 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\...
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1answer
70 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 ...
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1answer
159 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 $Y$ ...
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3answers
109 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 ...
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1answer
247 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.,...
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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 $\...
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1answer
168 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 ...
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1answer
136 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 ...
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24 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 ...
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45 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 ...
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1answer
155 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 ...
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0answers
52 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 ...
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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 ...
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94 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 $SO(3)$...
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1answer
578 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 ...
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1answer
35 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 ...
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42 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 ...
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0answers
63 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} ...
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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 ...
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1answer
76 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} | \...
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1answer
135 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 ...
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1answer
201 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\...
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4answers
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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^{...
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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(\...
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1answer
229 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 ...
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1answer
457 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 ...
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1answer
126 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 $(\...
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161 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 ...
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1answer
229 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\...
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1answer
118 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 ...
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2answers
618 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 $\rho$ ...
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1answer
86 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: $\vert 1 \rangle$ ...
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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 \...
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3answers
470 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 \...
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1answer
66 views

Population Decay Model and the Rotating Wave Approximation

The paper which I'm reading mentions the following: The Hamiltonian, $$ \frac{\varepsilon}{2}\sigma_z + \sum_k \omega_k b_k^\dagger b_k + \sum_k (\sigma^+ g_k^* b_k + \sigma^- g_k b_k^\dagger), $$...
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1answer
804 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 |\...
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1answer
434 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=\...
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1answer
259 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} \...
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1answer
4k 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 ...
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1answer
226 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 ...
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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?
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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 ...
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1answer
337 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 $\text{tr}\rho=1$. ...