Questions tagged [bloch-sphere]

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56 views

A potential well with 3-fold reflection symmetry

When we are talking about Bloch's theorem and also the tight-binding approximation, we can use them to help finding eigenstates of a system. However, I am so confused how to apply it in this case (...
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Time dependent perturbation theory on Bloch sphere

I have a Hamiltonian of the form: $ \begin{pmatrix} E_1 & W_1(t)+iW_2 \\ W_1(t)-iW_2 & E_2 \end{pmatrix} $ Where $E_1,E_2,W_1,W_2$ are real and only $W_1$ is time dependent (it is actually ...
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Visualizing dynamics of a two-level atom on the Bloch Sphere

I am trying to use the Bloch Sphere to understand how applying resonant sequences of $\pi/2$ and $\pi$ pulses changes the state of a two-level system initially in the ground state. Specifically, I am ...
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1answer
58 views

Trying to visualize the concept of qubits [closed]

Background Complete newbie, never taken a physics course. Question I'm trying to visualize the concept of qubits. Qubits can achieve a mixed state, called a "superposition" where they are ...
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1answer
38 views

Pole point on the Bloch sphere

If the state of the qubit is a point at a pole on the Bloch sphere, does this mean that the coefficient of the component corresponding to the other pole is zero?
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145 views

Vector from $SU(2)$ to $SO(3)$? [closed]

I know how to change the element(a matrix) in $SU(2)$ to the matrix in $SO(3)$, and I found one way to change the state(vector) from $2\times 1$ to $3\times 1$, but I don't know why. The method is ...
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1answer
42 views

Hamiltonian with identity operator: how to visualize the (time-evolution) rotation?

For the Hadarmard Hamiltonian, $\hat H = (\hat X+\hat Z)/\sqrt 2$, where $\hat X$ and $\hat Z$ are Pauli matrices. The time evolution of a state under this Hamiltonian could be visualized by a ...
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1answer
81 views

Bloch sphere representations for multi-qubit quantum systems

Just a short mixed quantum state representation question. Given a single qubit density matrix $\rho$, since the Pauli matrices form a basis for 2x2 complex matrices, the Bloch sphere representation ...
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1answer
66 views

Average over the Bloch sphere

Consider you have a function of a two-level wave quantum state $f(\vert \psi \rangle ) $, with $\vert \psi \rangle = \alpha \vert 0 \rangle + \beta e^{\rm i \phi} \vert 1 \rangle$. With no loss of ...
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How does the Bloch sphere indicate topology of 2-level $k\cdot p$ effective Hamiltonians?

It is known that the topology of some parameter space of a 2-level system (such as the Brillouin torus) may be found via the Gauss map to the Bloch sphere. The topology is indicated by the number of ...
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2answers
107 views

Why can we only perform rotations of the Bloch sphere (with unitary matrices), and not reflections?

It's easy to take a quantum state represented on the Bloch sphere and rotate it around an arbitrary ray emanating from the origin. On the other hand, we can never use a unitary matrix to get a ...
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Outcomes of quantum measurements

I'm pretty new to quantum computing, and I'm wondering how I can compute the outcome of a projective measurement of a spin along the +Z axis followed by a projective measurement along the -Z axis. I ...
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89 views

Why an element of $SU(2)$ acts as a rotation for the Majorana representation of states?

I recently asked a question in quantum computing stack exchange and as suggested by someone in the comments, I decided to ask my question here as well: I know that for a given spin-j quantum state, ...
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3answers
138 views

Prove that qubits can be represented on a unit sphere, avoiding the density matrix formalism [duplicate]

The Bloch Sphere is regarded as the most "intuitive" way of explaining a 2-level quantum system in computation and rotations of states described on Bloch sphere provides a really easy ...
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4answers
156 views

Quantum Computing without Complex Numbers

p.s. I am trying to get a handle on what actual computing operations a quantum computer program does. Any information on that would be appreciated [noting the issue that that might count as a ...
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2answers
63 views

Consider the following single-qubit state on the Block sphere [closed]

$$\left| \varphi \right>=\frac{1+i}{\sqrt{3}} \left| 0 \right> + {\sqrt{\frac{1}{3}}} \left| 1\right> $$ I need to find the coordinate 𝜃, and ϕ values of the quantum state.
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Time evolution of a qubit starting on the an x-axis eigenstate

Suppose I have starting qubit like the one described on the Bloch sphere below. Also, denote the state $|0\rangle \equiv |+\rangle$. So my starting state would be: $$ |\psi(0)\rangle=\frac{1}{\sqrt{...
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37 views

Terminology for the symmetric vector product on the Bloch sphere

I was reading the paper "The extended Bloch representation of quantum mechanics and the hidden-measurement solution to the measurement problem" by Diederik Aerts, Massimiliano Sassoli de Bianchi. ...
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1answer
75 views

Is there a geometrical representation of classical bits, like that of the Bloch Sphere for quantum bits?

Consider only a single qubit. We know that it can be accurately described using the so called Bloch Sphere. Is there a similar geometric construction used to describe classical bits?
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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
69 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
113 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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81 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
327 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
664 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
102 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
358 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
148 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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335 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
64 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
212 views

Are there natural geometrical representations for a qubit other than the 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
169 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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27 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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106 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
274 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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57 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
3k 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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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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884 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
38 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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54 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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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
2k 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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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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215 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
250 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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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
2k 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
279 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
642 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 ...