Questions tagged [bloch-sphere]
For questions related to the Bloch sphere. In quantum mechanics, the Bloch sphere is a geometrical representation of the pure state space of a two-level quantum mechanical system (qubit), named after the physicist Felix Bloch. (Wikipedia)
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Unital qubit channels as a convex combination/how to view transformations on the blochsphare
I am trying to show that for $T:B(\mathbb{C}^{2})\rightarrow B(\mathbb{C}^{2})$ a unital qubit channel, that T is a convex combination $T=pB+(1-p)Ad_{V}$, where B is a Entanglement-Breaking(EB) ...
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$R_x(\theta)$ and $R_y(\theta)$ implement rotations by an angle $\theta$ about the x and y axes of the Bloch sphere
Consider the operators (to be called rotations) :
$$R_x(\theta)= \begin{pmatrix} \cos(\theta/2) & -i\sin(\theta/2)\\
-i\sin(\theta/2) & \cos(\theta/2)\end{pmatrix}= e^{-iX\theta/2} \\
R_y(\...
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Represent Hadamard gate in terms of rotations and reflections in Bloch sphere
I read in a book that any single qubit operation can be decomposed as
$$
\bf{U} =e^{i\gamma}\begin{pmatrix}e^{-i\phi/2}&0\\ 0&e^{i\phi/2}\end{pmatrix}\begin{pmatrix}\cos{\theta/2}&-\sin{\...
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Resource for geometric representation of quantum channels
I was wondering if anyone knows about any good resources on representing unital/quantum channels by using rotations/pauli matrices. It is mentioned in Nielsen&Chuang on p774, but i feel it is ...
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Why does the Bloch sphere have a radius of 1?
Why does the Bloch sphere have a radius of 1?
Thank you so much! I am a quantum newbie, so forgive me if this is basic.
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Help with a lemma on the argument of a qubit after transformation
From:
King, R. (2023). An improved approximation algorithm for quantum max-cut on triangle-free graphs. Quantum, 7, 1180.
I have trouble understanding item 3 of the above lemma. Here $n_k \cdot \...
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What do the angles in a Poincaré sphere represent?
I understand the principle of the Bloch sphere. You write your state in the following way:
$$|\Psi\rangle = \cos(\theta/2)|0\rangle+e^{i\varphi}\sin(\theta/2)|1\rangle.$$
The angles $\varphi,\theta$ ...
- 2,356
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Converting $H$ gate to $R_x$ and $R_z$
EDIT: My solution is supposed to work for $|1\rangle$ state too. See https://imgur.com/a/7F1cHu4
Right of the bat the answer is $$H=R_z(\pi/2)R_x(\pi/2)R_z(\pi/2)\,.$$ My question is, I cannot reach ...
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Affine transformation of the Bloch sphere to Kraus representation of qubit channels
It is known that qubit channels can be written in the form:
$$
\begin{align}
\Phi(\rho) = \frac{1}{2}\left(I+(T\vec{r}+\vec{t})\cdot\sigma\right)\
\end{align}
$$
where $\vec{r}$ is the Bloch vector ...
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In the phase flip action on standard basis, why do we consider the $-1$ phase only for the $|1\rangle$?
Prof. Watrous in the first lecture of Qiskit summer school 2023, mentions:
"....the significance of putting a minus sign in front of the $|1\rangle$
basis vector and not $|0\rangle$ will be more ...
- 19
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quantum teleportation with shared state and teleported state in bloch representation
Suppose that two parties (Alice and Bob) share an entangled state
$$ \rho_F = F \lvert \phi^+ \rangle \langle \phi^+ \rvert + \frac{1-F}{3} \left( I \otimes I - \lvert \phi^+ \rangle \langle \phi^+ \...
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Why is it impossible to make a mixed-state qubit into a pure-state qubit?
How is it impossible to make a mixed-state qubit (having Bloch vector of length $l < 1$) into a pure-state qubit (having Bloch vector of length $l' = 1$) via a quantum operation that is invertible?
...
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How to find $p_x$ and $p_y$ components on the Bloch sphere?
Consider an arbitrary state:
$$|\psi\rangle = a|0\rangle+b|1\rangle,$$
where $a=\cos\left(\frac{\theta}{2}\right), b=\sin\left(\frac{\theta}{2}\right)e^{i\phi}$ (neglecting global phase), $\phi$ is ...
- 267
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Generalization of representing SU(2) as quaternions
I am familiar with the isomorphism between $SU(2)$ and the unit quaternions, and the group homomorphism from them to $SO(3)$. I am interested in knowing if there is a generalization for $SU(2^n)$. My ...
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Zeeman eigenstates and QuTiP Bloch sphere
One Hamiltonian corresponding to a qubit is the Zeeman Hamiltonian:
$$
\hat{H}_\mathrm{Zeeman} = \frac{\hbar\omega_0}{2}\hat{\sigma_z}$$
The eigenstate corresponding to the eigenvaue $+\hbar\omega_0/2$...
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