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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Bloch representation of a quantum channel acting on a 2-qubit density-matrix
A previous answer nicely show the relationship between the Pauli transfer matrix (PTM) and the Bloch representation of a quantum channel that acts on single-qubit density matrix.
In short, given a ...
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Visualizing the Effect of a Parameterized Gate on a Qubit Using the Bloch Sphere in Qiskit 1.0
I am seeking assistance with visualizing the effect of a parameterized gate on a qubit using the Bloch Sphere representation. I have a preliminary code snippet as follows:
...
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Can we find the quantum state using the Bloch vector in the Bloch sphere?
I had a question about qft. When we apply QFT in a circuit, we can display the qubits separately in the bloch sphere with the plot_bloch_multivector function. I realized that the bloch vectors do not ...
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For a Nqbit, prove that $\langle|\alpha_i|^2\rangle=1/2^N$ and that $\langle|\alpha_i \alpha_j|^2\rangle= 2^{\delta_{ij}-N}/(1+2^N) $
In the following article it is written that:
This is done by using spherical coordinates in a $2^N$ -dimensional real space,where $N$ is the number of qubits to be teleported. We thus find that $\...
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Unital qubit channels as a convex combination of entanglement-breaking and unitary channel
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 $\varphi,\theta$ ...
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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 ...
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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,$$
where $a=\cos\left(\frac{\theta}{2}\right), b=\sin\left(\frac{\theta}{2}\right)e^{i\phi}$ (neglecting global phase), $\phi$ is ...
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Generalization of representing SU(2) as quaternions
I am familiar with the isomorphism between $SU(2)$ and the unit quaternions, and the group homomorphism from them to $SO(3)$. I am interested in knowing if there is a generalization for $SU(2^n)$. My ...
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Zeeman eigenstates and QuTiP Bloch sphere
One Hamiltonian corresponding to a qubit is the Zeeman Hamiltonian:
$$
\hat{H}_\mathrm{Zeeman} = \frac{\hbar\omega_0}{2}\hat{\sigma_z}$$
The eigenstate corresponding to the eigenvaue $+\hbar\omega_0/2$...
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How to get $\theta$ and $\phi$ in a Bloch sphere for individual qubits in a quantum register?
I have a quantum register with some qubits (I'm just new to all of this). Is there a way to achieve $\theta$ and $\phi$ angle values in the Bloch sphere for each qubit separately? If I do a circuit ...
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What is the expression for $|\psi\rangle\!\langle\psi|$ if $|\psi\rangle=\cos(\theta/2)|0\rangle+\sin(\theta/2)e^{i\phi}|1\rangle$?
Let $|\psi\rangle = \alpha|0\rangle + \beta |1\rangle$. In Bloch sphere representation, this is
$\cos\frac{\theta}{2}|0\rangle + \sin\frac{\theta}{2}e^{i\phi}|1\rangle$.
In matrix representation:
$|\...
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Why does the point $(0,0,-1)$ on the bloch sphere correspond to the state $|1\rangle$ and not $-|1\rangle$ or $e^{i \phi}|1\rangle$?
In this representation for points on the $Z$-axis, $\phi$ is not defined. If the point $(0, 0, 1)$ is taken since $\theta$ is $0$ and $\sin(\theta/2)$ is zero, it doesn't matter what $\phi$ is. The ...
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How to integrate a function with the Haar measure over multiple qubits
I am starting with a product state over multiple qubits. That looks like the expression below.
$$
|\psi\rangle = \left(\cos\left(\frac{\theta_1}{2}\right)|0\rangle+e^{i\phi_1}\sin\left(\frac{\theta_1}{...
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Is a qubit passing through a gate a 3D rotation of the vector on the Bloch sphere?
When a qubit passes through a gate isnt it a 3D rotation of the vector on the Bloch sphere?
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Closeness between two unitaries on the Bloch sphere
The fidelity between two (single-qubit) quantum states can be easily translated into the euclidean distance between the two states on the Bloch sphere (hilbert-schidmit distance). I'm curious if this ...
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Sufficient conditions for a single-qubit unitary to be the identity
Say I have a unitary $U = e^{-iHt}$ where $H = \alpha X + Z$.
First, suppose $U = I$. Then it rotates a set of initial states to themselves. Say I'm working on a computational basis, then on the Bloch ...
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Can you twist a qubit?
Is it possible to operate on a single qubit by a map which has a nonzero degree?
Let $|c\rangle=c_0|0\rangle + c_1|1\rangle$ represent a qubit state where $c_0,c_1 \in \mathbb{C}$ and $|c_0|^2+|c_1|^2=...
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What does the product of two density matrices represent physically?
A quantum state, pure or mixed, can be described by a density matrix that encodes the Bloch vector $\hat{m}$ analog of a quantum state like
$\rho = \frac{1}{2}[\mathbb{I} + \hat{m}.\vec{\sigma}]$.
Let ...
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How to eliminate the global phase of a state vector?
Say that I have a qubit that began in the $|0\rangle$ state and then the
Hadamard gate is applied, resulting in the following state:
$ \begin{bmatrix}
\frac{1}{\sqrt{2}} \\
\frac{1}{\sqrt{2}}
\end{...
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Representing networks with qubits as edges
I am looking to take a classical non-negative real valued network and generalize it to the quantum case for processing. A network is given by an adjacency matrix, essentially edge weights $e_{ij}$ for ...
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How to represent an entangle state on Bloch Sphere [duplicate]
If states are not entangle, both bloch sphere represent the state of qubit but if they are entangle, why bloch sphere show nothing??
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Plotting the Bell state on a Bloch sphere
How can you plot the bell state on a Bloch sphere?
bell = QuantumCircuit(2)
bell.h(0)
bell.x(1)
bell.cx(0,1)
Is there any good reference for understanding how ...
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Is the Bloch sphere the same as the electron spin?
A single qubit is represented with a bloch sphere and implemented on an electron with only two energy values from lots of them.
So, I am confused about why an energy range is represented as a ...
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Show that for pure states the description of the Bloch vector we have given coincides with that in section 1.2
$\newcommand\bra[1]{\left\langle#1\right|}\newcommand\ket[1]{\left|#1\right\rangle} $
I am having a little bit of difficulty with part (4) of Exercises 2.72 from Nielsen and Chuang's "Quantum ...
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Why are rotations represented by exponentials of Pauli matrices?
I'm self-studying Quantum Computation from Nielsen and Chuang's book. In section 4.2 they discuss that for any unit vector $\hat n$, the rotation operator $R_{\hat n}(\theta) = \exp(-i\theta\hat n \...
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How to get a state vector from bloch sphere coordinates for 1 qubit?
I need to get a state vector for one qubit from bloch coordinates no matter if there could be many states that describes the same bloch coordinates, and it does not have to be normalized, because the ...
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Finding a unitary transformation for a qubit mixed state that projects onto a pure state
Suppose we have a single qubit mixed state described by a density matrix $\rho$, and we want to find a unitary transformation that brings $\rho$ to the pure state $|0\rangle\langle 0|$. Is there a ...
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What is $HTHTH\left| 0 \right>$?
I'm currently going through Introduction to Classical and Quantum Computing, by Thomas Wong, and I'm struggling with exercise 2.33 (page 108):
Exercise 2.33. Answer the following:
(a) Calculate $...
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Why are probabilities represented with alpha^2 and beta^2?
To preserve the Complementary Rule of probability, the sum of the probabilities of the outcomes (measured |0> or measured |1>) must equal 1 or 100%. That's why alpha^2+beta^2=1. However, why the ...
user24663
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Why is the conjugate being used when rewriting a qubit state in another basis?
I'm currently going through Introduction to Classical and Quantum Computing, by Thomas Wong, and I'm struggling with an example given on page 88 (point 4).
The author gives the two following states, ...
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Is there a criteria to ensure a one-qubit operator is exactly of the form $R_n(\theta)$ (i.e without a global phase $e^{i\alpha}$)?
Reading the Nielsen and Chuang, I saw that every unitary operator $U$ can be written as $e^{i\alpha} R_n(\theta)$ for some well chosen $n \in \mathbb{R}^3$ and $0 \leq \theta < 2\pi$.
I would like ...
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Is the function PU(2) and SO(3) induced by the Bloch sphere bijective?
I have difficulty understanding the fact that, as written in this reference,
every single-qubit
unitary corresponds to a unique rotation of R3 and vice versa.
If I understand well, this means there ...
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Finding the polar and azimuthal angles of the Bloch vector corresponding to $\frac1{\sqrt2}(1-i)|0\rangle-\frac i{\sqrt2}|1\rangle$ [duplicate]
I need to find the polar angles and azimuthal angles of the following bloch vector:
$$
\frac{1-i}{2}|0\rangle - \frac{i}{\sqrt{2}}|1\rangle
$$
I just couldn't figure it out, and I could also not find ...
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How many parameters do we need to characterize a pure state?
Suppose I have a pure qubit. I can think of starting with the state $\vert 0\rangle$ and apply some unitary to it. Such a unitary has three parameters according to this link. In $d$ dimensions, the ...
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Geometric representation of rotation operator on Bloch sphere
I'm studying Nielsen&Chuang Book and need some clarification on mapping arbitrary rotation operator onto geometric tranformation of vector on Bloch sphere. I thought that $R_{\vec{n}}(\theta)$ ...
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Where does 1/sqrt(2) come from in the state of i
I’m trying to learn about calculating coordinates for $\theta$ and $\varphi$ in a Bloch-sphere.
I came accross this book about it, including example questions.
At question 2.12b, they ask to give the ...
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Significance of angle $\phi$ on bloch sphere
So far I learned that a qubit can be written as $| \psi \rangle = \alpha | 0 \rangle + \beta | 1\rangle$ with $|\alpha|^2 + |\beta|^2 = 1$ and reparametrized as $| \psi \rangle = cos( \theta / 2) + e^{...
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What is the Bloch sphere representation of $\rho\to\mathcal{E}(\rho) = |+\rangle\langle+|ρ|+\rangle\langle+| + |−\rangle\langle−|ρ|−\rangle\langle−|$?
Suppose a projective measurement is performed on a single qubit in the basis $|+\rangle, |−\rangle$, where $|±\rangle \equiv (|0\rangle\pm |1\rangle)/\sqrt{2}$. In the event that we are ignorant of ...
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What do the values of θ, ɸ, and λ represent visually in the Bloch Sphere when defining a unitary gate? [duplicate]
In IBM Quantum Docs, it is stated that a unitary matrix can be defined as
$U = \begin{bmatrix} \cos(\theta/2) & -e^{j\lambda}\sin(\theta/2) \\ e^{j\phi}\sin(\theta/2) & e^{j\lambda+j\phi}\cos(\...
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Solving a Bloch sphere where alpha is imaginary
$$
\begin{array}{l}
|\psi\rangle=\frac{\sqrt{3 i}}{2}|0\rangle-\frac{1}{2}|1\rangle \\
|\psi\rangle=0.924|0\rangle-0.382 i |1\rangle
\end{array}
$$
Basically I’m trying to convert these to a standard ...
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