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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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$R_x(\theta)$ 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(\...
NotaChoice's user avatar
5 votes
2 answers
355 views

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{\...
A1Y's user avatar
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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 ...
Pink Elephants's user avatar
2 votes
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96 views

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.
Isha Bhadauria's user avatar
1 vote
1 answer
60 views

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 \...
Matteo's user avatar
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1 answer
56 views

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 $\varphi,\theta$ ...
Mauricio's user avatar
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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 ...
Minh Triet's user avatar
1 vote
1 answer
117 views

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 ...
JohnnyB's user avatar
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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 ...
Nash's user avatar
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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^+ \...
gehbiszumeis's user avatar
1 vote
2 answers
67 views

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? ...
Sudhir Kumar's user avatar
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1 answer
127 views

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 ...
Curious's user avatar
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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 ...
Chris Henson's user avatar
1 vote
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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 $+\hbar\omega_0/2$...
Len's user avatar
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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 ...
D.Giunchi's user avatar
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1 answer
139 views

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: $|\...
Physkid's user avatar
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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 ...
D Star Let's Explore's user avatar
2 votes
0 answers
101 views

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}{...
Endeavour 's user avatar
2 votes
1 answer
25 views

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?
Cerise's user avatar
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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 ...
Hailey Han's user avatar
3 votes
2 answers
301 views

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 ...
Hailey Han's user avatar
4 votes
1 answer
157 views

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=...
Jackson Walters's user avatar
1 vote
0 answers
94 views

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 ...
Physkid's user avatar
  • 520
2 votes
5 answers
423 views

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{...
bddicken's user avatar
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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 ...
Jackson Walters's user avatar
3 votes
1 answer
104 views

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??
Quantum_society's user avatar
0 votes
1 answer
447 views

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 ...
Khilesh Chauhan's user avatar
3 votes
0 answers
88 views

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 ...
Luis ALberto's user avatar
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0 answers
143 views

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 ...
am567's user avatar
  • 585
5 votes
2 answers
718 views

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 \...
slimmerikko's user avatar
1 vote
2 answers
288 views

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 ...
Luis ALberto's user avatar
3 votes
1 answer
202 views

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 ...
Zarathustra's user avatar
3 votes
2 answers
180 views

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 $...
Maxime Desalle's user avatar
-2 votes
2 answers
101 views

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 ...
user avatar
1 vote
1 answer
129 views

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, ...
Maxime Desalle's user avatar
0 votes
2 answers
104 views

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 ...
user8622655's user avatar
4 votes
1 answer
105 views

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 ...
user8622655's user avatar
2 votes
2 answers
338 views

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 ...
Spacetoon's user avatar
3 votes
1 answer
127 views

Clarification defining/finding the relative phase of a qubit

Let the vector $ |V\rangle = r_0 e^{i\theta_0} |0\rangle + r_1 e^{i\theta_1} |1\rangle $ correspond to the state of a qubit where $r_0,r_1,\theta_0,\theta_1 \in \mathbb{R}$. According to p. 22 of ...
RyRy the Fly Guy's user avatar
2 votes
3 answers
328 views

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 ...
Jimbo's user avatar
  • 23
2 votes
0 answers
101 views

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 $R_{\vec{n}}(\theta)$ ...
Михаил Горчаков's user avatar
1 vote
1 answer
297 views

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 ...
gggggggggg's user avatar
3 votes
1 answer
141 views

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^{...
maiT's user avatar
  • 143
2 votes
2 answers
317 views

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 ...
Sooraj S's user avatar
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2 votes
0 answers
39 views

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(\...
Satvik Duddukuru's user avatar
2 votes
1 answer
195 views

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 ...
Max Rush's user avatar
5 votes
1 answer
799 views

Why are orthogonal quantum states represented as collinear in the Bloch sphere?

We know that the angle between two orthogonal qubit states is 90 degrees. Why then, when we use the Bloch sphere, the angle becomes 180 degrees?
Rayhan's user avatar
  • 143
2 votes
1 answer
206 views

Writing a Density matrix in terms of the magnitude of the Bloch Vector

Working with the density matrix and the Bloch sphere, I have been attempting to complete an exercise in Entangled Systems; New Directions in Quantum Physics. If anyone has the book it is Question 4.3 ...
PGibbon's user avatar
  • 462
3 votes
1 answer
697 views

How to interpret QSphere visualisation

AFAIK, $n$ qubits together can be described by using a column vector (also called State vector) of $2^n$ complex amplitudes such that sum of the squares of mod of all these complex amplitudes will be $...
RSW's user avatar
  • 289
2 votes
1 answer
281 views

Decomposition of unitary operator into rotations around Bloch sphere

I apologize in advance for any mistakes as I am new to this field and come from a programming, rather than mathematical/physical background. I am looking for a way to decompose a given operator $U$ ...
Satvik Duddukuru's user avatar