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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Question on State Transfers [closed]

I am beginner to quantum physics and I came across this problem that I cannot figure out[![enter image description here][1]][1]
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Can a many-qubit quantum state be represented on Bloch spheres? [duplicate]

Can a general multi-qubit quantum state be represented on Bloch spheres? If not, what is the maximal number? Are there any corresponding constraints? Thanks.
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Rxx gate as a set of rotations

I'm trying to represent Rxx gate as a set of physical rotations of two qubits in 3D space (or as rotations of Bloch Spheres that is the same). In some simple cases it works well: If q0 is in the ...
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Find a set of vectors on the Bloch sphere such that $\langle \psi_i | \psi_j \rangle = \frac{1}{\sqrt{n}}$

How can I find a set of multiple vectors on the block sphere which satisfies $$\langle \psi_i | \psi_j \rangle = \frac{1}{\sqrt{n}}$$ where $n$ is any natural number greater than $2$? I think I have ...
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When using Qiskit's plot_bloch_vector function, how can I set colors for vectors?

I am able to visualize vectors on a bloch sphere using Qiskit as follows: ...
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Are qubits just analog, continuous classical bits? [duplicate]

Topologically, classical bits (cbits) are essentially special cases of qubits restricted to the poles of the Bloch sphere. However, this restriction doesn't seem to be classical per se, but is simply ...
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Is there a way to rotate an unknown state towards another known state?

I would like to rotate my $|\Psi\rangle$ state towards $|1\rangle$: $$ |\Psi\rangle= a|0\rangle + b|1\rangle \ \rightarrow \ |\Psi'\rangle= a'|0\rangle + b'|1\rangle$$ with $|a'| < |a|$, $|b'| > ...
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Direction of rotation operators

The rotation operators for a single qubit are defined as $R_{v}(\theta) = e^{-i \theta X/2}$, with $v \in \{ X,Y,Z\}$. If we look at the direction of rotation of $R_v$ w.r.t. the positive eigenvalue, ...
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why θ/2 in the Bloch sphere? [duplicate]

Describing the Restricted Qubit State is the best introduction I've seen on the Bloch sphere, but one thing still bugs me... Question: Why use $θ/2$ instead of $θ$ or $zθ$ for that matter (i.e. ANY ...
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Quantum Gate - Negating qubit whose state is known to lie in the equatorial x-y plane

Given a qubit whose state is known to lie in the equatorial x–y plane in the Bloch sphere is it possible to find a quantum gate that will always negate this qubit? If so, exhibit such a gate. If not, ...
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How do I implement $SU(2)$ rotation on the Bloch Sphere unsing qutip?

Using qutip I am trying to implement a qubit rotation according the formula $(25)$ provided in this document "Lecture notes: Qubit representations and ...
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Why does application of Hadamard gate on $|0\rangle$ inverts the state on bloch sphere?

I am trying to understand what is the affect of Hadamard gate on qubit. So far, I understand that applying $H$ gate on qubit, puts the qubit into superposition state where the probability of the qubit ...
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Efficient Representation of Qubits on a Digital Simulator

I was wondering about quantum simulators recently, and I was thinking about how a qubit could be represented on a digital machine. This Stack Overflow post seems to say that one will need at least $a2^...
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How do I stepwise rotate a qubit on a Bloch sphere using $SO(3)$ and $SU(2)$ group?

Let us first introduce two fundamental functions for coordinate transformations: ...
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How to represent the Hadamard gate as a rotations on the Bloch sphere?

I am new to Quantum Computing, and I have decided to try and learn the quantum gates. I am trying to understand how to represent some basic gates as rotations on the Bloch Sphere. I was able to ...
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Why is the angle of Rz(π/4) Rx(π/4) an irrational multiple of π

It is stated in the Qiskit tutorial section 2.4 that if you apply a rotation around the z-axis of π/4 and subsequently a rotation around the x-axis of π/4, the end result is an angle around some axis ...
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Why can a ZY decomposition not decompose an arbitrary single qubit gate?

To decompose an arbitrary single qubit gate, we need to do a "$e^{i \alpha}$ZYZ" decomposition. Intuitively however I do not understand why a "$e^{i \alpha}$ZY" decomposition is ...
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Can the Bloch sphere representation be applied to many-qubit states with an iterative approach?

By ignoring the global phase, we can represent a single qubit state as \begin{equation} |\psi\rangle = \cos(\theta)|0\rangle + e^{i\phi}\sin(\theta)|1\rangle \end{equation} which very much looks like ...
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Calculate qubit state in terms of two states that are opposite points on Bloch sphere

I am new to quantum computing and reading the book "Introduction to Classical and Quantum Computing", by Wong (link). I do not understand how to calculate the qubit state for the below ...
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How to prove that the coefficients of Pauli bases remain its length under unitary operation?

Formal problem statement. For a real vector $\vec r$ in bases of $\sigma_i\otimes \sigma_j$ with $i,j=0,1,2,3$, $i$ and $j$ not equal to $0$ at the same time, $\sigma_0=I$ and other indexes stands for ...
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How do we define qubit measurements in a plane?

When does $\vec{a} \cdot \vec{\sigma}$ define a measurement in x-y, y-z, and x-z planes?
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Why the Ry rotation matrix give counterclockwise rotation?

The y-axis centered rotation matrix is $R_{y}(\delta)=\left[\begin{matrix} \cos \frac{\delta}{2} & -\sin \frac{\delta}{2} \\ \sin \frac{\delta}{2} & \cos \frac{\delta}{2} \end{matrix} \right]$....
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How to change the probability of observation by some set amount when initial probability is unknown?

If I have some state $|\psi> = \alpha |0> + \beta|1>$, I know that the probability of observing $|0>$ is $p_1 = |\alpha|^2$. Is it possible to change the probability of observing $|0>$ ...
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Can we rotate Bloch vectors for qudits like we do with qubits in the Bloch sphere?

I have been looking into the Bloch vectors for qudits and have been wondering if we can do rotations that are similar to the rotations in the qubit Bloch sphere. Like, once we create a Bloch vector ...
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How to decompose Bloch sphere rotations $e^{\frac{i\theta}{2}(\cos(\phi)\sigma_x + \sin(\phi)\sigma_y)}$ in terms of $R_x,R_y,R_z$?

I learned a formula to represent the rotation around bloch sphere: $\theta_{\phi} = e^{\frac{i\theta}{2}(\cos(\phi)\sigma_x + \sin(\phi)\sigma_y)}$ So that $\pi_0$ is the gate $X$ and $\pi_{\frac{\pi}{...
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Evaluation of Wigner function representation of a Bloch Sphere

Consider Wigner function representation of a qubit in the basis labeled by $\sigma_z$ and $\sigma_x$ eigenvalues. A general single qubit mixed state has the Bloch representation,$\rho = 1/2 (I + r.\...
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Show that the trace of squared density matrix gives ${\rm tr}(\rho^2)=\frac12(1+\|\mathbf n\|^2)$ [duplicate]

Equation 7.7 is given below: $$\hat\rho = \frac12(I +n_x(\hat X)+n_y(\hat Y)+n_z(\hat Z)) $$ Where $I$ is the identity matrix and $\hat X,\hat Y,\hat Z$ are Pauli matrices. Now my attempt of this was ...
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How to calculate the coefficients of a qubit from the angles of its Bloch representation?

A quantum bit $|\psi\rangle=a|0\rangle+b|1\rangle$ is represented on the Bloch sphere as a point on the spherical surface with $\theta = 40^°$ and $\phi = 245^°$. Calculate the (complex) coefficients $...
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Why can the most general state of a qubit be written as $|\Psi\rangle=\cos(\frac\theta2)|0\rangle+e^{i\phi}\sin(\frac\theta2)|1\rangle$?

Why we can express a most general qubit as $|\Psi\rangle = \cos{\left(\frac{\theta}{2}\right)}|0\rangle + e^{i \phi} \sin{\left(\frac{\theta}{2}\right)} |1\rangle$? Is there any formal proof for this?
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How does Equation (9) follows from these definitions?

How does Equation (9) follows from these definitions? This is my working out for the first identity. It appears that the identity is not true. For the second identity, I used sympy to check. The ...
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Can I draw platonic solid inside a Bloch sphere?

I wonder if there's a tool available to draw platonic solids or other 3-D shapes inside a Bloch sphere. Here's an example I found on a research paper: In my previous visualizations, I used the ...
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How to derive the rotations caused by the H gate?

In Nielsen and Chuang, there's the following paragraph: The Hadamard operation is just a rotation of the sphere about the ˆy axis by 90◦, followed by a rotation about the ˆx axis by 180◦. I am ...
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Vector math of applying an X-gate on an $|i\rangle$ basis state

It is well known that the X-gate will apply a rotation about the x-axis on the bloch sphere. Knowing this, the $|i\rangle$ state should be converted to the $|-i\rangle$ state on the application of ...
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Meaning of a pound sign (#) on a Bloch sphere

For the following Bloch sphere representation of a qubit, what does the highlighted symbol mean? I'm not sure if it means anything or it's just for showing that it's a sphere, not a circle.
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Is every single-qubit unitary just a rotation around some unit vector on the Bloch sphere?

I remember reading this somewhere... Is there an elegant proof for this?
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What does the notation $|+\rangle,|−\rangle,|±i \rangle$ mean in Bloch sphere?

The axis in a a 2D diagram like the following, usually represent 2 quantities. Eg in pic below, $x$ represents time and $y$ represents velocity What gets measured along each axis of a Bloch sphere? ...
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Why is there no angle for the $z$ axis in the Bloch sphere?

I see that in Bloch spheres, there is an angle for the $x$ and $y$ axes but not for the $z$ axis. Why?
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How to write the eigenvectors of a mixture of two pure states?

Let $|\psi_1\rangle,|\psi_2\rangle$ be two pure states. Assume $\langle\psi_1|\psi_2\rangle\neq0$, and consider the convex combination $$\rho\equiv p_1 |\psi_1\rangle\!\langle\psi_1| + p_2 |\psi_2\...
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When does Hermitian operator with unit trace become a density operator?

The definition of density operators is that (i) positive semidefinite; and (ii) unit trace. Given a Hermitian matrix $\rho$ (say, the size is $2\times 2$) with unit trace, I know that such matrix may ...
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What is the difference between Bloch's sphere and IBM's Q-sphere?

I'm new to Quantum Computing and I've been trying to understand single-qubit operations, quantum phases etc through Bloch's Sphere visualization. However, in IBM's Circuit Simulator, they seem to be ...
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1 vote
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How to get Bloch sphere Cartesian coordinates from density matrix

I am vexed by a particular derivation. Given a state $\psi$ and corresponding density matrix $\rho = |\psi\rangle \langle \psi|$, or $\rho = \begin{bmatrix} a & c \\ b & d \end{bmatrix}$, I ...
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Can I understand mixed states using the Bloch sphere? [duplicate]

I'm a bit confused with the representation of mixed states in a Bloch sphere. Are they represented as points or vectors? For pure states, they're vectors on the surface of the Bloch sphere and have a ...
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Exercise 4.6 in Quantum Computing and Quantum Information Nielsen and Chuang

Question 4.6: One reason why the $R_\hat{n}(θ)$ operators are referred to as rotation operators is the following fact, which you are to prove. Suppose a single qubit has a state represented by the ...
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Are these two 'divided by two' terms related?

I have a question about the two equations: Any matrix in $SU(2)$ could be parametrized as $$ R_{\hat{n}}(\theta) = \cos\left(\frac{\theta}{2}\right)I-i\sin\left(\frac{\theta}{2}\right)(\hat{n}\cdot\...
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How do we determine the direction of a time evolution $e^{-itH}$ on the Bloch sphere?

I have a question about the direction of time evolution on a Bloch sphere: suppose I'm performing a unitary time-evolution $\exp(-iHt/\hbar)$ for a single qubit, then on the Bloch sphere it ...
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Question regarding bloch vector solutions for master equation on page 388 of N&C

On page 388 of N&C, you are asked to find the solution to a differential equation for a two-level atom coupled to a vacuum. However, I have no experience with differential equations, so I am ...
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time evolution around a 'reflective axis'

Here's a diagram illustrates my question: The pink and purple vectors are 'reflective' (just like a light ray hitting a surface, though the direction is not the same case). Two grey vectors are some ...
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Bloch Sphere of Qiskit logo

Trying to plot the Bloch Sphere of the IBM Qiskit logo ...
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How the arguments of $U_3$ gate are converted when they're not lying in the expected range?

From the qiskit documentation (here), a general form of a single qubit unitary is defined as $$ U(\theta, \phi, \lambda) = \begin{pmatrix} \cos\left(\frac{\theta}{2}\right) & -e^{i\lambda} \sin\...
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Why there're two axis of rotation when I'm trying to visualize this time-evolution?

This is a follow-up question of the problem I posted earlier. The following diagram illustrates my question: I'm trying to perform the time evolution of a random Hamiltonian. The green vector ...
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