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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Can I understand the mixed state using the Bloch sphere?

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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Direction of time evolution on a 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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1answer
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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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Why did I get two solutions to solve for the parameters of this $U_3$ gate? (I only expected one of them)

I have the following complex vector in $\mathbb{C}^2$: Vec= [[ 0.89741876+0.j] [-0.33540402+0.28660724j]] I'm trying to implement a $U_3$ gate to prepare this ...
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1answer
42 views

How to read a Q sphere representation?

I'm trying to understand the Q-sphere representation of a 3-qubit system. I get that the 3-qubits are in a superposition of 2 different states. The first qubit (rightmost) is in a superposition of <...
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154 views

Can I plot a bigger Bloch sphere using kaleidoscope?

I'm plotting a vector on the Bloch sphere using kaleidoscope: from kaleidoscope import bloch_sphere Is there a way I can make the plot bigger? (Instead of zooming ...
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Can I simultaneously plot 2 vectors on a single Bloch sphere and rotate the angle of visualization?

I'm trying to plot 2 vectors on the same Bloch sphere. From the qiskit documentation here, we can find ...
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Can I find the axis of rotation for any single-qubit $U_3$ gate?

Suppose I have an arbitrary qiskit $U_3$ gate: $U_3(\theta,\phi,\lambda)$. Is there a way I can find which axis the gate is rotating around? In other words, given any real numbers $\theta,\phi,\lambda$...
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Why can any density operator be written this way? (quantum tomography)

From page 24 of the thesis "Random Quantum States and Operators", where $(A,B)$ is the Hilbert-Schmidt inner product: \begin{aligned} \rho &=\left(\frac{1}{\sqrt{2}} I, \rho\right) \frac{...
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101 views

Which states can reached using single-qubit Clifford gates?

Starting with the qubit state $|0\rangle$, which single-qubit states can be obtained by applying single-qubit Clifford gates, i.e. Pauli + Hadamard + $S$ gates?
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How is a two qubit mixed state represented in the form of Bloch vector?

How is a two qubit mixed state represented in the form of Bloch vector?
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how to plot probability histogram and/or bloch sphere of single qubit in multi-qubit quantum circuit in Qiskit?

How can I get statevector and/or blochsphere representation of qubits of my choice. For example I have 3 qubits with different gates being applied on each qubit. the qiskit state_vector simulator ...
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1answer
119 views

Vector from SU(2) to SO(3)?

I know how to change the special unitary 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 ...
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What makes representing qubits in a 3D real vector space possible?

Qubits exist in a 2D complex vector space, but we can represent qubits on the Bloch sphere as a 3D real vector space. Mathematically, what makes this possible – why don't we need 4 real dimensions?
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Show that the Hadamard gate is equivalent to a 180 degree rotation of a certain axis

Show that the Hadamard gate is equivalent to a 180 degree rotation about the axis defined by $(\vec{e_x} - \vec{e_z}) / \sqrt{2}$ where $\vec{e_x}$ and $\vec{e_z}$ are unit vectors pointing along the ...
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Does this point-projection of a mixed state onto a pure state appear in the quantum information theory literature?

In my research, I stumbled on a smooth map: $$\pi_{\rho_0}: B \setminus \{\rho_0\} \to \partial B$$ where $B$ is the open Bloch ball, corresponding to the set of mixed states of a single qubit and $\...
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Numerical optimization of QRAC

I need to optimize a general version of 3$\rightarrow$1 QRAC where Bob is asked to retrieve one of the XOR combinations of the bits( If ABC is the given string to Alice, then Bob would be asked to ...
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54 views

How does the CPTP constraint reflect on the matrix representation of a qubit channel in the Pauli basis?

Let us write the possible states of a qubit in the Bloch representation as $$\newcommand{\bs}[1]{{\boldsymbol{#1}}}\rho_{\bs r}\equiv \frac{I+\bs r\cdot\bs \sigma}{2},$$ where $\bs\sigma=(\sigma_1,\...
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1answer
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$3 \rightarrow 1$ QRAC encoding for XOR functions

I'm currently working on QRAC and was wondering if there's an encoding protocol in $3 \rightarrow 1$ such that the receiver is able to retrieve any one of the XOR combinations of the bits, along with ...
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How does adding an identity to an Hamiltonian affect the corresponding time-evolution in the Bloch sphere?

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
85 views

What is the difference between the states $i|1\rangle$ and $|+i\rangle$?

I am new to Quantum computing. I see $|\mbox{+}i\rangle$ state maps to y-axis on bloch sphere ($\theta = 90$ degree and $\phi = 90$ degree) while $i|1\rangle$ maps on x-axis, $i|1\rangle$ is stated as ...
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1answer
147 views

How to sample from the uniform distribution over the tensor product of two Bloch spheres?

For some context, I am trying to assess the capacity that certain two qubit gates have to create entanglement. To do this I am using the idea of "entangling power", where one takes their ...
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3answers
250 views

Why the 3rd dimension of Bloch sphere?

I can understand the intuition behind a two dimensional bloch circle, as it represents the probability distribution of a certain state vector. However, I fail to grasp what the third dimension adds to ...
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2answers
85 views

Benefit of phase shift in quantum computing

I am new to quantum computing. I compare Pauli-X gate and Pauli-Y gate as equivalent to NOT gate in classical computers. Though I am not very sure when to use Pauli-X and Pauli-Y gates as the result ...
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1answer
79 views

Does composition of two single qubit rotations yield a single rotation around a unit vector?

$ \newcommand{\coefcos}[0]{c_1 c_2 - s_1 s_2 \hat{n}_1 \cdot \hat{n}_2} \newcommand{\coefsin}[0]{s_1 c_2 \hat{n}_1 + c_1 s_2 \hat{n}_2 - s_1 s_2 \hat{n}_2 \times \hat{n}_1}$This question relates to ...
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Deformation of the Bloch sphere and contraction of its planes under the action of channels

On pg 376-377 of N&C, it gives 3 different diagrams showing how the various axis of the Bloch sphere will be contracted under the action of the channels, limiting the possible states after it's ...
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54 views

What is the phase of state?

I am pretty new to Quantum Computing and am not exactly sure what the phase is. Could you please explain in terms of the Bloch sphere and point out how to mathematically calculate and represent the ...
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when we specify quantum states in $\mathbb C^2$, why do we only have $2$ basis states?

I am just starting to get up to speed with quantum computing via the Quiskit learning path: online tutorial Here they explain the Dirac notation and use it to describe quantum states as elements in $\...
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1answer
38 views

$N(\frac{1}{2},2)=3$ for vectors in a Hilbert Space

Came across This question regarding the maximum number of almost orthogonal vectors one can embed in a Hilbert space. They state that $N(\frac{1}{2},2)=3$, and that explicit construction of the ...
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2answers
332 views

How are eigenvectors and eigenvalues expressed in the Bloch sphere?

I'm relatively new to the subject of quantum computing, and I recently came across the idea of eigenvalues and eigenvectors. I believe I understand the relationship between the two, where eigenvalues ...
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Reduced Density Matrix Equation of Motion to describe an Ellipse

Given a pure two qubit state $|\psi_{AB}\rangle$. If we trace out system $B$, the remaining density matrix $\rho_A = Tr_B|\psi_{AB}\rangle\langle\psi_{AB}|$, can be represented as a point lying ...
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Why an element of SU(2) acts as a rotation for Majorana representation of states?

I know that for a given spin-j quantum state, say $\vert\psi\rangle = (\psi_0 , \psi_1 , \cdots , \psi_{2j})$, we can construct a polynomial as follows $ w(z) = \sum_{k = 0}^{2j} (-1)^k \psi_k \sqrt{\...
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1answer
66 views

Is $e^{i\beta} R_Z(-2\beta)$ equivalent to $U_1(2\beta)$?

As far as I know the single qubit gate $$ e^{i\beta\sigma_z} = \begin{bmatrix} e^{i\beta} & 0 \\ 0 & e^{-i\beta} \end{bmatrix} = e^{i\beta} \begin{bmatrix} 1 & 0 \\ 0 & e^{-i2\beta} \...
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Does normalizing a mixed state give a pure state?

According to bloch sphere interpretation, any point on the surface of the sphere corresponds to a pure state and any point inside the sphere corresponds to a mixed state. Suppose you have a point ...
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1answer
66 views

What is the intuition of the outer product of two states?

If it is possible give me an intuition both with vectors on the plane and on the Bloch-Sphere.
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97 views

Where is $i|0\rangle$ located on the bloch sphere?

I understand linear transformations on the plane but cannot understand the Bloch sphere. How can a three dimensional sphere be generated by two linearly dependent vecotrs (the basis states 0 and 1)?
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60 views

Cannot interpret transformations on the bloch sphere as matrix multiplications

I understand that X,Y and Z gates are rotations around the axes with the respective letters, but I cannot understand how can Y gate multiply the amplitude of 0 with unreal number and have it landing ...
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1answer
60 views

What kind of transformation does the Y-gate do on the bloch sphere?

I'm going through "Quantum Computation & Quantum Information" by Michael A. Nielsen and Isaac L. Chuang, and as a high school student with no previous knowledge, I cannot understand some things ...
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122 views

Effect of Pauli X gate on minus state using bloch sphere

As I understood, the X gate flips the state around : $X(|0\rangle) = |1\rangle$. It can also be visualized with a $\pi$ rotation around the $x$ axis in the Bloch sphere. I have no problem with that. ...
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173 views

Represent a pure state in terms of 2 antipodal points on the Bloch sphere

I recently had an assignment where the question is based on the assumption that we can write any pure state qubit $|\phi \rangle$ as: $$|\phi \rangle = \gamma |\psi\rangle + \delta |\psi^\perp ...
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59 views

Find local state and compute Bloch coordinates, like Quirk

In a multi-qubit system I can find the amplitudes of a state and compute probabilities, $\theta$, and $\varphi$. This falls out from simulation with simple numpy arrays. For example, after application ...
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1answer
155 views

$\lambda$ parameter for U3 gate in qiskit Bloch sphere visualisation

It is easy to see how $\theta$ (rotation from the positive z-axis) and $\varphi$ (rotation from the positive z-axis) affect the initial state of the qubit when looking at the Bloch sphere but I have ...
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1answer
113 views

Bloch sphere, where are magnitude and phase of a qbit?

Where are magnitude and phase of a qbit on Bloch sphere? Phase is angle φ. What do you mean by magnitude? Amplitudes? They are given by angle θ - amplitude of |0⟩ is cos(θ/2) and amplitude of |1⟩ ...
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How is a qubit represented on a bloch sphere?

A quantum state can be represented as linear combination of 2 states: In Chuang and Nielsens book, it states that because the squared amplitudes sum to 1: That the combination can be rewritten as: ...