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)
202 questions
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Can the Bloch sphere be generalized to two qubits?
The Bloch sphere is a nice visualization of single qubit states. Mathematically, it can be generalized to any number of qubits by means of a high-dimensional hypersphere. But such things are not easy ...
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Why are half angles used in the Bloch sphere representation of qubits?
Suppose we have a single qubit with state $| \psi \rangle = \alpha | 0 \rangle + \beta | 1 \rangle$. We know that $|\alpha|^2 + |\beta|^2 = 1$, so we can write $| \alpha | = \cos(\theta)$, $| \beta | ...
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Is there any online Bloch sphere simulator?
While writing this answer I realized it would be really helpful if I could show the OP a video or .gif of how qubit states in Bloch spheres transform under certain unitary operations. I googled up a ...
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How to think about the Z gate in a Bloch sphere?
I am confused about how to understand the $Z$ gate in a Bloch sphere.
Considering the matrix $Z = \begin{pmatrix}
1 & 0 \\
0 & -1
\end{pmatrix}$ it is understandable that $Z|0\rangle = |0\...
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Why is an entangled qubit shown at the origin of a Bloch sphere?
I'm unclear why the Bloch sphere representation of a maximally entangled qubit shows the state of the bit as being at the origin of the sphere.
For example, this illustration
shows the effect of ...
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Does the trace distance have a geometric interpretation?
Consider the trace distance between two quantum states $\rho,\sigma$, defined via
$$D(\rho,\sigma)=\frac12\operatorname{Tr}|\rho-\sigma|,$$
where $|A|\equiv\sqrt{A^\dagger A}$.
When $\rho$ and $\sigma$...
glS♦
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How many classical bits are needed to represent a qubit?
I have two question concerning information content of qubit.
Question 1: How many classical bits are needed to represent a qubit:
A qubit can be represented by a vector $q = \begin{pmatrix}\alpha \\\...
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Rotating about the y- or z-axis of the Bloch sphere
In order to rotate about an axis of the Bloch sphere we ususally use pulses e.g. in trapped ion quantum computing or superconducting qubits. Let's say we have rotation around the x-axis. What do I ...
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What is the relation between these two forms of a single-qubit unitary operation?
I want to understand the relation between the following two ways of deriving a (unitary) matrix that corresponds to the action of a gate on a single qubit:
1) HERE, in IBM's tutorial, they ...
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How to obtain Y rotation with only X and Z rotations gates?
Let's say you have a system with which you can perform arbitrary rotations around the X and Z axis. How would you then be able to use these rotations to obtain an arbitrary rotation around the Y axis?
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Do pure qudit states lie on a hypersphere in the Bloch representation?
It is known that every state $\rho$ of a $d$-level system (or if you prefer, qudits living in a $d$-dimensional Hilbert space) can be mapped into elements of $\mathbb R^{d^2-1}$ through the mapping ...
glS♦
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Can I find the axis of rotation for any single-qubit 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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Homeomorphism or stereographic projection corresponding to the set of mixed states within the Bloch sphere
The Bloch sphere is homeomorphic to the Riemann sphere, and there exists a stereographic projection $\Bbb S^2\to \Bbb C_\infty$. But this only holds for pure states. To quote Wikipedia:
Quantum ...
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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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How to prove that antipodal points on the Bloch sphere are orthogonal?
I started by assuming two antipodal states
$$
|(\theta,\psi)\rangle = \cos\dfrac{\theta}{2}|0\rangle + \sin\dfrac{\theta}{2}e^{i\psi}|1\rangle\\
|(\theta+\pi,\psi+\pi)\rangle= \cos\dfrac{\theta+\pi}{2}...
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From Q# measurements to Bloch sphere
I would like to represent the state of a qubit on a Bloch sphere from the measurements made with Q#.
According the documentation, it is possible to measure a qubit in the different Pauli bases (...
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Why is the Bloch sphere three-dimensional?
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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Maximum number of "almost orthogonal" vectors one can embed in Hilbert space
In a Hilbert space of dimension $d$, how do I calculate the largest number $N(\epsilon, d)$ of vectors $\{V_i\}$ which satisfies the following properties. Here $\epsilon$ is small but finite compared ...
user2669
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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?
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Drawing tangent vectors to the Bloch sphere with qutip
I need to plot drawings of qubit dynamics on the Bloch sphere. I know QuTip allows to do such drawings but I specifically need to represent evolution velocities on the Bloch sphere so I need to draw ...
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How to prove that Z operator rotates points on Bloch sphere about Z axis through 180°?
My idea was to apply $Z$ operator 𝐭𝐰𝐢𝐜𝐞, which leads us back to the point where we started from, and also show that after applying the $Z$ operator just 𝐨𝐧𝐜𝐞 we are not at the same point ...
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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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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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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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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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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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What's a vector in the Bloch sphere representation?
The Bloch sphere isn't so intuitive for me. But I am not sure how you are supposed to manipulate it using vectors and matrices.
How do you actually represent a vector on it? Is it
$(\cos(a) , e^i\...
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How is the surface of a Bloch sphere a Hilbert space?
In the linear algebra section of the Qiskit textbook appears the following claim regarding the Bloch sphere:
The surface of this sphere, along with the inner product between qubit
state vectors, is a ...
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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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How are measurements on $Z$ and $X$ axes interpreted in the Bloch sphere?
I'm having trouble understanding how the measurement on $z$ and $x$ axes can be interpreted in terms of the Bloch sphere representation.
.
I know that the state can be written as
$$∣𝜓⟩=\cos(𝜃/2)|...
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Show that the Hadamard gate is equivalent to a 180 degree rotation of a certain axis
Show that the Hadamard gate is equivalent to a 180 degree rotation about the axis defined by $(\vec{e_x} - \vec{e_z}) / \sqrt{2}$ where $\vec{e_x}$ and $\vec{e_z}$ are unit vectors pointing along the ...
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Why is the boundary of the set of states in the generalised Bloch representation comprised of singular matrices?
Consider an arbitrary qudit state $\rho$ over $d$ modes.
Any such state can be represented as a point in $\mathbb R^{d^2-1}$ via the standard Bloch representation:
$$\rho=\frac{1}{d}\left(\mathbb I +\...
glS♦
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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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Flying qubits compared with stationary qubits
In a previous question I asked how stationary bits could be processed by logic gates. I casually mentioned that I could visualize qubits traversing stationary logic gates, and @DaftWullie said “It ...
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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
$$
R_{\hat{n}}(\theta) = \cos\left(\frac{\theta}{2}\right)I-i\sin\left(\frac{\theta}{2}\right)(\hat{n}\cdot\...
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Which states can reached using single-qubit Clifford gates?
Starting with the qubit state $|0\rangle$, which single-qubit states can be obtained by applying single-qubit Clifford gates, i.e. Pauli + Hadamard + $S$ gates?
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Represent a pure state in terms of 2 antipodal points on the Bloch sphere
I recently had an assignment where the question is based on the assumption that we can write any pure state qubit $|\phi \rangle$ as:
$$|\phi \rangle = \gamma |\psi\rangle + \delta |\psi^\perp ...
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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
$$\rho\equiv p_1 |\psi_1\rangle\!\langle\psi_1| + p_2 |\psi_2\...
glS♦
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Reduced Density Matrix Equation of Motion to describe an Ellipse
Given a pure two qubit state $|\psi_{AB}\rangle$. If we trace out system $B$, the remaining density matrix $\rho_A = Tr_B|\psi_{AB}\rangle\langle\psi_{AB}|$, can be represented as a point lying ...
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Why is the angle of $R_z(\frac\pi4) R_x(\frac\pi4)$ an irrational multiple of $\pi$
It is stated in the Qiskit tutorial section 2.4 that if you apply a rotation around the z-axis of $\frac\pi4$ and subsequently a rotation around the x-axis of $\frac\pi4$, the end result is an angle ...
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Purity of mixed states as a function of radial distance from origin of Bloch ball
@AHusain mentions here that the purity of a qubit state can be expressed as a function of the radius from the center of a Bloch sphere. The state corresponding to the origin is maximally mixed whereas ...
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Why can the most general state of a qubit be written as $|\Psi\rangle=\cos(\frac\theta2)|0\rangle+e^{i\phi}\sin(\frac\theta2)|1\rangle$?
Why we can express a most general qubit as $|\Psi\rangle = \cos{\left(\frac{\theta}{2}\right)}|0\rangle + e^{i \phi} \sin{\left(\frac{\theta}{2}\right)} |1\rangle$? Is there any formal proof for this?
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Can I understand mixed states using the Bloch sphere? [duplicate]
I'm a bit confused with the representation of mixed states in a Bloch sphere. Are they represented as points or vectors? For pure states, they're vectors on the surface of the Bloch sphere and have a ...
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Find the $\theta$ and $\phi$ values on the Bloch sphere corresponding to the state $\frac{1+i}{2}|0\rangle+\frac1{\sqrt2}|1\rangle$
If I have the following state:
$$
\left| \varphi \right>=\frac{1}{\sqrt{2}}\left(\left(\frac{1+i}{\sqrt{2}} \right)\left| 0 \right> + \left| 1\right>\right)
$$
How can I find the $\theta$ ...
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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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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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How should I understand the change of qubit's basis as a rotation?
I have a little difficulty with understanding. How do I properly visualize the change of qubit's basis as a rotation?
Let's say that we have classical basis vectors, $|0\rangle$ and $|1\rangle$. Now, ...
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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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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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