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)
Questions tagged [bloch-sphere]
204 questions
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votes
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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 to visualize.
What attempts have been made to…
James Wootton
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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 $| \alpha | = \cos(\theta)$, $| \beta | = \sin(\theta)$ for some real number $\theta$. …
wanderingmathematician
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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 bit and could find only these two…
Sanchayan Dutta
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14
votes
1 answer
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 $\sigma$ are one-qubit states, the trace distance can be…
glS
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14
votes
2 answers
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 the simple circuit
over time, with $q_0$ on the…
orome
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14
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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 $Z|0\rangle = |0\rangle$ and $Z|1\rangle = -|1\rangle$.
It is explained here…
Bick
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11
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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 have to change in order to be able to rotate around…
Quasar
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11
votes
2 answers
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 \\\beta \end{pmatrix}, ~~ \alpha, \beta \in…
Martin Vesely
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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 represent the general unitary matrix acting on a qubit…
Mathist
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9
votes
1 answer
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 provided by the Bloch representation, by writing it…
glS
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9
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2 answers
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 mechanics is mathematically formulated in Hilbert…
Sanchayan Dutta
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9
votes
2 answers
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?
I have seen somewhere that rotation around an…
PhysicsMan
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8
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How to prove that antipodal points on the Bloch sphere are orthogonal?
I started by assuming two antipodal states
\begin{align*}
|(\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}|0\rangle +…
apen
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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 $\theta,\phi,\lambda$, can I find the vector $\hat n = (n_x,n_y,n_z)$…
ZR-
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8
votes
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Is it right to think of superposition as just angle?
Based on my current understanding, a qubit is represented as a vector $(a, b)$ which satisfies $a^2 + b^2 = 1$. Classical bit one can be represented as $(0, 1)$ and bit zero can be represented as $(1, 0)$.
And then we can have things that cannot be…
Wong Jia Hau
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