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8

For pure states, there is a reasonably simple way to make a "2 qubit bloch sphere". You basically use the Schmidt decomposition to divide your state into two cases: not entangled and fully entangled. For the not-entangled part, you just use two bloch spheres. And then the entangled part is isomorphic to the set of possible rotations in 3d space (the rotation ...


7

Let $(x,y,z)$ be a point in the unit sphere with $x^2+y^2+z^2 \leq 1$. The state associated with this point is \begin{eqnarray*} \rho &=& \frac{1}{2} (I_2 + x \sigma_x + y \sigma_y + z \sigma_z)\\ &=& \frac{1}{2} \begin{pmatrix} 1+z&x-iy\\ x+iy&1-z\\ \end{pmatrix} \end{eqnarray*} This is just a convenient way to parameterize all $2\...


7

Since a spin $j$ irreducible representation of $SU(2)$ has a dimension $2j+1$ ($j$ is half integer), any finite dimensional Hilbert space can be obtained as a representation space of $SU(2)$. Moreover, since all irreducible representations of $SU(2)$ are symmetric tensor products of the fundamental spinor representation, therefore every finite dimensional ...


6

It is a convention, chosen so that $\theta$ is the azimuthal angle of the point representing the state in the Bloch sphere. To see where this convention comes from, start from a state $|\psi\rangle=\alpha|0\rangle+\beta|1\rangle$. Remembering the normalisation constraint $|\alpha|^2+|\beta|^2=1$, and assuming without loss of generality $\alpha\in\mathbb R$,...


6

The Pauli-$Z$ gate maps $|0\rangle$ to $|0\rangle$ and $|1\rangle$ to $-|1\rangle$. For Bloch sphere representation, state of a qubit is written like (look at my previous answer for a detailed explanation) $$|\psi\rangle = \cos(\theta/2)|0\rangle + e^{i\phi}\sin(\theta/2)|1\rangle$$ Apply the Pauli-$Z$ gate on this and you get: $$|\psi'\rangle = \cos(\...


6

A density matrix $\rho$ has the properties of being Hermitian, non-negative and has trace 1. Any $2\times 2$ matrix can be written in the form $$ \rho=\frac{n_0\mathbb{I}+\vec{n}\cdot\vec{\sigma}}{2}. $$ The trace being 1 fixes that $n_0=1$, while the Hermitian property imposes that $\vec{n}\in\mathbb{R}^3$, where $\vec{\sigma}$ is the vector of the 3 Pauli ...


6

The Bloch sphere only represents the state of a single qubit. What you’re talking about is taking a multi-qubit state, and representing the state of just one of those qubits on the Bloch sphere. If the multi-qubit state is a product state (pure and separable), then the state of the single qubit is a pure state, and is represented as a point on the surface ...


5

Yes. The Bloch sphere formalism is used for geometrically representing pure and mixed states of two-dimensional quantum systems a.k.a qubits. Any pure state $|\Psi\rangle$ of a qubit can be written in the form: $$|\Psi\rangle = \cos\frac{\theta}{2}|0\rangle + e^{i\phi}\sin\frac{\theta}{2}|1\rangle$$ where $0\leq \theta\leq \pi$ and $0\leq \phi\leq 2\pi$. ...


5

Some people really do visualise single qubit operations by thinking about the Bloch sphere, and they work out everything based on that picture. Those who like thinking in that way are often incredibly quick at these sorts of manipulations, and have more intuitive understandings for why things work. But here's one concrete example where I use it: averaging ...


5

$|1\rangle$ and $-|1\rangle$ are assigned to the same point on the Bloch sphere because they are equal up to global phase. Algebraically: $|1\rangle \equiv -|1\rangle$ where $\equiv$ means "equal up to global phase". Meaning there is some $\theta$ such that $-|1\rangle = e^{i \theta} |1\rangle$. The thing that is confusing you is that, despite the fact that ...


5

The way to think about the Bloch sphere is in terms of the density matrix for the state. $Z$ acting on either $|0\rangle\langle 0|$ or $|1\rangle\langle 1|$ does nothing, as is true for any diagonal density matrix. To see the effect of the rotation, you need to look at how any non-diagonal density matrix is changed by $Z$, such as $|+\rangle\langle +|$.


4

Bloch spheres are a reasonable visualisation of individual qubits, so long as you don't fall in to the trap of thinking that's what they actually, physically look like. It's just a convenient mathematical mapping from a parametrisation of the state into a pretty picture. And it doesn't work so well once entanglement between multiple qubits comes into play. ...


4

Let $\hat{n}=(\cos\phi\sin\theta,\sin\phi \sin\theta,\cos\theta)$ i.e. the Cartesian coordinate vector for a point on the unit sphere with polar angle $\theta$ and azimuthal angle $\phi$. By sending a spin-1/2 particle through a Stern-Gerlach device with orientation $\hat{n}$, we can measure the observable \begin{align} S_n:=\vec{S}\cdot \hat{n} &=S_x \...


4

For more than 1-qubit visualization, we will need more complex visualizations than a Bloch sphere. The below answer from Physics Stack Exchange explains this concept quite authoritatively: Bloch sphere for 2 and more qubits In another article, the two qubit representation is described as a seven-dimensional sphere, S 7, which also allows for a Hopf ...


4

The problem you are describing (i.e. finding an approximation of some state given some number of identical copies of it and some set of measurements) is known as quantum state tomography or state tomography for short. In practise, the most efficient schemes for state tomography will depend on a specific experiment's setup and limitations, for which ...


4

The components of the Bloch vector of a state are the expectation values of the X,Y and Z Pauli matrices in that state and it has to be a full three-dimensional vector to capture the interior of the Bloch sphere as well, which represents mixed states. In general a state with density matrix $\rho$ of a single Qbit has Bloch vector $\vec{r}$ when $$ \rho = \...


4

Single-qubit unitaries are just 3D rotations, multiplied by a phase. So in order to find the actual angles, you can resort to the theory of rotation matrices, in particular to Euler's rotation theorem, which states that any rotation is a composition of 3 rotations (the theorem proof is constructive, so you get the actual angles).


3

We have some multiqubit visualizations within Q-CTRL's Black Opal package. These are all fully interactive and are designed to help build intuition about correlations in interacting two-qubit systems. The two Bloch spheres represent the relevant separable states of two qubits. The tetrahedra in the middle visually capture correlations between certain ...


3

First of all, qutip is not a visualisation library, even though it does provide some visualisation functionalities, mostly leveraging matplotlib. However, because qutip does provide handy functionalities to plot Bloch spheres and points on it, it does make sense to ask how one can tweak such functionalities to for example add tangent vectors to the bloch ...


3

For superconducting qubits, x and y rotations are usually both done with microwave pulses, and as you said the phase of the pulse determines the rotation axis. See mathematical details in this Physics Stack Exchange post: How do we perform transverse measurements in a two level system? Rotations about the z axis are quite different; they are done by ...


2

One can more generally show that $R_z(\theta)=e^{-i \theta Z/2}=\cos(\theta/2)-iZ\sin(\theta/2)$ rotates points on the Bloch sphere by an angle $\theta$ around the $z$-axis, and note that $Z=i R_z(\pi)$. Let $|\psi\rangle$ be an arbitrary pure state. The coordinates of the point representing $|\psi(\theta)\rangle\equiv R_z(\theta)|\psi\rangle$ on the Bloch ...


2

Whilst we normally talk about $\left|0\right>$ and $\left|1\right>$ as unchanging states in quantum computing, this is not usually the case in a physical realization where there tends to be an energy difference $\Delta E$ between these states such that $\left|1\right>_\mathrm{logical} = e^{-it \Delta E / \hbar} \left|1\right>_\mathrm{physical}$. ...


2

As per Wikipedia, we can write any pure state as $$|\psi\rangle = \cos\left( \frac{\theta}{2} \right) |0 \rangle + e^{i \phi} \sin\left( \frac{\theta}{2} \right) |1 \rangle$$ Where $\theta$ and $\phi$ are the angles on the Bloch sphere: Almost any point on the surface (i.e. pure state) has a unique representation in terms of the angles, except for the ...


2

If we use the convention $$| \psi \rangle = \cos(\theta) | 0 \rangle + e^{i \phi} \sin(\theta)| 1 \rangle$$ then the North ($\theta=0$) and the South ($\theta=\pi)$ are (physically) the same state $|0\rangle$; If we use the convention$$| \psi \rangle = \cos(\theta/2) | 0 \rangle + e^{i \phi} \sin(\theta/2)| 1 \rangle$$ then North is $|0\rangle$ and South ...


2

A simple way of proving that the purity (or any other property which only depends on the eigenvalues of $\rho$) can only depend on the distance from the center of the Bloch sphere is rotational invariance: Rotating a density matrix $\rho\mapsto U\rho U^\dagger$ corresponds to a rotation on the Bloch sphere. Thus, any property which only depends on the ...


2

Let me supplement the other answer by also showing what happens in the general case of the Bloch representation of generic qudits of dimension $d$. Let $\rho$ be an arbitrary state over $d$ modes, that is, a $d\times d$ positive semi-definite Hermitian matrix with unit trace. As any other Hermitian matrix, we can decompose it in terms of a basis of ...


2

Your intuition is correct, since rank-2 density matrices will be convex combinations of rank-1 density matrices, but they will be singular and hence will still be on the boundary. We can prove that non-singular $\rho$ are in the interior of $\Omega_3$ with a fairly direct proof. Let $\epsilon>0$ and let $\rho'$ be some matrix (the Bloch representation ...


2

Non-singular density matrices $\rho$ have no zero eigenvalues, $\lambda=\lambda_{\mathrm{min}}(\rho)>0$, with $\lambda_{\min}$ the smallest eigenvalue. Then, the ball $$ \{\rho+M|-\lambda I \le M \le \lambda I\} $$ will also be in the set of density operators, and thus, $\rho$ is an interior point.


2

I used this last time I needed to look up something about Bloch sphere. It's not perfect, since it doesn't allow entering the exact values of angles, let alone 2x2 matrices, but it has the benefit of being available online. This one looks promising in that it allows to enter matrices (and is also online), but I haven't tried it.


2

This doesn't really answer the question as it's not an online simulator. It might still be relevant though as it is a way to produce this sort of gifs if one has access to the software. It is relatively easy to do this sort of things using Wolfram Mathematica. As a quick and dirty example, if we just define a couple of relevant helper functions: pauliX = ...


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