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27 votes

Can the Bloch sphere be generalized to two qubits?

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. ...
Craig Gidney's user avatar
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19 votes
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Why are half angles used in the Bloch sphere representation of qubits?

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=\...
glS's user avatar
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12 votes

How to think about the Z gate in a Bloch sphere?

$|1\rangle$ 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 ...
Craig Gidney's user avatar
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12 votes
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Does the trace distance have a geometric interpretation?

There is a geometric interpretation that you certainly can take seriously, but the geometry that you get is not as clean as you might have hoped. Trace distance between operator states is an example ...
Greg Kuperberg's user avatar
12 votes

What makes representing qubits in a 3D real vector space possible?

Three real parameters are sufficient due to the constraint that $$ |\alpha|^2 + |\beta|^2 = 1\tag1 $$ where $\alpha$ and $\beta$ are the two components of a 2D complex vector describing the qubit ...
Adam Zalcman's user avatar
12 votes
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What makes representing qubits in a 3D real vector space possible?

Mathematically a qubit's coefficients $c_1$, $c_2$ must have the following properties: \begin{align} |c_1|^2 + |c_2|^2 =1 \tag{1}\\ |c_1|, |c_2| \in [0,1], \tag{2} \end{align} because Born's rule ...
user1271772 No more free time's user avatar
11 votes

Why can the most general state of a qubit be written as $|\Psi\rangle=\cos(\frac\theta2)|0\rangle+e^{i\phi}\sin(\frac\theta2)|1\rangle$?

The most general pure state of a qubit can be written as $|\Psi\rangle=a|0\rangle+b|1\rangle$ where $a,b\in\mathbb{C}$. The amplitudes $a$ and $b$ can be written in polar form as $a=re^{i\alpha}$ and $...
Adam Zalcman's user avatar
10 votes

Can the Bloch sphere be generalized to two qubits?

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 $SU(2)$. ...
David Bar Moshe's user avatar
10 votes

Why is an entangled qubit shown at the origin of a Bloch sphere?

Let $(x,y,z)$ be a point in the unit ball 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)\...
AHusain's user avatar
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10 votes

Is there any online Bloch sphere simulator?

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 ...
glS's user avatar
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9 votes
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What's a vector in the Bloch sphere representation?

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 ...
S.Move's user avatar
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9 votes

Why is an entangled qubit shown at the origin of a Bloch sphere?

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 ...
DaftWullie's user avatar
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9 votes

Why are half angles used in the Bloch sphere representation of qubits?

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\...
kludg's user avatar
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9 votes

Can I find the axis of rotation for any single-qubit gate?

A generic $2\times2$ (special) unitary matrix decomposes in terms of Pauli matrices as $$U = a_0 I + i \sum_{j=1}^3 a_j \sigma_j,\tag U$$ for $a_j\in\mathbb R$ such that $\sum_{j=0}^3 a_j^2=1$. One ...
glS's user avatar
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9 votes
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Can I understand mixed states using the Bloch sphere?

An arbitrary $2 \times 2$ Hermitian matrix $U$ can be decomposed into $$ U = n_I I + n_X X + n_Y Y + n_Z Z $$ with $n_X, n_Y, n_Z \in \mathbb{R}$ and $X, Y, Z$ are Pauli matrices and $I$ is the $2 \...
KAJ226's user avatar
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9 votes

Represent Hadamard gate in terms of rotations and reflections in Bloch sphere

but how about γ? Gamma doesn't show up on the Bloch sphere. It's a global phase. It's unobservable without conditioning the operation on a second qubit, in which case it turns into phase kickback ...
Craig Gidney's user avatar
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8 votes
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Drawing tangent vectors to the Bloch sphere with qutip

First of all, qutip is not a visualisation library, even though it does provide some visualisation functionalities, mostly leveraging ...
glS's user avatar
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8 votes
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Why are orthogonal quantum states represented as collinear in the Bloch sphere?

The Bloch representation of states is what you get when you decompose density matrices in terms of a basis of orthonormal Hermitian operators. That is, you write $\rho=\sum_k \sigma_k \langle\sigma_k,\...
glS's user avatar
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7 votes
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How to think about the Z gate in a Bloch sphere?

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 ...
DaftWullie's user avatar
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7 votes
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Purity of mixed states as a function of radial distance from origin of Bloch ball

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}....
DaftWullie's user avatar
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7 votes
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What is the meaning of writing a state in its Bloch representation?

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 ...
Sanchayan Dutta's user avatar
7 votes
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How are measurements on $Z$ and $X$ axes interpreted in the Bloch sphere?

The $\theta$ and the $\phi$ angles are not equivalent in the Bloch sphere. First, they have different ranges -- one is $\pi$ and the other is $2\pi$. More importantly, $\phi$ is a rotation around a ...
oleg's user avatar
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7 votes
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How many classical bits are needed to represent a qubit?

There are two types of information in physics: Classical information Quantum information Physics doesn't answer the question "What is (classical or quantum) information?". This is philosophic ...
kludg's user avatar
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7 votes
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Why is the Bloch sphere three-dimensional?

A qubit is a two-level quantum system and hence it can be written as: $$ |\psi \rangle = \alpha |0\rangle + \beta|1\rangle $$ where $|0 \rangle$ and $|1\rangle$ are the computational basis and they ...
KAJ226's user avatar
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7 votes

What makes representing qubits in a 3D real vector space possible?

For vector representation of any qubit it is true that: it has to be a unit vector global phase does not matter and can be fixed at any value As a result two degress of freedom are eliminated and as ...
Martin Vesely's user avatar
7 votes
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Can I find the axis of rotation for any single-qubit gate?

You can derive an expression for the rotation axis by combining (1) the decomposition of your unitary $O$ in the Pauli basis, and (2) by the representation of the general rotation operator $R_{\vec n}(...
Cryoris's user avatar
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7 votes
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Is every single-qubit unitary just a rotation around some unit vector on the Bloch sphere?

TL;DR: Yes, ignoring the unobservable global phase, every single-qubit unitary corresponds to a unique rotation of $\mathbb{R}^3$ and vice versa. Single-qubit unitaries and rotations Let us first pin ...
Adam Zalcman's user avatar
6 votes

How to obtain Y rotation with only X and Z rotations gates?

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,...
Ernesto Galvão's user avatar
6 votes

Can the Bloch sphere be generalized to two qubits?

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....
Michael Biercuk's user avatar
6 votes
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How to prove that Z operator rotates points on Bloch sphere about Z axis through 180°?

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 ...
Sanchayan Dutta's user avatar

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