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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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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
11 votes
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What is the difference between the "Fubini-Study distances" $\arccos|\langle\psi|\phi\rangle|$ and $\sqrt{1-|\langle\psi|\phi\rangle|}$?

Recall the law of cosines for two unit vectors $\mathbf{u}$ and $\mathbf{v}$ in $\mathbb R^2$: $$ \|\mathbf{u}-\mathbf{v}\|^2 = 2-2\cos\theta, $$ where $\theta$ is the angle between the vectors. ...
Chris Ferrie's user avatar
11 votes
Accepted

Is the set of all states with negative conditional Von Neumann entropy convex?

The conditional von Neumann entropy is a concave function: if $\rho$ and $\sigma$ are states of a pair of registers $(\mathsf{X},\mathsf{Y})$ and $\lambda\in[0,1]$ is a real number, then $$ \mathrm{H}(...
John Watrous's user avatar
  • 6,127
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
9 votes

What are useful resources about the geometric of qutrits and its relation with Gell-Mann matrices?

There are many ways to describe a qutrit or a general $N$ level system geometrically. There is also a large amount of references either explaining these geometries or applying them to various problems ...
David Bar Moshe's user avatar
9 votes

Is the set of all states with negative conditional Von Neumann entropy convex?

Geometric characterization (as any other characterization) of subsets of the quantum state space in relation with their locality and entanglement properties becomes very complicated as the number of ...
David Bar Moshe's user avatar
7 votes
Accepted

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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6 votes
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Do pure qudit states lie on a hypersphere in the Bloch representation?

Upon some reflection, the answer is that no, they most definitely do not. The easiest way to see this is to observe that there are $d^2-1$ orthogonal directions in the Bloch representation (i.e. ...
glS's user avatar
  • 25.6k
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
5 votes

Purity of mixed states as a function of radial distance from origin of Bloch ball

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, ...
glS's user avatar
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5 votes

Do states with the same purity always have the same rank?

The other answer already gave a counterexample. From a geometrical point of view, the question is about the intersection of hyperplanes with hyperspheres. Indeed, the purity of a state $\rho$ with ...
glS's user avatar
  • 25.6k
5 votes

Can the Bloch sphere be generalized to two qubits?

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 ...
Gokul Alex's user avatar
5 votes
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What mathematical object is the set of quantum states of a qudit? How should one write this?

The qudit Hilbert space is $\mathbb{C}^d$. But, as you say, the Hilbert space does not exactly correspond to qudit states because of global phases (and also the normalization requirement). The object ...
Nick Mertes's user avatar
4 votes

Homeomorphism or stereographic projection corresponding to the set of mixed states within the Bloch sphere

I'm late to the party, but here's my take: Pure qubit states As you said, the space of pure states of a single qubit can be described as a complex projective line $\mathbb{C}P^1$, which is ...
Guglielmo Lotti's user avatar
4 votes
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Homeomorphism or stereographic projection corresponding to the set of mixed states within the Bloch sphere

Almost. You get a manifold with boundary with the Bloch ball. The radius from the origin parameterizing how pure it is. The origin being maximally mixed. This isn't a manifold because a point on the ...
AHusain's user avatar
  • 3,663
4 votes
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Do states with the same purity always have the same rank?

$$ \rho = \begin{pmatrix} a & 0 & 0\\ 0 & 1-a & 0\\ 0 & 0 & 0\\ \end{pmatrix}\\ \sigma = \begin{pmatrix} b & 0 & 0\\ 0 & c & 0\\ 0 & 0 & 1-b-c\\ \end{...
AHusain's user avatar
  • 3,663
4 votes

How are orthogonal sets of pure states arranged in state space?

"Generalized Bloch" manifolds are synonyms to coherent state manifolds. The points of these manifolds do not correspond, in general, to orthonormal vectors, as there are much more points than the ...
David Bar Moshe's user avatar
4 votes

What is the difference between the "Fubini-Study distances" $\arccos|\langle\psi|\phi\rangle|$ and $\sqrt{1-|\langle\psi|\phi\rangle|}$?

I'll try to address the problem from the Riemannian geometry point of view. In this approach, the distances are identified as length of geodesics of Riemannian metrics on spaces of quantum states. The ...
David Bar Moshe's user avatar
4 votes
Accepted

What is the intuition behind Bures and angle metrics?

Filling out a number of details for the sake of a complete answer — Starting from the linked article, Distance measures to compare real and ideal quantum processes [arXiv:quant-ph/0408063], the ...
Niel de Beaudrap's user avatar
4 votes
Accepted

What is the connection between Bures metric and (finite) Bures distance?

The Bures metric is the limit of the Bures distance for two infinitesimally close density matrices $\rho$ and $\rho+d\rho$. The Bures distance however, is not unique. It depends on the space on ...
David Bar Moshe's user avatar
3 votes
Accepted

Constructing a state with constraints on reduced states

You could try solving this numerically using semidefinite programming. We know the trace norm of an operator $X$ can be formulated as $$ \begin{aligned} \|X\|_1 &= \min_{Y,Z}\quad \frac12\mathrm{...
Rammus's user avatar
  • 5,968
3 votes
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Quantum marginal problem - constructing a global state from reduced states

If perturbations are sufficiently small and $\rho_{AB}$ has sufficiently broad support then a desired global state $\rho_{AB}'$ exists. Define $$ \rho_{AB}' = \rho_{AB} + (\rho_A' - \rho_A) \otimes \...
Adam Zalcman's user avatar
3 votes

What are useful resources about the geometric of qutrits and its relation with Gell-Mann matrices?

I need some useful sources about the geometry of qutrit. The most useful resource I know on the geometries of qutrits is the paper Geometry of the generalized ...
user1271772 No more free time's user avatar
3 votes

How are orthogonal sets of pure states arranged in state space?

This other answer already gave a nice proof that orthogonal bases are mapped into vectors $r_i$ such that $\sum_i r_i=0$. Here I'll work out explicitly the coordinates in a few cases, to show what ...
glS's user avatar
  • 25.6k
3 votes
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How are orthogonal sets of pure states arranged in state space?

One natural generalization of that property is that Bloch vectors for a basis set must sum to 0 vector. Though, this property is not a criterion for basis sets in dimensions higher than 2. If $\rho_i$...
Danylo Y's user avatar
  • 7,452
3 votes
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Why is the boundary of the set of states in the generalised Bloch representation comprised of singular matrices?

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 ...
Sam Jaques's user avatar
  • 2,076
2 votes

Purity of mixed states as a function of radial distance from origin of Bloch ball

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 ...
Norbert Schuch's user avatar
2 votes
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What is the intuition of the outer product of two states?

The outer product of two states is a matrix. Here are some often used outer products: \begin{equation} |0\rangle \langle 0 | = \begin{pmatrix} 1&0\\0&0 \end{pmatrix} \qquad |0\rangle \langle ...
Davit Khachatryan's user avatar
2 votes

Why is the boundary of the set of states in the generalised Bloch representation comprised of singular matrices?

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 ...
Norbert Schuch's user avatar

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