# Tag Info

### 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. ...
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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 ...
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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. ...
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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. ...
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### 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....

### 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, ...
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### 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 ...
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### 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 ...
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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 ...
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### 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 ...
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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 ...
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