Can the Bloch sphere representation be applied to many-qubit states with an iterative approach?

Question

By ignoring the global phase, we can represent a single qubit state as \begin{equation} |\psi\rangle = \cos(\theta)|0\rangle + e^{i\phi}\sin(\theta)|1\rangle \end{equation} which very much looks like a generalization / nested version of the Euler formula where $i$ is replaced by $e^{i\phi}$ (therefore obtaining a sphere representation instead of a circle).

Can we generalize such a "nested Euler" formalism further (more nesting) to obtain a useful representation of a multi-qubit state (by having a high-dimensional sphere)?

I'm at the beginning of my QC studies and additionally lack the mathematical tools to answer this myself.

Answer 1

(Using (hyper)spherical coordinates to parametrise pure states) Indeed you can. This is similar to how you can use generalised spherical coordinates to parametrise hyperspheres. Similarly, to parametrise (pure) quantum states amounts to parametrise complex vectors defined up to norm and phase (more precisely, elements of complex projective spaces). You can do this by simply iterating the same thing you do for a qubit, introducing new angles and phases.

For example, you can parametrise states in a three-dimensional space as $$\cos(\theta_1)|0\rangle + \sin(\theta_1)\cos(\theta_2)e^{i\varphi_1}|1\rangle + \sin(\theta_1)\sin(\theta_2) e^{i(\varphi_1+\varphi_2)}|2\rangle,$$ and similarly for higher dimensional spaces. Note that I'm not talking about many-qubit states, but more generally about high-dimensional states. If you are dealing with $n$ qubits, that just means that you apply the above reasoning with your $2^n$-dimensional states.

Note also how the dimensions match: complex vectors in $\mathbb{C}^N$ defined up to phase and norm are characterised by $2N-2$ parameters, and $2N-2=2(N-1)$, consistently with the above reasoning introducing a pair of new angles for each added dimension.

(Caveats) It's also worth stressing that these representations are singular. You don't have a smooth bijection between quantum states and parameters in this "spherical" representation. Not even in the single-qubit case. This is true more generally for spherical coordinates, which only provide a homeomorphism for spheres if you remove a point from them.

This problem is even "worse" in the case of quantum states, because the representation is redundant for all states. Again, even restricting the attention to single qubits, $\theta$ and $\theta+\pi$ correspond to the same state, and similarly sending $\varphi\to-\varphi$ and $\theta\to \pi-\theta$ fixes the corresponding state. These redundancies are due to the fact that we are trying to parametrise complex projective spaces as if they were (hyper)spheres, which they are not.

This is not to say that you shouldn't use such representations. They are certainly useful in lots of situations. The fact that you don't get homeomorphisms between the space of states and the parameters isn't a real problem is most situations.

(Connection with Bloch sphere coordinates) Finally, note that this is not really a generalisation of the "Bloch sphere". In fact, what you get is not a sphere at all. Interpreting $\theta$ and $\phi$ as the spherical coordinates of a sphere doesn't translate into $\theta_1,...,\theta_N$ and $\varphi_1,...,\varphi_N$ being the $2N$ parameters parametrising a hypersphere in $\mathbb{R}^{2N+1}$.

See also

• How are orthogonal sets of pure states arranged in state space?
• Do pure qudit states lie on a hypersphere in the Bloch representation?
• Can we rotate Bloch vectors for qudits like we do with qubits in the Bloch sphere?

Answer 2

The generalization might be $|\psi\rangle=\exp(i \theta \mathbf{G}\cdot\mathbf{x})|0\rangle$ for some ground state $|0\rangle$, vector of Hermitian operators $\mathbf{G}=(G_1,G_2,\cdots)$, unit vector $\mathbf{x}$, and "angle" $\theta$. This follows because we can write pure qubits as rotations from the ground state $|0\rangle$ using exponentials of Pauli matrices.