Bloch Sphere generalization for more than one qubit

[pervect]

In https://web.cecs.pdx.edu/~mperkows/june2007/bloch-sphere.pdf, Ian Glendinning describes a derivation of the Bloch sphere for one qubit. To paraphrase his basic argument, one qubit in a pure state can be represented by 2 complex numbers, $\alpha= a + bi$ and $\beta = c + di$. This yields the equation of a 3-sphere, a^2 + b^2 + c^2 + d^2, but noting that multiplying by a random phase which has no physical effects, one can choose such a phase multiplier to set b=0, giving rise to a 2-sphere representation, a^2+c^2+d^2=0.

This is still a double cover of the qubit, so further steps needed to be taken to recover the actual Bloch sphere, which involves a half-angle transformation.

The author mentions extending this to multiple qubits in "future topics", but I have not been able to find a publication of such extension. I believe he has retired, from the little bit of internet research I tried.

Attempting to pursue the approach on my own, two entangled qubits, the inital phase of the argument yields 4 complex numbers and a 6-sphere, and 3 entangled qubits result in 8 complex numbers and a 14-sphere.

I'd expect the 6-sphere to be a 4 fold cover of two entangled qubits, and the 14-sphere to be an 8-fold cover, but I don't really have any idea how to proceede further.

I'm ultimately interested in the representation problem, ideally is simply as possible, of how to geometrically represent n entangled qubits (in a pure state). However, I'm also interested in comments about my incomplete attempt to extend the author's argument to this case (do they make sense), and / or other authors approaches to the problem.

For instance, Wikipedia's approach to the Bloch sphere, https://en.wikipedia.org/wiki/Bloch_sphere, starts with talking about projective spaces, which I am only mildly familiar with. To me, it seems basically equivalent to me to Ian's approach.

 

[Demystifier]

You might be interested in this book:
https://www.amazon.com/Geometry-Qua...-Entanglement/dp/1107026253?tag=pfamazon01-20

 

[Hornbein]

Check out

Solving The Travelling Salesman Problem Using A Single Qubit
Kapil Goswami,1, ∗ Gagan Anekonda Veereshi,1 Peter Schmelcher,1, 2 and Rick Mukherjee1, †

It seems to me impractical to scale this up to any meaningful value of n, but I think it is a clever idea.

 

[pervect]

I ran across the very interesting https://arxiv.org/abs/quant-ph/0302081, "Geometry of the 3-Qubit State, Entanglement and Division Algebras" which goes a long way to answering my own question.

We present a generalization to 3-qubits of the standard Bloch sphere representation for a single qubit and of the 7-dimensional sphere representation for 2 qubits presented in Mosseri {\it et al.}\cite{Mosseri2001}. The Hilbert space of the 3-qubit system is the 15-dimensional sphere , which allows for a natural (last) Hopf fibration with as base and as fiber.

The extra phase factor is not eliminated

 

[anuttarasammyak]

The extra phase factor is not eliminated

n qubit has 2^n base with complex coefficients so they have 2^(n+1) real parameters. Normalization and the extra phase factor reduce 2 so we should have 2(2^n-1) dimension real manifolds or 2^n-1 complex manifolds to express n qubit states. For coefficients $\{\ r_je^{i\phi_j}\}$, the manifold is

$$\{r_j \geq 0 \ |\ \sum_{j=1}^{2^n}r_j^2=1\} \cap \{0 \leq \phi_j < 2\pi\ |\ \phi_1=0 \} $$

It means that a model we seek may be combination of a point on a unit sphere of 2^n - 1 dimension
, and a set of 2^n-1 angles. How about it ?

I am not sure that Broch way to halve/double the angle $\theta$ to form a sphere from hemisphere where two bases are orthogonal visually, would be beneficial in higher n cases.