How can multiple entangled Qubits be represented and visualized?

[pervect]

One Qubit can be represented on the Bloch sphere. How would multiple entangled Qubits (say 2 or 3) be represented? Classically, one would think that if one Qubit is represented on a Bloch sphere, 2 Qubits would be represented on 2 Bloch spheres, but I'm pretty sure it doesn't work this way.

I believe it takes 2 complex numbers to represent one Qubit, ##\alpha_0 |0> + \alpha_1 |1>##, where the ##\alpha_a## are complex numbers, and this winds up maps to the 2-sphere because SU(2) is a double cover of SO(3). It takes 4 complex numbers to represent 2 (entangled) Qubits, and 8 complex numbers to represent 3 (entangled) Qubits, i.e. for the last case, ##\alpha_0 |000> + \alpha_1 |001> + ... \alpha_7 |111>##. So it I don't think 3 Bloch spheres can possibly represent 3 entangled Qubits, as it doesn't seem like there are enough degrees of freedom. Is there a reasonably simple geometric figure of higher dimension that can? Or some other way to represent or visualize 3 Qubits?

 

[LittleSchwinger]

There are several ways of representing the space of quantum states by some geometric object and this topic can lead one into some deep algebraic geometry. A great book for this kind of thing is:
I. Bengtsson and K. Życzkowski, Geometry of Quantum States 2nd Edition, Cambridge University Press, Cam-
bridge, 2017

 

[LittleSchwinger]

To give you an example, a density matrix in ##\mathbb{C}^{n}## can be written using the generators of ##SU(N)## as:
##\rho = \frac{1}{N}\mathbb{I} + \sum_{i}^{N^2 - 1}\tau_{i}\sigma_{i}##
and the ##\tau_{i}## give the coordinates of your state in a generalised Bloch space. Pure states are then picked out by the condition:
$$
\begin{align*}
\tau^{2} &= \frac{N - 1}{2N}\\\\
\left(\vec{\tau}\star\vec{\tau}\right)_{i} &= \frac{N-2}{N}\tau_{i}
\end{align*}
$$
So the first condition tells you the pure states lie on a sphere of dimension ##N^{2} - 2## and the second condition states they lie on a subset of that sphere that form the complex projective space ##\mathbb{CP}^{N - 1}##.

Your question is just the case ##N = 2^{m}, m \in \mathbb{N}##.

There are several other representations besides this Lie Algebra one and you led into questions like what is the geometric characterisation of the space of entangled, discordant and so on states in each representation. Very quickly you find links to major theorems in algebraic geometry and even one of Hilbert's open problems.