# Is there a way to represent a 3 qubit system using 3 Bloch Spheres?

I am relatively new to the Quantum Computing world and was wondering if representing a 3 qubit system using 3 Bloch Spheres feasible and if so what would the correct way to do it?

I understand a Bloch sphere can represent a single qubit.

$$\phi = \alpha |0> + \beta |1>$$ (i.e the $|0> + |1>$ states along with it's amplitudes)

In a 2- qubit system you could represent it using two bloch spheres.

$$\phi = \alpha_1 |00> + \beta_1 |01> + \alpha_2 |10> + \beta_2 |11>$$

So one sphere for the $|00>$ and $|01>$ amplitudes and another sphere for the $|10>$ and $|11>$ amplitudes

But I am a little confused about a 3-qubit system as from what I can gather it is represented by a structure like this (granted I realise different textbooks have different notation for the amplitudes):

$$/phi = \alpha_1 |000> + \beta_1 |001> + \gamma_1 |010> + \delta_1 |011> + \alpha_2 |100> + \beta_2 |101> + \gamma_2|110> + \delta_2|111>$$

How would one divide the amplitudes with their computational basis states in order to represent phi as what I expect should be 3 Bloch spheres (as it is a 3 qubit system) Since Bloch spheres represent a single $|0>+|1>$ ? Or should there be more Bloch spheres?

For example should it actually be represented by 4 Bloch spheres in the manner: $sphere1 = \alpha_1 |000> + \beta_1 |001>$

$sphere2 = \alpha_1 |000> + \beta_1 |001>$

$sphere3 = \alpha_2 |100> + \beta_2 |101>$

$sphere4= \gamma_2|110> + \delta_2|111>$

and if this is the correct manner to represent, why is that?

Apologies if this is a silly questions, I am still very much a beginner in this field!

It is not possible two represent an $n$-qubit system with $n$ (3D) Bloch spheres. You would need instead a $(2^{n+1}-1)$ dimensional hypersphere.

That can be simply deduced from the degrees of freedom of the system you want to represent. A $n$-qubit system has $2^{n}$ base vectors. For every base vector there is one complex amplitude, which makes $2^{n+1}$ degrees of freedom. Because of the normalization and the global phase we have to substract two degrees of freedom again, which makes $2^{n+1}-2$ DoF.

$n$ 3D standard bloch spheres on the other hand have just $2n$ free parameters, which is clearly not sufficient to represent the whole system. The linear growth of degrees of freedom would only suffice to represent a classical system.

A simple example, when your proposal would suffice is when the many-body state is just a product state of single particle wavefunctions (qubits), i.e. \begin{align} \left | \Psi\right\rangle = \prod_{n}\left( \cos\theta_n\left |0\right\rangle_n+\exp(-i\phi_n)\sin\theta_n \left |1\right\rangle_n \right) \end{align}

• Thank you for your explanation I realised they I might be wrong but was a little confused as to why. If Bloch spheres aren't the way to go, what would be the best way to represent a 2 or 3 quit systems to do something like compare a before and after of certain operation on a qubit system. E.g 2 qubit system showing before and after using Hadamard gate or the Quantum Fourier Transform etc. Feb 16, 2017 at 15:25
• Often one plots the density matrix, see e.g. here qutip.org/docs/3.1.0/guide/guide-visualization.html Feb 16, 2017 at 15:36
• You can ignore global phase to save another DOF. Feb 17, 2017 at 0:38
• Yes exactly. I edited my answer. Feb 17, 2017 at 9:11
• Also all states in this higher dimensional ball don't represent valid density matricies in general. The equivalent of the Bloch sphere in higher dimensions is a complicated convex body. Feb 18, 2017 at 6:35