# Can the Bloch sphere be generalized to two qubits?

The Bloch sphere is a nice visualization of single qubit states. Mathematically, it can be generalized to any number of qubits by means of a high-dimensional hypersphere. But such things are not easy to visualize.

What attempts have been made to extend visualizations based on the Bloch sphere to two qubits?

## 5 Answers

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. For the not-entangled part, you just use two bloch spheres. And then the entangled part is isomorphic to the set of possible rotations in 3d space (the rotation is how you translate measurements on one qubit into predictions on the other qubit). This gives you a representation with eight real parameters:

1) A real value w between 0 and 1 indicating the weight of not-entangled vs fully-entangled.

2+3) The not-entangled unit bloch vector for qubit 1.

4+5) The not-entangled unit bloch vector for qubit 2.

6+7+8) The fully-entangled rotation.

Here's what it looks like if you show the rotation part as "where X Y and Z axes get mapped", and additionally scale the axes by w so that it gets bigger the more entangled you are:

(The bouncing in the middle one is due to a numerical degeneracy in my code.)

For mixed states, I've had a bit of success showing the envelope of bloch vectors predicted for qubit 2 given every possible measurement of qubit 1. That looks like this:

But note that a) this 'envelope' representation is not symmetric (one of the qubits is the control and the other is the target) and b) although it looks pretty it's not algebraically compact.

This display is available in the alternate dev-entanglement-display branch of Quirk. If you're able to follow the build instructions, then you can play with it directly.

Since a spin $$j$$ irreducible representation of $$SU(2)$$ has a dimension $$2j+1$$ ($$j$$ is half integer), any finite dimensional Hilbert space can be obtained as a representation space of $$SU(2)$$. Moreover, since all irreducible representations of $$SU(2)$$ are symmetric tensor products of the fundamental spinor representation, therefore every finite dimensional Hilbert space can be thought of as a symmetric tensor product of fundamental $$SU(2)$$ fundamental representation spaces.

This is the basis of the Majorana stellar representation construction. A state of a qudit living in a Hilbert space of dimension $$2j+1$$ can be represented by $$2j$$ points on the Bloch sphere. The state vector can be reconstructed from the $$2j$$ (2-dimensional) spin vectors of the $$2j$$ points by a symmetrized tensor product.

Given a state vector in a $$2j+1$$ dimensional Hilbert space (Please see Liu, Fu and Wang, section 2.1) $$|\psi\rangle = \sum_{m=-j}^{j} C_m |j, m\rangle,$$ The locations of the corresponding points (the Majorana stars) on the Bloch sphere are given by the roots of the equation: $$\sum_{k=0}^{2j} \frac{(-1)^k C_{j-k}}{(2j-k)! k!} z^{2j-k}=0.$$

(The parametrization is by means of the stereographic projection coordinate $$z = \tan \theta e^{i\phi}$$ ($$\theta$$, $$\phi$$ are the spherical coordinates))

One application of this representation to quantum computation, is in the visualization of the trajectories giving rise to geometric phases, which serve as the gates in holonomic quantum computation. These trajectories are reflected as trajectories of the Majorana stars on the Bloch spheres and the geometric phases can be computed from the solid angles enclosed by these trajectories. Please see Liu and Fu's work on Abelian geometric phases. A treatment of some non-Abelian cases is given by Liu Roy and Stone.

Finally, let me remark that there are many geometric representations relevant to quantum computation, but they are multidimensional and may be not useful in general as visualization tools. Please see for example Bernatska and Holod treating coadjoint orbits which can serve as phase spaces of the finite dimensional Hilbert spaces used in quantum computation. The Grassmannian which parametrizes the ground state manifold of adiabatic quantum Hamiltonians is a particular example of these spaces.

• I know they are time consuming to find or make, but is there any chance you could illustrate this answer with such visualisations? Perhaps an example of a CNOT gate? Nov 15, 2018 at 8:42
• In general, a unitary transformation of a state will move its constellation to new locations such that the coordinate of a star in the final state depends algebraically on all the coordinates of all stars in the initial state. However, in simple cases, we can perform the computation by a simple inspection. Please see for example Bengtsson and Życzkowski: researchgate.net/profile/Karol_Zyczkowski/publication/… page 103, figure 4.7, Nov 15, 2018 at 15:09
• cont. where for example, the CNOT gate action on a state with three stars at the north pole shifts one of the stars to the south pole while keeping the other two stars in place. Nov 15, 2018 at 15:09

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.

The two Bloch spheres represent the relevant separable states of two qubits. The tetrahedra in the middle visually capture correlations between certain projections of the two qubits. When there is no entanglement, the Bloch vectors live entirely on the surfaces of the respective spheres. However, a fully entangled state lives exclusively in the space of correlations in this representation. The extrema of these spaces will always be maximally entangled states like Bell states, but maximally entangled states can also reside within multiple tetrahedra simultaneously.

• Would you be able to describe these representations? It would be nice if you could expand this into a self-contained answer. Nov 13, 2018 at 23:39
• edited to add further material. Nov 14, 2018 at 0:39
• Thanks @MichaelBiercuk, and good to see you here. Nov 14, 2018 at 6:17

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 sphere for 2 and more qubits

In another article, the two qubit representation is described as a seven-dimensional sphere, S 7, which also allows for a Hopf fibration, with S 3 fibres and a S 4 base. The most striking result is that suitably oriented S 7 Hopf fibrations are entanglement sensitive.

Geometry of entangled states, Bloch spheres and Hopf fibrations

Having said that, a Bloch sphere based approach is quite useful even to model the behavior of qubits in a noisy environment. There has been analysis of the two-qubit system by use of the generalized Bloch vector to generate tractable analytic equations for the dynamics of the four-level Bloch vectors. This is based on the application of geometrical concepts from the well-known two-level Bloch sphere.

We can find that in the presence of correlated or anti-correlated noise, the rate of decoherence is very sensitive to the initial two-qubit state, as well as to the symmetry of the Hamiltonian. In the absence of symmetry in the Hamiltonian, correlations only weakly impact the decoherence rate:

Bloch-sphere approach to correlated noise in coupled qubits

There is another interesting research article on the representation of the two-qubit pure state parameterized by three unit 2-spheres and a phase factor.For separable states, two of the three unit spheres are the Bloch spheres of each qubit with coordinates (A,A) and (B,B). The third sphere parameterises the degree and phase of concurrence, an entanglement measure.

This sphere may be considered a ‘variable’ complex imaginary unit t where the stereographic projection maps the qubit-A Bloch sphere to a complex plane with this variable imaginary unit. This Bloch sphere model gives a consistent description of the two-qubit pure states for both separable and entangled states.

As per this hypothesis, the third sphere (entanglement sphere) parameterizes the nonlocal properties, entanglement and a nonlocal relative phase, while the local relative phases are parameterized by the azimuthal angles, A and B, of the two quasi-Bloch spheres.

Bloch sphere model for two

• Would it be possible to expand a bit on these remarks? Rather than linking to these articles, it would be good to describe the relevant ideas in some detail to keep the answer self-contained. (Also, in your third answer in this post, the symbols are not rendering properly...) Nov 13, 2018 at 23:44
• Near "the azimuthal angles": What is it before "A" and "B"? Firefox shows it as "F066". Also near "qubit with coordinates", before A and B (four in total), two of them "F071"? Nov 14, 2018 at 6:57

Pure two-qubit states may be represented by just two Bloch Spheres. This article explains how to do it. The division into separable and maximally entangled part is done as in the first answer. The maximally entangled part gets represented by the relative directions of coordinate axes of two Bloch Spheres. The key point is that the coordinate axes of the Bloch Spheres must have different handedness. Here is how Bell states are represented in this paper.

There are illustrations of the CNOT gate action (Figures 9, 10, 11) in the paper and also an illustration of entanglement and disentanglement process (Figure 14). Visually the process of entanglement looks like a gradual reflection of one of the Bloch Sphere's coordinate axes, given some plane of reflection. Here is an illustration of a partially entangled state. Note how the x-axis of the second BS is partially inverted. Here the representations of separable and maximally entangled state have been combined into two Bloch Spheres of radius 1 each.

The work employs concepts from Geometric Algebra - a powerful framework that provides unified language for various areas of Mathematics, Engineering and Physics. But the main concept employed here is the concept of "plane of rotation". It is basically what it says. The rotations are defined as happening in the plane (e.g. $$x_1 y_1$$ plane) as opposed to being around an axis (e.d. around $$z_1$$ axis). This allows to define passive rotations - rotations of coordinate axes instead of rotations of the statevector arrow - which are useful for rotations of maximally entangled states which have no statevector arrow to rotate.