¿Se puede generalizar la esfera Bloch a dos qubits?

La esfera de Bloch es una buena visualización de estados de un solo qubit. Matemáticamente, se puede generalizar a cualquier número de qubits por medio de una hiperesfera de alta dimensión. Pero tales cosas no son fáciles de visualizar.

¿Qué intentos se han hecho para extender las visualizaciones basadas en la esfera Bloch a dos qubits?

Para los estados puros, hay una forma razonablemente simple de hacer una "esfera de 2 qubit bloch" . Básicamente, utiliza la descomposición de Schmidt para dividir su estado en dos casos: no enredado y completamente enredado. Para la parte no enredada, solo usa dos esferas bloch. Y luego la parte enredada es isomorfa al conjunto de rotaciones posibles en el espacio 3d (la rotación es cómo se traducen las mediciones en un qubit en predicciones en el otro qubit). Esto le brinda una representación con ocho parámetros reales:

1) Un valor real w entre 0 y 1 que indica el peso de no enredado versus totalmente enredado.

2 + 3) El vector de bloque unitario no enredado para el qubit 1.

4 + 5) El vector de bloque de unidad no enredado para el qubit 2.

6 + 7 + 8) La rotación completamente entrelazada.

Esto es lo que parece si muestra la parte de rotación como "donde se asignan los ejes XY y Z", y además escala los ejes por w para que se agrande cuanto más enredado esté:

(El rebote en el medio se debe a una degeneración numérica en mi código).

Para los estados mixtos, he tenido un poco de éxito al mostrar la envoltura de los vectores bloch predichos para el qubit 2 dadas todas las medidas posibles de qubit 1. Esto se ve así:

Pero tenga en cuenta que a) esta representación 'envolvente' no es simétrica (uno de los qubits es el control y el otro es el objetivo) yb) aunque parece bonito, no es algebraicamente compacto.

Esta pantalla está disponible en la rama alternativa de despliegue-entrelazado-despliegue de Quirk. Si puede seguir las instrucciones de compilación, puede jugar con él directamente.

Como una representación irreducible de spin $j$ de $SU(2)$ tiene una dimensión $2j+1$ ( $j$ es medio entero), cualquier espacio de Hilbert de dimensión finita se puede obtener como un espacio de representación de $SU(2)$ . Además, dado que todas las representaciones irreducibles de $SU(2)$ son productos tensoriales simétricos de la representación fundamental del spinor, por lo tanto, cada espacio de Hilbert de dimensión finita puede considerarse como un producto tensor simétrico de S U ( 2 ) fundamental$SU(2)$ Espacios de representación fundamentales.

$2j+1$$2j$$2j$$2j$ puntos mediante un producto tensor simétrico.

$2j+1$

(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.

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

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.

A paper has been published on the subject, called "Bloch sphere model for two-qubit pure states"

https://arxiv.org/abs/1403.8069