Bloch-sphere colorings and Bell inequalities

We consider the quantum and local hidden variable (LHV) correlations obtained by measuring a pair of qubits by projections defined by randomly chosen axes separated by an angle $\theta$. Local hidden variables predict binary colorings of the Bloch sphere with antipodal points oppositely colored. We prove Bell inequalities separating the LHV predictions from the singlet quantum correlations for $\theta \in (0,\frac{\pi }{3})$. We raise and explore the hypothesis that, for a continuous range of $\theta >0$, the maximum LHV anticorrelation is obtained by assigning to each qubit a coloring with one hemisphere black and the other white.

Figure 1

Some antipodal coloring functions $a_x$ on the sphere, see Appendix pp3 for definitions. Their correlations $C_x\left(\theta \right)$, computed from Eq. (4), subject to the constraint (6), are plotted in Appendix pp3.

Figure 2

Diagram of the measurements performed by Alice and Bob that are used in the proof of Lemma 2. Alice's and Bob's measurements $k$ are the same for $k=0,1,...,N-1$ and $N\ge 2$; these are projections onto the states $|{\xi }_k\rangle$ and correspond to points in the Bloch sphere with label $k$. These points form a zigzag path crossing the dashed great circle. The state $|{\xi }_N\rangle$ is antipodal to $|{\xi }_0\rangle$ and represents the measurement $k=0$ with reversed outcomes. The solid lines represent arcs of great circles with the same angle $\theta >\frac{\pi }{N}$ that connect adjacent points. If $\theta =\frac{\pi }{N}$, all these points are on the same great circle.

Figure 3

Diagram of the measurements performed by Alice and Bob that are used in the proof of Theorem 1. Alice's and Bob's measurements $A$ and $B$ are projections onto the state $|{\xi }_A\rangle$ and $|{\chi }_B\rangle$ and correspond to points in the Bloch sphere with labels $A$ and $B$, respectively, for $A,B\in \{0,1,...,N-1\}$ and $N\ge 2$. These points form a zigzag path crossing the dashed great circle. The state $|{\xi }_N\rangle$ is antipodal to $|{\xi }_0\rangle$ and represents Alice's measurement $A=0$ with reversed outcomes. The solid lines represent arcs of great circles with the same angle $\theta >\frac{\pi }{2N}$ that connect adjacent points. If $\theta =\frac{\pi }{2N}$, all these points are on the same great circle.

Figure 4

Alice's and Bob's measurement axes $\vec{a}$ and $\vec{b}$ form an angle $\theta$. The spherical coordinates of $\vec{a}$ and $\vec{b}$ are $(\epsilon ,\varphi )$ and $(\alpha ,\beta )$, respectively, related by Eqs. (C1) and (C2). Equation (4) computes the correlation $C_x\left(\theta \right)$ by (i) integrating the coloring function $b_x\left(\vec{b}\right)$ over the circle on the sphere generated by $\vec{b}$ [parametrized by the angle $\omega$ in Eqs. (C1) and (C2)] and (ii) integrating the coloring function $a_x\left(\vec{a}\right)$ over the sphere generated by $\vec{a}$. A general correlation $C\left(\theta \right)={\int }_{\mathcal{X}}dx\mu \left(x\right)C_x\left(\theta \right)$ is computed by integrating over the probability distribution $\mu \left(x\right)$ of the colorings satisfying the antipodal property (2).

Figure 5

Correlations computed with (4), subject to the constraint (6), for the coloring functions $a_x$ shown schematically in Fig. 1 and defined in Appendix pp3 1. The correlations for coloring 2, 3, and 4 are blue long-dash–short-dashed, red solid, and green long-dashed curves, respectively. The black long-dash–short-dashed curve represents the singlet-state quantum correlation $Q\left(\theta \right)$. The dark red short-dashed and dark green long-dash–short-dashed curves show the coloring 1 correlation $C_1\left(\theta \right)$ and anticorrelation $-C_1\left(\theta \right)$, respectively. The gray solid straight lines show the bounds given by Theorem 1 for $\theta \ge \frac{\pi }{12}$.

Figure 6

Correlations obtained for coloring $3_{\delta }$, defined in Appendix pp3 1, for $\delta =-0.038\pi$ (a, green dashed curve) and $\delta =-0.046\pi$ (b, blue long-dash–short-dashed curve) and for colorings 3, 1, and the singlet-state quantum correlation $Q\left(\theta \right)$ (red solid, dark red short-dashed, and black long-dash–short-dashed curves, respectively).