Entanglement-assisted communication in the absence of shared reference frame

Alice wants to convey the value of a parameter to Bob with whom she does not share a reference frame. What physical object can she use for this task? Shall she encode this value into the angle between two physical vectors such as the angle between two spins? Can she benefit from using entanglement? We investigate these questions here and show that an entangled state of two qubits has three parameters that are invariant under changes of the reference frame. We also calculate for specific examples the average information gain for different circumstances, where one of these parameters is used for communication. We compare our result with the special case of separable states and find that entanglement enhances the information gain.

Figure 1

In the absence of a reference frame, a pure product state of two qubits (left) can only carry information in the angle between the two vectors representing the two qubits in the Bloch sphere. An entangled state (right) [Eq. (12)] can carry two more parameters. The questions are which parameter is a better carrier and what are the conditions for achieving this optimality.

Figure 2

Alice's state preparation circuit. She uses this circuit to independently adjust each of the relative parameters in a suitable way to provide Bob with the highest information gain possible.

Figure 3

Average information gain (bits) for estimation of $\theta$ in terms of ${\alpha }_0$ at ${\psi }_0=0$ for two different prior distributions over $\theta$. The plot shows that the more the entanglement the higher the average information gain. This also shows that product states are the worst choice for encoding a message in $\theta$.

Figure 4

Average information gain (bits) for estimation of $\psi$ in terms of ${\alpha }_0$ at ${\theta }_0=\pi$ for two different prior distributions over $\psi$. It is seen that for no entanglement no information is gained since $\psi$ vanishes for product states. On the other hand, as entanglement increases, so does the average information gain.

Figure 5

Average information gain (bits) for estimation of $\alpha$ in terms of ${\theta }_0$ at ${\psi }_0=0$ for two different prior distributions over $\alpha$. Increasing ${\theta }_0$ increases the average information gain; therefore, the best states to encode a message in their $\alpha$ parameter are the ones with ${\theta }_0=\pi$ and ${\psi }_0=0$.

Figure 6

Average information gain (bits) for estimation of $\theta , \psi$, and $\alpha$ in terms of the two other parameters in each encoding scenario for the uniform prior distributions over the message parameter. Contours are included in the plots for further clarification. (a) $I_{\text{avg}}^{\left(\theta \right)}({\alpha }_0,{\psi }_0)$. (b) $I_{\text{avg}}^{\left(\psi \right)}({\alpha }_0,{\theta }_0)$. (c) $I_{\text{avg}}^{\left(\alpha \right)}({\theta }_0,{\psi }_0)$.

Figure 7

Average information gain (bits) for estimation of $\theta , \psi$, and $\alpha$ in terms of the two other parameters in each encoding scenario for the discrete prior distributions over the message parameter. Contours are included in the plots for further clarification. (a) $I_{\text{avg}}^{\left(\theta \right)}({\alpha }_0,{\psi }_0)$. (b) $I_{\text{avg}}^{\left(\psi \right)}({\alpha }_0,{\theta }_0)$. (c) $I_{\text{avg}}^{\left(\alpha \right)}({\theta }_0,{\psi }_0)$.