Secure alignment of coordinate systems using quantum correlation

We show that two parties far apart can use shared entangled states and classical communication to align their coordinate systems with a very high fidelity. Moreover, compared with previous methods proposed for such a task, i.e., sending parallel or antiparallel pairs or groups of spin states, our method has the extra advantages of using single-qubit measurements and also being secure, so that third parties do not extract any information about the aligned coordinate system established between the two parties. The latter property is important in many other quantum information protocols in which measurements inevitably play a significant role.

Figure 1

Two-dimensional coordinate sharing scheme shown in the coordinate system of Bob: Alice and Bob agree on the $\mathbf{z}$ direction and they want to share the direction $\mathbf{m}$ which has been located in the $x-y$ plane. Bob estimates the desired direction ${\mathbf{m}}_e$ due to the value of the correlation function.

Figure 2

The average fidelity for different values of $N$ when the direction $\mathbf{m}$ which is estimated by Bob is located in a specific plane that he is aware of.

Figure 3

Alice and Bob want to share the direction $\mathbf{m}$, which in the coordinate system of Bob makes the angles $\alpha , \beta$, and $\gamma$ with $x, y$, and $z$ axes, respectively.

Figure 4

The blue circles show the behavior of the average fidelity for different values of $N$ when $3N$ singlet pairs are shared between Alice and Bob and they use method A of Sec. 3b to share a direction. The upper red squares show the optimal fidelity $\frac{3N+1}{3N+2}$ which is achievable only by collective measurements [8]. We use $3N$ in the formula of optimal fidelity in order to have a comprehensive comparison.

Figure 5

Average fidelity of estimation for two different methods as a function of the effective number of singlet pairs used in each run of the protocols. Blue circles correspond to the fidelities of method A, and green stars correspond to the fidelities of method B. For more explanation see the paragraph after Eq. (32).