Quantum-phase synchronization

We study mechanisms that allow one to synchronize the quantum phase of two qubits relative to a fixed basis. Starting from one qubit in a fixed reference state and the other in an unknown state, we find that, contrary to the impossibility of perfect quantum cloning, the quantum phase can be synchronized perfectly through a joined unitary operation. When both qubits are initially in a pure unknown state, perfect quantum-phase synchronization through unitary operations becomes impossible. In this situation we determine the maximum average quantum-phase synchronization fidelity and the distribution of relative phases and fidelities, and we identify optimal quantum circuits that achieve this maximum fidelity. A subset of these optimal quantum circuits enable perfect quantum-phase synchronization for a class of unknown initial states restricted to the equatorial plane of the Bloch sphere.

Figure 1

Quantum circuit for the most general unitary transformation $U\in \text{SU}\left(4\right)$ on two qubits (see Fig. 7 in Ref. [40]). The circuit $U_c$ is obtained by setting $V_1=V_2=W_1=W_2={\mathbb{1}}_2$.

Figure 2

Quantum circuit $U_{\text{max}}$ that maximizes the mean PSF for initial pure product states.

Figure 3

Distribution of the relative phase $P\left(\mathrm{\Delta }{\phi }^{\prime }\right)$ after the optimal quantum circuit $U_{\text{max}}$ (radial coordinate, arbitrary units) as a function of $\mathrm{\Delta }{\phi }^{\prime }\in (-\pi ,\pi ]$ (azimuthal coordinate), i.e., the phase of the second Bloch vector measured relative to the first one, with $\mathrm{\Delta }{\phi }^{\prime }=0$ corresponding to the $x$ axis (left panel). Corresponding distribution $P\left(F\right)$ of the PSF, $F=cos\left(\mathrm{\Delta }{\phi }^{\prime }\right)$ (right panel). Both distributions are generated numerically.

Figure 4

PSF for different latitudes $\theta \equiv {\theta }_1={\theta }_2$ as a function of $\theta$ and $\mathrm{\Delta }\phi$. While for $\theta =\pi /2$ perfect QPS is achieved, the greater the distance from the equator, the worse $U_{\text{max}}$ synchronizes phases.