Degrees of Freedom of the 3-User Rank-Deficient MIMO Interference Channel

We provide the degrees of freedom (DoF) characterization for the $3$-user $M_T\times M_R$ multiple-input multiple-output (MIMO) interference channel (IC) with \emph{rank-deficient} channel matrices, where each transmitter is equipped with $M_T$ antennas and each receiver with $M_R$ antennas, and the interfering channel matrices from each transmitter to the other two receivers are of ranks $D_1$ and $D_2$, respectively. One important intermediate step for both the converse and achievability arguments is to convert the fully-connected rank-deficient channel into an equivalent partially-connected full-rank MIMO-IC by invertible linear transformations. As such, existing techniques developed for full-rank MIMO-IC can be incorporated to derive the DoF outer and inner bounds for the rank-deficient case. Our result shows that when the interfering links are weak in terms of the channel ranks, i.e., $D_1+D_2\leq \min(M_T, M_R)$, zero forcing is sufficient to achieve the optimal DoF. On the other hand, when $D_1+D_2> \min(M_T, M_R)$, a combination of zero forcing and interference alignment is in general required for DoF optimality. The DoF characterization obtained in this paper unifies several existing results in the literature.Comment: 28 pages, 7 figures. To appear in IEEE transactions on wireless communication

Fitting magnetic field gradient with Heisenberg-scaling accuracy

We propose a quantum fitting scheme to estimate the magnetic field gradient with $N$-atom spins preparing in W state, which attains the Heisenberg-scaling accuracy. Our scheme combines the quantum multi-parameter estimation and the least square linear fitting method to achieve the quantum Cram\'{e}r-Rao bound (QCRB). We show that the estimated quantity achieves the Heisenberg-scaling accuracy. In single parameter estimation with assumption that the magnetic field is strictly linear, two optimal measurements can achieve the identical Heisenberg-scaling accuracy. Proper interpretation of the super-Heisenberg-scaling accuracy is presented. The scheme of quantum metrology combined with data fitting provides a new method in fast high precision measurements.Comment: 7 pages, 2 figure