A modified least squares-based tomography with density matrix perturbation and linear entropy consideration along with performance analysis

Quantum state tomography identifies target quantum states by performing repetitive measurements on identical copies. In this paper, we have two key contributions aimed at improving traditional post-processing computational complexity and sample complexity of quantum tomography protocols. In the first case, we propose a new low-cost positivity constraint method based on density matrix perturbation after the least squares (LS) estimation of the density matrix. In the second case, we propose a new cost function with the maximum linear entropy and LS method to improve the sample average trace distance with reasonably low sample complexity. We call it the LS with the maximum entropy (LSME) method. Our proposed algorithm does not follow the iterative optimization technique, which is true for existing maximum likelihood and entropy-based ones. Performance analysis is conducted for our proposed methods by studying how they compare to the existing techniques for different sample complexities and dimensionalities. Extensive numerical simulations have been conducted to demonstrate the advantages of the proposed tomography algorithms.