Two-Stage Estimation for Quantum Detector Tomography: Error Analysis, Numerical and Experimental Results

Abstract

Quantum detector tomography is a fundamental technique for calibrating quantum devices and performing quantum engineering tasks. In this paper, a novel quantum detector tomography method is proposed. First, a series of different probe states are used to generate measurement data. Then, using constrained linear regression estimation, a stage-1 estimation of the detector is obtained. Finally, the positive semidefinite requirement is added to guarantee a physical stage-2 estimation. This Twostage Estimation (TSE) method has computational complexity O(nd 2 M), where n is the number of d-dimensional detector matrices and M is the number of different probe states. An error upper bound is established, and optimization on the coherent probe states is investigated. We perform simulation and a quantum optical experiment to testify the effectiveness of the TSE method.

Optimal and two-step adaptive quantum detector tomography

“…In this paper we address the question of finding an optimal QDT and an adaptive QDT method based on linear regression. Here we present the background knowledge and briefly introduce the two-stage QDT reconstruction algorithm in [25], which will be employed as a critical part for developing optimal and adaptive QDT.…”

“…such that Pi = P † i , Pi ≥ 0 for 1 ≤ i ≤ n and n i=1 Pi = I. We review the solving procedure in [25]. Let {Ω i } d 2 i=1 be a complete basis set of orthonormal operators with d-dimension.…”

“…Ref. [25] proposed a two-stage solution with an analytical computational complexity and error upper bound. Ref.…”

“…For optimal QDT , we only consider Step 1 where we obtain measurement data and then use the two-stage reconstruction algorithm in [25] to identify the unknown detectors. A natural question would be which input probe states are optimal, according to some requirements or criteria.…”

“…A natural question would be which input probe states are optimal, according to some requirements or criteria. Here we use the upper bound of the mean squared error (UMSE) [25] and the robustness described by the condition number against measurement errors as two criteria. We prove that the minimum UMSE is (n−1)(d 4 +d 3 −d 2 )…”