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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 Two-stage 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. [ABSTRACT FROM AUTHOR]

This paper introduces the covering dimension of a linear code over a finite field, which is analogous to the critical exponent of a representable matroid and, thus, generalizes invariants that lie at the heart of several fundamental problems in a coding theory. An upper bound on the covering dimension is proved, improving Kung’s classical bound for the critical exponent. In addition, a construction is given for linear codes that attain equality in this covering dimension bound. [ABSTRACT FROM AUTHOR]

Characterising the capacity region for a network can be extremely difficult. Even with independent sources, determining the capacity region can be as hard as the open problem of characterising all information inequalities. The majority of computable outer bounds in the literature are relaxations of the linear programming bound, which involves entropy functions of random variables related to the sources and link messages. When sources are not independent, the problem is even more complicated. Extension of linear programming bounds to networks with correlated sources is largely open. Source dependence is usually specified through a joint probability distribution, and one of the main challenges in extending linear program bounds is the difficulty (or impossibility) of characterising arbitrary dependences via entropy functions. This paper tackles the problem by answering the question of how well entropy functions can characterise correlation among sources. We show that by using carefully chosen auxiliary random variables, the characterisation can be fairly “accurate”. Using such auxiliary random variables, we also give implicit and explicit outer bounds on the capacity of networks with correlated sources. The characterisation of correlation or joint distribution via Shannon entropy functions is also applicable to other information measures, such as Rényi entropy and Tsallis entropy. [ABSTRACT FROM AUTHOR]

Motivated by the recent resolution of asymptotic quantum birkhoff conjecture (AQBC), we attempt to estimate the distance between a given unital quantum channel and the convex hull of unitary channels. We provide two lower bounds on this distance by employing techniques from quantum information and operator algebras, respectively. We then show how to apply these results to construct some explicit counterexamples to AQBC. We also point out an interesting connection between the Grothendieck’s inequality and AQBC. [ABSTRACT FROM AUTHOR]