2015/08/31 by Jeongwan Haah, Aram W. Harrow, Zhengfeng Ji +2 · 4 citations
Physics and Astronomy · Computer Science · Mathematics · #quant-ph #cs.IT #math.IT
paper · pdf · doi:10.1109/tit.2017.2719044
published as IEEE Transactions on Information Theory 63(9), 5628-5641 (2017) · revtex, 16 pages, 3 figures. (v1) in STOC 2016, 913-925. (v2) improved lower bound for independent measurements
arxiv created 2017/01/24 · arxiv updated 2017/09/01
It is a fundamental problem to decide how many copies of an unknown mixed quantum state are necessary and sufficient to determine the state. Previously, it was known only that estimating states to error ε in trace distance required O(dr2/ε2) copies for a d-dimensional density matrix of rank r. Here, we give a theoretical measurement scheme (POVM) that requires O (dr/ δ) ln (d/δ) copies of ρ to error δ in infidelity, and a matching lower bound up to logarithmic factors. This implies O( (dr / ε2) ln (d/ε) ) copies suffice to achieve error ε in trace distance. We also prove that for independent (product) measurements, Ω(dr2/δ2) / ln(1/δ) copies are necessary in order to achieve error δ in infidelity. For fixed d, our measurement can be implemented on a quantum computer in time polynomial in n.