Adaptive experimental design for one-qubit state estimation with finite data based on a statistical update criterion

We consider 1-qubit mixed quantum state estimation by adaptively updating measurements according to previously obtained outcomes and measurement settings. Updates are determined by the average-variance-optimality (A-optimality) criterion, known in the classical theory of experimental design and applied here to quantum state estimation. In general, A-optimization is a nonlinear minimization problem; however, we find an analytic solution for 1-qubit state estimation using projective measurements, reducing computational effort. We compare numerically two adaptive and two nonadaptive schemes for finite data sets and show that the A-optimality criterion gives more precise estimates than standard quantum tomography.Comment: 15 pages, 7 figure

Trinuclear μ₃‑Silyl Complexes of Ruthenium and Group 9 Metals Having 3c-2e Interactions and Transformation of a μ₃‑Silyl Complex of Ru₂Ir into μ‑Silyl and μ₃‑Silylene Complexes

μ₃-Silyl complexes (Cp*Ru)₂(Cp*M)­(μ₃-H₂SiR)­(μ-H)₃ (<b>4</b>, M = Co; <b>5</b>, M = Rh; <b>6</b>, M = Ir) were synthesized by the reaction of trinuclear heterometallic clusters of Ru and group 9 metals, (Cp*Ru)₂(Cp*M)­(μ-H)₃(μ₃-H) (<b>1</b>, M = Co; <b>2</b>, M = Rh; <b>3</b>, M = Ir), with primary silanes. The unprecedented coordination mode of primary silanes on a trimetallic plane was unambiguously confirmed by means of multinuclear NMR spectroscopy, IR spectroscopy, and X-ray diffraction studies. The μ-silane complex (Cp*Ru)₂(Cp*Ir)­(μ-H₂SiH^<i>t</i>Bu)­(μ-H)₄ (<b>7</b>), which is a plausible intermediate en route to the μ₃-silyl complex, was obtained by the reaction of <b>3</b> with ^<i>t</i>BuSiH₃. The μ₃-silyl complex <b>6a</b> was transformed into the μ-silyl complex (Cp*Ru)₂(Cp*Ir)­(μ-H₂SiPh)­(^<i>t</i>BuNC)­(μ-H)₃ (<b>8</b>) upon treatment with ^<i>t</i>BuNC. Complex <b>6a</b> also reacted with CO to afford the μ₃-silylene complex (Cp*Ru)₂(Cp*Ir)­(μ₃-HSiPh)­(μ-CO)­(μ-H)₂ (<b>10</b>) via the formation of the μ-silyl complex (Cp*Ru)₂(Cp*Ir)­(μ-H₂SiPh)­(CO)­(μ-H)₃ (<b>9</b>)

Trinuclear μ₃‑Silyl Complexes of Ruthenium and Group 9 Metals Having 3c-2e Interactions and Transformation of a μ₃‑Silyl Complex of Ru₂Ir into μ‑Silyl and μ₃‑Silylene Complexes

μ₃-Silyl complexes (Cp*Ru)₂(Cp*M)­(μ₃-H₂SiR)­(μ-H)₃ (<b>4</b>, M = Co; <b>5</b>, M = Rh; <b>6</b>, M = Ir) were synthesized by the reaction of trinuclear heterometallic clusters of Ru and group 9 metals, (Cp*Ru)₂(Cp*M)­(μ-H)₃(μ₃-H) (<b>1</b>, M = Co; <b>2</b>, M = Rh; <b>3</b>, M = Ir), with primary silanes. The unprecedented coordination mode of primary silanes on a trimetallic plane was unambiguously confirmed by means of multinuclear NMR spectroscopy, IR spectroscopy, and X-ray diffraction studies. The μ-silane complex (Cp*Ru)₂(Cp*Ir)­(μ-H₂SiH^<i>t</i>Bu)­(μ-H)₄ (<b>7</b>), which is a plausible intermediate en route to the μ₃-silyl complex, was obtained by the reaction of <b>3</b> with ^<i>t</i>BuSiH₃. The μ₃-silyl complex <b>6a</b> was transformed into the μ-silyl complex (Cp*Ru)₂(Cp*Ir)­(μ-H₂SiPh)­(^<i>t</i>BuNC)­(μ-H)₃ (<b>8</b>) upon treatment with ^<i>t</i>BuNC. Complex <b>6a</b> also reacted with CO to afford the μ₃-silylene complex (Cp*Ru)₂(Cp*Ir)­(μ₃-HSiPh)­(μ-CO)­(μ-H)₂ (<b>10</b>) via the formation of the μ-silyl complex (Cp*Ru)₂(Cp*Ir)­(μ-H₂SiPh)­(CO)­(μ-H)₃ (<b>9</b>)