Characterization of equivariant maps and application to entanglement detection

We study equivariant linear maps between finite-dimensional matrix algebras, as introduced by Bhat. These maps satisfy an algebraic property which makes it easy to study their positivity or k-positivity. They are therefore particularly suitable for applications to entanglement detection in quantum information theory. We characterize their Choi matrices. In particular, we focus on a subfamily that we call (a, b)-unitarily equivariant. They can be seen as both a generalization of maps invariant under unitary conjugation as studied by Bhat and as a generalization of the equivariant maps studied by Collins et al. Using representation theory, we fully compute them and study their graphical representation, and show that they are basically enough to study all equivariant maps. We finally apply them to the problem of entanglement detection and prove that they form a sufficient (infinite) family of positive maps to detect all k-entangled density matrices.Comment: 16 pages, 4 figure

同変写像の研究とその量子情報理論への応用

京都大学0048新制・課程博士博士(理学)甲第21531号理博第4438号新制||理||1638(附属図書館)京都大学大学院理学研究科数学・数理解析専攻(主査)准教授 COLLINS Benoit Vincent Pierre, 教授 堤 誉志雄, 教授 泉 正己学位規則第4条第1項該当Doctor of ScienceKyoto UniversityDFA

Child neurodevelopment after multidomain interventions from preconception through early childhood

ImportanceMultidomain interventions in pregnancy and early childhood have improved child neurodevelopment, but little is known about the effects of additional preconception interventions.ObjectiveTo evaluate the effect of a multifaceted approach including health; nutrition; water, sanitation, and hygiene (WASH); and psychosocial support interventions delivered during the preconception period and/or during pregnancy and early childhood on child neurodevelopment.Design, Setting, and ParticipantsIn this randomized trial involving low- and middle-income neighborhoods in Delhi, India, 13 500 participants were assigned to preconception interventions or routine care for the primary outcome of preterm births and childhood growth. Participants who became pregnant were randomized to pregnancy and early childhood interventions or routine care. Neurodevelopmental assessments, the trial’s secondary outcome reported herein, were conducted in a subsample of children at age 24 months, including 509 with preconception, pregnancy, and early childhood interventions; 473 with preconception interventions alone; 380 with pregnancy and early childhood interventions alone; and 350 with routine care. This study was conducted from November 1, 2020, through February 25, 2022.InterventionsHealth, nutrition, psychosocial care and support, and WASH interventions delivered during preconception, pregnancy, and early childhood periods.Main Outcomes and MeasuresCognitive, motor, language, and socioemotional performance at age 24 months, assessed using the Bayley Scales of Infant and Toddler Development 3 tool.ResultsThe mean age of participants at enrollment was 23.8 years (SD, 3.0 years). Compared with the controls at age 24 months, children in the preconception intervention groups had higher cognitive scores (mean difference [MD], 1.16; 98.3% CI, 0.18-2.13) but had similar language, motor, and socioemotional scores as controls. Those receiving pregnancy and early childhood interventions had higher cognitive (MD, 1.48; 98.3% CI, 0.49-2.46), language (MD, 2.29; 98.3% CI, 1.07-3.50), motor (MD, 1.53; 98.3% CI, 0.65-2.42), and socioemotional scores (MD, 4.15; 98.3% CI, 2.18-6.13) than did controls. The pregnancy and early childhood group also had lower incidence rate ratios (RRs) of moderate to severe delay in cognitive (incidence RR, 0.62; 98.3% CI, 0.40-0.96), language (incidence RR, 0.73; 98.3% CI, 0.57-0.93), and socioemotional (incidence RR, 0.49; 98.3% CI, 0.24-0.97) development than did those in the control group. Children in the preconception, pregnancy, and early childhood intervention group had higher cognitive (MD, 2.60; 98.3% CI, 1.08-4.12), language (MD, 3.46; 98.3% CI, 1.65-5.27), motor (MD, 2.31; 98.3% CI, 0.93-3.69), and socioemotional (MD, 5.55; 98.3% CI, 2.66-8.43) scores than did those in the control group.Conclusions and RelevanceMultidomain interventions during preconception, pregnancy and early childhood led to modest improvements in child neurodevelopment at 24 months. Such interventions for enhancing children’s development warrant further evaluation