Cai　and　Zhang　(2018)　established　separate　perturbation　upper　bound　estimators　for　canonical　correlation　directions　under　centered　Gaussian　population　and　some　conditions　on　the　minimum　singular　value　sigma(r)(S)　of　a　correlation　matrix　S　.　They　posed　an　open　problem　for　the　optimality　of　their　estimators.　In　this　paper,　the　optimality　of　Cai　and　Zhang＇s　estimation　is　firstly　proved　up　to　some　multiplicated　constants.　Then　motivated　by　Ma　and　Li＇s　work　(Ma　and　Li,　2020),　we　give　an　upper　bound　estimation　for　centered　sub-Gaussian　population,　and　a　better　estimate　for　bounded　sub-Gaussian　population.　Finally,　all　estimates　are　extended　from　centered　population　to　non-centered　one.　(C)　2021　Elsevier　Inc.　All　rights　reserved.