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Recursive construction of a regular simplex", "Subtitle", CellGroupingRules->{GroupTogetherGrouping, 10001.}, CellChangeTimes->{{3.4207126208015327`*^9, 3.420712636333867*^9}, { 3.4207126542796717`*^9, 3.4207126544899745`*^9}, 3.42071269253468*^9, { 3.42071273509588*^9, 3.4207127351359377`*^9}, {3.420712840046792*^9, 3.420712842099744*^9}, {3.4207141710807247`*^9, 3.420714171441243*^9}, { 3.4207167936217537`*^9, 3.420716797667571*^9}, {3.4212319262839055`*^9, 3.4212319295385857`*^9}, {3.4212333975194387`*^9, 3.421233397559496*^9}, { 3.4212334397301345`*^9, 3.421233447250949*^9}, {3.4212360503139696`*^9, 3.421236050394085*^9}, 3.4212382614634466`*^9, 3.4212456816331263`*^9}], Cell[TextData[{ "There is a recursive way to construct a regular simplex that is centered at \ the origin and inscribed in a unit sphere. For example, regular simplex in 1 \ dimension is just a line segment from -1 to 1. 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Construction in the non-negative orthant", "Subtitle", CellGroupingRules->{GroupTogetherGrouping, 10002.}, CellChangeTimes->{{3.4207126208015327`*^9, 3.420712636333867*^9}, { 3.4207126542796717`*^9, 3.4207126544899745`*^9}, 3.42071269253468*^9, { 3.42071273509588*^9, 3.4207127351359377`*^9}, {3.420712840046792*^9, 3.420712842099744*^9}, {3.4207141710807247`*^9, 3.420714171441243*^9}, { 3.4207167936217537`*^9, 3.420716797667571*^9}, {3.4212319262839055`*^9, 3.4212319295385857`*^9}, {3.4212333975194387`*^9, 3.421233397559496*^9}, { 3.4212334397301345`*^9, 3.421233447250949*^9}, {3.421235999921509*^9, 3.421236013020344*^9}, {3.4212360541494846`*^9, 3.4212360542296*^9}, { 3.4212361606626434`*^9, 3.4212361613836803`*^9}, 3.421238271788293*^9, 3.4212457029437695`*^9}], Cell[TextData[{ "This is a construction of ", Cell[BoxData[ FormBox["n", TraditionalForm]]], " vectors in the non-negative orthant of ", Cell[BoxData[ FormBox[ SuperscriptBox["\[DoubleStruckCapitalR]", "n"], TraditionalForm]]], ", sucht that the inner product between any two of them is ", Cell[BoxData[ FormBox[ FractionBox["1", SqrtBox[ RowBox[{"n", "+", "1"}]]], TraditionalForm]]], ". Sometimes this set of vectors can be extended to a SIC-POVM. The \ construction works as follows. The first vector we take to be ", Cell[BoxData[ FormBox[ RowBox[{"(", GridBox[{ {"1", "0", "0", "0", "0", "\[TripleDot]"} }], ")"}], TraditionalForm]]], ". Then the next must be ", Cell[BoxData[ FormBox[ RowBox[{"(", GridBox[{ { SubscriptBox["\[Alpha]", "1"], SubscriptBox["\[Beta]", "1"], "0", "0", "0", "\[TripleDot]"} }], ")"}], TraditionalForm]]], ", where ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["\[Alpha]", "1"], "=", FractionBox["1", SqrtBox[ RowBox[{"n", "+", "1"}]]]}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["\[Beta]", "1"], "=", SqrtBox[ FractionBox["n", RowBox[{"n", "+", "1"}]]]}], TraditionalForm]]], ". The third vector we take to be of the form ", Cell[BoxData[ FormBox[ RowBox[{"(", GridBox[{ { SubscriptBox["\[Alpha]", "1"], SubscriptBox["\[Alpha]", "2"], SubscriptBox["\[Beta]", "2"], "0", "0", "\[TripleDot]"} }], ")"}], TraditionalForm]]], ", where ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubsuperscriptBox["\[Alpha]", "1", "2"], "+", RowBox[{ SubscriptBox["\[Alpha]", "2"], SubscriptBox["\[Beta]", "1"]}]}], "=", FractionBox["1", SqrtBox[ RowBox[{"n", "+", "1"}]]]}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["\[Beta]", "2"], "=", SqrtBox[ RowBox[{"1", "-", SubsuperscriptBox["\[Alpha]", "1", "2"], "+", SubsuperscriptBox["\[Alpha]", "2", "2"]}]]}], TraditionalForm]]], ". Next we take ", Cell[BoxData[ FormBox[ RowBox[{"(", GridBox[{ { SubscriptBox["\[Alpha]", "1"], SubscriptBox["\[Alpha]", "2"], SubscriptBox["\[Alpha]", "3"], SubscriptBox["\[Beta]", "3"], "0", "\[TripleDot]"} }], ")"}], TraditionalForm]]], " and so on. 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Then the first component of the remaining vectors can be taken equal to \ ", Cell[BoxData[ FormBox[ FractionBox["1", SqrtBox[ RowBox[{"n", "+", "1"}]]], TraditionalForm]]], ", since we can ignore the global phase for each vector. The second \ component of the second vector can be taken equal to ", Cell[BoxData[ FormBox[ SqrtBox[ FractionBox["n", RowBox[{"n", "+", "1"}]]], TraditionalForm]]], ", since we can multiply the ", Cell[BoxData[ FormBox["i", TraditionalForm]]], "-th component of each vector by the same phase without affecting the inner \ products. Hence we can always transform our SIC-POVM so that the second \ vector becomes ", Cell[BoxData[ FormBox[ RowBox[{"(", GridBox[{ { FractionBox["1", SqrtBox[ RowBox[{"n", "+", "1"}]]], SqrtBox[ FractionBox["n", RowBox[{"n", "+", "1"}]]], "0", "0", "\[TripleDot]"} }], ")"}], TraditionalForm]]], ". 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Zero in the 3rd component", "Subtitle", CellGroupingRules->{GroupTogetherGrouping, 10001.}, CellChangeTimes->{{3.4207126208015327`*^9, 3.420712636333867*^9}, { 3.4207126542796717`*^9, 3.4207126544899745`*^9}, 3.42071269253468*^9, { 3.42071273509588*^9, 3.4207127351359377`*^9}, {3.420712840046792*^9, 3.420712842099744*^9}, {3.4207141710807247`*^9, 3.420714171441243*^9}, { 3.4207167936217537`*^9, 3.420716797667571*^9}, {3.4212319262839055`*^9, 3.4212319295385857`*^9}, {3.4212333975194387`*^9, 3.421233397559496*^9}, { 3.4212334397301345`*^9, 3.421233447250949*^9}, {3.421235999921509*^9, 3.421236013020344*^9}, {3.4212360541494846`*^9, 3.4212360542296*^9}, { 3.4212361606626434`*^9, 3.4212361613836803`*^9}, 3.421238271788293*^9, 3.4212420818869457`*^9, {3.421242095546587*^9, 3.4212421210132065`*^9}, { 3.4212445568958335`*^9, 3.4212445577170143`*^9}, 3.421245247368685*^9, 3.4215142940097456`*^9, {3.4215143235221825`*^9, 3.421514323532197*^9}, { 3.4215145375399246`*^9, 3.4215145577189407`*^9}}], Cell["\<\ Observe that the first three vectors of the SIC-POVM described in Section \ 3.1. have the 3rd component equal to 0. One might ask if this could happen in \ higher dimensions too. 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