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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 19607, 638]*) (*NotebookOutlinePosition[ 20294, 662]*) (* CellTagsIndexPosition[ 20250, 658]*) (*WindowFrame->Normal*) Notebook[{ Cell[BoxData[ \(Projection\ Operators\)], "Input", Evaluatable->False], Cell[BoxData[{\(According\ to\ our\ postulates\ if\ a\ measurement\ is\ made\ \ of\), "\[IndentingNewLine]", \(the\ physical\ system\ with\ a\ device, \ lets\ say\ \ \[Sigma]\_z, \ then\ the\ system\ state\ function\ immediatly\ collapses\ into\), "\ \[IndentingNewLine]", RowBox[{\(one\ of\ the\ eigenstates\ of\ \[Sigma]\_z\), ",", " ", RowBox[{ RowBox[{"either", " ", RowBox[{"(", "\[NegativeThinSpace]", GridBox[{ {"1"}, {"0"} }], "\[NegativeThinSpace]", ")"}]}], ";", " ", RowBox[{"or", " ", RowBox[{ RowBox[{"(", "\[NegativeThinSpace]", GridBox[{ {"0"}, {"1"} }], "\[NegativeThinSpace]", ")"}], " ", "."}]}]}]}]}], "Input", Evaluatable->False], Cell[BoxData[{ RowBox[{\(Our\ \ Qubit\ is\ in\ state\ | \(\(\[Psi]\)\(>\)\)\), " ", "=", " ", RowBox[{ RowBox[{ RowBox[{"c1", " ", RowBox[{"(", "\[NegativeThinSpace]", GridBox[{ {"1"}, {"0"} }], "\[NegativeThinSpace]", ")"}]}], "+", " ", RowBox[{"c2", " ", RowBox[{"(", "\[NegativeThinSpace]", GridBox[{ {"0"}, {"1"} }], "\[NegativeThinSpace]", ")"}]}]}], "=", RowBox[{"(", "\[NegativeThinSpace]", GridBox[{ {"c1"}, {"c2"} }], "\[NegativeThinSpace]", ")"}]}]}], "\[IndentingNewLine]", RowBox[{"where", "\[IndentingNewLine]"}], "\[IndentingNewLine]", \(\(\(|\)\(c1\)\( | \^2\ \)\(+\(\(|\)\(c2\)\( | \^2\)\)\)\) = 1\)}], "Input", Evaluatable->False], Cell[BoxData[ \(According\ to\ our\ postulates\ the\ probability\ to\ obtain\ \ eigenvalue\ \ + 1\ in\ a\ measurement\ of\ \ \ \[Sigma]\_z\ \ is\ \ | c1\( | \^2\), \ whereas\ | c2\( | \^2\)\ is\ the\ probability\ of\ obtaining\ eigenvalue\ - 1\)], "Input", Evaluatable->False], Cell[BoxData[{ \(According\ to\ our\ rules\[IndentingNewLine]\), "\[IndentingNewLine]", \(c1 = \(\(<\)\(+\)\) | \[Psi] > \ \ which\ can\ be\ re - written\)}], "Input", Evaluatable->False], Cell[BoxData[ \(\(\(<\)\(+\)\)\ \ | \(+\(\(>\)\(<\)\(+\)\)\) | \[Psi] > \(+\ \ \(\(<\)\(+\)\)\)\ \ | -> \(\(<\)\(-\)\) | \(\(\[Psi]\)\(>\)\)\ \ = \ \[IndentingNewLine]\(\(\(<\)\(+\)\)\ | \ \(+\(\(>\)\(<\)\(+\)\)\)\ | \(\(\ \[Psi]\)\(\ \)\(>\)\)\ = \ \(\(<\)\(+\)\) | \(M\_+\) | \[Psi]\ > \ \[IndentingNewLine]\[IndentingNewLine]\(M\_+\) \[Congruent] | \ \(+\(\(>\)\(<\)\(+\)\)\) | \)\)], "Input", Evaluatable->False], Cell[BoxData[ \(where\ we\ have\ used\ the\ important\ closure, \ or\ completeness\ assumption\ for\ our\ basis\ states, \ i . e . \)], "Input", Evaluatable->False], Cell[BoxData[ \(\(\(\(\(|\)\(+\(\(>\)\(<\)\(+\)\)\)\(|\)\(\ \)\(+\ | \)\) -> \(\(\(<\)\ \(-\)\)\(|\)\) = Identity\)\(\[IndentingNewLine]\) \)\)], "Input", Evaluatable->False], Cell[BoxData[ \(\((Can\ you\ prove\ the\ above\ \(\(relation\)\(\ \)\(?\)\))\)\)], \ "Input", Evaluatable->False], Cell[BoxData[{ \(So\ \), "\[IndentingNewLine]", \(p \((+)\)\ = \ \(\(\(<\)\(+\)\) | \ \(M\_+\)\ | \[Psi] > \ \ \ \ p \ \((-)\) = \(\(<\)\(+\)\) | \ \(M\_-\)\ | \(\(\[Psi]\)\(>\)\)\)\)}], "Input", Evaluatable->False], Cell[BoxData[ \(\(\(\(M\_-\)\(\[Congruent]\)\)\(|\)\) -> \(\(\(<\)\(-\)\)\(|\)\)\)], \ "Input", Evaluatable->False], Cell[BoxData[{ \(For\ any\ state\ | \(\(\[CurlyPhi]\)\(>\)\), \ that\ is\ normalized\ to\ unity, \ \(\(operators\ of\ \ the\ \ \ Form\ \ \ M\_\[CurlyPhi]\)\(\[Congruent]\)\) | \[CurlyPhi] > < \ \[CurlyPhi] | \ are\ called\ projection\ operators\), "\[IndentingNewLine]", \(they\ have\ the\ property\)}], "Input", Evaluatable->False], Cell[BoxData[ \(M\_\[CurlyPhi]\ \ M\_\[CurlyPhi] = \(\(M\^\[Dagger]\)\_\[CurlyPhi]\ \ M\ \_\[CurlyPhi] = \(\(M\_\[CurlyPhi]\) \(M\^\[Dagger]\)\_\[CurlyPhi]\ = \ \ \ \(M\_\[CurlyPhi] = \(M\^\[Dagger]\)\_\[CurlyPhi]\)\)\)\)], "Input", Evaluatable->False], Cell[BoxData[ \(\(\((Can\ you\ prove\ \(\(this\)\(\ \)\(?\)\))\)\(.\)\)\)], "Input", Evaluatable->False], Cell[BoxData[ \(Matrix\ \ representation; 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