Pauli Operator | \n", + "Matrix | \n", + "Eigenvalue | \n", + "Eigenvector/post-measurement state | \n", + "Measurement Result in Q# | \n", + "
---|---|---|---|---|
$X$ | \n", + "$\\begin{bmatrix} 0 & 1 \\\\ 1 & 0 \\end{bmatrix}$ | \n", + "+1 | \n", + "$\\ket{+}$ | \n", + "Zero | \n", + "
-1 | \n", + "$\\ket{-}$ | \n", + "One | \n", + "||
$Y$ | \n", + "$\\begin{bmatrix} 0 & -i \\\\ i & 0 \\end{bmatrix}$ | \n", + "+1 | \n", + "$\\ket{i}$ | \n", + "Zero | \n", + "
-1 | \n", + "$\\ket{-i}$ | \n", + "One | \n", + "||
$Z$ | \n", + "$\\begin{bmatrix} 1 & 0 \\\\ 0 & -1 \\end{bmatrix}$ | \n", + "+1 | \n", + "$\\ket{0}$ | \n", + "Zero | \n", + "
-1 | \n", + "$\\ket{1}$ | \n", + "One | \n", + "
Eigenvalue | \n", + "Measurement Result in Q# | \n", + "Eigenbasis | \n", + "Measurement Projector | \n", + "
---|---|---|---|
$+1$ | \n", + "Zero | \n", + "$\\{ \\ket{0,+}, \\ket{1,-} \\}$ | \n", + "$P_{+1} = \\ket{0,+}\\bra{0,+} + \\ket{1,-} \\bra{1,-}$ | \n", + "
$-1$ | \n", + "One | \n", + "$\\{ \\ket{0,-}, \\ket{1,+} \\}$ | \n", + "$P_{-1} = \\ket{0,-}\\bra{0,-} + \\ket{1,+} \\bra{1,+}$ | \n", + "
Measurement outcome | \n", + "Probability of outcome | \n", + "
---|---|
$00$ | \n", + "$\\left( \\frac{1}{3}\\right)^2 = \\frac{1}{9}$ | \n", + "
$01$ | \n", + "$\\left( \\frac{2}{3}\\right)^2 = \\frac{4}{9}$ | \n", + "
$10$ | \n", + "$\\left( 0\\right)^2 = 0$ | \n", + "
$11$ | \n", + "$\\left( \\frac{2}{3}\\right)^2 = \\frac{4}{9}$ | \n", + "
Measurement outcome | \n", + "Probability of outcome | \n", + "
---|---|
$++$ | \n", + "$\\left( \\frac{5}{6}\\right)^2 = \\frac{25}{36}$ | \n", + "
$+-$ | \n", + "$\\left( -\\frac{1}{6}\\right)^2 = \\frac{1}{36}$ | \n", + "
$-+$ | \n", + "$\\left( \\frac{1}{2}\\right)^2 = \\frac{1}{4}$ | \n", + "
$++$ | \n", + "$\\left( \\frac{1}{6}\\right)^2 = \\frac{1}{36}$ | \n", + "
Measurement outcome | \n", + "Probability of outcome | \n", + "Post-measurement state | \n", + "
---|---|---|
$0$ | \n", + "$\\frac{5}{6}$ | \n", + "$\\frac{1}{\\sqrt{10}} \\left( 3\\ket{00} + \\ket{01}\\right)$ | \n", + "
$1$ | \n", + "$\\frac{1}{6}$ | \n", + "$\\frac{1}{\\sqrt{2}} \\left(\\ket{10} + \\ket{11}\\right)$ | \n", + "