Student handout: Outer Product of a Vector on Itself

Quantum Fundamentals 2022
Students compute the outer product of a vector on itself to product a projection operator. Students discover that projection operators are idempotent (square to themselves) and that a complete set of outer products of an orthonormal basis is the identity (a completeness relation).
What students learn
  • Outer products yield projection operators
  • Projection operators are idempotes (they square to themselves)
  • A complete set of outer products of an orthonormal basis is the identity (a completeness relation)
Outer Product of a Vector on Itself

  1. Your group will be given a pair (or triple) of vectors below, find the matrix that is the outer product of each vector on itself (i.e., \(\left|{v_1}\right\rangle \left\langle {v_1}\right|\))? All the vectors are written in the \(S_z\) basis. \begin{align*} 1)\qquad \left|{+}\right\rangle &\doteq \begin{bmatrix} 1\\0 \end{bmatrix} &\left|{-}\right\rangle &\doteq \begin{bmatrix} 0\\1 \end{bmatrix} \\[10pt] 2)\qquad\left|{+}\right\rangle _x &\doteq \frac{1}{\sqrt{2}}\begin{bmatrix} 1\\1 \end{bmatrix} &\left|{-}\right\rangle _x &\doteq \frac{1}{\sqrt{2}}\begin{bmatrix} 1\\-1 \end{bmatrix}\\[10pt] 3)\qquad\left|{+}\right\rangle _y &\doteq \frac{1}{\sqrt{2}}\begin{bmatrix} 1\\i \end{bmatrix} &\left|{-}\right\rangle _y &\doteq \frac{1}{\sqrt{2}}\begin{bmatrix} 1\\-i \end{bmatrix}\\[10pt] 4)\qquad\left|{v_7}\right\rangle &\doteq \frac{1}{5}\begin{bmatrix} 3\\4 \end{bmatrix} &\left|{v_8}\right\rangle &\doteq \frac{1}{5}\begin{bmatrix} 4\\-3 \end{bmatrix}\\[10pt] 5)\qquad\left|{v_9}\right\rangle &\doteq \begin{bmatrix} a\\be^{i\phi} \end{bmatrix} &\left|{v_{10}}\right\rangle &\doteq \begin{bmatrix} b\\-ae^{i\phi} \end{bmatrix}\\[10pt] 6)\qquad\left|{1}\right\rangle _x &\doteq \frac{1}{\sqrt{2}}\begin{bmatrix} \frac{1}{\sqrt{2}}\\1\\\frac{1}{\sqrt{2}} \end{bmatrix} &\left|{0}\right\rangle _x &\doteq \frac{1}{\sqrt{2}}\begin{bmatrix} 1\\0\\-1 \end{bmatrix} &\left|{-1}\right\rangle _x &\doteq \frac{1}{\sqrt{2}}\begin{bmatrix} \frac{1}{\sqrt{2}}\\-1\\\frac{1}{\sqrt{2}} \end{bmatrix} \end{align*}

  2. What is the square of each of your outer products?

  3. What is the product of each pair of your outer products?

  4. For each row of vectors, add all of the outer products.

  5. What is the determinant of each of your outer products?

  6. What is the transformation caused by each of your outer products?

    Bonus: How would you answer questions (2), (3), (4) staying purely in Dirac bra-ket notation?

Keywords
Projection Operators Outer Products Matrices
Learning Outcomes