0.8 Outer products

Apart from the inner product \langle u|v\rangle, which is a complex number, we can also form the outer product |u\rangle\langle v|, which is a linear map (operator) on \mathcal{H} (or on \mathcal{H}^\star, depending how you look at it). This is what physicists like (and what mathematicians dislike!) about Dirac notation: a certain degree of healthy ambiguity.

  • The result of |u\rangle\langle v| acting on a ket |x\rangle is |u\rangle\langle v|x\rangle, i.e. the vector |u\rangle multiplied by the complex number \langle v|x\rangle.
  • Similarly, the result of |u\rangle\langle v| acting on a bra \langle y| is \langle y|u\rangle\langle v|, i.e. the linear functional \langle v| multiplied by the complex number \langle y|u\rangle.

The product of two maps, A=|a\rangle\langle b| followed by B=|c\rangle\langle d|, is a linear map BA, which can be written in Dirac notation as BA = |c\rangle\langle d|a\rangle\langle b| = \langle d|a\rangle|c\rangle\langle b| i.e. the inner product (complex number) \langle d|a\rangle times the outer product (linear map) |c\rangle\langle b|.

Any operator on \mathcal{H} can be expressed as a sum of outer products. Given an orthonormal basis \{|e_i\rangle\}_{i=1,\ldots,n}, any operator which maps the basis vectors |e_i\rangle to vectors |f_i\rangle can be written as \sum_{i=1}^n|f_i\rangle\langle e_i|. If the vectors \{|f_i\rangle\} also form an orthonormal basis then the operator simply “rotates” one orthonormal basis into another. These are unitary operators which preserve the inner product. In particular, if each |e_i\rangle is mapped to |e_i\rangle, then we obtain the identity operator: \sum_i|e_i\rangle\langle e_i|=\mathbf{1}. This relation holds for any orthonormal basis, and it is one of the most ubiquitous and useful formulas in quantum theory, known as completeness.16 For example, for any vector |v\rangle and for any orthonormal basis \{|e_i\rangle\}, we have \begin{aligned} |v\rangle &= \mathbf{1}|v\rangle \\&= \sum_i |e_i\rangle\langle e_i|\;|v\rangle \\&= \sum_i |e_i\rangle\;\langle e_i|v\rangle \\&= \sum_i v_i|e_i\rangle \end{aligned} where v_i=\langle e_i|v\rangle are the components of |v\rangle.

Finally, note that calculating the adjoint of an outer product boils down to just swapping the order: (|a\rangle\langle b|)^\dagger = |b\rangle\langle a|.

This whole package of stuff and properties and structure (i.e. finite dimensional Hilbert spaces with linear maps and the dagger) bundles up into an abstract framework called a dagger compact category. We will not delve into the vast world of category theory in this book, and to reach an understanding of all the ingredients that go into the one single definition of dagger compact categories would take more than a single chapter. But it’s a good idea to be aware that there are researchers in quantum information science who work entirely from this approach, known as categorical quantum mechanics.

  1. Not to be confused with “completeness” in the sense of Hilbert spaces.↩︎