『Quantum Computing: A Gentle Introduction』の演習問題を解く 3.10

Quantum Computing: A Gentle Introduction (Scientific and Engineering Computation) (English Edition)

作者: Eleanor G. Rieffel,Wolfgang H. Polak
出版社/メーカー: The MIT Press
発売日: 2011/03/04
メディア: Kindle版
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Exercise 3.10. Show that for any orthonormal basis ${ B = \{ |\beta_1\rangle,\,|\beta_2\rangle,\, ... \,|\beta_n\rangle \} }$ and vectors ${ |v\rangle = a_1|\beta_1\rangle + a_2|\beta_2\rangle + \cdots + a_n|\beta_n\rangle }$ and ${ |w\rangle = c_1|\beta_1\rangle + c_2|\beta_2\rangle + \cdots + c_n|\beta_n\rangle }$

a. the inner product of ${ |v\rangle }$ and ${ |w\rangle }$ is ${ \bar{c}_1a_1 + \bar{c}_2a_2 + \cdots + \bar{c}_na_n }$ , and

b. the length squared of ${ |v\rangle }$ is ${ \left||v\rangle\right|^2 = \langle v|v \rangle = |a_1|^2 + |a_2|^2 + \cdots + |a_n|^2 }$ .

Write all steps in Dirac's bra/ket notation.

準備

${ |v\rangle,\,|w\rangle }$ はそれぞれ

　　 ${ \displaystyle\begin{align*} |v\rangle &= \sum_{i=1}^n a_i|\beta_i\rangle, & |w\rangle &= \sum_{i=1}^n c_i|\beta_i\rangle \end{align*}}$

また、 ${ \{|\beta_i\rangle\} }$ は正規直交基底なので

　　 ${ \displaystyle\begin{align*} \langle \beta_i | \beta_j \rangle = \delta_{ij} \end{align*}}$

を満たします。

a.

　　 ${ \displaystyle\begin{align*} \langle w | v\rangle &= \left(\sum_{i=1}^n c_i|\beta_i\rangle\right)^\dagger\sum_{j=1}^n a_j|\beta_j\rangle \\ &= \sum_{i,j=1}^n \bar{c}_ia_j \langle \beta_i | \beta_j \rangle \\ &= \sum_{i,j=1}^n \bar{c}_ia_j \delta_{ij} \\ &= \sum_{i=1}^n \bar{c}_ia_i \end{align*}}$