在阅读该页内容之前,我们向量子计算的开创者费曼和Deutsch致敬,同时向三位量子信息学的奠基人Charles H. Bennett, David Deutsch, Peter Shor表示敬意.

问题:

Show that the identity operator on a vector space $V$ has a matrix representation which is one along the diagonal and zero everywhere else, if the matrix representation is taken with respect to the same input and output bases. This matrix is known as the identity matrix.

说明空间$V$上的单位算子在输入基与输出基相同时,有一个对角线元素为1,其余元素为0的矩阵表示,即单位矩阵.

解答

算子表达式为$\left.A | v_{i}\right\rangle=\left|v_{i}\right\rangle$.

展开为

$\begin{array}{c}
A\left|v_{1}\right\rangle=1\left|v_{1}\right\rangle+0\left|v_{2}\right\rangle+\cdots+0\left|v_{n}\right\rangle \\
A\left|v_{2}\right\rangle=0\left|v_{1}\right\rangle+1\left|v_{2}\right\rangle+\cdots+0\left|v_{n}\right\rangle \\
\vdots \\
A\left|v_{n}\right\rangle=0|v_{1}\rangle+0\left|v_{2}\right\rangle+\cdots+1\left|v_{n}\right\rangle
\end{array}$.

故矩阵表示为

$\left(\begin{array}{cccc}
1 & 0 & \cdots & 0 \\
0 & 1 & \cdots & 0 \\
\vdots & \vdots & & \vdots \\
0 & 0 & \cdots & 1
\end{array}\right)$.

参考

[1]www.qtumist.com

参与者

作者:HKL, W65

贡献者:Dingyan, Wjw,Wxw

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