New answers tagged quantum-operation
3
This is a neat idea. However, having individual overlaps being small isn't sufficient in a quantum system. For example, imagine the set of overlaps
$$
\langle V_1|V_N\rangle=0,\qquad \langle V_1|V_n\rangle=\langle V_N|V_n\rangle=\epsilon \quad \forall n=2,\ldots,N-1, \langle V_n|V_m\rangle=0 \quad \forall n,m=2,\ldots,N-2
$$
Remember that these are supposed ...
2
Given the constraints of the problem, we have for $U$ unitary,
$$
\newcommand{\ketbra}[1]{\lvert#1\rangle\!\langle #1\rvert}
\newcommand{\ket}[1]{\lvert#1\rangle}
\newcommand{\bra}[1]{\langle#1\rvert}
\newcommand{\sqoverlap}[2]{|\langle #1|#2\rangle|^2}
U=\ketbra{\lambda}+\sum_k e^{i\varphi_k}\ketbra{\lambda_k},$$
for some orthonormal basis $\{\ket\lambda\}\...
2
You certainly need to know all the other eigenvectors and eigenvalues. It's also important to know $|\mu\rangle$ and the nature of the operator $U$. Say, if $U$ is unitary, then the eigenvectors form an orthonormal set and their eigenvalues have modulus $1$. It's easy to resolve $|\mu\rangle$ along these orthonormal vectors.
To be concrete, suppose the ...
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