\( \DeclareMathOperator{\tr}{tr} \DeclareMathOperator{\sgn}{sgn} \DeclareMathOperator{\Id}{Id} \)Horn’s inequality states that for any two compact operators \(\sigma,\tau\) on a Hilbert space \(E\), $$\prod_{k=1}^n s_k(\sigma\tau) \le \prod_{k=1}^n s_k(\sigma)s_k(\tau)$$ where \(s_1(\tau),s_2(\tau),\dots\) are the singular values of \(\tau\) arranged in descending order. Alfred Horn’s original 1950 paper provides a short proof that relies on the following result: