Arbitrary-rank selected-sum bound

Prove that every tuple of arbitrary-rank projective measurements with k settings and n outcomes satisfies the selected-sum operator-norm bound d53cd53dd52f_{a_1,d52ed52f_1,d52ed52f_2,d52ed52f_3} \left\|\sum_{x=1}^{k}P_{j_x|x}\right\|_\infty\geq\lambda_{k,n}, where \lambda_{k,n} is the largest zero of \mu_{k,n}(t)=(1-n^{-1}\frac{d}{dt})^k t^n, extending the result proved for rank-one measurements and the established arbitrary-rank cases.

Background

The paper proves that the largest selected-sum norm is at least \lambda_{k,n} for arbitrary rank in several regimes, including pairs, dichotomic tuples, and triples. For rank-one tuples, the result follows from an average characteristic-polynomial identity, while the arbitrary-rank triple case requires a separate harmonic-reduction argument. The authors do not establish the corresponding statement for general arbitrary-rank tuples, leaving open whether the polynomial \mu_{k,n} universally controls the selected-sum norm beyond these regimes.