Explain the ineffectiveness of UTNPWs with an isolated (1−x) factor

Explain why a unit-trace non-positive witness (UTNPW) constructed with exponents m_1=0 and m_2=1 is ineffective, specifically why whenever such a UTNPW certifies the non-positivity of a Hermitian matrix, the corresponding UTNPW with m_1=1 and m_2=0 also certifies it.

Background

The paper develops UTNPWs for detecting whether a unit-trace Hermitian matrix has at least one negative eigenvalue. These witnesses are generated from the polynomial G_NN(x)=g(x)x{m_1}(1-x){m_2}\prod_i(c_i-x){n_i}, with the parity and values of m_1 and m_2 affecting the resulting trace-power inequality.

Simulation results show that UTNPWs with m_1=0 and m_2=1 are ineffective. The paper observes a stronger relationship: any certification obtained from this parameter choice is also obtained by the corresponding witness with m_1=1 and m_2=0. The author explicitly states that the reason for this behavior is not understood, leaving an unresolved explanatory problem about the structure and redundancy of these witness families.

References

Interestingly, UTNPW{s} corresponding to $m_1 = 0$ and $m_2 = 1$ are ineffective. Precisely, it means that whenever a UTNPW corresponding to $m_1 = 0$ and $m_2 = 1$ can certify the non-positivity of a Hermitian matrix, so does the UTNPW corresponding to $m_1 = 1$ and $m_2 = 0$. Unfortunately, I do not have a good explanation.

— From Certifying Rank $k$ Projectors To Non-Positivity, Entanglement, And Non-Hermitian Witnesses Through Traces Of Matrix Powers  (2609.09557 - Chau, 9 Sep 2026) in Section 4.2.2, Subsection “Non-Positive Witness” (discussion following Tables 2 and 3)