Simultaneous Estimation of Partial-Transpose Moments with Active Memory Independent of the Moment Order

We study the simultaneous estimation of partial-transpose moments p_j\(ρ_AB\)=Tr\[\(ρ_AB^T_B\)^j\], j=2,ldots,K, of an unknown bipartite n-qubit state from independent copies under an explicit active-memory constraint. We give a sequential qubit-reuse realization of the partial-transpose permutation that uses at most 2n+1 active qubits, independent of K, and estimates all moments p₂,ldots,p_K to uniform additive error ε with total copy complexity O\(Klog K/ε²\). We also prove two converse bounds. First, any uniformly accurate simultaneous estimator requires Ω\(K/ε²\) copies in the worst case. Second, the same scaling holds on an explicit isospectral two-qubit negative-partial-transpose (NPT) family whose ordinary moments are constant while the partial-transpose moments vary. These results characterize the copy complexity of the partial-transpose moment hierarchy up to a logarithmic factor and extend simultaneous nonlinear-functional estimation from ordinary state powers to partial-transpose spectral data under active quantum memory independent of the target moment order.