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

Abstract

We study the simultaneous estimation of partial-transpose moments $p_j(ρ_{AB})=\mathrm{Tr}[(ρ_{AB}^{T_B})^j]$, $j=2,\ldots,K$, 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 $p_2,\ldots,p_K$ to uniform additive error $ε$ with total copy complexity $O(K\log K/ε^2)$. We also prove two converse bounds. First, any uniformly accurate simultaneous estimator requires $Ω(K/ε^2)$ 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.