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

Abstract

We study the simultaneous estimation of partial-transpose moments $p_j(\ddot{I}{}_{AB})=\mathrm{Tr}[(\ddot{I}{}_{AB}^{T_B}{)}^j]$, $j=2,\dots ,K$, 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 $p_2,\dots ,p_K$ to uniform additive error $\hat{I}\mu$ with total copy complexity $O(K\log K/\hat{I}{\mu }^2)$. We also prove two converse bounds. First, any uniformly accurate simultaneous estimator requires $\hat{I}\text{c◯}(K/\hat{I}{\mu }^2)$ 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.

24 pages, 6 figures, 1 table