Streaming the partial-transpose moment hierarchy with order-independent quantum memory

Abstract

Partial transposition reveals mixed-state entanglement through the spectrum of ${\rho }_{AB}^{T_B}$, but this operator is not generally a physical state. Its power moments are accessible through collective permutations, whose coherent width grows with moment order. Here we introduce a streaming protocol that processes fresh copies sequentially while retaining one copy and a reusable ancilla. By realizing opposite cyclic permutations on the two subsystems, each measurement record yields unbiased estimators of all moments $p_2,\dots ,p_K$ through successive outcome products. For an $n$-qubit state, the protocol uses $2n+1$ active qubits independent of $K$ and $O(K\log K/{\epsilon }_{\mathrm{mom}}^2)$ copies, matching $\Omega (K/{\epsilon }_{\mathrm{mom}}^2)$ lower bounds up to a logarithmic factor, including a PT-specific isospectral construction. Higher-order moments strengthen thermal-state entanglement certification toward the exact PPT threshold. Simulations cover pure and mixed states, and a cloud quantum processor implements a three-qubit instance.

arXiv commentMain text: 16 pages, 5 figures, 1 table; supplementary materials: 16 pages, 2 figures, 1 table; references: 3 pages. Total: 35 pages, 7 figures, 2 tables