Optimal finite-copy readout of three-qubit entanglement with state retention

Abstract

For unknown three-qubit pure states, we characterize the optimal trade-off among GHZ-class certification, binary probability readout of the three-tangle, and conditional state retention. The vanishing ideal of the null class specifies the optimal certification subspace, while a polynomial multiplication operator gives the spectral limit for binary readout. Four copies are necessary for a nonzero certificate. A fifth copy raises the optimal success probability from ${\tau }^2/48$ to $5{\tau }^2/72$ and permits the original unknown state to be retained, where $\tau$ is the three-tangle. We obtain this gain as an instance of a general symmetry-based additional-copy theorem. At six copies the optimal certification probability is $(39+2S){\tau }^2/296$, with $S={\sum }_X\mathrm{Tr}{\rho }_X^2$, whereas the largest exact binary probability is ${\tau }^2/8$. Local Schur measurements and conditional local operations with classical communication attain these bounds. Retaining the full six-copy branch record yields a weighted estimator with lower variance than binary thinning of the same data. Explicit circuits and fixed-budget comparisons connect the success probabilities to input-copy consumption, retained quantum output, and coherent gate cost.

arXiv comment19 pages, 5 figures