Invariants for multiple qubits

DAVID A. MEYER, Noland Wallach

Abstract The problem of quantifying entanglement in multiparticle quantum systems can be addressed using techniques from the invariant theory of Lie groups. We briefly review this theory, and then develop these techniques for application to entanglement of multiple qubits.

Trevor A. Welsh, Peter Jarvis, R.C. King

Karol Bartkiewicz, Grzegorz Chimczak

We describe a direct method to experimentally determine local two-qubit invariants by performing interferometric measurements on multiple copies of a given two-qubit state. We use this framework to analyze two different kinds of two-qubit invariants of Makhlin and Jing et al. These invariants allow us to fully reconstruct any two-qubit state up to local unitaries. We demonstrate that measuring three invariants is sufficient to find, e.g., the optimal Bell inequality violation. These invariants can be measured with local or nonlocal measurements. We show that the nonlocal strategy that follows from Makhlin's invariants is more resource efficient than local strategy following from the invariants of Jing et al. To measure all of the Makhlin's invariants directly one needs to use both two-qubit singlets and three-qubit $W$-state projections on multiple copies of the two-qubit state. This problem is equivalent to a coordinate system handedness measurement. We demonstrate that these three-qubit measurements can be performed by utilizing Hong--Ou--Mandel interference, which gives significant speedup in comparison to the classical handedness measurement. Finally, we point to potential applications of our results in quantum secret sharing.

Dafa Li, Xiangrong Li, Hongtao Huang +1 more

Coffman et al. [Phys. Rev. A 61, 052306 (2000)] introduced the residual entanglement for three qubits. In this paper, we present the entanglement measure τ(ψ) for even n qubits; for odd n qubits, we propose the residual entanglement τ(i)(ψ) with respect to qubit i and the odd n-tangle R(ψ) by averaging the residual entanglement with respect to each qubit. In this paper, we show that these measures are LU-invariant, entanglement monotones, invariant under permutations of the qubits, and multiplicative in some cases.

Jean-Gabriel Luque, Jean‐Yves Thibon, Frédéric Toumazet