Generalized spin-squeezing inequalities in $N$-qubit systems: Theory and experiment

We present detailed derivations, various improvements and application to concrete experimental data of spin squeezing inequalities formulated recently by some of us [Phys. Rev. Lett. 95, 120502 (2005)]. These inequalities generalize the concept of the spin squeezing parameter and provide necessary and sufficient conditions for genuine two-, or three-qubit entanglement for symmetric states, and sufficient condition for general $N$-qubit states. We apply our method to theoretical study of Dicke states, and, in particular, to $\mathrm{W}$ states of $N$ qubits. Then, we analyze the recently experimentally generated seven- and eight-ion $\mathrm{W}$ states [Nature 438, 643 (2005)]. we also present some details concerning this experiment. Finally, we improve criteria for detection of genuine tripartite entanglement based on entanglement witnesses.

Figure 1

(Color online) The (interpolated) plots of the left-hand side of the inequality (30) corresponding to the eigenvectors $\mid \psi ⟩$ (top) and $\mid {\psi }^{\prime }⟩$ (bottom).

Figure 2

The plot of the parameter $X$ as a function of the amount of noise for noisy W states of $N=7$ qubits (solid line) and $N=8$ qubits (dashed line).

Figure 3

(a) Level scheme of ${\mathrm{Ca}}^{+}$. (b) Schematics of the two lowest levels of the harmonic oscillator describing the bus mode. (c) Joint energy level diagram of the electronic qubit levels {∣1,∣0⟩} and the phonon numbers of the ion’s motional mode used for entanglement generation $\{{\mid 0⟩}_m,{\mid 1⟩}_m\}$. Carrier transitions are marked as solid arrows, the blue sideband transition as a dashed arrow. Note that the ${\mid 0⟩}_m$ ∣0⟩-level does not couple to the blue sideband.

Figure 4

Absolute values of the reconstructed density matrix of a $\mid W_7⟩$ state as obtained from quantum state tomography. Ideally, the dark entries should all have the same height of $\frac{1}{7}$, the bright bars should vanish.