Partial separability and entanglement criteria for multiqubit quantum states

We explore the subtle relationships between partial separability and entanglement of subsystems in multiqubit quantum states and give experimentally accessible conditions that distinguish between various classes and levels of partial separability in a hierarchical order. These conditions take the form of bounds on the correlations of locally orthogonal observables. Violations of such inequalities give strong sufficient criteria for various forms of partial inseparability and multiqubit entanglement. The strength of these criteria is illustrated by showing that they are stronger than several other well-known entanglement criteria (the fidelity criterion, violation of Mermin-type separability inequalities, the Laskowski-Żukowski criterion, and the Dür-Cirac criterion) and also by showing their great noise robustness for a variety of multiqubit states, including $N$-qubit Greenberger-Horne-Zeilinger states and Dicke states. Furthermore, for $N⩾3$ they can detect bound entangled states. For all these states, the required number of measurement settings for implementation of the entanglement criteria is shown to be only $N+1$. If one chooses the familiar Pauli matrices as single-qubit observables, the inequalities take the form of bounds on the antidiagonal matrix elements of a state in terms of its diagonal matrix elements.

Figure 1

Schematic representation of the ten partial separability classes of three-qubit states.

Figure 2

The maximum value for ${⟨X_0⟩}^2+{⟨Y_0⟩}^2$ obtainable by entangled quantum states (dots), by separable quantum states (crosses), and by LHV models (squares), plotted as a function of the number of qubits $N$. Note the exponential divergence between both the maxima obtained for entangled states as well as for separable states compared to the LHV value, where the former maximum is exponentially increasing and the latter maximum is exponentially decreasing.

Figure 3

The threshold noise ratios $p_0$ for detection of full $N$-qubit entanglement when admixing white noise to the $N$-qubit GHZ state for the criterion (52) derived here (plus signs) and for the stabilizer witness of Ref. 11 (squares). The noise robustness for detecting inseparability under all splits as given in (i) in Eq. (84) is also plotted (crosses).