Applications of the Hillery-Zubairy entanglement criteria to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi></mml:math>-qubit systems: The Tavis-Cummings model

We consider the application of the entanglement criteria derived by Hillery and Zubairy [Phys. Rev. Lett. 96, 050503 (2006)] to the detection of entanglement in $N$-qubit systems. For $N=2$ qubits we show that, with the natural choice of operators, one of the criteria never detects entanglement; we also derive conditions for the other criterion to work and for it to have a simple relation to the negativity when it does. For general angular momenta we show the Hillery-Zubairy relations can always detect the entanglement of the (pure) states of well-defined total (J, ${J}_{z}$) if the ``test'' operators are chosen appropriately. We then show how this may be used, in particular, to develop useful criteria to detect entanglement in a system of $N$ two-level atoms interacting with a field initially in a number state (the Tavis-Cummings model).