Consider a cavity-spin system and assume we want to quantify the correlation between them. The cavity system can be described using a bosonic mode whereas the spin system requires 4 bosons to describe it after Holstein-Primakoff. We can assume the states are gaussian, but in the literature that I found for a covariance matrix as shown here:

[![Covariance matrix][1]][1]


the associated logarithmic negativity is:

[![Logarithmic Negativity][2]][2]

You can see extra information in https://arxiv.org/abs/1301.0146.

In my case:

[![Covariance atom-cavity][3]][3]

So I cannot obtain the determinant of $C$ because it is not a square matrix. Under these conditions, how can I calculate the logarithmic negativity or any other entanglement measure of the cavity-atoms system taking into account that the system is gaussian and we can determine the mean and covariance of the system for any time. Thanks.


  [1]: https://i.sstatic.net/f8hNF.png
  [2]: https://i.sstatic.net/XFFGF.png
  [3]: https://i.sstatic.net/tdWvU.png