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:

the associated logarithmic negativity is:

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

In my case:

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.

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 2 bosons to describe it after Holstein-Primakoff. We can assume that the states are gaussian, and in the literature that I found for a covariance matrix as shown here:

the associated logarithmic negativity is:

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

In my case:

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.