# How to interpret zero eigenvalue(s) of the interaction matrix of a reduced density operator ?

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I am working with a circular lattice of N oscillators, where each oscillator have an interaction with its adjacent oscillators. After tracing out n from N of the oscillators I ended up with the following reduced density operator, where $A$ is complex symmetric, $A^*$ is the complex conjugate(not to be confused with Hermitian Conjugate), $B$ is Hermitian $x=\left(x_{n+1},\ldots,x_{N}\right)$ (the position coordinates of the oscillators)

$$\rho_{out}\left(x,x'\right)=\exp\left(x^TAx+x'^TA^*x'+x^TBx'\right)$$

The eigenvalues of this operator is proportional to the determinant of B. If one of the eigenvalues of B is zero, then the entanglement entropy is also zero. Is there any physical interpretation to a zero eigenvalue of B ?

Yes, it does, but I have observed that the determinant of B is very small when I have traced around half of the oscillators for a large $N$. Is this behavior expected?
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