Proof of the expression for entanglement entropy from correlation matrix

Let a quantum system be described by a correlation matrix $$C_{ij} = \langle c_i^\dagger c_j \rangle\ ,$$ which we can split in components $$A$$ and $$\bar{A}$$. I have read that we can calculate the entanglement entropy between these components by restricting the correlation matrix to the rows and columns corresponding to $$A$$, calculating its eigenvalues $$\{\nu_p\}$$, and then $$S = -\sum_p \left( \nu_p \log \nu_p + (1-\nu_p)\log(1-\nu_p) \right)\ .$$

I'm trying to find any source where this statement is proven. I would appreciate if anyone can share any source about this.

• Section 2.2 in this review should be helpful: arxiv.org/abs/0905.2562 Jun 2 at 16:19
• Bosons or fermions? Jun 3 at 2:57

It should, of course, be cautioned that the quantum state corresponding to the correlation matrix $$C$$ is a Gaussian state -- for a pure state, this simply means it's a Slater determinant, but more generally it could be a thermal density matrix of a free fermion system (I'm assuming your $$c_i$$ operators are fermionic). That is to say, not only are two-point correlations of your state described by this correlation matrix (which one could of course write down for any state), but also all higher moment correlation functions can be determined from $$C$$ using Wick's theorem.

To give a bit more direction on top of Prof. Schuch's answer, here is the general logical flow:

1. first, note that from a correlation matrix, you can find the density matrix corresponding to that correlation matrix quite simply: just ask, how do I find a free fermion Hamiltonian $$H = \sum_{ij} h_{ij} c^{\dagger}_i c_j$$ whose two-point correlations are described by $$C$$? as per Prof. Schuch's hint, it helps to work in the basis in which $$C$$ is diagonal, and think of the density matrix $$\rho \propto e^{-\beta H}$$ as a thermal density matrix. Once you find an expression for $$h_{ij}$$ in terms of $$C_{ij}$$ in the diagonal basis, you should easily be able to find the general relation.

2. Given a Gaussian state, or its associated correlation matrix (after step 1, you can now obtain one from the other), how do you obtain the reduced density matrix of a subsystem $$A$$? As a hint, note that all correlation functions within the subsystem of interest can be computed from a submatrix of $$C$$.

3. Once you have the reduced density matrix $$\rho_A$$, which will be described by some free fermion Hamiltonian $$h'_{ij}$$ (NOT the same one as in the total state), you can trivially write down all of the eigenvalues of $$\rho_A$$ simply by diagonalizing $$h'$$. With these eigenvalues in hand, it is simple to compute the entropy of $$\rho_A$$. Note that the eigenvalues of $$h'$$ will be related to eigenvalues $$\nu_p$$ of the subsystem correlation matrix in a simple way.

The general strategy for non-interacting bosons or fermions in this case is as follows.

First, note that the quantity you are interested (entropy) is basis independent. Thus, you can move into a basis where things are most convenient, which is a direct sum of 2x2 blocks. (How you get there depends on whether you have bosons or fermions.) The entries of this block can be determined in a simple way from the properties of $$C$$.

The resulting system describes a number of independent modes, each described by one of the 2x2 matrices. For each of these modes, it should be an easy exercise to compute the desired quantity (certainly the entropy for fermions).

As your quantity is additive, the total entropy is just the sum of the entropy of the 2x2 blocks.