Tensor Inverse for Gaussian Quantum Fisher Information

Question

I'm trying to calculate the Quantum Fisher Information (QFI) of a Gaussian state. In this case, I'm following the prescription outlined in this paper https://arxiv.org/abs/1407.7352, which was published here https://link.springer.com/article/10.1140/epjd/e2014-50560-1. In general, my question is about the definition of the tensor inverse for calculating the QFI matrix and how to deal with singular matrices, which seem to come up.

In the Gaussian case, the QFI matrix can be written as $$ \mathfrak{F}_{ij}=\frac{1}{2}(\mathfrak{m}^{-1})_{\mu\nu,\alpha\beta}(\nabla_{\theta_j}\Sigma^{\alpha\beta})(\nabla_{\theta_i}\Sigma^{\mu\nu}) $$

where $\mathfrak{m}$ is defined as $$ \mathfrak{m}=\Sigma\otimes\Sigma+\frac{1}{4}\Omega\otimes\Omega, $$ $\Sigma$ is the covariance matrix of the state, the indices represent the listing of the vector of lowering and raising operators of the state (over the various modes) $a^{\mu}=\left(\hat{a}_1,\hat{a}_1^\dagger,\hat{a}_2,\hat{a}_2^\dagger\right)$, $\theta_i$ is the i-th element of the parameter list, and $\Omega$ is defined by $\Omega=i \sigma_y\oplus i\sigma_y$ where $\sigma_y$ is the Pauli-y matrix.

One part of this is to understand what the "inverse" of the tensor $\mathfrak{m}$ should be. To my knowledge, the authors seem to define it as below: $$ (\mathfrak{m}^{-1})_{ijkl}\mathfrak{m}_{klmn}=\delta_{im}\delta_{jn} $$ So that we can arrive at their equation 38 using an analog of equation 27 $$ (\mathfrak{m}^{-1})_{\gamma\epsilon,\alpha\beta}\mathfrak{m}^{\alpha\beta,\mu\nu}\mathfrak{A}_{\mu\nu}=\frac{1}{2}(\mathfrak{m}^{-1})_{\gamma\epsilon,\alpha\beta}(\partial_{\theta_i}\Sigma^{\alpha\beta})\\ \delta^\mu_\gamma\delta^\nu_\epsilon\mathfrak{A}_{\mu\nu}=\frac{1}{2}(\mathfrak{m}^{-1})_{\gamma\epsilon,\alpha\beta}(\partial_{\theta_i}\Sigma^{\alpha\beta})\\ \mathfrak{A}_{\gamma\epsilon}=\frac{1}{2}(\mathfrak{m}^{-1})_{\gamma\epsilon,\alpha\beta}(\partial_{\theta_i}\Sigma^{\alpha\beta}). $$

However, in going through the motions of calculating this inverse tensor for several of their examples (see section 4.1 for example), I keep running into issues where the method to calculate the inverse fails as the matrix is singular. Is there some other way of calculating the inverse (specifically of the singular "matrix") that I'm unaware of or is there a separate definition for taking the tensor inverse (I know that they are not usually unique)?

Below I list my methods for calculating the inverse, which typically work for a generic matrix with symbolic elements using mathematica. Neither of which yield an inverse of the tensor examples given in the text due to their singular nature.

1. I used the code found in this post https://mathematica.stackexchange.com/questions/244973/inverse-of-the-matrix-with-several-indices

2. solving for the inverse directly from the definition as a system of linear equations

numModes=4;
b = Table[
   Symbol["b" <> ToString[i] <> ToString[j] <> ToString[k] <> 
     ToString[l]], {i, 1, numModes}, {j, 1, numModes}, {k, 1, numModes}, {l, 1, numModes}];

Solve[And @@ 
   Flatten[
      Table[
         Sum[b[[i, j, l, k]]*M[[l, k, m, n]], {l, 1, numModes}, {k, 1, numModes}] 
             == KroneckerDelta[i, l]KroneckerDelta[j, p], 
       {i, 1, numModes}, {j, 1,numModes}, {m, 1, numModes}, {p, 1, numModes}]
    ], 
Flatten[b]]

This post imported from StackExchange Physics at 2024-10-19 20:42 (UTC), posted by SE-user ModeQuanta

Comments

You probably want to use a regularization procedure (add a tiny bit of a state that is full rank, do the calculation, then take a limit where that parameter vanishes) arxiv.org/abs/1801.00299v4

This post imported from StackExchange Physics at 2024-10-19 20:42 (UTC), posted by SE-user Quantum Mechanic

---

@QuantumMechanic thanks for the reference! Indeed that was incredibly helpful. It seems like the vectorization procedure is fairly specific. So, even if my matrix inverse program worked on a general matrix, it may not work in any case. Still a bit surprised that the straight sum method didn't work, but in any case the vectorization seems to give me consistent results with the paper.

This post imported from StackExchange Physics at 2024-10-19 20:42 (UTC), posted by SE-user ModeQuanta