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There are several definitions of a (bosonic) gaussian state (related question).

One of them is given in the paper "Gaussian quantum channels" by J. Eisert and M.M. Wolf on page 3, equation 4. Here it is:

Gaussian states are exactly those having a Gaussian characteristic function $$\chi_\rho(\xi) = e^{-\xi^T\Gamma\xi/4 + D^T\xi}.$$ Here, the $2n \times 2n$-matrix $\Gamma$ and the vector $D \in \mathbb{R}^{2n}$ are essentially the first and second moments

The definition of the characteristic function is given in the same paper - equation 3.

But I also found some slides of Dr. Alessio Serafini for ICPT Winter College on Optics. Here on page 4 is the following definition

Let us define the set of Gaussian states as all the ground and thermal states of quadratic Hamiltonians with positive definite Hamiltonian matrix $H > 0$.

Are they really equivalent? How to prove it?

glS
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Alina Kuznetsova
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1 Answers1

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I have the same question. After reading the [slides][1], my interpretation is as follows:

  1. The Wigner distribution uniquely determines a quantum state.
  2. A Gaussian state is a state which the Wigner function is Gaussian. It is determined solely by the covariance matrix and mean.
  3. The ground and thermal states of quadratic Hamiltonians are Gaussian states from (2.48) and (2.49).
  4. The ground and thermal states of quadratic Hamiltonians can be characterized by three parameters: r , S , and {ν_j} from (1.51) and (1.61).
  5. For any given mean and covariance, it is always possible to find suitable r, S, and {ν_j} such that the mean and covariance of some ground or thermal state of a quadratic Hamiltonian match the desired values from (1.51) and (1.28).

Therefore, the ground and thermal states of quadratic Hamiltonians can represent all Gaussian states.

Elderitch Lin
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