content_image/405761.jpg content_image/405765.jpg content_image/405768.jpg content_image/405769.jpg content_image/405766.jpg content_image/405760.jpg content_image/405762.jpg content_image/405764.jpg content_image/405767.jpg
form (2) in the Gaussian regime would require that — just as in the qubit case — the separability after the one- sided Gaussian channel depends only on the channel and not on the input state. Before this fact is proved, some basic concepts of the theory of Gaussian states have to be introduced. 

Consider a mode of the electro-magnetic field with an- nihilation and creation operators ˆ a, ˆ a $^{†}$. The quadratures ˆ q, ˆ p are defined by 

Every quantum state of the field corresponds uniquely to a Wigner function which can be considered as a quasi- probability distribution over the possible measurement results for the quadratures. Gaussian states are states whose Wigner function is a Gaussian distribution. In par- ticular, they are completely characterized by the expec- tation values and variances of the quadrature operators. The expectation values can be set to zero by local opera- tions and are therefore irrelevant for the entanglement of the state. All the interesting information is contained in the variances which are conveniently summarized in the covariance matrix (CM). The CM of a two-mode state ρ$_{AB}$ on modes A and B is a real symmetric 4 × 4 matrix σ$_{AB}$ with entries 

where ˆ R = (ˆ q$_{A}$, ˆ p$_{A}$, ˆ q$_{B}$, ˆ p$_{B}$ ) is the vector containing the quadratures and 〈 ˆ X 〉 is the expectation value of the op- erator ˆ X in the state ρ$_{AB}$ . Note that not every real sym- metric 4 × 4 matrix is the CM of a quantum state since the variances have to fulfill the Robertson-Schr¨ odinger inequality. If we write σ$_{AB}$ in block form 

(with real 2 × 2 matrices α, β, γ ), the Robertson- Schr¨ odinger inequality is fulfilled exactly if (see [5]) 

For future reference we also note that the separability of σ$_{AB}$ is equivalent to 

The most important example of an entangled two-mode Gaussian state is the two-mode squeezed state with CM 

## Gaussian Channels and Covariance Matrices 

<img src='content_image/405761.jpg'>

<img src='content_image/405765.jpg'>

<img src='content_image/405768.jpg'>

<img src='content_image/405769.jpg'>

<img src='content_image/405766.jpg'>

( 1$_{n}$ denotes the n × n identity matrix) 

A Gaussian quantum channel is defined as a quan- tum channel which maps Gaussian states onto Gaussian states. To calculate the evolution of the entanglement, we only need to consider the action of the channel on the CM. For future convenience, let us define the abbre- viation A [ B ] ≡ ABA T if A and B are matrices. If a Gaussian channel G acts only on mode B of a two-mode Gaussian state, the initial CM σ$_{AB}$ evolves as (see [6]) 

where f and g are two real 2 × 2 matrices which com- pletely characterize the channel G , O represents the zero matrix, and ⊕ denotes the matrix direct sum. Since the g -contribution is independent of the state σ$_{AB}$ , it can be thought of as a noise term. Note that not every pair of matrices f, g describes a Gaussian quantum channel since the output state must again respect the Robertson- Schr¨ odinger inequality as embodied by (6). Channel ma- trices f, g which describe a genuine Gaussian channel ful- fill (see [6]) 

## Separability Criterion for Gaussian Channels 

Proposition 1: If a one-sided Gaussian quantum channel G , which is characterized by channel matrices f and g , acts on mode B of a pure and entangled two-mode Gaussian state with covariance matrix σ$_{AB}$ , then the out- put state ( 1 ⊕ f )[ σ$_{AB}$ ] + ( O ⊕ g ) is separable if and only if 

The same condition (11) is also equivalent to the separa- bility of the output state if the channel acts on a single mode of a Gaussian state on more than two modes. 

This means that, for initially pure and entangled states, the separability of the output state depends only on the properties of the channel and not on the prop- erties of the initial state. Accordingly, every one-sided Gaussian channel can easily be classified as “disentan- gling” or “non-disentangling” via the separability condi- tion (11). Note that this is quite counterintuitive: For any two Gaussian states X and Y — no matter how slightly entangled X and how strongly entangled Y — there is no one-sided Gaussian channel that destroys the 

<img src='content_image/405760.jpg'>

<img src='content_image/405762.jpg'>

<img src='content_image/405764.jpg'>

<img src='content_image/405767.jpg'>

