Approximate reversal of quantum Gaussian dynamics. (arXiv:1702.04737v2 [quant-ph] UPDATED)

Recently, there has been focus on determining the conditions under which the
data processing inequality for quantum relative entropy is satisfied with
approximate equality. The solution of the exact equality case is due to Petz,
who showed that the quantum relative entropy between two quantum states stays
the same after the action of a quantum channel if and only if there is a
\textit{reversal channel} that recovers the original states after the channel
acts. Furthermore, this reversal channel can be constructed explicitly and is
now called the \textit{Petz recovery map}. Recent developments have shown that
a variation of the Petz recovery map works well for recovery in the case of
approximate equality of the data processing inequality. Our main contribution
here is a proof that bosonic Gaussian states and channels possess a particular
closure property, namely, that the Petz recovery map associated to a bosonic
Gaussian state $\sigma$ and a bosonic Gaussian channel $\mathcal{N}$ is itself
a bosonic Gaussian channel. We furthermore give an explicit construction of the
Petz recovery map in this case, in terms of the mean vector and covariance
matrix of the state $\sigma$ and the Gaussian specification of the channel

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