2008

A No-Go Theorem for Gaussian Quantum Error Correction

Niset, Julien, Fiurasek, Jaromir, Cerf, Nicolas J.

Understand

It is proven that Gaussian operations are of no use for protecting Gaussian states against Gaussian errors in quantum communication protocols.

Specifically, we introduce a new quantity characterizing any single-mode Gaussian channel, called entanglement degradation, and show that it cannot decrease via Gaussian encoding and decoding operations only.
The strength of this no-go theorem is illustrated with some examples of Gaussian channels.

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Note that in distil3 , the impossibility is proven using V ⁡ ( γ A B ) V(\gamma_{AB}) as the entanglement measure, where V ⁡ ( γ A B ) V(\gamma_{AB}) is defined as the largest value p ≤ 1 p\leq 1 such that γ A B ≥ p ⁡ ( γ A ⊕ γ B ) \gamma_{AB}\geq p(\gamma_{A}\oplus\gamma_{B}) for some γ A \gamma_{A} and γ B \gamma_{B} . However, it is shown in the appendix of distil3 that for a 1 × N 1\times N Gaussian state ρ A B \rho_{AB} , V ⁡ ( γ A B ) V(\gamma_{AB}) is exactly equal to the smallest symplectic eigenvalue of the covariance matrix of the partially transposed state ρ A B T A \rho_{AB}^{T_{A}} . It follows that E N ( γ A B ) E_{N}(\gamma_{AB}) is a monotonic function of V ⁡ ( γ A B ) V(\gamma_{AB}) , so that the result of distil3 is equivalent to the impossibility of increasing the logarithmic negativity of 1 × N 1\times N Gaussian states with Gaussian operations only
Cited in the paper.

Then

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Closest in time.
J. Niset, U. L. Andersen, and N. J. Cerf, Phys. Rev. Lett. 101
2008
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