On almost randomizing channels with a short Kraus decomposition

Guillaume Aubrun (1)

1 ICJ - Institut Camille Jordan

For large $d$, we study quantum channels on $\C^d$ obtained by selecting randomly $N$ independent Kraus operators according to a probability measure $\mu$ on the unitary group $\mU(d)$. When $\mu$ is the Haar measure, we show that for $N \succcurlyeq d/\e^2$, such a channel is $\e$-randomizing with high probability, which means that it maps every state within distance $\e/d$ (in operator norm) of the maximally mixed state. This slightly improves on a result by Hayden, Leung, Shor and Winter by optimizing their discretization argument. Moreover, for general $\mu$, we obtain a $\e$-randomizing channel provided $N \succcurlyeq d (\log d)^6/\e^2$. For $d=2^k$ ($k$ qubits), this includes Kraus operators obtained by tensoring $k$ random Pauli matrices. The proof uses recent results on empirical processes in Banach spaces.

Domaines: Probabilités [math.PR]; Analyse fonctionnelle [math.FA]; Physique Quantique [quant-ph]

Guillaume Aubrun. On almost randomizing channels with a short Kraus decomposition. 2008. ⟨hal-00280769v1⟩