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 U(d). When mu is the Haar measure, we show that for N>d/epsilon^2$, such a channel is epsilon-randomizing with high probability, which means that it maps every state within distance epsilon/d (in operator norm) of the maximally mixed state. This slightly improves on a result by Hayden, Leung, Shor and Winter by optimizing their discretization argument. Moreover, for general mu, we obtain a epsilon-randomizing channel provided N > d (\log d)^6/epsilon^2$. 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. Communications in Mathematical Physics, 2009, 288, pp.1103-1116. ⟨hal-00280769v2⟩