Summary: ON ALMOST RANDOMIZING CHANNELS WITH A SHORT KRAUS
Abstract. For large d, we study quantum channels on Cd obtained by selecting randomly N inde-
pendent Kraus operators according to a probability measure µ on the unitary group U(d). When µ
is the Haar measure, we show that for N d/2, such a channel is -randomizing with high proba-
bility, which means that it maps every state within distance /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 µ, we obtain a -randomizing channel provided
N d(log d)6/2. For d = 2k (k qubits), this includes Kraus operators obtained by tensoring k
random Pauli matrices. The proof uses recent results on empirical processes in Banach spaces.
The completely randomizing quantum channel on Cd
maps every state to the maximally mixed
state . This channel is used to construct perfect encryption systems (see  for formal definitions).
However it is a complex object in the following sense: any Kraus decomposition must involve at least
operators. It has been shown by Hayden, Leung, Shor and Winter  that this "ideal" channel
can be efficiently emulated by lower-complexity channels, leading to approximate encryption systems.
The key point is the existence of good approximations with much shorter Kraus decompositions. More