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ON ALMOST RANDOMIZING CHANNELS WITH A SHORT KRAUS DECOMPOSITION
 

Summary: ON ALMOST RANDOMIZING CHANNELS WITH A SHORT KRAUS
DECOMPOSITION
GUILLAUME AUBRUN
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.
1. Introduction
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 [1] for formal definitions).
However it is a complex object in the following sense: any Kraus decomposition must involve at least
d2
operators. It has been shown by Hayden, Leung, Shor and Winter [12] 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

  

Source: Aubrun, Guillaume - Institut Camille Jordan, Université Claude Bernard Lyon-I

 

Collections: Mathematics