On almost randomizing channels with a short Kraus decomposition

TL;DR

This paper investigates how randomly chosen quantum channels with a short Kraus decomposition can approximate the completely depolarizing channel, providing probabilistic bounds and improvements over prior results.

Contribution

It offers new bounds on the number of Kraus operators needed for epsilon-randomization, optimizing previous discretization methods and extending to general measures.

Findings

Channels with N > d/epsilon^2 are epsilon-randomizing under Haar measure.

For general measures, N > d (log d)^6/epsilon^2 suffices for epsilon-randomization.

Tensor products of random Pauli matrices form epsilon-randomizing channels for qubits.

Abstract

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.