On almost randomizing channels with a short Kraus decomposition

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$,suchachannelisepsilon-randomizingwithhighprobability,whichmeansthatitmapseverystatewithindistanceepsilon/d(inoperatornorm)ofthemaximallymixedstate.ThisslightlyimprovesonaresultbyHayden,Leung,ShorandWinterbyoptimizingtheirdiscretizationargument.Moreover,forgeneralmu,weobtainaepsilon-randomizingchannelprovidedN>d(\log d{)}^6/epsilo{n}^2$. 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.

We added some background on geometry of Banach spaces

References in corpus (2)

• Randomizing quantum states: Constructions and applications
• Approximate Randomization of Quantum States With Fewer Bits of Key

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