An application of quantum Solovay randomness
Martin-Löf randomness and Solovay randomness are equivalent notions of randomness for infinite sequences of bits: elements of 2ω. These notions can be extended to sequences of quantum bits (qubits). We show that these notions remain equivalent in the quantum setting. A sequence of n qubits is modelled by a density matrix on C2n. Nies and Scholz modelled an infinite sequence of qubits by a state. A state is a sequence, ρ=(ρn)n∈ω where for each n, ρn is a density matrix on C2n=C2⊗C2n−1 and the partial trace of ρn over C2 is ρn−1. They defined quantum Martin-Löf randomness (q-MLR) for states and asked if the set of quantum Martin-Löf random states is closed under taking finite convex sums. They also suggested that one could define Solovay randomness for states and asked if it is equivalent to q-MLR. We define a notion of quantum Solovay randomness and show that it is equivalent to q-MLR. Using this equivalence, we show that the set of quantum Martin-Löf random states is closed under taking finite convex sums.