# An application of quantum Solovay randomness

Martin-Löf randomness and Solovay randomness are equivalent notions of randomness for infinite sequences of bits: elements of $2^{\omega}$. 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 $\mathbb{C}^{2^n}$. Nies and Scholz modelled an infinite sequence of qubits by a *state*. A state is a sequence, $\rho=(\rho_n)_{n\in\omega}$ where for each $n$, $\rho_n$ is a density matrix on $\mathbb{C}^{2^n}= \mathbb{C}^{2}\otimes \mathbb{C}^{2^{n-1}}$ and the partial trace of $\rho_n$ over $\mathbb{C}^2$ is $\rho_{n-1}$. 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.