Nies and Scholz introduced the notion of a state to describe an infinite sequence of qubits and defined quantum-Martin-Lof randomness for states, analogously to the well known concept of Martin-L\"of randomness for elements of Cantor space (the space of infinite sequences of bits). We formalize how 'measurement' of a state in a basis induces a probability measure on Cantor space. A state is 'measurement random' (mR) if the measure induced by it, under any computable basis, assigns probability one to the set of Martin-L\"of randoms. Equivalently, a state is mR if and only if measuring it in any computable basis yields a Martin-L\"of random with probability one. While quantum-Martin-L\"of random states are mR, the converse fails: there is a mR state, x which is not quantum-Martin-L\"of random. In fact, something stronger is true. While x is computable and can be easily constructed, measuring it in any computable basis yields an arithmetically random sequence with probability one. I.e., classical arithmetic randomness can be generated from a computable, non-quantum random sequence of qubits.
The outcomes of local measurements made on entangled systems can be certified to be random provided that the generated statistics violate a Bell inequality. This way of producing randomness relies only on a minimal set of assumptions because it is independent of the internal functioning of the devices generating the random outcomes. In this context it is crucial to understand both qualitatively and quantitatively how the three fundamental quantities -- entanglement, non-locality and randomness -- relate to each other. To explore these relationships, we consider the case where repeated (non projective) measurements are made on the physical systems, each measurement being made on the post-measurement state of the previous measurement. In this work, we focus on the following questions: For systems in a given entangled state, how many nonlocal correlations in a sequence can we obtain by measuring them repeatedly? And from this generated sequence of non-local correlations, how many random numbers is it possible to certify? In the standard scenario with a single measurement in the sequence, it is possible to generate non-local correlations between two distant observers only and the amount of random numbers is very limited. Here we show that we can overcome these limitations and obtain any amount of certified random numbers from an entangled pair of qubit in a pure state by making sequences of measurements on it. Moreover, the state can be arbitrarily weakly entangled. In addition, this certification is achieved by near-maximal violation of a particular Bell inequality for each measurement in the sequence. We also present numerical results giving insight on the resistance to imperfections and on the importance of the strength of the measurements in our scheme.
We investigate protocols for generating a state
t-design by using a fixed separable initial state and a diagonal-unitary
t-design in the computational basis, which is a
t-design of an ensemble of diagonal unitary matrices with random phases as their eigenvalues. We first show that a diagonal-unitary
t-design generates a
O(1/2N)-approximate state
t-design, where
N is the number of qubits. We then discuss a way of improving the degree of approximation by exploiting non-diagonal gates after applying a diagonal-unitary
t-design. We also show that it is necessary and sufficient to use
O(log2t)-qubit gates with random phases to generate a diagonal-unitary
t-design by diagonal quantum circuits, and that each multi-qubit diagonal gate can be replaced by a sequence of multi-qubit controlled-phase-type gates with discrete-valued random phases. Finally, we analyze the number of gates for implementing a diagonal-unitary
t-design by {\it non-diagonal} two- and one-qubit gates. Our results provide a concrete application of diagonal quantum circuits in quantum informational tasks.