Measuring an infinite sequence of qubits

Tejas Bhojraj (University of Wisconsin-Madison)

Nies and Scholz (https://arxiv.org/abs/1709.08422) introduced the notion of a state to describe an infinite sequence of qubits and defined quantum-Martin-Löf randomness for states, analogously to the well known concept of Martin-Löf 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-Löf randoms. Equivalently, a state is mR if and only if measuring it in any computable basis yields a Martin-Löf random with probability one. While quantum-Martin-Löf random states are mR, the converse fails: there is a mR state which is not quantum-Martin-Löf random. In fact, something stronger is true. While this state is computable and can be easily constructed, measuring it in any computable basis yields an arithmetically random sequence with probability one.