Fig. 1: The two steps of the protocol.

a MMSS preparation: Bringing up a maximally mixed qubit to an MMSS \({\rho }_{K}^{sym}\) of K qubits, and measuring its total angular momentum using a procedure from¹ implements a biased random walk that probabilistically generates an MMSS of increasing size. b Relative Localisation: Begin with two MMSS states of M + N qubits. Each can be interpreted as a random mixture of many copies of an unknown pure state. By pairing up and measuring M qubits from each using the s/t measurement (red ovals) we build up information about the relative angle the Bloch vectors of these unknown pure states make against each other. This allows us to estimate the unknown pure states to high accuracy, up to an unimportant global unitary transformation. The remaining qubits can hence be used, together with s/t measurements, to implement cluster state quantum computation, as detailed in ref. ⁸.