Realizing the Optimal Tomography through a Sequence of Collective Weak Measurements

In their seminal paper, Massar and Popescu showed that in tomography, given N-copies of a pure qubit, the optimal fidelity one can achieve (on average) is (N+1)/(N+2) [1]. Without adaptive measurement, reaching this bound requires a collective measurement on the entire ensemble, and this can be achieved through a POVM whose measurement outcomes are spin coherent states of the collective spin, J=N/2. In this work, we prove that one can realize this POVM through a sequence of weak measurements of the collective spin along random directions on the sphere. We give numerical evidence that supports this result, and show that we saturate the optimal fidelity for quantum state tomography averaged over all unknown states. We describe the connection between this protocol and tomography via continuous weak measurement in the presence of time-dependent control [2].

[1] S. Massar and S. Popescu, Phys. Rev. Lett. 74, 1259 (1995).
[2] A. Silberfarb, P. S. Jessen, and I. H. Deutsch Phys. Rev. Lett. 95, 030402 (2005); C. A. Riofrio, P. S. Jessen, and I. H. Deutsch, J.Phys. B: At. Mol. Opt. Phys. 44, 154007 (2011).