Strictly-complete measurements for bounded-rank quantum-state tomography

Abstract

We consider the problem of quantum-state tomography under the assumption that the state is pure, and more generally that its rank is bounded by a given value. In this scenario, new notions of informationally complete POVMs emerge, which allow for high-fidelity state estimation with fewer measurement outcomes than are required for an arbitrary rank state. We study this in the context of matrix completion, where the POVM outcomes determine only a few of the density matrix elements. We give an analytic solution that fully characterizes informational completeness and elucidates the important role that the positive-semidefinite property of density matrices plays in tomography. We show how positivity can impose a stricter notion of information completeness and allow us to use convex optimization programs to robustly estimate bounded-rank density matrices in the presence of statistical noise.

Quantum State Tomography via Reduced Density Matrices

“…On the other hand, the exponential resources QST requires make scaling it to large systems infeasible in practice. In the past decade, tremendous effort has been devoted to boosting the efficiency of QST [6][7][8][9][10][11][12]. QST via reduced density matrices (RDMs) [13][14][15][16][17][18] is one especially promising approach, as it is significantly less resource-intensive and many experimental setups are able to perform local measurements conveniently and accurately.…”

Experimental Study of Optimal Measurements for Quantum State Tomography

“…The most efficient fully IC POVM is the SIC POVM, which has the minimal number of POVM outcomes d 2 [12]. Other examples of fully IC POVMs include the d þ 1 mutually unbiased bases (MUB) [13], and the 2d − 1 generalized Gell-Mann bases (GMB) [17], with d 2 þ d and 2d 2 − d outcomes, respectively. Additional notions of IC become relevant for QST on restricted subsets of states.…”

“…We refer to this POVM as pure state informationally complete (PSI), and it can be shown to be R1S-IC by the method proposed in Ref. [17]. The PSI POVM has the best known scaling with d of any R1S-IC POVM that consists of rank-1 elements.…”

“…[19] and proven to be R1S-IC in Ref. [17], and the second of which we refer to as 5 polynomial bases (5PB) [20]. We further implement two R1-IC POVMs, which we refer to as 4 Gell-Mann bases (4GMB) [19] and 4 polynomial bases (4PB) [21].…”

“…Notably, quantum information processing tends to employ pure states, and diagnostic tools such as randomized benchmarking [34,35] can verify that an experiment operates close to this regime. We thus test several efficient strategies for QST of rank-1 density operators: rank-1 IC (R1-IC) POVMs, which uniquely identify a pure state only from other pure states [15][16][17], and rank-1 strictly IC (R1S-IC) POVMs, which uniquely identify a pure state from all physical density matrices of any rank [16][17][18]. Strictly IC POVMs are particularly useful because they allow state estimation via convex optimization [17].…”