Robust certification of non-projective measurements: theory and experiment

TL;DR

This work addresses the boundary between simulable and non-simulable POVMs, introducing a hierarchical semidefinite-programming framework that yields upper bounds on the critical visibility required for a POVM to be simulated by projective measurements. By exploiting dual SDPs, it derives non-simulability witnesses that can be measured experimentally, and it further strengthens robustness to state-preparation errors with an experimental-data–aware extension. The authors demonstrate the approach experimentally on a trapped-ion qudit processor for a qubit SIC-POVM and a real-space IC-POVM in $d=3$, certifying non-simulability with high confidence, and extend the methodology to ancilla-assisted scenarios. Together, these results provide practical tools for certifying genuinely high-dimensional quantum measurements and offer insights into the geometry of simulable POVMs and the role of ancillas in measurement complexity.

Abstract

Determining the conditions under which positive operator-valued measures (POVMs), the most general class of quantum measurements, outperform projective measurements remains a challenging and largely unresolved problem. Of particular interest are projectively simulable POVMs, which can be realized through probabilistic mixtures of projective measurements, and therefore offer no advantage over projective schemes. Characterizing the boundary between simulable and non-simulable POVMs is, however, a difficult task, and existing tools either fail to scale efficiently, provide limited experimental feasibility or work only for specific POVMs. Here, we introduce and demonstrate a general method to certify non-simulability of a POVM by introducing a hierarchy of semidefinite programs. It provides upper bounds on the non-simulability measure of critical visibility of arbitrary POVMs which are tight in many cases and outperform previously known criteria. We experimentally certify the non-simulability of two- and three-dimensional POVMs using a trapped-ion qudit quantum processor by constructing non-simulability witnesses and introduce a modification of our framework that makes them robust against state preparation errors. Finally, we extend our results to the setting where an additional ancilla system is available.

Figures (6)

• Figure 1: Faithful representation of the set of POVMs constructed from the fiducial vector in Eq. \ref{['eq:fid_vec_two_parameters']}, where $\vartheta$ denotes the polar and $\phi$ the azimuthal angle. Each point on the transparent unit sphere corresponds to the fiducial vector of one of the POVMs. The SIC-POVMs, constructed from Eq. \ref{['eq:sic3fid']}), and their (anti-)unitary equivalents are located on the equator. The origin corresponds to the maximally noisy measurement, such that noisy versions $\Phi_t(M)$ of POVM $M$ on the surface of the sphere lie on a straight line to the origin. The surface of the inner convex set corresponds then to the noisy POVMs at their critical visibilities, which in this picture corresponds to the distance to the origin. This distance is highlighted by the color coding. On the equator, we highlight some special choices of equivalence classes of SIC-POVMs which correspond to those having lowest (Hesse SIC, orange, corresponding to the choice of $\phi = \pi$, i.e., $\varphi = 0$ in Eq. \ref{['eq:sic3fid']}), highest (green) and intermediate (purple) critical visibility.
• Figure 2: Schematic representation of the set of projectively simulable POVMs (red region) as a convex subset of the set of all POVMs. A non-simulability witness (solid line) separates simulable POVMs from non-simulable POVMs (blue region). For the definition of $\expval{M_i}$ see Eq. \ref{['eq:witness']}.
• Figure 3: Number of shots $N$ versus the required fidelity $F$ of the probe states for the qubit SIC and real-space qutrit IC3 POVM, to obtain a violation with confidence intervals of $3\sigma$ and $5\sigma$. Thresholds correspond to fidelities at which no certification is possible anymore.
• Figure 4: Schematic view of the trap and the course level structure of the $^{40}\text{Ca}^+$ ion. A combination of RF-potential on the quadrupole blades and a DC potential on the endcaps confines the ion in the trap centre. Detection and readout are performed using a 397n m laser, computation or state manipulation using a tightly focused 729n m laser, coming in at a 67.5 $^\circ$ angle to the trap axis.
• Figure 5: Certified fidelity for the different states to be measured by the POVMs. The top plot shows the states used for certifying the 2d-POVM, the bottom one the states used for the 3d POVM certification. Each state represents the average over 50000 shots.

Theorems & Definitions (4)

• Theorem 1: Higher level of the hierarchy leads to a better bound
• proof
• Conjecture 2
• Theorem 3