Semidefinite Programming Detects Non-Projective Measurements Beyond Qubit Systems

The quest to understand the limits of measurement in quantum mechanics drives ongoing research into positive operator-valued measures, or POVMs, which represent the most general type of quantum measurement. Gabriele Cobucci from Lund University, Raphael Brinster from Heinrich Heine University Düsseldorf, and Shishir Khandelwal, along with their colleagues, investigate how well standard, projective measurements, combined with classical data processing, can replicate the effects of these more complex POVMs. Their work challenges a long-held assumption that the most strongly non-projective measurements are always symmetric informationally complete, or SIC, particularly as quantum systems grow beyond the simple qubit. By developing a new method to identify genuinely non-projective measurements and determine how accurately they can be simulated, the team provides crucial insights into the fundamental nature of quantum information and proposes a conjecture regarding the most non-projective measurements in larger quantum systems, such as qutrits and ququarts.

This investigation addresses a practical challenge, as complex measurements are significantly more difficult and expensive to implement in real-world quantum technologies. The core of their methodology involves a systematic evaluation of “positive operator-valued measures”, or POVMs, which represent the most general type of quantum measurement. Rather than directly comparing POVMs, the team focused on determining the threshold of “noise” that a POVM could withstand before it became indistinguishable from a projective measurement.

This establishes a benchmark for how genuinely “non-projective” a measurement is; a measurement requiring less noise to become projective is considered more strongly non-projective. A key innovation lies in the development of a mathematical criterion formulated as a “semidefinite program”, a powerful tool for identifying genuinely non-projective measurements and determining their simulability thresholds. This computational approach enabled them to explore a wider range of possibilities beyond known measurements. Interestingly, the team discovered that commonly studied “symmetric informationally complete” (SIC) POVMs, previously thought to be the most non-projective measurements, do not consistently hold this title in higher dimensions. Their results indicate that some SIC-POVMs are easier to simulate than others, and in four-dimensional systems, none of them represent the most non-projective measurement. This led them to identify a new type of measurement, resembling SIC-POVMs, that exhibits greater resistance to projective simulation, suggesting that our understanding of non-projective measurements is incomplete.

POVM Simulability and Projective Measurement Limits

This research investigates the simulability of quantum measurements (POVMs) and the limits of how well they can be approximated by simpler, projective measurements. Researchers used metrics like “visibility” to quantify how well a POVM can be simulated by a projective measurement, and considered different types of noise, including depolarising noise and a more challenging “worst-case” noise scenario. The research demonstrates that the worst-case noise model provides a more stringent test of simulability than the depolarising noise model, meaning a POVM that can withstand worst-case noise is likely to be more robust. The researchers developed a numerical search algorithm to find the most non-projective POVMs in a given dimension, helping to identify measurements that are particularly challenging to simulate. This work provides a deeper understanding of the limitations of quantum measurement and the trade-offs between accuracy and simplicity, with implications for quantum information processing and the development of quantum technologies. The results can guide the experimental implementation of quantum measurements and help benchmark the performance of quantum devices.

SIC Measurements Not Always Most Robust

The research demonstrates that symmetric informationally complete (SIC) measurements, previously considered fundamentally non-projective, are not necessarily the measurements most resistant to noise when simulating quantum measurements. Using a novel semidefinite programming approach, the team quantified the robustness of various measurements and found that, beyond simple two-level systems (qubits), SIC measurements do not consistently exhibit the strongest non-projective properties. These findings challenge the long-held assumption that SIC measurements represent the most extreme form of non-projectivity. The researchers observed an unexpected trend: the robustness of SIC measurements does not always decrease as the complexity of the quantum system increases.

Furthermore, the study explored different models of noise, including both depolarising noise and a “worst-case” noise scenario, and consistently found that flagged Hesse SIC measurements outperformed standard SIC measurements in higher dimensions. The authors acknowledge that determining whether this represents a genuine decaying non-projective property of SIC measurements requires further investigation. Future work could focus on extending this analysis to even higher-dimensional systems and exploring the implications of these findings for quantum information processing and the foundations of quantum measurement.