Distance-based resource quantification for sets of quantum measurements

Lucas Tendick1, Martin Kliesch1,2, Hermann Kampermann1, and Dagmar Bruß1

1Institute for Theoretical Physics, Heinrich Heine University Düsseldorf, D-40225 Düsseldorf, Germany
2Institute for Quantum-Inspired and Quantum Optimization, Hamburg University of Technology, D-21079 Hamburg, Germany

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The advantage that quantum systems provide for certain quantum information processing tasks over their classical counterparts can be quantified within the general framework of resource theories. Certain distance functions between quantum states have successfully been used to quantify resources like entanglement and coherence. Perhaps surprisingly, such a distance-based approach has not been adopted to study resources of quantum measurements, where other geometric quantifiers are used instead. Here, we define distance functions between sets of quantum measurements and show that they naturally induce resource monotones for convex resource theories of measurements. By focusing on a distance based on the diamond norm, we establish a hierarchy of measurement resources and derive analytical bounds on the incompatibility of any set of measurements. We show that these bounds are tight for certain projective measurements based on mutually unbiased bases and identify scenarios where different measurement resources attain the same value when quantified by our resource monotone. Our results provide a general framework to compare distance-based resources for sets of measurements and allow us to obtain limitations on Bell-type experiments.

Quantum technologies allow for dramatic improvements over conventional approaches in different tasks in the fields of computation, sensing, and cryptography. Identifying what properties make quantum systems more powerful than their classical counterparts promises further future improvements. Unlike for classical systems, the state of a quantum system cannot be directly fully observed. Instead, a quantum measurement changes the state of a quantum system and only yields probabilistic outcomes. In order to achieve the desired quantum advantages, one often needs to carefully design sophisticated measurement schemes, which involve sets of different measurement settings. Therefore, it is important to characterize how useful a given set of measurement settings is for a given task. The goal of resource theories is to quantify such task-dependent usefulness in a systematic way. One of the most famous features of quantum measurements, first noticed by Heisenberg, is that certain sets of measurement settings, in stark contrast to classical physics, cannot be measured simultaneously. Initially thought of as a drawback, this incompatibility of quantum measurements lies at the heart of many quantum information processing tasks. It is for instance necessary to employ these incompatible quantum measurements to reveal that quantum systems can exhibit much stronger correlations than any classical system, which allows for quantum advantages in communication and cryptography devices. Our work provides new methods to quantify resources for sets of measurements in a unified way. This allows us not only to quantify the incompatibility of sets of quantum measurements but also to establish a hierarchy that relates this incompatibility to several other important measurement resources.

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[1] Lucas Tendick, Hermann Kampermann, and Dagmar Bruß, "Distributed Quantum Incompatibility", Physical Review Letters 131 12, 120202 (2023).

[2] Yu Luo, Fanxu Meng, and Youle Wang, "Epsilon measures of state-based quantum resource theory", Physical Review A 109 5, 052413 (2024).

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