On the simulation of quantum multimeters

Andreas Bluhm; Leevi Leppäjärvi; Ion Nechita

doi:10.22331/q-2025-01-27-1608

On the simulation of quantum multimeters

Andreas Bluhm¹, Leevi Leppäjärvi², and Ion Nechita³

¹Univ. Grenoble Alpes, CNRS, Grenoble INP, LIG, 38000 Grenoble, France
²RCQI, Institute of Physics, Slovak Academy of Sciences, Dúbravská cesta 9, 84511 Bratislava, Slovakia
³Laboratoire de Physique Théorique, Université de Toulouse, CNRS, UPS, France

Published:	2025-01-27, volume 9, page 1608
Editor:	Remigiusz Augusiak
Eprint:	arXiv:2402.18333v2
Doi:	https://doi.org/10.22331/q-2025-01-27-1608
Citation:	Quantum 9, 1608 (2025).

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Abstract

In the quest for robust and universal quantum devices, the notion of simulation plays a crucial role, both from a theoretical and from an applied perspective. In this work, we go beyond the simulation of quantum channels and quantum measurements, studying what it means to simulate a collection of measurements, which we call a multimeter. To this end, we first explicitly characterize the completely positive transformations between multimeters. However, not all of these transformations correspond to valid simulations, as otherwise we could create any resource from nothing. For example, the set of transformations includes maps that always prepare the same multimeter regardless of the input, which we call trash-and-prepare. From the perspective of an experimenter with a given multimeter as part of a complicated setup, having to discard the multimeter and using a different one instead is undesirable. We give a new definition of multimeter simulations as transformations that are triviality-preserving, i.e., when given a multimeter consisting of trivial measurements they can only produce another trivial multimeter. In the absence of a quantum ancilla, we then characterize the transformations that are triviality-preserving and the transformations that are trash-and-prepare. Finally, we use these characterizations to compare our new definition of multimeter simulation to three existing ones: classical simulations, compression of multimeters, and compatibility-preserving simulations.

Featured image: The most general multimeter transformation where the multimeter $M$ is transformed using a multi-instrument $\Lambda$ and a multimeter $B$. Quantum systems are depicted by solid lines, while classical systems are represented by dotted wires. Note the quantum ancilla wire connecting the multi-instrument $\Lambda$ and the multimeter $B$.

Popular summary

Imagine that you have built a complicated experiment, with a fixed set of measurements that you can perform, switching between different measurements by turning a knob. To get the most out of this setup, you could ask yourself what other measurements you could potentially perform without changing the setup too much, for example, by selecting measurements from your fixed setup at random and by post-processing the results of the measurement on a classical computer. In analogy to the multimeters we encounter when measuring currents and voltages, we call a set of quantum measurements that you can choose from a quantum multimeter. The question above could therefore be formulated as finding all multimeters that can be simulated with a fixed multimeter by classical means. This is of course not the only way we could want to simulate: simulation by compression means that the dimension of the input system can be changed by measuring parts of the system (i.e., performing a quantum instrument), whereas we could also allow all operations that do not create incompatible measurements from compatible ones, in a similar spirit to LOCC operations not creating entanglement out of nothing.

In this article, our aim is to unify existing notions of simulation of a multimeter by another one. To this end, we mathematically characterize which kind of maps transform a multimeter into another one. However, these maps are too general to all be considered simulations. For example, they include the possibility of taking your measurement setup, throwing it into the bin, and replacing it by another one. If you have spent years on building your experiment, that is certainly not what you would like to do when thinking about what else the setup could be used for. In the absence of a quantum auxiliary system, we manage to characterize these undesired operations, which we call trash-and-prepare.

Finally, we propose a property that any transformation between multimeters should have in order to be called a simulation. For this, we consider multimeters that are trivial: That is, these multimeters always output their results according to the same probability distribution,
regardless of which quantum state they are given as an input. In the spirit of not creating something non-trivial out of thin air, we then require that our transformations are triviality-preserving to be called a simulation, i.e., given a trivial multimeter, we can only simulate other trivial measurements. In the absence of an auxiliary system, we then give a mathematical characterization of all triviality-preserving transformations between multimeters. We conclude by comparing our new notion of simulation with the existing ones mentioned in the first paragraph.

► BibTeX data

@article{Bluhm2025simulationofquantum,
  doi = {10.22331/q-2025-01-27-1608},
  url = {https://doi.org/10.22331/q-2025-01-27-1608},
  title = {On the simulation of quantum multimeters},
  author = {Bluhm, Andreas and Lepp{\"{a}}j{\"{a}}rvi, Leevi and Nechita, Ion},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1608},
  month = jan,
  year = {2025}
}

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Cited by

[1] Tim Achenbach, Andreas Bluhm, Leevi Leppäjärvi, Ion Nechita, and Martin Plávala, "Factorization of multimeters: a unified view on nonclassical quantum phenomena", Letters in Mathematical Physics 116 3, 56 (2026).

[2] Davide Rolino, Marco Erba, Alessandro Tosini, and Paolo Perinotti, "Minimal operational theories: classical theories with quantum features", New Journal of Physics 27 2, 023004 (2025).

[3] Robert Allen and Dominic Verdon, "Supermaps between channels of any type", arXiv:2410.01389, (2024).

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