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On the simulation of quantum multimeters

Andreas Bluhm, Leevi Leppäjärvi, Ion Nechita

TL;DR

The paper formalizes the simulation of a collection of quantum measurements (multimeters) through the lens of quantum supermaps, arguing that only transformations that are triviality-preserving should count as simulations. It provides a complete ancilla-free (and classical-ancilla) characterization of such simulations, distinguishing them from trash-and-prepare maps that replace a multimeter by a fixed one. By connecting the framework to classical simulations, compression, and compatibility-preserving simulations, it shows how these existing schemes emerge as special cases within a unified theory based on Choi matrices and 2-combs. The work lays groundwork for future generalizations to quantum ancillas and potential resource-theoretic treatments of multimeter simulability, with implications for designing robust and universal quantum devices. Overall, it advances a broad, mathematically grounded understanding of how complex measurement collections can be simulated and controlled in quantum experiments.

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.

On the simulation of quantum multimeters

TL;DR

The paper formalizes the simulation of a collection of quantum measurements (multimeters) through the lens of quantum supermaps, arguing that only transformations that are triviality-preserving should count as simulations. It provides a complete ancilla-free (and classical-ancilla) characterization of such simulations, distinguishing them from trash-and-prepare maps that replace a multimeter by a fixed one. By connecting the framework to classical simulations, compression, and compatibility-preserving simulations, it shows how these existing schemes emerge as special cases within a unified theory based on Choi matrices and 2-combs. The work lays groundwork for future generalizations to quantum ancillas and potential resource-theoretic treatments of multimeter simulability, with implications for designing robust and universal quantum devices. Overall, it advances a broad, mathematically grounded understanding of how complex measurement collections can be simulated and controlled in quantum experiments.

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.
Paper Structure (25 sections, 16 theorems, 95 equations, 16 figures, 1 table)

This paper contains 25 sections, 16 theorems, 95 equations, 16 figures, 1 table.

Table of Contents

  1. Introduction
  2. Main results
  3. Fundamental quantum devices
  4. States, measurements, transformations
  5. Quantum devices as channels
  6. Multimeters as quantum channels
  7. Transformations between multimeters
  8. Quantum superchannels
  9. Channels between multimeters
  10. Previously considered simulation schemes
  11. Realizations with a classical ancilla
  12. Pre- and postprocessings
  13. Quantum preprocessings
  14. Classical preprocessings and postprocessings
  15. Classical simulation
  16. ...and 10 more sections

Key Result

Theorem 1

For any transformation $\Psi$ which maps multimeters of $g$ POVMs each with $k$ outcomes on a $d$-dimensional quantum system to multimeters of $r$ POVMs each with $l$ outcomes on an $n$-dimensional quantum system, there exist an ancillary system $\mathbb{C}^s$, completely positive maps $\Lambda_{x|y In the Schrödinger picture, this means that the simulated POVMs $\{N_{\cdot|y}\}_{y \in [r]}$ arise

Figures (16)

  • Figure 1: A multimeter $M$ is transformed using instruments $\Lambda_{\cdot|y}$ and a postprocessing $B_{\cdot|a,x,y}$. 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 $B$.
  • Figure 2: The simulation of multimeter $N$ by the multimeter $M$ admits a realization with a classical ancilla represented by $\lambda \in [s]$. Compare with the general case in Fig. \ref{['fig:multimeter-simulation-intro']}, and notice that in this case the postprocessing $\nu$ and the ancilla $\lambda$ are classical.
  • Figure 3: A multi-instrument $\Lambda$ that factorises and induces a triviality-preserving multimeter transformation $\Psi$.
  • Figure 4: Pictorial representation of quantum devices. From left to right: a quantum state (preparator), a quantum channel, a measurement, and an instrument. Diagrams are to be read from left to right. Quantum systems are depicted by solid lines, while classical systems are represented by dotted lines.
  • Figure 5: A multimeter $M$ is transformed using instruments $\Lambda_{\cdot|y}$ and a postprocessing $B_{\cdot|a,x,y}$. 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 $B$.
  • ...and 11 more figures

Theorems & Definitions (45)

  • Theorem
  • Theorem
  • Definition : Simulation of multimeters
  • Theorem
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Theorem 4.1
  • Remark 4.2
  • Example 4.3
  • ...and 35 more