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Quantum Field Measurements in the Fewster-Verch Framework

Jan Mandrysch, Miguel Navascués

Abstract

The Fewster-Verch (FV) framework provides a local and covariant approach for defining measurements in quantum field theory (QFT). Within this framework, a probe QFT represents the measurement device, which, after interacting with the target QFT, undergoes an arbitrary local measurement. Remarkably, the FV framework is free from Sorkin-like causal paradoxes and robust enough to enable quantum state tomography. However, two open issues remain. First, it is unclear if the FV framework allows conducting arbitrary local measurements. Second, if the probe field is interpreted as physical and the FV framework as fundamental, then one must demand the probe measurement to be itself implementable within the framework. That would involve a new probe, which should also be subject to an FV measurement, and so on. It is unknown if there exist non-trivial FV measurements for which such an ``FV-Heisenberg cut" can be moved arbitrarily far away. In this work, we advance the first problem by proving that Gaussian-modulated measurements of locally smeared fields fit within the FV framework. We solve the second problem by showing that any such measurement admits a movable FV-Heisenberg cut. As a technical byproduct, we establish that state transformations induced by finite-rank perturbations of the classical phase space underlying a linear scalar field preserve the Hadamard property.

Quantum Field Measurements in the Fewster-Verch Framework

Abstract

The Fewster-Verch (FV) framework provides a local and covariant approach for defining measurements in quantum field theory (QFT). Within this framework, a probe QFT represents the measurement device, which, after interacting with the target QFT, undergoes an arbitrary local measurement. Remarkably, the FV framework is free from Sorkin-like causal paradoxes and robust enough to enable quantum state tomography. However, two open issues remain. First, it is unclear if the FV framework allows conducting arbitrary local measurements. Second, if the probe field is interpreted as physical and the FV framework as fundamental, then one must demand the probe measurement to be itself implementable within the framework. That would involve a new probe, which should also be subject to an FV measurement, and so on. It is unknown if there exist non-trivial FV measurements for which such an ``FV-Heisenberg cut" can be moved arbitrarily far away. In this work, we advance the first problem by proving that Gaussian-modulated measurements of locally smeared fields fit within the FV framework. We solve the second problem by showing that any such measurement admits a movable FV-Heisenberg cut. As a technical byproduct, we establish that state transformations induced by finite-rank perturbations of the classical phase space underlying a linear scalar field preserve the Hadamard property.

Paper Structure

This paper contains 18 sections, 10 theorems, 99 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Technical setting for quantum field theory
  3. Quantum field measurements and instruments
  4. Gaussian-modulated measurements
  5. State tomography
  6. The FV framework and its "Heisenberg cut"
  7. Implementation of Gaussian measurements within the FV framework
  8. Interaction:
  9. Probe state:
  10. Probe measurement:
  11. Implementing the measurement scheme:
  12. Moving the "FV Heisenberg cut":
  13. Projective measurements and recovery of the "projection postulate":
  14. Movable "Heisenberg cut" in the purely *-algebraic framework:
  15. Conclusion
  16. ...and 3 more sections

Key Result

Theorem 1

Suppose a QFT $\mathcal{T}$ induced by $T$ as described in sec:alg is given, and consider its Gaussian-modulated field measurements $M^{f,\epsilon}$ as given in gaussian_POVM and associated dephased Gaussian-modulated instruments $I^{f,\epsilon,\delta}$ as given in eq:dephasedgauss_instr. Then, $M^{

Figures (2)

  • Figure 1: Schematic setup for the spacetime regions involved in the measurement scheme.
  • Figure 2: Schematic setup for the involved spacetime regions in the iterated scheme.

Theorems & Definitions (20)

  • Theorem 1
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Theorem 4
  • Remark 5
  • Definition 6
  • Lemma 7
  • proof
  • ...and 10 more