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Classifying Measurement Incompatibility under Classical Pre- and Post-Processing Operations

Arun Kumar Das, Saheli Mukherjee, Debashis Saha, Debarshi Das, A. S. Majumdar

TL;DR

This work develops an operational, theory-agnostic framework to classify quantum measurement incompatibility under two classical operations: coarse-graining of outcomes and disjoint-convex-mixing of inputs. It provides analytical criteria for full or k-incompatibility in various settings (two rank-one measurements, three-qubit triples, and more) and derives noise-robustness thresholds using MUBs, with prime-dimension results for mutually unbiased bases. The paper also constructs operational witnesses of incompatibility, both device-independent (Bell-type tests and CGLMP scenarios) and semi-device-independent (random access codes), enabling certification from statistics alone. Overall, the work clarifies how classical post- and pre-processing reveal hierarchical layers of incompatibility, offering practical benchmarks for experimental verification and task-oriented selection of measurements.

Abstract

Measurement incompatibility has proved to be an important resource for quantum information processing. In this work, we present an operational approach that leverages classical operations on the inputs (pre-processing) and outputs (post-processing) of measurement devices to explore different layers of incompatibility among the measurements performed by the device. We study classifications of measurement incompatibility with respect to these two types of classical operations, viz., post-processing or coarse-graining of measurement outcomes and pre-processing or convex-mixing of different measurements. We derive analytical criteria for determining when a set of projective measurements is fully incompatible with respect to coarse-graining or convex-mixing. Robustness against white noise for different layers of incompatibility for mutually unbiased bases is investigated. Furthermore, we study operational witnesses for incompatibility subject to these classical operations, using the input-output statistics of Bell-type experiments as well as experiments in the prepare-and-measure scenario.

Classifying Measurement Incompatibility under Classical Pre- and Post-Processing Operations

TL;DR

This work develops an operational, theory-agnostic framework to classify quantum measurement incompatibility under two classical operations: coarse-graining of outcomes and disjoint-convex-mixing of inputs. It provides analytical criteria for full or k-incompatibility in various settings (two rank-one measurements, three-qubit triples, and more) and derives noise-robustness thresholds using MUBs, with prime-dimension results for mutually unbiased bases. The paper also constructs operational witnesses of incompatibility, both device-independent (Bell-type tests and CGLMP scenarios) and semi-device-independent (random access codes), enabling certification from statistics alone. Overall, the work clarifies how classical post- and pre-processing reveal hierarchical layers of incompatibility, offering practical benchmarks for experimental verification and task-oriented selection of measurements.

Abstract

Measurement incompatibility has proved to be an important resource for quantum information processing. In this work, we present an operational approach that leverages classical operations on the inputs (pre-processing) and outputs (post-processing) of measurement devices to explore different layers of incompatibility among the measurements performed by the device. We study classifications of measurement incompatibility with respect to these two types of classical operations, viz., post-processing or coarse-graining of measurement outcomes and pre-processing or convex-mixing of different measurements. We derive analytical criteria for determining when a set of projective measurements is fully incompatible with respect to coarse-graining or convex-mixing. Robustness against white noise for different layers of incompatibility for mutually unbiased bases is investigated. Furthermore, we study operational witnesses for incompatibility subject to these classical operations, using the input-output statistics of Bell-type experiments as well as experiments in the prepare-and-measure scenario.
Paper Structure (17 sections, 8 theorems, 63 equations, 1 figure, 5 tables)

This paper contains 17 sections, 8 theorems, 63 equations, 1 figure, 5 tables.

Table of Contents

  1. introduction
  2. Classifying incompatibility under classical operations of the inputs and the outputs
  3. Classical operations on outputs
  4. Classical operation on inputs
  5. Classical operations on both inputs and outputs
  6. Physical motivations for considering the two specific classical operations
  7. Classifying measurement incompatibility under coarse-graining and disjoint-convex-mixing
  8. Coarse-graining of outcomes
  9. Disjoint-convex-mixing of measurements
  10. Robustness under noise for different levels of incompatibility
  11. Robustness of incompatibility under coarse-graining of outcomes
  12. Robustness of incompatibility under disjoint-convex-mixing of measurements
  13. Robustness of incompatibility under disjoint-convex-mixing of measurements followed by coarse-graining of outcomes
  14. Operational witnesses of incompatibility
  15. Coarse-graining of outcomes
  16. ...and 2 more sections

Key Result

Theorem 1

The following condition is necessary but not sufficient for two rank-one projective measurements of dimension $\ge 4$, defined by $\{|\psi_i\rangle\}$ and $\{|\phi_j\rangle\}$, to be fully incompatible w.r.t. coarse-graining: However, the above condition is necessary and sufficient for two 3-dimensional rank-one projective measurements.

Figures (1)

  • Figure 1: Classical operations on the outputs (left) and classical operations on the inputs (right) to realize a new set of measurements are depicted.

Theorems & Definitions (25)

  • Definition 1: Fully incompatible measurements w.r.t. coarse-graining
  • Definition 2: $k$-incompatible measurements w.r.t. coarse-graining
  • proof
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Corollary 1
  • proof
  • ...and 15 more