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Quantum reversal: a general theory of coherent quantum absorbers

Mankei Tsang

TL;DR

This work generalizes the coherent quantum absorber paradigm by introducing reversal conditions that allow a second system to coherently undo any effect of the first system on a traveling field. The core method expresses the condition in concise, exact terms using CPTP maps and the Petz recovery map, with antiunitary and Kraus-operator structure providing the essential ingredients. A central result is a necessary-and-sufficient relation, $\\mathcal{G} = \\\mathcal{W}\\Theta \\\mathcal{F}^{Petz} \\\Theta \\\mathcal{W}^{-1}$, that ties the two-system dynamics through purification data; this unifies and streamlines previous absorber formalisms, removing the reliance on quantum Markov semigroups. The SQDB-$\\theta$ detailed-balance condition further simplifies the reversal criterion, and a special reversal condition with explicit Kraus relations offers practical design guidance. The paper includes concrete examples—random-unitary channels and a continuous-time Markov model—to illustrate the theory and connect to metrological applications that exploit time-reversal-based measurements.

Abstract

The fascinating concept of coherent quantum absorber - which can absorb any photon emitted by another system while maintaining entanglement with that system - has found diverse implications in open quantum system theory and quantum metrology. This work generalizes the concept by proposing the so-called reversal conditions for the two systems, in which a "reverser" coherently reverses any effect of the other system on a field. The reversal conditions are rigorously boiled down to concise formulas involving the Petz recovery map and Kraus operators, thereby generalizing as well as streamlining the existing treatments of coherent absorbers.

Quantum reversal: a general theory of coherent quantum absorbers

TL;DR

This work generalizes the coherent quantum absorber paradigm by introducing reversal conditions that allow a second system to coherently undo any effect of the first system on a traveling field. The core method expresses the condition in concise, exact terms using CPTP maps and the Petz recovery map, with antiunitary and Kraus-operator structure providing the essential ingredients. A central result is a necessary-and-sufficient relation, , that ties the two-system dynamics through purification data; this unifies and streamlines previous absorber formalisms, removing the reliance on quantum Markov semigroups. The SQDB- detailed-balance condition further simplifies the reversal criterion, and a special reversal condition with explicit Kraus relations offers practical design guidance. The paper includes concrete examples—random-unitary channels and a continuous-time Markov model—to illustrate the theory and connect to metrological applications that exploit time-reversal-based measurements.

Abstract

The fascinating concept of coherent quantum absorber - which can absorb any photon emitted by another system while maintaining entanglement with that system - has found diverse implications in open quantum system theory and quantum metrology. This work generalizes the concept by proposing the so-called reversal conditions for the two systems, in which a "reverser" coherently reverses any effect of the other system on a field. The reversal conditions are rigorously boiled down to concise formulas involving the Petz recovery map and Kraus operators, thereby generalizing as well as streamlining the existing treatments of coherent absorbers.
Paper Structure (13 sections, 18 theorems, 115 equations, 1 figure)

This paper contains 13 sections, 18 theorems, 115 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Reversal condition
  3. Model
  4. Definition of the reversal condition
  5. A precise formula for the reversal condition
  6. Detailed balance
  7. Special reversal condition
  8. Examples
  9. Random unitary channel
  10. Continuous-time Markov model
  11. Conclusion
  12. Antiunitary operators and maps
  13. Partial ket and bra operators

Key Result

Lemma 2.1

The reversal condition is satisfied if and only if the $\mathcal{F}$ and $\mathcal{G}$ maps defined by Eqs. (F) and (G) obey where $\mathcal{I}_{xy\dots}$ is the identity map on $\mathcal{O}(\mathcal{H}_x\otimes\mathcal{H}_y\otimes\dots)$.

Figures (1)

  • Figure 1: A quantum circuit illustrating the interactions between system A, system B, and the field mode.

Theorems & Definitions (41)

  • Remark 2.1
  • Definition 2.1: Reversal condition
  • Lemma 2.1
  • proof
  • Definition 2.2: Hilbert-Schmidt inner product and adjoint
  • Definition 2.3: Connes inner product connes74 and adjoint
  • Definition 2.4: Petz recovery map petz84petzwilde
  • Lemma 2.2
  • proof
  • Lemma 2.3: Ref. duvenhage15
  • ...and 31 more