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The resource theory of interactive quantum instruments

Chung-Yun Hsieh, Armin Tavakoli, Huan-Yu Ku, Paul Skrzypczyk

Abstract

Quantum instruments describe both the classical output and the updated quantum state in a measurement process. To do this in a non-trivial way, instruments must have the capability to interact coherently with the state that they measure. Here, we develop a resource theory for instruments. We consider a relevant quantifier of the separation between interactive and non-interactive instruments and show that it admits three distinct operational interpretations in terms of quantum information tasks. These concern (i) the preservation of maximally entangled states after a local measurement, (ii) the average ability to preserve random states after measurement, and (iii) the ability to recover the classical information generated from measuring half of a maximally entangled state. We also introduce a natural set of allowed operations and show that the third task fully characterises the resource content of instruments. Our general framework reproduces as special cases established resource theories for channels and measurements.

The resource theory of interactive quantum instruments

Abstract

Quantum instruments describe both the classical output and the updated quantum state in a measurement process. To do this in a non-trivial way, instruments must have the capability to interact coherently with the state that they measure. Here, we develop a resource theory for instruments. We consider a relevant quantifier of the separation between interactive and non-interactive instruments and show that it admits three distinct operational interpretations in terms of quantum information tasks. These concern (i) the preservation of maximally entangled states after a local measurement, (ii) the average ability to preserve random states after measurement, and (iii) the ability to recover the classical information generated from measuring half of a maximally entangled state. We also introduce a natural set of allowed operations and show that the third task fully characterises the resource content of instruments. Our general framework reproduces as special cases established resource theories for channels and measurements.

Paper Structure

This paper contains 7 sections, 3 theorems, 98 equations, 3 figures.

Table of Contents

  1. acknowledgements
  2. End Matter
  3. Supplementary Material Section I: Semi-Definite Programming for Interactive Instrument Robustness
  4. Supplementary Material Section II: Proof of Result \ref{['Result:F_ave R']}
  5. Supplementary Material Section III: Proof of Result \ref{['Result: three operational meanings']}
  6. Supplementary Material Section IV: interactive instrument robustness is non-increasing under allowed operations
  7. Supplementary Material V: Proof of Result \ref{['Result:conversion iff conditions']}

Key Result

Lemma 1

$\tr_A(\omega_a^{AB})\preceq d_A\mathds{1}^B$$\forall\,a$ if and only if

Figures (3)

  • Figure 1: A non-interactive instrument that discards the input and prepares both the classical outcome $a$ and updated state $\hat{\tau}_a$ based on a pre-fixed distribution $r(a)$.
  • Figure 2: Schematic representations of the tasks for (a) measuring maximally entangled fraction, (b) benchmarking average quantum communication fidelity, and (c) entanglement-assisted unambiguous discrimination. Through the interactive instrument robustness, these three tasks are shown equivalent.
  • Figure 3: An allowed operation $\Pi$ acting on ${\bm{\mathcal{E}}}$. Solid/dashed lines represent quantum/classical processing.

Theorems & Definitions (3)

  • Lemma 1
  • Lemma 2
  • Theorem 3