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Detection of nonabsolute separability in quantum states and channels through moments

Bivas Mallick, Saheli Mukherjee, Nirman Ganguly, A. S. Majumdar

TL;DR

This work addresses the challenge of certifying non-absolutely separable states and non-absolutely separating channels without full state or spectral tomography. It introduces a moment-based framework that derives necessary (and in some cases sufficient) inequalities on sequences of moments, such as $s_2^2 \le s_3$ and Hankel-determinant tests of $H_m$, to detect non-absolute properties. The approach extends to non-absolutely PPT states and to covariant channels through analogous moment criteria, including $q_n$ and $\mathbf{R}_{\tilde{\Lambda}}$ constructions. The authors further demonstrate the operational relevance by proving that every non-absolutely separable state provides an advantage in quantum channel discrimination tasks and by giving explicit examples with qubit and qutrit systems. Overall, the method offers a scalable, tomography-light tool for identifying non-absolute resources in high-dimensional quantum systems with potential experimental applicability.

Abstract

In quantum information and computation, the generation of entanglement through unitary gates remains a significant and active area of research. However, there are states termed as absolutely separable, from which entanglement cannot be created through any non-local unitary action. Thus, from a resource-theoretic perspective, non-absolutely separable states are useful as they can be turned into entangled states using some appropriate unitary gates. In this work, we propose an efficient method to detect non-absolutely separable states. Our approach relies on evaluating moments that can bypass the need for full state tomography, thereby enhancing its practical applicability. We then present several examples in support of our detection scheme. We also address a closely related problem concerning states whose partial transpose remains positive under any arbitrary non-local unitary action. Furthermore, we examine the effectiveness of our moment-based approach in the detection of quantum channels that are not absolutely separating, which entails the detection of resource preserving channels. Finally, we demonstrate the operational significance of non-absolutely separable states by proving that every such state can provide an advantage in a quantum-channel discrimination task.

Detection of nonabsolute separability in quantum states and channels through moments

TL;DR

This work addresses the challenge of certifying non-absolutely separable states and non-absolutely separating channels without full state or spectral tomography. It introduces a moment-based framework that derives necessary (and in some cases sufficient) inequalities on sequences of moments, such as and Hankel-determinant tests of , to detect non-absolute properties. The approach extends to non-absolutely PPT states and to covariant channels through analogous moment criteria, including and constructions. The authors further demonstrate the operational relevance by proving that every non-absolutely separable state provides an advantage in quantum channel discrimination tasks and by giving explicit examples with qubit and qutrit systems. Overall, the method offers a scalable, tomography-light tool for identifying non-absolute resources in high-dimensional quantum systems with potential experimental applicability.

Abstract

In quantum information and computation, the generation of entanglement through unitary gates remains a significant and active area of research. However, there are states termed as absolutely separable, from which entanglement cannot be created through any non-local unitary action. Thus, from a resource-theoretic perspective, non-absolutely separable states are useful as they can be turned into entangled states using some appropriate unitary gates. In this work, we propose an efficient method to detect non-absolutely separable states. Our approach relies on evaluating moments that can bypass the need for full state tomography, thereby enhancing its practical applicability. We then present several examples in support of our detection scheme. We also address a closely related problem concerning states whose partial transpose remains positive under any arbitrary non-local unitary action. Furthermore, we examine the effectiveness of our moment-based approach in the detection of quantum channels that are not absolutely separating, which entails the detection of resource preserving channels. Finally, we demonstrate the operational significance of non-absolutely separable states by proving that every such state can provide an advantage in a quantum-channel discrimination task.

Paper Structure

This paper contains 14 sections, 8 theorems, 83 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Absolutely separable states
  4. Absolutely separating channel
  5. Moment criteria
  6. Moment-based detection of non-absolutely separable states and non-absolutely PPT states
  7. Detection of non-absolute separability
  8. Examples:
  9. Detection of non-absolutely PPT states
  10. Detection of non-absolutely separating covariant quantum channels
  11. Example
  12. Operational advantage of non-absolutely separable states
  13. Conclusions
  14. Acknowledgements

Key Result

Theorem 1

Let $\rho_{AB}$ be an absolutely separable state. Then, for all global unitary operators $\mathcal{U} \in \mathbb{U}$, the following inequality holds: where $s_2$ and $s_3$ are defined as in equation moments.

Figures (2)

  • Figure 1: A flow diagram illustrating the basic distinction between the problem of entanglement detection, and the detection of non-absolutely separable states using our moment-based technique.
  • Figure 2: Detection of the non-absolutely PPT state $\rho_4$ using Theorem \ref{['theorem3']} and Theorem \ref{['theorem4']}. The plot shows that $\det[H_{2}(\mathbf{R}_{\mathcal{T}})] < 0$ for $p > 0.65$.

Theorems & Definitions (18)

  • Definition 1
  • Theorem 1
  • Theorem 2
  • Example 1
  • Example 2
  • Example 3
  • Definition 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • ...and 8 more