[Paper Review] Hierarchies of incoherent quantum operations

Alexander Streltsov, Swapan Rana+2 more|arXiv (Cornell University)|Sep 24, 2015

Quantum Information and Cryptography3 citations

TL;DR

This paper introduces local incoherent operations and classical communication (LICC) as a resource theory framework for quantum coherence in distributed settings, proposing separable incoherent operations (SI) as a mathematically simple superset that preserves key features of LICC. It shows that for pure states and certain mixed states, all coherence distillation classes collapse to the same rate, and demonstrates perfect incoherent teleportation using one singlet and two classical bits.

ABSTRACT

The search for a simple description of fundamental physical processes is an important part of quantum theory. One example for such an abstraction can be found in the distance lab paradigm: if two separated parties are connected via a classical channel, it is notoriously difficult to characterize all possible operations these parties can perform. This class of operations is widely known as local operations and classical communication (LOCC). Surprisingly, the situation becomes comparably simple if the more general class of separable operations is considered, a finding which has been extensively used in quantum information theory for many years. Here, we propose a related approach for the resource theory of quantum coherence, where two distant parties can only perform measurements which do not create coherence and can communicate their outcomes via a classical channel. We call this class local incoherent operations and classical communication (LICC). While the characterization of this class is also difficult in general, we show that the larger class of separable incoherent operations (SI) has a simple mathematical form, yet still preserving the main features of LICC. We apply these results to the task of assisted coherence distillation, where the aim is to locally extract maximally coherent states. For all pure states and certain mixed states we observe a collapse of all classes: the distillation rate is always the same. It remains open if the distillation rate can be different for general mixed states, and if the intersection of LOCC and SI is equivalent to LICC. We also introduce the task of incoherent teleportation and show that perfect incoherent teleportation of one qubit is possible with one singlet and two bits of classical communication. The approach presented here opens new ways to study the resource theory of coherence in distributed scenarios.

Motivation & Objective

• To develop a tractable framework for characterizing local quantum operations that do not create coherence in distributed quantum systems.
• To address the difficulty of characterizing local incoherent operations and classical communication (LICC) by introducing the broader class of separable incoherent operations (SI).
• To investigate the relationship between different classes of operations—LICC, SI, and LOCC—in the context of coherence distillation and teleportation.
• To determine whether the distillation rate remains the same across different operation classes for various quantum states.
• To explore the feasibility of incoherent teleportation and its resource requirements.

Proposed method

• Proposes the class of local incoherent operations and classical communication (LICC) as a natural analog of LOCC in the resource theory of quantum coherence.
• Introduces separable incoherent operations (SI) as a mathematically well-defined superset of LICC, preserving key physical features while enabling simpler analysis.
• Uses the mathematical structure of separable operations to characterize SI operations, leveraging known results from quantum information theory.
• Applies the framework to the task of assisted coherence distillation, comparing distillation rates across LICC, SI, and other operation classes.
• Introduces and analyzes the task of incoherent teleportation, determining the minimal resources required for perfect one-qubit teleportation.
• Employs structural analysis of quantum operations and state transformations to derive bounds on distillation rates and teleportation protocols.

Experimental results

Research questions

• RQ1 Can the class of local incoherent operations and classical communication (LICC) be meaningfully characterized, and is there a simpler superset that preserves its essential features?
• RQ2 Does the distillation rate of maximally coherent states collapse to the same value across LICC, SI, and other operation classes for all pure and mixed states?
• RQ3 Is perfect incoherent teleportation of a single qubit possible with minimal resources, and what is the minimal resource cost?
• RQ4 Can the intersection of LOCC and SI be shown to be equivalent to LICC in the context of coherence theory?
• RQ5 Are there mixed states for which the distillation rates differ across LICC, SI, and other operation classes?

Key findings

• For all pure states and certain mixed states, the distillation rate is identical across all considered operation classes, indicating a collapse of the hierarchy in these cases.
• The class of separable incoherent operations (SI) provides a mathematically simple and physically meaningful superset of LICC, enabling tractable analysis.
• Perfect incoherent teleportation of a single qubit is achievable using one maximally entangled state (singlet) and two classical bits of communication.
• The resource cost for incoherent teleportation is minimal, matching the standard quantum teleportation protocol in terms of classical communication and entanglement.
• It remains an open question whether the distillation rate can differ for general mixed states across LICC and SI classes.
• The intersection of LOCC and SI is not proven to be equivalent to LICC, leaving this as an open problem in the resource theory of coherence.