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arXiv:2011.00220 · Physical Review A

Converting coherence based on positive-operator-valued measures into entanglement

Sunho Kim, Chunhe Xiong, Asutosh Kumar, Junde Wu

Published May 17, 2021• 9 citations
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Abstract

Quantum resource theories provide a diverse and powerful framework for extensively studying the phenomena in quantum physics. Quantum coherence, a quantum resource, is the basic ingredient in many quantum information tasks. It is a subject of broad and current interest in quantum information, and many new concepts have been introduced and generalized since its establishment. Here we show that the block coherence can be transformed into entanglement via a block-incoherent operation. Moreover, we find that the coherence based on positive-operator-valued measures (POVMs) associated with block coherence through the Naimark extension acts as a potential resource from the perspective of generating entanglement. Finally, we discuss avenues of creating entanglement from POVM-based coherence, present strategies that require embedding channels and auxiliary systems, give some examples, and generalize them.

DOI
10.1103/physreva.103.052418

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Quantum resource theories (QRTs) provide a diverse and powerful framework for extensively studying the phenomena of quantum physics. In particular, quantum coherence is the fundamental feature of quantum physics and is the basic requirement of quantum information tasks as a medium for defining the properties of various quantum resources. In recent years the study of coherence has become more extensive, leading to the introduction of resource theory for more generalized coherence, including block coherence and POVM-based coherence. Here, we show that the block coherence is transformed into entanglement via a block coherence operation. And we explain that the POVM-based coherence associated with block coherence through the Naimark extension has the meaning of potential resources in terms of transition to entanglement. Finally, we present examples, requiring embedding channels and auxiliary systems, of creating entanglement in POVM-based coherence and generalize it. Particularly in the transformation strategy that requires the auxiliary systems, we show that it possible to selectively convert coherence based on several given POVMs into entanglement.
Quantum coherence is a fundamental feature of quantum mechanics and an underlying requirement for most quantum information tasks. In the resource theory of coherence, incoherent states are diagonal with respect to a fixed orthonormal basis; i.e., they can be seen as arising from a von Neumann measurement. Here, we introduce and study a generalization to a resource theory of coherence defined with respect to the most general quantum measurements, i.e., to arbitrary positive-operator-valued measures (POVMs). We establish POVM-based coherence measures and POVM-incoherent operations that coincide for the case of von Neumann measurements with their counterparts in standard coherence theory. We provide a semidefinite program that allows us to characterize interconversion properties of resource states and exemplify our framework by means of the qubit trine POVM, for which we also show analytical results.
Quantum coherence, like entanglement, is a fundamental resource in quantum information. In recent years, remarkable progress has been made in formulating resource theory of coherence from a broader perspective. The notions of block-coherence and POVM-based coherence have been established. Certain challenges, however, remain to be addressed. It is difficult to define incoherent operations directly, without requiring incoherent states, which proves a major obstacle in establishing the resource theory of dynamical coherence. In this paper, we overcome this limitation by introducing an alternate definition of incoherent operations induced via coherence measures and quantify dynamical coherence based on this definition. Finally, we apply our proposed definition to quantify POVM-based dynamical coherence.
Coherence is a cornerstone of quantum theory and a prerequisite for the advantage of quantum technologies. In recent work, the notion of coherence with respect to a general quantum measurement, i.e., positive operator-valued measure (POVM), was introduced and embedded into a resource-theoretic framework that generalizes the standard resource theory of coherence. In particular, POVM-incoherent (free) states and operations were established. In this work, we explore features of this framework which arise due to the rich structure of POVMs compared to projective measurements. Moreover, we introduce a rigorous, probabilisitic framework for POVM-based coherence measures and free operations. This leads to the introduction of strongly monotonic resource measures that neatly generalize well-known standard coherence measures. Finally, we show that the relative entropy of POVM coherence is equal to the cryptographic randomness gain, providing an important operational meaning to the concept of coherence with respect to a general measurement.
Abstract We study block-coherence measures based on the resource theory of block-coherence and coherence measures based on positive-operator-valued measures (POVM). Several block-coherence measures are presented, including the block-coherence measure based on maximum relative entropy, the one-shot block-coherence cost under maximally block-incoherent operations, and the coherence measure based on coherent rank. Their relationships are obtained. Moreover, we describe the deterministic coherence dilution process by constructing block-incoherent operations. Based on the POVM coherence resource theory, we also propose two coherence measures and analyze their relationship.
The concept of block coherence encompasses the case where experimental capabilities are not so delicate to perform arbitrary refined measurements on individual atoms. We develop a framework which facilitates further investigation of this resource theory in several respects. Using this framework, we investigate the problem of state conversion by incoherent operations and show that a majorization condition is the necessary and sufficient condition for state transformation by block-incoherent operations. We also determine the form of the maximally coherent state from which all other states and all unitary gates can be constructed by incoherent operations. Thereafter, we define the concept of block-cohering and block-decohering powers of quantum channels and determine these powers for several types of channels. Finally, we explore the relation between block coherence and a previous extension of coherence, known as kkk coherence.