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The intrinsic quantum coherence measure with the basis-independence

Weichen Wang, Mao-Fa Fang

Published Jan 18, 2017• 0 citations
OpenAlex

Abstract

Quantum coherence is a key resource in quantum information processing scenarios, and quantifying coherence is an important task for both quantum foundation and quantum technology. However, until now, all most of coherence measures are basis-dependent that does not accord with physical reality, since the physical properties of the physical system should not be changed with the different choice of coordinate systems. Here, we propose an \textit{intrinsic basis-independent quantum coherence measure} which satisfies all conditions for quantifying coherence. This measurement not only reveals physical essence of quantum coherence of the quantum state itself clearly, but also simplifies the measurement procedure by avoiding the optimization procedure of distance measure.

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Quantum coherence is a key resource in quantum information processing scenarios, and quantifying coherence is an important task for both quantum foundation and quantum technology. However, until now, all most of coherence measures are basis-dependent that does not accord with physical reality, since the physical properties of the physical system should not be changed with the different choice of coordinate systems. Here, we propose an \textit{intrinsic basis-independent quantum coherence measure} which satisfies all conditions for quantifying coherence. This measurement not only reveals physical essence of quantum coherence of the quantum state itself clearly, but also simplifies the measurement procedure by avoiding the optimization procedure of distance measure.
Quantum coherence is a key resource in quantum information processing scenarios, and quantifying coherence is an important task for both quantum foundation and quantum technology. However, until now, all most of coherence measures are basis-dependent that does not accord with physical reality, since the physical properties of the physical system should not be changed with the different choice of coordinate systems. Here, we propose an \textit{intrinsic basis-independent quantum coherence measure} which satisfies all conditions for quantifying coherence. This measurement not only reveals physical essence of quantum coherence of the quantum state itself clearly, but also simplifies the measurement procedure by avoiding the optimization procedure of distance measure.
In the quantitative theory of quantum coherence, the amount of coherence is defined as the distance between the given state to the closest incoherent state. The set of incoherent states is conventionally defined as any state with a diagonal density matrix. One of the objections to this formulation is that the incoherent states are intrinsically basis-dependent, which makes the amount of coherence also a basis-dependent quantity. Basis-independent measures have recently been proposed where the incoherent state is taken as the maximally mixed state. We show that this is the only possible choice of reference incoherent state, without modifying the original definition of coherence. We find a relation between the two formulations by defining a contribution to the coherence due to the basis choice. The hierarchical relationship between quantum coherence and the various quantum correlations is explored in detail. Finally, we illustrate some operational uses of the basis-independent quantum coherence in quantum information theory tasks.
We propose a novel approach to quantify quantum coherence which, contrary to the previous ones, does not rely on resource theory but rather on ontological considerations. In this framework, coherence is understood as the ability of a quantum system’s statistics to deviate from the total probability formula. After motivating the importance of the total probability formula in quantum foundations, we propose a new set of axioms that a measure of coherence should satisfy, and show that it defines a class of measures different from the main previous proposal. Finally, we prove a general result about the dependence of the [Formula: see text]-coherence norm on the basis of interest, and show that it is well approximated by the square root of the purity in most bases.
Recently, how to measure quantum coherence is a hot topic in the quantum resource theory. In 2014, Baumgratz et al. proposed a rigorous framework to quantify coherence. In this framework, a particular basis of the Hilbert space is required to define the coherence measure, and hence, the proposed coherence measure of the same quantum state does not remain the same under the basis transformations. In this paper, a measure of coherence based on the normalized modulus of the coherent vector is proposed in coherent vector representation, where the property of the coherent vector under basis transformation is considered. The measure of coherence proposed here is independent of the choice of the basis. The incoherent states and incoherent operators based on this measure can also be well defined as the classical maximal mixed states and the unitary operators respectively. And three important properties can be obtained, namely, non-negativity, convexity and invariance on the incoherent operators.
Quantum coherence is an essential ingredient in quantum information processing and plays a central role in emergent fields such as nanoscale thermodynamics and quantum biology. However, our understanding and quantitative characterization of coherence as an operational resource are still very limited. Here we show that any degree of coherence with respect to some reference basis can be converted to entanglement via incoherent operations. This finding allows us to define a novel general class of measures of coherence for a quantum system of arbitrary dimension, in terms of the maximum bipartite entanglement that can be generated via incoherent operations applied to the system and an incoherent ancilla. The resulting measures are proven to be valid coherence monotones satisfying all the requirements dictated by the resource theory of quantum coherence. We demonstrate the usefulness of our approach by proving that the fidelity-based geometric measure of coherence is a full convex coherence monotone, and deriving a closed formula for it on arbitrary single-qubit states. Our work provides a clear quantitative and operational connection between coherence and entanglement, two landmark manifestations of quantum theory and both key enablers for quantum technologies.
The coherence of an individual quantum state can be meaningfully discussed only when referring to a preferred basis. This arbitrariness can, however, be lifted when considering sets of quantum states. Here we introduce the concept of set coherence for characterizing the coherence of a set of quantum systems in a basis-independent way. We construct measures for quantifying set coherence of sets of quantum states as well as quantum measurements. These measures feature an operational meaning in terms of discrimination games and capture precisely the advantage offered by a given set over incoherent ones. Along the way, we also connect the notion of set coherence to various resource-theoretic approaches recently developed for quantum systems.