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Quantum Information and Computation

Two novel pure-state coherence measures in quantifying coherence

Manis Hazra, Debabrata Goswami

Published Oct 1, 2024• 0 citations
Publisher

Abstract

Quantification is one of the primary goals of quantum coherence resource theory. Here, we put forward two coherence measures, analytically prove their validity in the pure-state regime following the axiomatic definition of a legitimate pure-state coherence measure (PSCM), and provide their general normalized forms. We further discuss their extensions in the domain of mixed states with the assistance of the convex roof of coherence, and top coherence (quantification in terms of pure state coherence) and define their classes (coherence \emph{monotone} or \emph{measure}); in this regard, we thoroughly investigate the top coherence class. For the quantitative demonstration, we pick up different laser pulse-two-qubit interaction scenarios and compare the evolutions of these two coherence measures along with \(l_{1}\)-norm of coherence and relative entropy of coherence; this study also signifies the importance of coherence in probing an atomic or molecular system.

DOI
10.26421/qic24.11-12-4

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Quantification is one of the primary goals of quantum coherence resource theory. Here, we put forward two coherence measures, analytically prove their validity in the pure-state regime following the axiomatic definition of a legitimate pure-state coherence measure (PSCM), and provide their general normalized forms. We further discuss their extensions in the domain of mixed states with the assistance of the convex roof of coherence, and top coherence (quantification in terms of pure state coherence) and define their classes (coherence \emph{monotone} or \emph{measure}); in this regard, we thoroughly investigate the top coherence class. For the quantitative demonstration, we pick up different laser pulse-two-qubit interaction scenarios and compare the evolutions of these two coherence measures along with (l_{1})-norm of coherence and relative entropy of coherence; this study also signifies the importance of coherence in probing an atomic or molecular system.
In the resource theory of coherence, the quantification of quantum-state coherence is an important task. In this regard, the key ingredients are the various coherence monotones (or measures). There are few coherence-monotone classes that solely depend on other coherence measures defined for all the pure states; in other words, they rely on the pure state coherence measures (PSCM). Here, we set forth two such novel PSCMs, and validate each of them through the fulfillment of all four necessary conditions. In addition, we delve into the most recent (as per our knowledge) coherence-monotone class based on the innovative idea of quantifying coherence in terms of pure-state coherence, further redefine it, and, through the study of convexity under mixing, justify why this coherence monotone class cannot be treated as a coherence-measure class in general.
Quantifying quantum coherence is a key task in the resource theory of coherence. Here we establish a good coherence monotone in terms of a state conversion process, which automatically endows the coherence monotone with an operational meaning. We show that any state can be produced from some input pure states via the corresponding incoherent channels. It is found that the coherence of a given state can be well characterized by the least coherence of the input pure states, so a coherence monotone is established by only effectively quantifying the input pure states. In particular, we show that our proposed coherence monotone is the supremum of all the coherence monotones that give the same coherence for any given pure state. We also prove that our coherence monotone is continuous. Considering the convexity, we prove that our proposed coherence measure is a subset of the coherence measure based on the convex roof construction. The similarities and differences between our coherence monotone and coherence cost are studied in detail. As applications, we give a concrete expression of our coherence measure by employing the geometric coherence of a pure state. We also give a thorough analysis of the states of the qubit and finally obtain a series of analytic coherence measures. The numerical examples are also given to show the difference between our coherence monotone and that based on the convex roof construction.
Being the key resource in quantum physics, the proper quantification of coherence is of utmost importance. Amid complex-looking functionals in quantifying coherence, we set forth a simple and easy-to-evaluate approach: principal diagonal difference of coherence (CPDD), which we prove to be non-negative, self-normalized, and monotonic (under any incoherent operation). In an attempt to extend CPDD in the domain of mixed states, we propose a simple, generic expression (applicable to other pure state coherence measures as well). To validate this theory, we consider a fictitious two-qubit system (both interacting and non-interacting) and, through the laser pulse–system interaction (semi-classical approach), compare the coherence evolution of CPDD with the relative entropy of coherence (Cr⋅e) and the l1-norm of coherence (Cl1), in both pure- and mixed-state regimes. The results from both the numerical and analytical (for a two-level system) studies clearly demonstrate a better sensitivity of CPDD in the close proximity of the maximally coherent states (MCS) than Cr⋅e or Cl1. To the best of our knowledge, CPDD is the only quantifier (among all existing coherence quantifiers) that is a linear function of density-matrix elements.
Quantifying coherence is an essential endeavor for both quantum mechanical foundations and quantum technologies. We present a bona fide measure of quantum coherence by utilizing the Tsallis relative operator (α,β)-entropy. We first prove that the proposed coherence measure fulfills all the criteria of a well defined coherence measure, including the strong monotonicity in the resource theories of quantum coherence. We then study the ordering of the Tsallis relative operator (α,β)-entropy of coherence, Tsallis relative α-entropies of coherence, Rényi α-entropy of coherence and l1 norm of coherence for both pure and mixed qubit states. This provides a new method for defining new coherence measure and entanglement measure, and also provides a new idea for further study of quantum coherence.
Abstract Since a rigorous framework for quantifying quantum coherence was established by Baumgratz et al (2014 Phys. Rev. Lett. 113 140401), many coherence measures have been found. For a given coherence measure C , extending the values of C on pure states to mixed states by the convex roof construction, we will get a valid coherence measure C ¯ , we call C ¯ the corresponding convex roof quantifier of C . Whether C = C ¯ for a given coherence measure is an important question. In this work, we show that for the widely used coherence measure, l 1 norm of coherence C l 1 , it holds that C l 1 ≠ C l 1 ¯ .
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.