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arXiv:2610.34784

Measure of set coherence

Jiabo Panel, Yu Guo, Yuqin Wang

quant-phSubmitted Oct 26, 2026• Announced Oct 27, 2026• 0 citations
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Abstract

Set coherence is a quantum resource for a set of states that characterizes whether a collection of quantum states contains coherence, namely whether these states fail to be simultaneously incoherent in a common basis. In this work, we show that the free operations of set coherence are common commutativity-preserving quantum channels, from which we then establish an axiomatic framework for set-coherence measure (SCM). Within this framework, we investigate the robustness based measures deduced by taking, respectively, the average, maximum, and sum of the single-state robustnesses over a common reference basis, where the first two were previously introduced in the literature and we check here whether they satisfy our SCM axioms. Consequently, we show that the averaged robustness is an SCM but it is neither unified nor complete, the maximal robustness is a unified SCM but is neither complete nor strongly superadditive, whereas the sum robustness is a complete SCM. Furthermore, we propose several classes of SCMs based on pairwise commutators. We show that the Schatten $p$-norm based SCMs are complete and strongly superadditive. We also find that the universal fourth-order Bargmann gap measure proposed in the literature is a complete and strongly superadditive quantifier of set coherence. Finally, we discuss the operational meaning of the SCM in the context of optimal approximate quantum broadcasting, and show that the amount of set coherence can closely reflect the performance of this task.

Comments
19 pages, 3 figures. Comments are welcome

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Set coherence is a quantum resource for a set of states that characterizes whether a collection of quantum states contains coherence, namely whether these states fail to be simultaneously incoherent in a common basis. In this work, we show that the free operations of set coherence are common commutativity-preserving quantum channels, from which we then establish an axiomatic framework for set-coherence measure (SCM). Within this framework, we investigate the robustness based measures deduced by taking, respectively, the average, maximum, and sum of the single-state robustnesses over a common reference basis, where the first two were previously introduced in the literature and we check here whether they satisfy our SCM axioms. Consequently, we show that the averaged robustness is an SCM but it is neither unified nor complete, the maximal robustness is a unified SCM but is neither complete nor strongly superadditive, whereas the sum robustness is a complete SCM. Furthermore, we propose several classes of SCMs based on pairwise commutators. We show that the Schatten ppp-norm based SCMs are complete and strongly superadditive. We also find that the universal fourth-order Bargmann gap measure proposed in the literature is a complete and strongly superadditive quantifier of set coherence. Finally, we discuss the operational meaning of the SCM in the context of optimal approximate quantum broadcasting, and show that the amount of set coherence can closely reflect the performance of this task.
Recent studies have shown that Bargmann invariants provide effective detectors of set imaginarity. In this paper, we investigate set imaginarity as a quantum resource in qubit systems. By exploiting the structure of Bargmann invariants, we show that the free operations for qubit set imaginarity consist precisely of common unital operations and common planarized operations. Based on this characterization, we introduce an axiomatic framework for set-imaginarity measures (SIMs). In particular, we propose two refined notions, namely unified SIMs and complete SIMs, which allow a more fine-grained quantification of set imaginarity. To make these notions concrete, we construct two qubit SIMs from the Bargmann invariants of three-state subsets. We prove that one of them is a unified SIM, while the other satisfies the stronger requirements of a complete SIM. Furthermore, we revisit the robustness of set imaginarity previously introduced in the literature. We show that, although this robustness is a valid SIM for qubit systems, it is neither a unified SIM nor a complete SIM. To overcome this limitation, we propose an improved robustness-type measure and rigorously prove that it defines a complete qubit SIM.
Abstract In this work we investigate how to quantify the coherence of quantum measurements. First, we establish a resource theoretical framework to address the coherence of measurement and show that any statistical distance can be adopted to define a coherence monotone of measurement. For instance, the relative entropy fulfills all the required properties as a proper monotone. We specifically introduce a coherence monotone of measurement in terms of off-diagonal elements of positive-operator-valued measure components. This quantification provides a lower bound on the robustness of measurement-coherence that has an operational meaning as the maximal advantage over all incoherent measurements in state discrimination tasks. Finally, we propose an experimental scheme to assess our quantification of measurement-coherence and demonstrate it by performing an experiment using a single qubit on IBM Q processor.
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.
Informationally complete (IC) measurements are a useful class of measurements, as their outcome statistics uniquely determine an unknown quantum state. Hence, they are important for certain tasks such as quantum state tomography, quantum process tomography, etc. In this work, we study the quantification of informational completeness for arbitrary quantum measurements by introducing and characterizing a faithful measure for it. We explicitly evaluate the informational completeness of qubit symmetric informationally complete (SIC) measurements and show that it is an upper bound for all qubit minimal informationally complete measurements. Furthermore, by introducing a faithful measure, we try to quantify and characterize the ability of an arbitrary quantum channel to preserve the informational completeness of any IC measurement when the channel acts on it in the Heisenberg picture. We call this measure informational completeness-preservability (IC preservability) of quantum channels. After studying its properties, we establish its relation to another quantity, namely, the absolute output coherence of a quantum channel, which quantifies the minimum amount of coherence (w.r.t. an arbitrary incoherent basis) that can always be obtained from the output of that channel. Finally, we illustrate our results with an example from light-matter interaction scenarios. Thus, in this work, not only do we try to provide a quantitative framework for studying both the informational completeness of quantum measurements and the ability of quantum channels to preserve it, but we also try to offer key insight into the conceptual relation between informational completeness and quantum coherence.
Quantum coherence is a fundamental aspect of quantum physics and plays a central role in quantum information science. This essential property of the quantum states could be fragile under the influence of the quantum operations. The extent to which quantum coherence is diminished depends both on the channel and the incoherent basis. Motivated by this, we propose a measure of nonclassicality of a quantum channel as the average quantum coherence of the state space after the channel acts on, minimized over all orthonormal basis sets of the state space. Utilizing the squared l1{l}_{1}l1​-norm of coherence for the qubit channels, the minimization can be treated analytically and the proposed measure takes a closed form of expression. If we allow the channels to act locally on a maximally entangled state, the quantum correlation is diminished making the states more classical. We show that the extent to which quantum correlation is preserved under local action of the channel cannot exceed the quantumness of the underlying channel. We further apply our measure to the quantum teleportation protocol and show that a nonzero quantumness for the underlying channel provides a necessary condition to overcome the best classical protocols.
Set coherence is a basis-independent relational form of quantum coherence: a finite family of quantum states is set incoherent exactly when all its members are diagonal in one common basis. We determine how much low-order Bargmann data are needed to decide this property. For two states, second-order data are complete for qubits but fail for qutrits, while complete third-order data are sufficient for qutrits but fail already in dimension four. We then show that fourth-order, ordering-sensitive Bargmann invariants give the first universal pairwise criterion for set coherence. Applied to all unordered pairs, this criterion yields a complete test for arbitrary finite families. The result provides a low-order hierarchy connecting cyclic trace invariants with the noncommutativity that prevents a common incoherent basis.