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arXiv:2512.12342

Quantum Coherence Dispersion

Fernando Parisio

quant-phphysics.bio-phSubmitted Dec 13, 2025• Announced Dec 16, 2025• 0 citations• Revised Apr 23, 2026 (v2)
v1 · Dec 13, 2025v2 · Apr 23, 2026
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Abstract

We investigate how quantum coherence can be distributed among the several off-diagonal elements of an arbitrary density matrix. An easily computable quantity that captures this variability notion is proposed and it is argued that it presents marked features of complexity quantifiers. It turns out that this coherence dispersion ($Δ_{\rm c}$) is maximized for intermediate values of an appropriate entropy (the relative entropy of coherence), a prevalent signature of complexity quantifiers across different fields, from evolutionary biology to linguistics and information science. The final part of the manuscript is dedicated to the connection between the proposed framework and non-equilibrium systems in the quantum regime.

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We investigate how quantum coherence can be distributed among the several off-diagonal elements of an arbitrary density matrix. An easily computable quantity that captures this variability notion is proposed and it is argued that it presents marked features of complexity quantifiers. It turns out that this coherence dispersion (ΔcΔ_{\rm c}Δc​) is maximized for intermediate values of an appropriate entropy (the relative entropy of coherence), a prevalent signature of complexity quantifiers across different fields, from evolutionary biology to linguistics and information science. The final part of the manuscript is dedicated to the connection between the proposed framework and non-equilibrium systems in the quantum regime.
Quantum coherence is an essential ingredient in quantum information processing and plays an important role in quantum computation. Therefore, it is a hot issue about how to quantify the coherence of quantum states in theoretical framework. The coherence effect of a state is usually described by the off-diagonal elements of its density matrix with respect to a particular reference basis. Recently, based on the established notions from quantitative theory of entanglement, a resource theory of coherence quantification has been proposed[1,2]. In the theory framework, a proper measure of coherence should satisfy three criteria: the coherence should be zero for all incoherent state; the coherence should not increase under mixing quantum states; the coherence should not increase under incoherent operations. Then, a number of coherence measures have been suggested, such as l1 norm of coherence and the relative entropy of coherence[2]. Wigner function is known as an important tool to study the non-classical property of quantum states for continuous-variable quantum systems. It has been generalized to finite-dimensional Hilbert spaces, and named as discrete Wigner function[9-16]. The magic property of quantum states, which promotes stabilizer computation to universal quantum computation, can be generally measured by the absolute sum of the negative items (negativity sum) in the discrete Wigner function of the observed quantum states. In this paper we investigate quantum coherence from the view of discrete Wigner function. From the definition of the discrete Wigner function of the quantum systems with odd prime dimensions, for a given density matrix we analyze in phase space the performance of its diagonal and off-diagonal items. We find that, the discrete Wigner function of a quantum state contains two aspects: the true quantum coherence and the classical mixture, where the part of classical mixture can be excluded by only considering the discrete Wigner function of the diagonal items of the density matrix. Thus, we propose a possible measure method for quantum coherence from the discrete Wigner function of the off-diagonal items of the density matrix. We show that the proposed measure method satisfies the criteria (C1) and (C2) of coherence measure perfectly. For the criteria (C3), we give a numerical proof in three-dimensional quantum system. Meanwhile, we compare the proposed coherence measure with l1 norm coherence, and get an inequality relationship between them. Finally, an inequality is obtained to discuss the relation between quantum coherence and the negativity sum of discrete Wigner function, which shows that the quantum coherence is only necessary but not sufficient for quantum computation speed-up.
We introduce a quantifier of phase-space complexity for discrete-variable (DV) quantum systems. Motivated by a recent framework developed for continuous-variable systems, we construct a complexity measure of quantum states based on the Husimi Q-function defined over spin coherent states. The quantifier combines into a single scalar quantity two complementary information-theoretic quantities, the Wehrl entropy, which captures phase-space spread, and the Fisher information, which captures localization. We derive fundamental properties of this measure, including its invariance under SU(2) displacements. The complexity is normalized such that coherent states have unit complexity, while the completely mixed state has zero complexity, a feature distinct from the continuous-variable case. We provide analytic expressions for several relevant families of states, including Gibbs and Dicke states, and perform a numerical analysis of spin-squeezed states, NOON states, and randomly generated states. Numerical results reveal a monotonic, but not deterministic, relationship between complexity and purity, leading us to conjecture that maximal complexity is attained by pure states, thereby connecting the problem to the optimization of Wehrl entropy via Majorana constellations. Finally, we extend the framework to quantum channels, defining measures for both the generation and breaking of complexity. We analyze the performance of common unitary gates and the amplitude damping channel, showing that while low-dimensional systems can achieve maximal complexity via spin squeezing or NOON states, this becomes impossible in higher dimensions. These results highlight dimension-dependent limitations in the generation of phase-space complexity and establish a unified phase-space approach to complexity across both continuous and discrete variable regimes.
We introduce a quantifier of phase-space complexity for discrete-variable (DV) quantum systems. The quantifier combines into a single scalar quantity two complementary information-theoretic quantities, the Wehrl entropy, which captures phase-space spread, and the Fisher information, which captures localization. We provide analytic expressions for several relevant families of states, including Gibbs and Dicke states, and perform a numerical analysis of spin-squeezed states, NOON states, and randomly generated states. We conjecture that maximal complexity is attained by pure states, thereby connecting the problem to the optimization of Wehrl entropy via Majorana constellations. Finally, we extend the framework to quantum channels, defining measures for both the generation and breaking of complexity. Our results highlight dimension-dependent limitations in the generation of phase-space complexity and establish a unified phase-space approach to complexity across both continuous and discrete variable regimes.
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.