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. 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.
Abstract The notion of complexity of quantum states is quite different from uncertainty or information contents, and involves the tradeoff between its classical and quantum features. In this work, we introduce a quantifier of complexity of continuous-variable states, e.g. quantum optical states, based on the Husimi quasiprobability distribution. This quantity is built upon two functions of the state: the Wehrl entropy, capturing the spread of the distribution, and the Fisher information with respect to location parameters, which captures the opposite behavior, i.e. localization in phase space. We analyze the basic properties of the quantifier and illustrate its features by evaluating complexity of Gaussian states and some relevant non-Gaussian states. We further generalize the quantifier in terms of s -ordered phase-space distributions and illustrate its implications.
The statistical complexity of continuous-variable quantum states can be characterized with a quantifier defined in terms of information-theoretic quantities derived from the Husimi [Formula: see text]-function. In this work, we utilize this complexity quantifier of quantum states to study the complexity of single-mode bosonic quantum channels. We define the complexity of quantum channels as the maximal amount of complexity they can generate from an initial state with the minimal complexity. We illustrate this concept by evaluating the complexity of Gaussian channels and some examples of non-Gaussian channels.
Abstract The Wehrl entropy of a quantum state is the Shannon entropy of its coherent-state distribution function, and remains non-zero even for pure states. We investigate the relationship between this entropy and the many-particle quantum entanglement, for N spin-1/2 particles. Explicitly, we numerically calculate the Wehrl entropy of various N-particle ( 2 ⩽ N ⩽ 20 ) entangled pure states, with respect to the SU(2) ⊗ N coherent states. Our results show that for the large-N ( N ≳ 10 ) systems the Wehrl entropy of the highly chaotic entangled states (e.g. 2 − N / 2 ∑ s 1 , s 2 , … , s N =↑ , ↓ | s 1 , s 2 , … , s N ⟩ e − i ϕ s 1 , s 2 , … , s N , with ϕ s 1 , s 2 , … , s N being random angles) are substantially larger than that of the very regular entangled states (e.g. the Greenberger–Horne–Zeilinger state). Therefore, the Wehrl entropy can reflect the complexity of the quantum entanglement of many-body pure states, as proposed by Sugita (2003 J. Phys. A: Math. Gen. 36 9081). In particular, the Wehrl entropy per particle (WEPP) can be used as a quantitative description of this entanglement complexity. Unlike other quantities used to evaluate this complexity (e.g. the degree of entanglement between a subsystem and the other particles), the WEPP does not necessitate the division of the total system into two subsystems. We further demonstrate that many-body pure entangled states can be classified into three types, based on the behavior of the WEPP in the limit N → ∞ : states approaching that of a maximally mixed state
We introduce a quantum convolution and a conceptual framework to study states in discrete-variable (DV) quantum systems. All our results suggest that stabilizer states play a role in DV quantum systems similar to the role that Gaussian states play in continuous-variable systems. Hence we suggest the name ``discrete quantum Gaussians'' for stabilizer states. For example, we prove that the convolution of two stabilizer states is another stabilizer state, and that stabilizer states extremize both quantum entropy and Fisher information. We establish a ``maximal entropy principle,'' a ``second law of thermodynamics for quantum convolution,'' and a ``quantum central limit theorem.'' The latter is based on iterating the convolution of a zero-mean quantum state, which we prove converges to a stabilizer state. We bound the exponential rate of convergence of the quantum central limit theorem by the ``magic gap,'' defined in terms of the support of the characteristic function of the state. We elaborate on our general results with a discussion of some examples, as well as extending many of them to quantum channels.
The rapid advancement of quantum information science has spurred research into the ultimate performance limits of quantum systems in communication, sensing, and computation. This thesis adopts a decision-theoretic and information-theoretic viewpoint to systematically study fundamental problems in quantum statistical inference, with a focus on continuous-variable and bosonic platforms. The central theme is to uncover the interplay among physical structure (e.g., Gaussianity and phase-space geometry), operational restrictions (e.g., postselection and covariant measurements), and inference objectives (e.g., risk, error probability, and detection delay), and their joint impact on fundamental limits and attainability. The thesis first advances the understanding of phase-space entropy and the Wigner entropy conjecture. For Wigner-nonnegative quantum states, we develop verifiable sufficient criteria based on moment functionals and purity constraints, thereby extending the range of states for which the conjecture holds. We also establish explicit purity thresholds that guarantee the conjecture in the low-purity regime. These results reveal how Wigner nonnegativity constrains non-Gaussian structure and suggest pathways for incorporating further quantum-realizability conditions. Second, the thesis investigates the role of postselection in quantum state discrimination. We prove that, on average, environment-assisted partial postselection cannot outperform the Helstrom bound, yet conditional improvements are possible in selected branches. This clarifies the decision-theoretic trade-off between conditional performance and success probability, providing a rigorous basis for evaluating postselected protocols. In parameter estimation, we study phase-shift sensing within the covariant measurement framework. By linking the Bayesian risk to the Fourier coefficients of the input state, we derive an explicit characterization of optimal probe states via the Perron–Frobenius theorem and show that Heisenberg scaling is attainable in the high-energy limit. This offers constructive guidance for designing optimal sensing states under symmetry constraints. Finally, the thesis extends classical quickest change detection to sequential quantum inference. We establish the asymptotic relationship between quantum relative entropy and detection delay, and develop protocol architectures tailored to bosonic sensing and joint communication–detection tasks. In particular, for phase-change detection, we provide an analytic framework for computing quantum relative entropy with Gaussian probes, supplying a theoretical toolkit for practical system design. By integrating phase-space methods, information measures, and sequential decision theory, this thesis provides a cohesive analytical framework and delivers several substantive advances in quantum statistical inference for continuous-variable systems, with implications for quantum sensing, quantum communication, and quantum signal processing.