A finite-step entropy hierarchy generated by compatible stabilizer dephasings is established, characterized by the entropy hierarchy and refines the previous mean-state bound of Bu, Gu, and Jaffe.
Abstract
Quantum convolution provides a discrete-variable analogue of classical convolution, with the mean state capturing the stabilizer structure preserved under repeated convolution. We establish a finite-step entropy hierarchy generated by compatible stabilizer dephasings. Along every compatible isotropic flag, the entropy increases toward the mean-state entropy ceiling, while the relative-entropy distance to the mean state decomposes exactly into successive coherence losses and a terminal classical nonuniformity. Optimizing over compatible subspaces yields an intrinsic entropy profile of the state. For quantum convolutional channels, Weyl covariance reduces the one-shot classical communication problem to minimal output entropy. A spectral-transfer argument shows that suitable stabilizer inputs reproduce stabilizer-measurement distributions of the environment as channel-output spectra. This gives a computable Holevo lower bound over all complete stabilizer measurements; its compatible restriction is characterized by the entropy hierarchy and refines the previous mean-state bound of Bu, Gu, and Jaffe. The bound is exact for stabilizer-diagonal environments, for which the Holevo capacity is strongly additive, and yields a single-letter formula for a nonstabilizer qutrit family. The same family also exhibits a coexistence region with simultaneously positive classical and quantum communication rates
How a quantum state responds to small parameter changes is central to quantum criticality, state distinguishability, and metrological sensitivity. Pauli-string statistics provide a natural many-body representation of quantum states, but it is not evident how much of their local quantum geometry is retained by the corresponding classical probability distribution. We show that, for any smooth family of pure $N$-qubit states, the classical Fisher matrix of the complete labeled Pauli distribution is exactly twice the quantum Fisher information matrix (QFIM). The Pauli spectrum is therefore locally metric-complete, while scalar R\'enyi functionals retain only compressed information about its parameter-dependent motion. The same distribution has a direct physical realization: pairwise Bell measurements between corresponding sites of a state and its complex conjugate sample the Pauli spectrum and attain the full QFIM of the conjugate-pair state with a fixed transversal readout. For generic mixed states, physical Bell statistics separate from Hilbert--Schmidt-normalized squared Pauli coordinates, and we derive an exact positive decomposition of the resulting Bell-information gap. These results identify the labeled Pauli spectrum as a common statistical structure underlying many-body response, nonstabilizerness, and multiparameter quantum geometry.
Stabilizer convolution channels in odd-prime dimensions have a natural description in finite-field phase space. We analyze these channels using Clifford symmetry and show that, for a fixed environmental state, the maximal single-letter coherent information is unchanged under Clifford-equivalent choices of the stabilizer convolution matrix. This reduces the apparently quadratic family of admissible stabilizer convolutions, at the one-use level, to sign-orbit classes of a single finite-field parameter. We also isolate quadratic cases in which the induced Clifford symmetry becomes either the identity or phase-space inversion; these cases give symmetry-protected families with zero maximal single-letter coherent information. To complement this symmetry analysis, we introduce a minimal weighted magic entropy and prove that it sharpens the magic-relative-entropy upper bound for positive convolution channels at the single-letter level. Finally, we define the single-letter coherent-information power of a convolution and show that the optimizing environmental state may be taken to be pure. Small-dimensional numerical examples illustrate the sign-orbit classification and show that coherent-information enhancement is governed by the finite-field convolution parameter rather than by dimension alone.
Coherent states bridge the gap between quantum and classical physics, but their overcomplete and nonorthogonal nature makes it difficult to identify the minimal discrete set needed to reconstruct quantum information. Finite spin-coherent tomography and discrete coherent-state operator bases are known, but here we address the more specific rank-resolved problem of preserving the canonical contravariant-symbol representation. We show that the canonical finite coherent-state formula reconstructs every operator in the rank-$S$ sector exactly if and only if the sampling points form a spherical $(2J+S)$-design. We call the associated configurations spin-coherent quantum designs. We further give a fully explicit positive-weight Gauss-Legendre construction that avoids the need for an equal-weight spherical design. Together, these results establish a unified framework for reading out physical observables from a handful of measurement samples, playing for spin systems the role that the so-called von Neumann lattice plays for canonical coherent states. Finally, we derive practical protocols for estimating moments of spin operators from these constructions, with direct applications to polarimetry, magnetometry, and quantum state tomography.
Marcin Rudzinski, A. Goldberg, A. B. Klimov et al.· 0 citations
Absolutely maximally entangled states represent a highly constrained form of multipartite entanglement and play an important role in quantum information theory. We investigate a weaker form of uniformity of entanglement for four-party systems of local dimension $d>2$ that requires the three balanced bipartitions to have equal but not necessarily maximal linear entropy. We introduce a linear map $\Xi$ that enforces exact equality of entropies under reshuffling and partial transposition. The transformation arises as the asymptotic limit of an iterative averaging procedure and admits a group-theoretic description in terms of permutations of tensor indices. For Haar-random unitary inputs, a leading-moment analysis supported by numerical simulations predicts highly entangled outputs whose common entropy approaches the maximal value as the local dimension grows. We characterize the algebraic structure, fixed points, and asymptotic behavior of this map and its relation to two-unitary matrices and orthogonal Latin squares.
Coupling between a quantum system and its environment causes decoherence by transferring information from the system to environmental degrees of freedom. When discretized in time, such interactions can be interpreted as sequences of weak measurements that provide an effective model of noisy quantum dynamics. Motivated by this picture, we propose an AI-assisted error-mitigation framework for quantum diffusion processes generated by sequential local weak measurements. The forward process progressively erases information from the input state through weak measurements performed in randomly selected Pauli bases, producing basis-dependent local dephasing and locally depolarizing dynamics on average. Machine-learning models are trained on exact synthetic density matrices to learn a channel- and distribution-specific denoising map and estimate the corresponding pre-noise state. We benchmark the approach on single-qubit states and separable and entangled multi-qubit registers. We also study distribution-dependent local-to-global reconstruction, in which local reduced density matrices are used to reconstruct the global state. This experimentally motivated setting relies on locally accessible information and is therefore compatible with noisy and distributed quantum systems. More broadly, the framework provides a hybrid classical-quantum approach for approximating non-unitary dynamics and mitigating coherence loss.
Yuval Idan, Ofek Nourian, E. Mentovich et al.· 0 citations
The quantum capacity of the bosonic thermal attenuator, which is given by the regularization of its coherent information, is unknown. The seminal work of Holevo and Werner established in 1999 the standard one-use lower bound obtained from input thermal states. We first prove that this long-standing lower bound is the exact supremum over all single-mode Gaussian states and then show that, crucially, a non-Gaussian state can do better. As a consequence, we prove positivity of the quantum capacity in a parameter region where the channel is not antidegradable, yet its coherent information optimized over single-mode Gaussian states vanishes. For example, with one thermal photon in the environment and at transmissivity $\eta=0.8$, the coherent information is non-positive for every single-mode Gaussian input. We give an explicit rank-two non-Gaussian state, supported on only six Fock levels, whose coherent information is certified to be at least $4.7\times10^{-4}$ qubits per channel use. This short witness is far from numerically optimal: a numerical optimization over fixed non-Gaussian families reaches at least $8.4\times 10^{-3}$ qubits per channel use at the same point. More generally, at $\nu=1$, using non-Gaussian inputs we certify positivity of the coherent information, and therefore of the quantum capacity, down to $\eta=0.7841$; by contrast, the channel is antidegradable, and hence has zero quantum capacity, for $\eta\leq0.75$. Overall, our work identifies new high-noise regimes in which bosonic quantum communication is possible.
Francesco Anna Mele, Giuseppe Catalano, Marco Fanizza et al.· 0 citations
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