This work introduces a unified measure, the magic R\'enyi entropy (MRE), to quantify computational resources in spins, bosons, and fermions on an equal footing and shows that the MRE is a resource monotone under stabilizer and Gaussian protocols involving measurements and feedforward operations.
Abstract
Characterizing a quantum state through the lens of quantum resources provides an information-theoretic perspective on many-body systems. While quantum entanglement serves as the paradigmatic example of a quantum resource, recent studies have shown that quantum magic, a resource for universal quantum computation, captures aspects of many-body states complementary to those described by entanglement. For instance, in spin systems, conformal field theory (CFT) analysis of the stabilizer R\'enyi entropy has revealed universal features of nonstabilizerness qualitatively distinct from entanglement. In bosonic and fermionic systems, however, a comparable formulation for their computational resource, non-Gaussianity, has yet to be established. In this work, we introduce a unified measure, the magic R\'enyi entropy (MRE), to quantify computational resources in spins, bosons, and fermions on an equal footing. We show that the MRE is a resource monotone under stabilizer and Gaussian protocols involving measurements and feedforward operations. The MRE reveals common universal aspects of nonstabilizerness and non-Gaussianity in critical many-body states. In particular, our CFT analysis shows that the universal contribution to the MRE appears as the size-independent term determined by the Affleck-Ludwig boundary entropy. We find that non-Gaussianity can continuously renormalize this universal contribution or drive a boundary transition through bulk-induced boundary renormalization-group flows. As a concrete example, we present a CFT analysis of non-Gaussianity in interacting spinless fermions described by the Tomonaga-Luttinger liquid, showing boundary transitions at the Luttinger parameters $K=1/3$ and $K=3$. Our field-theoretical predictions are confirmed by numerical calculations. These results provide a unified field-theoretical understanding of many-body magic across spins, bosons, and fermions.
Recent work introduced Interval Quantum Mechanics (IQM), a finite-precision framework in which physical information is represented by sets of possible states rather than exact states. We extend this approach to algebraic quantum field theory (AQFT) by introducing Interval Algebraic Quantum Field Theory (IAQFT), whose basic objects are quantum parcels: weak* open convex regions of state space encoding finite-precision information obtained from finitely many local observations. We develop parcel reduction and measurement update, establish finite-dimensional information-contraction results, and formulate locality through compatible parcel nets. Major structural features of AQFT are recast in parcel-theoretic terms, including the Reeh--Schlieder property, Haag's theorem, modular theory, the KMS condition, and the Unruh effect. We show that spacelike vacuum correlations and strict Bell violations persist under finite precision, and that lattice approximations are compatible with the parcel framework. Finally, we recover the Murray--von Neumann classification of factors together with the trace and projection equivalence in Type~$\mathrm{II}_1$ factors from the geometry of limiting parcels. IAQFT thus provides a unified finite-precision formulation of the operational, modular, and operator-algebraic structures of relativistic quantum theory.
Quantum Fisher Information (QFI) is a powerful spectroscopic tool to witness many-body quantum entanglement in solid state materials, but it is not always obvious how to relate it to other characteristics of condensed matter systems. In this study we elaborate on the meaning and interpretation of QFI in condensed matter by examining simple theoretical spin systems. We use finite sized spin systems to illustrate that QFI quantifies the momentum- dependent degree of quantum entanglement (that is the entanglement depth at a given wave vector) within a wave function. We subsequently use semiclassical frustrated spin models to show that, in the context of linear spin wave theory (LSWT), QFI quantifies the momentum-dependent degree of magnon squeezing in the ground state. In antiferromagnets with a zero-energy Goldstone mode, LSWT breaks down and QFI (and entanglement depth) diverges at the magnetic ordering wave vector. We also show examples of emergent quantum phases in frustrated spin systems that do not appear in classical phase diagrams. When these emergent phases are approached, QFI diverges across multiple wave vectors in momentum space. Taken together, QFI is not only helpful as a lower bound for entanglement depth, but serves as a momentum-resolved probe of entanglement that offers a novel perspective on quantum critical phenomena.
Quantum information theory is built on entropic quantities; among them, the sandwiched R\'enyi relative entropy is a fundamental divergence with various applications, and its data processing inequality (DPI) under quantum channels is a cornerstone result. In this work, we present a Lean 4 library for quantum information, designed as a reusable formal infrastructure for theoretical analysis. As a central demonstration of the library, we formalize the DPI for the sandwiched R\'enyi relative entropy for positive semidefinite operators on finite-dimensional quantum systems. The library provides a basis-independent operator-theoretic framework for finite-dimensional quantum mechanics compatible with the standard mathematical library Mathlib, including reusable interfaces for finite-dimensional systems, states, channels, tensor products, partial traces, Choi operators, Kraus representations, and Stinespring representations. It also builds infrastructure for noncommutative trace inequalities, including operator monotonicity and convexity via the real continuous functional calculus, block-operator positivity, Hilbert-Schmidt operator spaces, Jensen's operator inequality, generalized perspectives, operator power means, and Lieb-Ando trace inequalities. On top of this framework, we formalize entropy-specific ingredients for the DPI: variational formulas for the sandwiched quasi-entropy via Young and reverse-Young inequalities, tensor-product compatibility of real powers, and Haar measures on unitary groups. Together, these components yield a Lean formalization of the DPI, give strong subadditivity as a corollary, and provide the last missing component needed to complete the Lean formalization of the generalized quantum Stein's lemma. More broadly, the development provides machine-checkable foundations for future formalized and AI-assisted research in quantum information theory.
Kazumi Kasaura, Kei Tsukamoto, Kento Mori et al.· 2 citations
Quantum resource theories characterize distinct forms of nonclassicality in many-body quantum states, raising the question of whether these resources evolve independently under generic ergodic dynamics. Considering diagnostics quadratic in the state, we show that the dynamics of different resource measures become mutually interdependent and are governed by a few common degrees of freedom. For Haar-random circuits, the purity together with a single resource witness suffices to reconstruct the remaining resource measures, as we demonstrate for coherence, various asymmetries, and number entropies. The same relations hold, to a good approximation, under chaotic Floquet and continuous-time Hamiltonian dynamics, establishing that this dynamical resource equivalence extends beyond random circuits. We further derive an exact coherence--imaginarity relation, remarkably accurate also beyond Haar-random circuits. Our results reveal an emergent simplification of many-body dynamics, in which sufficiently strong scrambling reduces seemingly distinct quantum resources to a few common dynamical degrees of freedom.
Sreemayee Aditya, X. Turkeshi, P. Sierant· 0 citations
Quantum computing provides a promising framework for the simulation of quantum field theories, where the computational cost depends both on the quantum algorithm employed and on the representation of the Hamiltonian. We investigate a formulation of the $\phi^4$ model based on the harmonic-oscillator basis in coordinate space. We derive the lattice $\phi^4$ Hamiltonian in this representation and analyze the structure of the resulting one-body matrix and interaction tensor. We show that both exhibit an effective band-diagonal structure, allowing controlled truncations of the Hamiltonian while preserving the low-energy spectrum. We validate this formulation by comparing low-energy observables obtained from numerical diagonalization with those computed in the standard harmonic-oscillator momentum-space representation. Finally, we estimate the resources required to encode the Hamiltonian on a quantum computer using both binary and unary boson-to-qubit mappings. By exploiting effective locality, the coordinate-space representation reduces the resources required for quantum simulation over a broad range of parameters.
Gaétan Bardy, M. Saubanère, A. Tanasa· 0 citations
Quantum magic is a fundamental resource that quantifies to what extent quantum states can be efficiently simulated on a classical computer. We study it for states constructed from the Sachdev-Ye-Kitaev (SYK) Hamiltonian with $N$ Majoranas by the fermionic anti-flatness (FAF). We show analytically that, in the large $N$ limit, the quantum magic of pure Kourkoulou-Maldacena (KM) states, dual to a quantum black hole with an end-of-world particle behind the horizon, is linear in $N$ with a slope, depending on the black hole temperature, that can be tuned between zero and $1/2$. By contrast, the FAF of Gaussian states evolved in real time with the SYK Hamitonian approaches $\approx N/2$ exponentially at a rate given by a multiple of the leading Ruelle-Pollicot resonance. Subleading corrections in $N$ for SYK energy eigenstates, computed numerically for $N \leq 54$ by combining Krylov subspace with GPU acceleration techniques, decay exponentially with $N$, but power-law if the SYK couplings are sparsified, and are order of magnitude larger for states close to the ground state, a region with an established gravity analogue. Our results offer new insights about the relation between quantum information, quantum chaos and low-dimension quantum gravity.
Antonio M. Garc'ia-Garc'ia, Xianlong Liu, Jie-ping Zheng· 0 citations