Papers

We study a non-Abelian Moore-Read fractional Chern insulator under a translation-preserving, nonreciprocal deformation that generates the non-Hermitian skin effect under open boundaries. The model combines the imaginary-gauge Hatano-Nelson deformation with a kagome-lattice three-body interaction designed to stabilize Moore-Read order at $\nu=1/2$. Our primary diagnostic is the biorthogonal $(2,4)$-admissible particle-entanglement counting of the sixfold Moore-Read manifold, the standard Moore-Read fingerprint. Across three sizes ($N=16,20,24$), the counting locks to the clean references 1308, 3965, and 9282 over finite nonreciprocity windows through $\gamma\le0.55$, $0.65$, and $0.74$, respectively, with positive reference-rank entanglement gaps. Within every reported window the count is unchanged by the spectral readings tested; at $N=16$ it is also unchanged across three reduced density operators, with all 15 combinations returning 1308. The sixfold pattern for even $N_f$ and the adiabatically tracked Ising-odd doublet remain separated over the tested range $\gamma\le0.6$. Beyond a geometry-dependent threshold the instantaneous-lowest-six reference-rank gap drops sharply and its counting destabilizes. At $N=24$ a sector-0 branch pair becomes complex conjugate over a narrow interval inside the delocking bracket; both continuations through the interval are delocked at the tested PES points $\gamma=0.76$, $0.77$, and $0.80$. A same-lattice Abelian $\nu=1/3$ Laughlin realization retains its counting to $\gamma=1.0$, so its counting is the more robust. On the torus the eigenstates remain extended; under open boundaries the right and left states skin-localize at opposite edges while the biorthogonal particle-entanglement spectrum is invariant under the imaginary-gauge similarity, so the torus and the open cylinder probe the same deformation under periodic and open boundaries.

Physics is usually done from a third-person perspective, as if the world were described from outside. Yet, observers are themselves physical systems within the world they observe. Here, we investigate this tension using a toy model of classical particles. Observers are physical systems characterised by a choice of (i) a manifest variable, whose value constitutes their empirical record, and (ii) ready states, which provide initial information. We ask what such observers (subjects) can learn about the world through interactions that establish a correlation with another system (the object). For each combination of the three types of learning (about the past, the future, or both), manifest variables, and ready states, we determine what the subject can learn. This shows that many subjects face an epistemic horizon—a limitation to what they can learn about the world—even though the model is classical and deterministic. For example, a subject with a complete manifest variable—one whose ontic state is the empirical record—and partial initial information can learn at most half the object's variables, recovering Spekkens' knowledge-balance principle, i.e. an analogue of Heisenberg's uncertainty principle. More generally, we find that subjects face more severe epistemic horizons when predicting than when retrodicting, and that learning by repeatable measurements can be more limited than either prediction or retrodiction alone. Our work provides language and tools to study how the first-person perspective can differ from the third-person perspective beyond the toy model studied here, and invites explanations of the inherent uncertainty in quantum theory from the standpoint of embedded observers.

Magic-state injection is a standard route to realize universal fault-tolerant quantum computation. Whereas the set of Clifford gates in combination with the non-Clifford T gate is a widely used universal gate set, extending the available set of non-Clifford primitives can reduce compilation overhead, provided that the additional primitives can be prepared fault-tolerantly with competitive resource costs and at sufficiently low logical noise rates. In this work, we introduce flag fault-tolerant protocols for preparing logical $|\sqrt{\mathrm{T}} \rangle $ magic states on the 3D tetrahedral color code and its smaller morphed variant. Our simulations under circuit-level noise verify fault tolerance, quantify acceptance and logical error rates, and we reconstruct the effective logical channels of the corresponding circuits for gate injection via logical process tomography. We find that access to $\sqrt{\mathrm{T}}$ reduces the average space-time cost of synthesizing Haar-random single-qubit unitaries by approximately $20$-$30$% relative to the Clifford+T gate set across practically relevant approximation regimes. These results demonstrate how expanding the set of fault-tolerant non-Clifford primitives can improve computational efficiency and broaden the design space for universal quantum computation in the early fault-tolerant era.

Boson sampling is one of the most prominent methods to verify quantum advantage, harnessing the computational complexity of the permanent of a network matrix. Similar quantum sampling problems utilize other expressions for the output probabilities, such as the fermion-based analog applying determinants. In this work, we study sampling setups of interacting light-matter systems, combining fermion and boson sampling. The mixed network input consists of fermionic and bosonic excitations. Sampling the output probability of the modes then yields contributions that are neither purely determinant nor permanent but general immanants. Specifically, we prove that these immanant contributions are linearly independent from determinants and permanents, rendering our mixed sampling network an ideal candidate for light-matter quantum application beyond pure fermion and boson sampling. Furthermore, quantum sampling with statistics that are neither bosonic nor fermionic results in expressions which even exceed immanants, further extending the functionality of quantum samplers.

Driven-dissipative quantum oscillators lock their phase, deform their limit cycles, and undergo dissipative phase transitions, yet these behaviors are read from unrelated quantities defined on the same steady-state density matrix. We show that a single geometric quantity organizes them. Winding the phase of the drive generates a closed loop of nonequilibrium steady states, and because the Liouvillian is covariant under number rotations, the kinematic mixed-state geometric phase of this loop reduces exactly to an eigensystem functional of a single steady state. Under weak driving, it is governed by the same nearest-neighbor coherences that produce phase locking and inherits the Arnold tongue of synchronization. Near the Hopf threshold, it registers the nonperturbative reorganization of the steady-state eigenvectors. And in the squeezing-driven Kerr resonator, it develops distinct signatures at the first and second order dissipative phase transitions. The geometric phase thus provides a unified and experimentally accessible characterization of steady-state reorganization.

Glueball spectroscopy and real-time production with quantum computing require three distinct ingredients: a correlated gauge vacuum, a controlled construction of pure-gauge excitations, and a dynamical detector. We develop a physics-informed quantum-algorithm toolbox for these tasks in a $(2+1)$-dimensional $\mathbb{Z}_2$ lattice gauge theory. We use the term glueball-like for localized closed-flux excitations on the confining side of this Abelian model, without identifying them with the non-Abelian glueballs of QCD. A loop-gas circuit and Hamiltonian variational ansatz prepare the gauge vacuum, Wilson-loop quantum subspace expansion constructs and characterizes low-lying excitations, and eigenvector continuation imports parameter-dependent dressing without a rapidly enlarged explicit loop basis. A Bethe–Salpeter-type transition amplitude quantifies the spatial broadening of the lightest state. For dynamics, a vacuum-dressed contractible-loop counter measures excess production of localized glueball-like structures. Although demonstrated in an Abelian model, the toolbox separates preparation, construction, compression, characterization, and dynamical detection in a form that is naturally extensible to non-Abelian lattice gauge theories.

Motility-induced phase separation (MIPS) describes a central clustering phenomenon in active matter systems where particles spontaneously separate into dense and dilute phases even in the absence of interparticle attractive forces. Recent theoretical and experimental efforts have taken the first steps to extend active matter concepts to the quantum level. However, whether a genuine quantum analog of MIPS exists and how quantum coherence would compete or cooperate with the dissipative self-propulsion has remained open so far. Here, we provide evidence that quantum MIPS can occur in a model of active hard-core bosons in one dimension. Our model yields superlinear number fluctuations characteristic of MIPS, leading to microphase separation with large but finite cluster size. Adding nearest-neighbor repulsive interactions, we find numerical evidence for restoring genuine phase separation with a divergent correlation length. Crucially, the clustered steady states maintain a long-distance quantum coherence, revealing a genuinely quantum feature with no classical counterpart. These results provide a foundation for exploring quantum MIPS and suggest that the coherence-activity interplay can generate new types of nonequilibrium quantum states.

Phase-sensitive parametric devices enable quadrature-selective amplification with the potential for sub-quantum-limited noise performance. In this work, we investigate the operation of a SQUID-based Josephson Parametric Amplifier (JPA), comparing its performance in the phase-preserving and phase-sensitive regimes. The device, fabricated using VTT SWAPS technology, is driven in a three-wave mixing configuration and characterized in a reflection-based measurement setup at millikelvin temperatures. To directly probe the noise performance at low JPA gains, we employ a cascaded amplification scheme in which a Traveling-Wave Parametric Amplifier (TWPA) provides low-noise pre-amplification of the JPA output. In a phase-preserving operation, the JPA exhibits near-quantum-limited performance with a system noise temperature of $351\pm53$ mK at 6 GHz. In contrast, phase-sensitive operation yields a minimum system noise temperature of $94\pm12$ mK, well below the standard quantum limit of 288 mK. Our results demonstrate that a JPA-TWPA amplifier cascade opens the door to direct, high-fidelity probing of quantum devices without the need for background noise subtraction.

Global navigation satellite systems (GNSS) are often disrupted or unavailable at sea, and unaided inertial navigation systems (INS) drift without correction. Quantum-sensing-based gravity map matching offers a passive, infrastructure-free aid, but field demonstrations of GNSS-free quantum gravimetric navigation have not been reported. Here we perform gravity map matching and fine-resolution gravity survey with a mobile quantum gravimeter aboard a 29 m surface vessel. We hybridize an atomic sensor with a classical accelerometer for bias stabilization and independently mechanize a navigation-grade IMU, all installed in an uncontrolled cabin with no environmental stabilization or calibration. Operated in both gimbaled and strapdown configurations over identical traversals, the hybrid sensor corrected the inertial solution over an 83 km maritime trajectory by referencing locally measured gravity to a satellite-derived anomaly map. Gravity-aiding constrains INS drift and delivers bounded positioning at nautical-mile-level accuracy, with GNSS excluded throughout the measurement chain. In a separate GNSS-referenced mode, the same system surveyed coastal routes up to Sea State 4, achieving mGal-level agreement with gravimetric maps and sub-mGal repeatability and stability, with gimbaled and strapdown operation performing comparably. Resolved anomalies reach an along-track scale of ~300 m, 50X finer than the satellite map's half-power wavelength. A 56 h stationary test shows atom referencing lowers long-term drift ~70X versus the classical channel alone. These results provide the first same-instrument comparison of gimbaled and strapdown mobile quantum gravimetry and the first fully GNSS-independent gravity-map-matching navigation demonstration using a quantum gravimeter, pointing toward compact, autonomous-platform-ready quantum sensing for GNSS-denied maritime navigation and survey.

Quantum low-density parity-check (qLDPC) codes reduce the qubit overhead of fault-tolerant quantum computation by an order of magnitude, but their decoding is harder than its classical counterpart: because many physical errors are equivalent up to stabilizers, the degenerate maximum-likelihood (ML) decoder must compare the probabilities of entire equivalence classes of errors, that is, partition functions, rather than single errors. The workhorse decoder BP+OSD sidesteps degeneracy heuristically and offers no guarantees. We treat degenerate decoding as probabilistic inference in an undirected graphical model: the probability of each logical class is the partition function of an unconstrained, strictly positive Markov random field over the code's check variables, a construction that generalizes the random-bond Ising mapping of the surface code to arbitrary CSS codes and to spacetime decoding with measurement errors and circuit-level noise. On this model we build two decoders. The first estimates all class partition functions by annealed importance sampling with common random numbers and attaches to every decision a certificate of optimality: a paired bootstrap test, or, composed with constant-factor estimators such as WISH, an exact optimality proof. The second is region-based: the Bethe free energy, whose bias cancels between classes, reproduces exact ML decoding on every tested surface-code instance at millisecond cost, and enlarging the regions to elimination clusters makes exact degenerate ML decoding of the [[72,12,6]] bivariate bicycle code feasible. Across surface codes and the bivariate bicycle codes [[72,12,6]] and [[144,12,12]], under code-capacity, phenomenological, and circuit-level noise, the sampling decoder matches or exceeds BP+OSD while certifying the bulk of its decisions, and the certificate flags exactly the syndromes on which any fast decoder should be distrusted.

Multi-pass schemes send a photon through a sample several times to learn more about it. When the sample rather than the light is scarce, the natural figure of merit is the information gained per photon the object absorbs. We show that one constant fixes the loss-limited optimum of every fixed scheme in which a single photon passes repeatedly through the sample and is detected once at the end. Three properties suffice: information that grows as the square of the pass number, a fixed survival probability per pass, and no dose from a photon already lost. They force a single trade-off function $h(x)=x^2/(e^x-1)$, where $x$ is the number of passes times the loss per pass. Its maximum, 0.648, sits at $x_{\mathrm{opt}}=1.594$. The loss-limited ceiling of interaction-free interrogation and the multi-pass phase optimum of Yu et al. are two instances. Phase sensing of a weakly absorbing object is a third, optimal at $m_{\mathrm{opt}}=x_{\mathrm{opt}}/(\epsilon+\alpha)$ passes; the absorption $\alpha$ and the parasitic loss $\epsilon$ enter the optimum only through their sum, but the damage counts only $\alpha$. N00N states and their loss-robust generalisations do worse per absorbed photon, and optimising over photon number and pass number together returns a single recycled photon. Two further problems are priced in the same measure. Absorption estimation gains nothing, for any scheme. Detecting a faint companion below the Rayleigh limit costs the companion a dose that does not depend on its faintness, while under direct imaging the dose grows without bound. For a fragile sample the gentlest measurement is also the simplest one: a single photon, recycled.

The idea that information processing is optimised near the boundary between order and chaos has emerged as a recurring principle across neuroscience, complex systems, and machine learning. Here we test this hypothesis in quantum many-body systems, numerically simulating two-dimensional Ising networks of up to $N=20$ spins, operated as quantum reservoirs for time-series forecasting. Using out-of-time-order correlators (OTOCs), we locate the onset of information scrambling as the input strength is swept, separating regimes where information is frozen and scrambled across the whole reservoir state. We show that prediction precision peaks at the onset of scrambling, along with the number of computational-basis states that the reservoir actively populates. We then show that prediction precision grows as a power law $\sim N^{\alpha}$ in the reservoir size, superextensively $(\alpha>1)$ at the onset of scrambling and only sublinearly $(\alpha<1)$ in either neighbouring regime. Finally, we show that scrambling enhances the nonlinear components of the reservoir memory while reducing its linear capacity. At the onset, the total memory capacity grows superextensively, provided that the necessary “forgetting” mechanism is supplied by a collective relaxation channel. These results consolidate the role of information scrambling in learning systems, turning it from an operating point into a scaling law for the performance of quantum reservoirs.

A central challenge in fluxonium-based quantum processors is the extension of the qubit connectivity to two-dimensional lattices compatible with quantum code-error correction. Here, we present a fluxonium quantum processor that employs lumped-element resonator couplers which realizes, for the first time, a connectivity-four unit cell with suppressed parasitic interactions. We achieve parallel single-qubit gate fidelities exceeding 99.9 % in simultaneous randomized benchmarking experiments, while maintaining residual static ZZ interactions below 1 kHz across all coupled qubit pairs. We implement resonator-induced phase (RIP) gates and benchmark two-qubit gate fidelities exceeding 99 % using interleaved randomized benchmarking. To cancel spectator errors observed in two-qubit operations, we implement a refocused RIP gate, recovering coherent control in the presence of multi-qubit connectivity. Furthermore, we prepare Greenberger-Horne-Zeilinger states of up to five qubits with a tomographic fidelity of 90 %, verifying multi-qubit entanglement within the unit cell. These results establish the fluxonium-resonator-fluxonium architecture as a viable approach to realizing densely connected fluxonium processors and provide a scalable path toward quantum error-correction-compatible processor architectures.

Fast, accurate, and low-leakage qubit measurement is a key requirement for quantum error correction. Here, we demonstrate measurement of a superconducting transmon qubit with a total duration of 97(1) ns, defined as the time from the start of the measurement pulse until the measurement-induced error on a subsequent $\pi$-pulse operation falls below $10^{-4}$. By combining a large state-averaged resonator decay rate of $\kappa_\mathrm{eff}/2\pi$ = 30.8 MHz with a dispersive shift close to the optimal SNR-per-photon condition, we achieve an assignment error of 0.17(1)% using a 58-ns measurement pulse, with residual readout photons depleting passively in tens of nanoseconds without an active depletion pulse. Using a repeated-measurement sequence together with a leakage-sensitive measurement, we benchmark the measurement-induced state transitions, finding a per-measurement leakage rate of $2.7(2) \times 10^{-5}$, only twice the background rate and two orders of magnitude below the measurement-induced relaxation rate, which dominates the assignment error. Floquet simulations indicate that the multiphoton resonances present at the operating point are weakly coupled and traversed diabatically, without causing leakage. These results demonstrate that a large resonator decay rate, combined with a dispersive shift close to the optimal SNR-per-photon condition, can enable fast, high-fidelity, low-leakage dispersive readout at small qubit-resonator detuning.

Analytical approximations to eigenvalues of parameter-dependent Hamiltonians can provide physical insight that is not readily apparent from numerical diagonalization alone. Here, we introduce an activation-selection network (ASN), a differentiable symbolic-regression architecture in which each input node learns a sparse combination of predefined analytical functions, and the trained network can be converted directly into an explicit expression. Before regression, the Hamiltonian parameters and eigenvalues are expressed as dimensionless ratios. This normalization enforces dimensional homogeneity, reduces the number of independent variables, and ensures that the extracted expressions do not depend on the choice of energy units. Using the library {0, x, x^2}, compositions across successive hidden layers generate polynomial expansions of progressively higher degree; polynomial expansions of arbitrary finite degree can therefore be obtained in principle by increasing the network depth. We apply the ASN to effective three- and four-site spin-chain Hamiltonians relevant to zero-quantum nuclear magnetic resonance. Comparisons with degenerate perturbation theory show that the extracted expressions capture the expected constant, linear, and quadratic structure. Fixed-basis least-squares models match or slightly outperform the ASN when an appropriate quadratic basis is specified in advance, while inclusion of a radial feature improves the local approximation near the degeneracy. These results establish the ASN as a differentiable framework for selecting compact symbolic representations when several functional forms are plausible, while showing that adaptive activation selection does not provide an intrinsic accuracy advantage over a suitable predefined basis.

Multicritical phenomena play a central role in quantum many-body systems, yet their microscopic origin in light–matter platforms remains largely unexplored. Here we investigate a generalized two-mode quantum Rabi model with independently tunable rotating- and counter-rotating-wave couplings. We demonstrate that anisotropy lifts the parent U(1)-symmetric superradiant manifold with a gapless Goldstone-like mode by phase locking the complex superradiant order parameter. This phase-locking mechanism provides the common microscopic origin of the coordinate- and momentum-like soft-mode instabilities, the emergence of symmetry-related two- and four-triple-point multicritical topologies, and the corresponding thermodynamic responses. Within a unified Bogoliubov framework, we further show how the dominant soft mode is redistributed between the two symmetry-related instability channels, thereby determining the multicritical phase structure. The resulting phase boundaries are determined by collective soft-mode softening and coincide exactly with the mean-field instability conditions. We further show that the multicritical topology is directly encoded in the full quantum ground-state energy and its response functions, establishing a unified connection between phase topology, competing collective excitations, and thermodynamic observables. Our results identify competing soft-mode channels as the microscopic origin of tunable multicriticality in strongly coupled light–matter systems.

This study shows that the usual time evolution operator can act as a U(1) gauge transformation on the initial wavefuntion. When considering the spin freedom in the system, the time evolution operator corresponds to a non-Abelian SU(2) gauge transformation for spin-1/2 particles and a SU(3) gauge transformation for spin-1 particles. If we adopt the adiabatic approximation in the magnetization dynamics of a ferromagnetic system driven by a spin-polarized current, a temperature-dependent non-Abelian thermal gauge potential will appear. We can employ a temperature-dependent Landau-Lifshitz-Gilbert (LLG) equation to describe the magnetization dynamics in the semi-classical limit, a linear approximated solution for this equation is given analytically.

Quantum skyrmions are topological structures that have garnered significant interest due to their demonstrated robustness and versatility across diverse optical platforms. However, existing approaches for their generation are limited to producing pre-determined two dimensional qubit states with a single topology. Here we create multi-dimensional topological states by introducing a non-local interaction between high-dimensional photonic entanglement and a metasurface, where the topological transformation induced by the metasurface is made non-deterministic by the probabilistic nature of the interfacing entangled state. Within this framework, we demonstrate that individual quantum states can host multiple co-existing topologies that are only revealed upon measurement, allowing for their parallel transport within distinct spatial mode channels. We confirm this by revealing the rich topological landscape within our modified Hilbert space while controlling the desired output topology by orbital angular momentum (OAM) projections on one of the entangled photons, producing multiple non-local polarization-OAM entangled states characterized by distinct topological classes, all from a single metasurface device. Our results reveal new capability when structured high-dimensional entanglement is interfaced with structured matter capable of coupling photonic degrees of freedom, establishing a new pathway for the compact generation of complex quantum states.

Codeword-stabilized quantum codes give a unified graph-state description of stabilizer and nonadditive quantum error-correcting codes. Although each individual CWS word state is stabilizer, coherent superpositions of different word states can be nonstabilizer. We develop a CWS-adapted magic-witness framework that isolates this codeword coherence and converts it into certified lower bounds on robustness of magic. The main result is an exact reduction of the stabilizer threshold of a natural CWS coherence witness to a finite-geometric problem over the classical CWS word set. For general weighted superpositions, the threshold is computed by enumerating affine intersections and affine-quadratic phases. For equal-weight superpositions, the phase optimization collapses, and the threshold is determined entirely by how many CWS words can lie in affine flats of each dimension. Thus a quantum optimization over stabilizer states becomes a classical affine-incidence problem. This reduction yields a fixed-parameter algorithm, an analytic lower bound for an infinite union-stabilizer family, and exact rational certificates for several standard nonadditive CWS examples. The framework also clarifies why exact enumeration fails for large structured families and identifies the remaining task as an affine-intersection problem. The result provides a geometric mechanism by which nonlinear CWS word sets generate certifiable magic.

Topological Euler insulators emerge in multiband systems with real Bloch Hamiltonians and wavefunctions. Their fragile topologies are characterized by the Euler class of degenerate bands and protected by the $PT$ or $C_2T$ symmetry in two dimensions, which go beyond the tenfold $K$-theory classification of topological matter. In this work, we extend the conception of topological Euler insulators to non-Hermitian systems and propose a theoretical framework to unlock their nontrivial Euler topology. Focusing on two-dimensional, three-band non-Hermitian lattice models with symmetric Hamiltonians, we formulate a comprehensive description of their topological Euler bands, entanglement spectrum and bulk-boundary correspondence. Three typical models of non-Hermitian Euler insulators are constructed and investigated explicitly to illustrate our theory. Unique topological phase transitions and anomalous edge-band overlaps with non-Hermitian origins are further identified. Our study establishes the presence of topological Euler bands in non-Hermitian systems and unveils their intriguing physical characteristics, thereby broadening the existing territory of topological matter in non-Hermitian open systems.

We introduce the wavemap: a spatial portrait of noise effects that assigns each site a per-noise-level arrival delay l_\gamma (v) and cross-entropy loss L_\gamma(v). These observables are exact at the lightcone frontier, where bond dimension \chi is small and the simulation is most faithful. Eigenvalue analysis of the composed gate-plus-noise Pauli transfer matrices confirms that the studied noise is pure amplitude damping: the spatial propagation pattern is entirely determined by the gate, making the wavemap a model-free noise diagnostic. We apply the multi-product formula (MPF) to recover the noiseless Pauli weight field from the noisy samples, subject to the Lieb-Robinson causal constraint nMPF <= nnl . Fitting time-adaptive coefficients \alpha(t) over the frontier recovers up to 55% of the information loss relative to the best noisy sample, exploiting the fact that the frontier is where truncation error is smallest. On an IBM heavy-hex lattice with heterogeneous hardware noise the method identifies an information-starved regime, pointing to calibrated synthetic noise as the next required experiment.

We study estimation of the von Neumann entropy of a $d$-dimensional quantum state from independent outcomes of a fixed rank-one measurement. Let $\kappa_\nu\in[1,d+1]$ denote the normalized trace-zero frame spread; $\kappa_\nu=1$ for exact complex projective $2$-designs. For any fixed rank-one measurement and $n\le c_2d^2/\kappa_\nu$, the minimax mean-squared error is at least $c_1\log^2\{d^2/(\kappa_\nu n)\}$. For bounded frame spread and every fixed $0<a<2$, the minimax risk is of order $\log^2 d$ throughout $n\lesssim d^{2-a}$; hence fixed uniform accuracy requires $\Omega(d^2)$ observations. Existing tomography bounds give bounded risk from $O(d^3\log^3 d)$ observations for finite equal-weight exact designs and the Haar covariant measurement, while the intermediate minimax regime remains open. The results concern a fixed measurement repeated independently across copies; adaptive and collective protocols, and protocols that reuse common randomness across copies, are not covered.

The entanglement of purification $E_P$, introduced by Terhal, Horodecki, Leung, and DiVincenzo in 2002, measures the total correlations of a bipartite state by the entanglement needed to purify it; whether it is additive on tensor powers has been open since its introduction. In 2012 Chen and Winter argued "beyond reasonable doubt", from numerical evidence, that the two-qubit Werner state at singlet fraction $f = 1/200$ is a counterexample, and explicitly left a completely rigorous proof to future work. To our knowledge we give the first rigorous proof: for this state $W$, $E_P^\infty(W) \le 0.9663\ldots < 97/100 < E_P(W)$, hence the entanglement of purification is strictly nonadditive at some finite tensor power. The lower bound is a finite certificate: analytic reductions send the Schmidt spectrum of every purification in the complete complex Stiefel domain to a point satisfying explicit necessary inequalities in a three-dimensional box, and an exact-rational subdivision of that box has 25,383 leaves, each of which either excludes nonphysical points or proves the entropy bound. The upper bound combines an exact symmetric-support endpoint, an explicit feasible isometry, and a convexity theorem for the regularized quantity, closed by two directed rational logarithm certificates. Every computer-assisted certificate reduces to integer comparisons; a dependency-free program verifies all of them from the supplementary artifact.

Quantum software is increasingly built on fast-moving Python SDKs such as Qiskit, PennyLane, and Cirq. When these SDKs evolve, user programs can fail because execution helpers are removed, import paths change, simulator abstractions shift, device names are deprecated, or circuit export interfaces are revised. Although some failures surface as simple missing-symbol errors, repairing them is not only a syntactic problem: developers must identify the replacement workflow, preserve the quantum-facing intent, and validate behavior under pinned framework versions. This paper presents QEVOLVE-Bench, a small executable seed benchmark for quantum SDK evolution. The current public release contains 14 controlled migration tasks across Qiskit, PennyLane, and Cirq. Each task is a self-contained Python project with pinned dependencies, failing and passing outputs, pytest acceptance checks, metadata, and benchmark notes. The artifact also includes a manifest, sanity checker, representative smoke-test logs, Zenodo archive, screencast, and a lightweight QEVOLVE-Agent prototype that converts task evidence into structured repair proposals. QEVOLVE-Bench is intentionally not a statistically powered benchmark; instead, it is an extensible seed artifact that lowers the setup cost for studying automated repair, LLM-based software engineering, and maintenance of quantum SDK-based projects.

We establish a lower bound for estimating the von Neumann entropy from independent outcomes of any fixed rank-one POVM. A rotation-averaged van Trees argument gives a global minimax risk of at least $(d/n)\log^2\{n/(4d)\}$ when $d\ge C$ and $n\ge Cd$, without a projective-design assumption. We also characterize risk on an operator-norm ball of radius $r$ around the maximally mixed state. We allow an approximate second moment: on the trace-zero Hermitian subspace, the measurement frame may differ by $\varepsilon<1$ from the tight projective frame. A clipped estimator based on canonical dual shadows and the complete U-statistic for purity has risk at most $d^3r^2/n+d^4/n^2+d^6r^6$. Lower bounds under the same frame control yield the local minimax rate $d^3r^2/n+d^4/n^2$ when $n\ge Cd^2$ and $r$ lies in an explicit matching range. For every fixed upper bound on $\varepsilon$, approximation changes only the constants, not the powers of $d,n,r$. At the critical radius $r=n^{-1/2}$, the local risk is asymptotically negligible relative to the global risk when $n\log^2(n/d)\gg d^3$. This separation holds for projective 2-designs, including global Clifford measurements in qubit dimensions, and for their uniformly well-conditioned frame approximations. Finite-sample experiments in dimension four illustrate the critical-radius benchmark and the effect of a nonexact frame.

Let $F_k(G)$ be the $k$-token graph of a finite graph with nonnegative edge weights, and let $A_k$ and $D_k$ be its weighted adjacency and degree matrices. For every $-1\leq\vartheta\leq1$, we prove that $k\mapsto\lambda_{\max}(A_k+\vartheta D_k)$ is discretely concave. Complement symmetry then makes this sequence nondecreasing up to the middle level. At $\vartheta=1$ and $\vartheta=0$, this gives the signless-Laplacian and adjacency spectral-radius monotonicity conjectures of Apte, Parekh, and Sud. The spectral result follows from a finite-time theorem: for every $t\geq0$, the heat contents $\binom{n}{k}^{-1}\mathbf{1}^{*}e^{t(A_k+\vartheta D_k)}\mathbf{1}$ are log-concave in $k$. We encode all token levels in one Lorentzian polynomial. A four-variable operator symbol proves preservation by each edge heat gate, and the Lie-Trotter formula passes this preservation to the full semigroup. Large-time growth rates recover the top eigenvalues. The same construction also yields Lorentzian polynomials from top spectral projections. Finally, we show that the local symbol certifies exactly the parameter range $[-1,1]$.

Let $W$ be a representation of the Weyl relations on a separable Hilbert space and write $\mathcal R_W(K)=\{W(f):f\in K\}''$. If the symplectic pairing of $K_1$ and $K_2$ is nonzero, there is a self-adjoint unitary $B\in\mathcal R_W(K_2)$ such that, for every normal state $\rho$, one can choose a self-adjoint unitary $A_\rho\in\mathcal R_W(K_1)$ with $\operatorname{Var}_\rho(-i[A_\rho,B])=4$. Hence the optimized mean-input-energy-constrained coefficient equals $4$ at every admissible energy threshold. For the fixed trigonometric Weyl witness $h$, an exact identity for $4-h^2$ reduces the deficit to a phase-fixed problem for commuting squared Weyl translations. For the one-mode quadratic energy $G_M=\frac12 R^TMR-\frac12\sqrt{\det M}$, with $M>0$ and $u^T\Omega v=\pi$, we prove $4-\gamma_{G_M,E}(h)=\frac{u^TMu+v^TMv+2\pi\sqrt{\det M}}{4E}+O(E^{-2})$. The lower bound holds over all normal states satisfying the mean-energy constraint, and a localized Zak construction attains the same coefficient. For $G=d\Gamma(H_1)$ and witness directions $u,v\in\operatorname{dom}H_1^{1/2}$, the fixed-witness deficit is $\Theta(E^{-1})$ whenever at least one direction lies outside $\ker H_1$; if both directions are zero modes, the constrained supremum equals the endpoint at every positive threshold.

We develop a holomorphic representation of quantum error correction codes (QECCs) within the Segal–Bargmann space. By encoding qubits into Schwinger boson modes $(z_{a_j}, z_{b_j})$ subject to a degree-one homogeneity constraint, we derive closed-form differential operator representations for stabilizers, syndrome extraction, and recovery for fundamental codes (three-, five-, seven-, and nine-qubit codes). Quantum errors are characterized as holomorphic perturbations violating this constraint, while syndrome measurement projects onto eigenspaces of commuting differential operators. Restricting to unit-magnitude variables ($|z|=1$) reveals a toroidal space $\T^{2n}$ where error syndromes manifest as discrete translations in winding number space $\Z^{2n}$, and recovery acts as Hamiltonian flows restoring the winding configuration. In the full Segal–Bargmann space, the code space is a holomorphic submanifold of $\CP^{2^n-1}$, with correctable errors as transverse normal directions. Consequently, the Knill–Laflamme condition becomes a Fubini–Study orthogonality condition between the code submanifold and its error-translated images. Topological protection emerges from the $U(1)^n$ fiber bundle structure: global phase noise along fibers is unobservable, while base-space errors require active correction. Finally, we establish a path-integral formulation for semiclassical error correction dynamics and show that geometric entanglement via the Segre embedding naturally quantifies code distance. This framework unifies algebraic, geometric, and topological perspectives on fault-tolerant quantum protocols.

Femtoscopic correlation functions (CFs) have been increasingly used to extract strong interactions between pairs of unstable particles, but their physical soundness has recently been questioned. To answer this, we formulate CFs at the operator level, with the observed subsystem described by a reduced density operator and subsequent dynamics absorbed into an effective measurement operator. The Koonin-Pratt form is recovered under four well-motivated reductions. The formulation makes explicit that the source-side and interaction-side representations must be consistently matched, and motivates an operational convention in which a measured reference correlation establishes a compatible source–interaction pairing that can be extended to other pairs for CF-to-CF predictions.

We study a class of Hamiltonians with a structure that we call "dynamical splitting": the Hamiltonian terms can be divided into two sets acting on the same degrees of freedom such that every term in one set commutes with every term in the other, although terms within either set do not all commute. This structure yields an algebraic duality to effective degrees of freedom on which the two parts of the Hamiltonian act disjointly, enabling exact diagonalization on lattices with approximately twice as many spins as usually accessible. We exploit it in the "chiral XYZ model", a geometrically frustrated spin-$1/2$ model on the triangular lattice which was previously introduced as a special limit of a Majorana-Hubbard model. This model also possesses anticommuting noncontractible line symmetries, which enforce an exact, topology-dependent degeneracy between locally indistinguishable states. We first study a $\mathbb{Z}_N$ clock generalization and find, at large $N$, a gapless ground state with three subsystem-symmetry-protected nodal lines. Exploiting dynamical splitting and the subsystem symmetries, we carry out exact diagonalization of the $N=2$ model on lattices up to $9\times9$ spins. The many-body gap and bipartite entanglement provide strong evidence for a gapless state consistent with the large-$N$ nodal structure: the entanglement scales as $L\log L$ and exhibits $1+1$-dimensional CFT-like chord scaling on cylinders. Finally, we study instabilities and proximate phases. In particular, we find evidence that a subsystem-symmetry-preserving deformation drives a finite coupling transition to a gapped phase with $\mathbb Z_2 \times \mathbb Z_2$ topological order.