Papers

We determine the two-way quantum, entanglement-distribution, private, and secret-key capacities of the pure-loss bosonic channel under an unconditional mean transmitted-photon-number constraint. For transmissivity $\eta$ and mean photon number $N$, all four capacities coincide and are given by $g(N)-g((1-\eta)N)$, where $g$ is the entropy of a thermal mode. This resolves the longstanding energy-constrained capacity problem by establishing the optimality of the reverse-coherent-information rate introduced in 2009. Our central tool is sector teleportation simulation, which we develop specifically to transfer the input-energy constraint to the shared entanglement resource, thereby preserving the constraint throughout converse arguments for general adaptive protocols. For every $N>0$, we further show that the corresponding strong-converse thresholds coincide with the unconstrained value $-\log_2(1-\eta)$. For parallel protocols with a hard total-photon-number cutoff, the same energy-constrained capacity formula instead satisfies an exponential strong converse. As an extension, we determine the energy-constrained two-way capacities of the quantum-limited amplifier, showing that classical assistance does not improve its unassisted quantum and private rates. Together, these results establish a unified finite-energy framework for adaptive bosonic communication and reveal how energy constraints fundamentally reshape both attainable rates and converse bounds.

In this paper, we study the weak limit of the invariant measures attached to an analytic family $\{f_t\}_{t\in\mathbb{D}^*}$ of regular polynomial automorphisms over $\mathbb{C}^N$ parametrized by the unit punctured disk which may degenerate at the origin, by using the techniques called hybrid spaces developed by Boucksom–Favre–Jonsson. For each $t$, the invariant measure $\mu_t$ is constructed by Sibony, and we show that the weak limit of $\{\mu_t\}_t$ over the hybrid space is given by the invariant measure of the regular polynomial automorphism $f_{\mathbb{C}((t))}^{\operatorname{an}}$ over $\mathbb{C}((t))$ induced from the original family. This is a generalization of the author's previous result for degenerating families of dynamics of Hénon mappings.

We study agnostic tomography of pure bosonic Gaussian states: given copies of an arbitrary $n$-mode bosonic state $\rho$, the goal is to output a pure Gaussian state whose infidelity with $\rho$ is at most $\mathrm{opt} + \epsilon$, where $\mathrm{opt}$ is the minimum infidelity achievable by any pure Gaussian state. We give efficient protocols achieving this in both the high and low fidelity regimes. When $\mathrm{opt}$ is below some universal constant, our protocol has runtime and copy complexity which is strongly polynomial in $n, 1/\epsilon$ and $\log \log E$, where $E$ is the energy of the closest pure Gaussian state. For arbitrary $\mathrm{opt}$, our protocol uses $(n+1)^{\mathrm{poly}(1/\epsilon)} \mathrm{poly}\left(1+\log\log(E)\right)$ copies and runtime. As a corollary, we obtain the first truly tolerant Gaussianity testing protocol for distinguishing whether $\mathrm{opt} > c + \epsilon$ or $\mathrm{opt} < c - \epsilon$, for any threshold $c\in(0,1)$. We also prove $\mathrm{poly}(n,1/\epsilon)$ runtime is impossible, unless $\mathrm{NP}\subseteq\mathrm{BQP}$. Our protocols follow a shared paradigm: first, we iteratively use general Gaussian measurements combined with techniques from classical robust statistics to obtain a good warm start estimate, then we leverage non-Gaussian measurements to refine this warm start using convex and non-convex optimization methods. Interestingly, we prove that non-Gaussian measurements are necessary to match the strong agnostic guarantees we obtain, and in fact these guarantees are provably superior to what is possible for robustly estimating classical Gaussians.

The Sachdev-Ye-Kitaev model is a strongly interacting fermionic system that has been well-studied in condensed matter and high energy physics. It is highly quantum: Gaussian states are far from the thermal state (Hastings and O'Donnell, STOC'22) and representing the thermal state requires large polynomial-size quantum circuits (Anschuetz et al., QIP'25). Very recently, it was nonetheless proven that classical algorithms can estimate local thermal expectations at sufficiently high temperature in quasipolynomial time (Zlokapa, FOCS'26). We show that classical algorithms can in fact estimate local observables at all constant temperatures in polynomial time. Our techniques also extend straightforwardly to classical systems: we resolve an open question about computing thermal expectations of a classical spin glass up to its phase transition (Bencs et al., STOC'26). Our proof develops a fully rigorous quantum cavity method. Due to the success of the classical cavity method in optimization, sampling, inference and learning, we expect the quantum cavity method to find further applications of independent interest. As an example, we give a quantum algorithm that learns SYK Hamiltonians from the Gibbs state at any constant temperature with polynomial time and sample complexity.

We settle the long-standing Godbersen conjecture that the mixed volume of a convex body of fixed positive volume with its reflection about the origin is maximized precisely by simplices. For the proof, we establish a new characterization of simplices by a natural incidence relation on the face lattice. Moreover, we establish a higher-order generalization of the Godbersen inequality and show that simplices are the only nontrivial equality cases, confirming a conjecture of Schneider from 2000.

Descriptive set theorists often refer to elements of the Baire space, $\mathcal{N} = {}^\omega\omega$, as “reals.” In this expository article, we present mathematical justification for this practice and highlight the advantages of working with $\mathcal{N}$ over $\mathbb{R}$ in the context of descriptive set theory.

Let $\Bbbk$ be an algebraically closed field of characteristic zero and let $f$ be a nonconstant polynomial in finitely many freely noncommuting variables. We prove the asymptotic Makar-Limanov rank conjecture, namely that the normalized rank of a value of $f$ can be made arbitrarily small by choosing matrices over $\Bbbk$ of finite size.

Machine-learning emulators of chaotic and stochastic systems are usually validated on forecast skill and long-run statistics. Neither certifies that an emulator responds correctly to forcing, the property that projection and attribution studies rely on. Linear response theory makes this testable: the forced response follows from unperturbed correlations through a generalized fluctuation-dissipation relation, and decomposes over the stochastic Ruelle-Pollicott resonances of the Koopman generator. Building on the Koopmanism Response framework, we turn this into a calibrated, mode-resolved test for learned surrogates: each surrogate rollout passes or fails each check, and failure rates are compared with those of independent realizations of the true system. On stochastic Lorenz-63, a three-variable toy model, we evaluate SINDy, an MLP, a reservoir computer, a neural ODE and a neural SDE with learned diffusion, over up to 80 rollouts each. A sparse-regression model with the correct library passes every check at rates consistent with the true system. Invariant-statistics fidelity and response fidelity dissociate in both directions: a quarter of reservoir-computer rollouts pass every invariant-statistics check and match the static susceptibility $\chi(0)$, yet misrepresent the slow relaxation modes, while the neural ODE and SDE rarely meet the invariant-statistics floor but recover those modes in three quarters of rollouts. As expected of a time-integrated quantity dominated here by fast relaxation, $\chi(0)$ does not separate these cases. For a fixed network, the training formulation (one-step drift, flow map, or multi-step through the integrator) decides which of these properties it gets right.

Let $1\le k\le n$ and let $A$ be the adjacency matrix of a uniformly random $d$-regular directed graph on $n$ vertices. Suppose that $\lambda n\le d \le (1-\lambda)n$ for a fixed $0<\lambda\le 1/2$. We show that there exists $c_{\lambda}>0$ depending only on $\lambda$ such that $$ \mathbb{P}[\operatorname{rank}(A)\le n-k]\le 2e^{-c_{\lambda} kn}. $$ This gives a large corank extension of the exponential singularity bound of Jain, Sah, and Sawhney.

Gaussian elimination is the oldest and most popular method for solving a linear system. Its numerical stability for a given matrix is controlled by the growth factor, a measure of how large entries can become during elimination. In this work, we prove a number of new results regarding the growth factor of random matrices. First, we provide tight estimates for the growth factor of a matrix preconditioned by a Haar orthogonal matrix without pivoting and characterize the asymptotic distribution of the growth factor of Gaussian matrices without pivoting. Second, and most notably, we prove that the growth factor of an $n \times n$ Gaussian matrix under partial pivoting is rarely much larger than $\sqrt{n}$, resolving an old conjecture of Nick Trefethen. The same techniques used to prove this conjecture also provide an improved smoothed analysis of the growth factor under partial pivoting.

We solve two long outstanding problems here. We first solve in ZFC a problem that the first author and Weaver posed ten years ago: a state on a von Neumann algebra is countably additive on orthogonal projections if and only if it is sequentially weak* continuous, equivalently, sequentially normal. Such states may be considered the quantum (countably additive) probability measures, or rather their extension to a noncommutative integral. The principal tools are known ideas from direct integral theory from the 1970s. Indeed we prove a disintegration theorem for states on products of sigma-finite von Neumann algebras. We give several applications. For example we give a new variant of Gleason theorem, extending countably additive probability measures on projections to sequentially weak* continuous states when the algebra has no type $I_2$ direct summand. We also establish normality and sequential normality criteria for weights, and give several other applications of our main result to weights. In forthcoming work we consider many applications to `quantum measure and integration theory', such as variants of Lebesgue's dominated convergence theorem for von Neumann algebras. The analytic tools and product-quotient construction developed here are used to prove in an Appendix by the first author that normality and sequential normality of arbitrary weights are determined by their restrictions to abelian von Neumann subalgebras, resolving Haagerup's Problem 1.11. (This may be moved to a companion paper.) That is, normality of weights is commutatively determined.

Neural solvers for partial differential equations (PDEs) can approach an accurate solution while their parameters grow without bound. In such cases, the limiting solution may have no finite representation in the chosen model, leaving the best loss unattained. Our analysis connects missing limits in deep neural tanh- networks to unbounded hidden parameters or increasingly redundant neurons. For a class of models built from translated kernels, we describe the missing functions and recover them by adding kernel derivatives to the model. This completion makes the best approximation attainable under standard assumptions. Numerical studies follow the associated parameter growth and explore how completion affects PDE optimization.

In a previous work, we showed that real quantum theory (RQT) cannot be experimentally distinguished from ordinary quantum theory (QT). In this paper, we investigate whether this indistinguishability survives two natural extensions of the setting: allowing separated measurements to be represented by commuting rather than tensor-product observables, and imposing superselection rules, as in fermionic information theory. We show that it does in both cases. Our first result is a positive answer to the Tsirelson problem in RQT: commuting and tensor-product measurements generate the same finite-dimensional correlations. This is surprisingly not a trivial consequence of the standard complex proof, as it does not carry over to the reals. The proof obstruction becomes particularly relevant in the presence of superselection rules, which is why we first establish the equivalence in RQT before extending the framework to superselected RQT. Our second result is a local, real quantum formulation of superselected theories. We extend the RQT framework to accommodate a representation of superselection over the reals and show, in particular, that it admits a fermionic version whose predictions are experimentally indistinguishable from those of fermionic information theory. More generally, the construction extends to all superselected QTs.

We prove that the knowledge of local Cauchy data for a fractional Schrödinger equation on a smooth closed and connected Riemannian manifold of dimension at least two determines the manifold, its metric, and a smooth complex- valued potential, up to an isometry fixing the observation set. The observation set may be any nonempty open subset and no further geometric assumptions are required. The proof uses functional calculus identities to recover the local action of the resolvent of the Laplace-Beltrami operator.

We derive rates of exponential relative-entropy decay for a broad class of infinite-dimensional quantum Gibbs samplers. A key technical ingredient is a perturbative stability theory for hypercontractive estimates and logarithmic Sobolev inequalities. For suitably chosen dissipative couplings, we obtain exponential entropy decay for full multimode Bose-Hubbard samplers with arbitrary hopping and repulsive density interactions. For harmonically confined quantum Coulomb gases and molecular systems, we prove exponential entropy decay for every fixed finite-rank low-energy truncation of the interaction. Finally, we exhibit a periodic charging model with a positive spectral gap but no finite-time $L^2$-to-$L^p$ bound for any $p>2$.

For simple, nonelementary C*-algebras, we show uniform density of functions realized as ranks of operators. It follows that strict comparison by traces forces almost divisibility, and hence pureness. As an application, we show that strict comparison implies $\mathcal{Z}$-stability for separable, simple, nonelementary C*-algebras with locally finite nuclear dimension. In particular, the Toms–Winter conjecture holds for approximately subhomogeneous C*-algebras. More generally, we obtain uniform density of ranks for C*-algebras with the Global Glimm Property. Consequently, an exact C*-algebra is pure if and only if it has strict comparison together with the Global Glimm Property.

We study collective state preparation and channel encoding in fermionic systems under parity superselection and composition governed by the canonical anticommutation relations. For a two-mode Gaussian state $\tau$, we obtain the exact entanglement of formation, $E_{\mathrm F}^{\mathrm{phys}}(\tau^{\otimes m})=\lceil m/2\rceil\log2$ for every $m\geq1$. Two copies have the same formation value as one, violating additivity and strong superadditivity, while regularization halves the single-copy value. For an associated Gaussian channel, we determine the minimum output entropy and Holevo information at every blocklength and prove nonadditivity of both. Together, these results give exact answers to the fermionic counterparts of the four additivity questions associated with Shor and Hastings. One physical use carries no classical information, whereas two uses transmit one bit perfectly. The unassisted classical and zero-error classical capacities both equal one half bit per mode. A sharp finite-block bound gives a strong converse: the decoding success probability vanishes at every rate above capacity. These optimizations cover all physical pure-state decompositions and channel inputs, with Gaussian ensembles and classical codes attaining the optima. The formation and minimum output entropy formulas hold for every Rényi order $\alpha\in(0,\infty]$. We also determine the unassisted quantum and zero-error quantum capacities, both equal to one half logical qubit per mode, and prove a strong converse: entanglement fidelity vanishes at every rate above capacity. For a noisy Gaussian family, we determine the one- and two-copy formation values and prove strict nonadditivity at full rank. For $\tau$, Bell violation under local parity-preserving measurements first appears at three copies.

We study continuous-time supercritical branching random walks in space-inhomogeneous random environment on the Hamming cube $\{0,1,\dots,b-1\}^d$, where the reproduction laws at each site are i.i.d. sampled. This serves as an idealized model for RNA sequence evolution with mutation–selection balance. The reproduction and mutation events are decoupled, and we assume that the reproduction law has essential supremum strictly below the deterministic mutation rate. Our main results provide tight asymptotics of the first-passage times for the model, uniformly in the origin–target distance $1\le m\le d$ as $d\to\infty$, conditional upon survival. We prove both quenched tightness in environment probability and annealed tightness around a deterministic center. Moreover, we identify the leading order of the deterministic center and develop an expansion in the sparse-distance regime $m=o(d)$. Our proof technique derives quantitative approximations of the model using an independent-visit variant, where revisits still resample the environment. As an application, we show that on a macroscopic scale, increasing the reproduction law in convex order decreases the first-passage times, and increasing the mutation rate increases the first-passage times for the sparse regime.

Self-testing certifies quantum states and measurements from observed correlations alone, without requiring a description of the devices that produce them. In this work we establish, to the best of our knowledge, the first robust self-test of the full continuous family of measurements generated by Majorana operators. We achieve this by showing that the Grothendieck game, a two-player nonlocal game in which the players receive directions on a sphere and return binary answers, is a self-test of Majorana observables. That is, any near-optimal performance certifies an entire continuously parameterized family of quantum measurements. We further extend this certification to products of Majorana observables, obtaining a robust self-test of the spin representation of the continuous pin group.

We study the emergent randomness in projected ensembles generated by dual-unitary circuits. Specifically, we consider evolution by solvable dual-unitary brickwork circuits followed by measurements on part of the system, which induce an ensemble of states on the unmeasured subsystem. Previous work has established that this projected ensemble of states forms a quantum state design in the limit where the bath size is large, but the rate of convergence at finite bath size has not been rigorously established. Under three inverse-polynomial assumptions, local permutation mixing, a bound on recurrent backward motion, and contraction at the measured boundary, we fill this gap and prove that a bath of polynomial size suffices to form approximate $k$-designs. In establishing our results, we develop a microscopic transport interpretation of emergent randomness in the projected ensemble, in which deviations from local permutation operators appear as defects, and make connections with the theory of quantum walks.

We study alternateness of the Jones-Kauffman polynomial of virtual links. In particular, we identify a class of virtual links with alternating Jones-Kauffman polynomial that will be called the class of mock-alternating virtual links. Then, we explain how every mock-alternating virtual link can give rise to an infinite family of mock-alternating virtual links using the twisting technique introduced in [4] and later generalized in [5,18]. Finally, we give a connected state expansion of the mock-determinant for checkerboard-colorable links.

We completely determine the possible parities of the Euler characteristic of closed $Pin^{\pm}$-manifolds of a given dimension. As an application we partially calculate the controlled cut-and-paste groups $SKK_{2k+1}^{Pin^{\pm}}$, which were previously undetermined.

This paper is primarily concerned with the existence of a symplectic $4$-manifold, homeomorphic but not diffeomorphic to $CP^2$, which admits a nontrivial finite order symplectomorphism. We reduced the existence of such an exotic $CP^2$ to a question in contact geometry. More concretely, we showed that there are precisely $12$ plumbed manifolds, each equipped with a canonical contact structure associated to the plumbing, such that an exotic $CP^2$ with a nontrivial finite order symplectomorphism exists if and only if one of the $12$ contact plumbed manifolds admits a certain specific symplectic filling. As the main technical result, we completely determined the order and local representations of a pseudofree symplectic $Z_p$-action for an odd prime $p$ on a rational homology $CP^2$ with positive canonical class. Our result suggests that potentially, there are symplectic $Z_p$-actions which are not a finite automorphism of a fake projective plane. Moreover, we introduced a construction, which, under the minimal assumptions on the symplectic fillings of the $12$ contact plumbed manifolds, produces a symplectic $4$-manifold on the Bogomolov-Miyaoka-Yau line via a reverse-engineering process.

We study a variant of Schumacher compression. The task asks for the minimal memory cost for compressing many copies of a density matrix with known spectrum and unknown eigenbasis, without preserving its purification. For fixed dimension and distinct positive eigenvalues, we obtain the cost through its additive constant. Achievability uses a generalization of Werner's cloning map to $\mathrm{GL}(d,\mathbb C)$ irreducible representations, with finite trace-distance bounds controlled by highest-weight differences and row gaps. The matching converse follows from a quantitative form of Koashi–Imoto incompressibility for irreducible group orbits under a spectral-gap assumption. We also identify the relation between this quantum minimum description length and universal lossless coding overhead, and its connection to free entropy is explained in a companion letter. We provide a Lean certificate for our proofs.

We study the dimension defect of finitely many subspaces over an arbitrary field. We derive a symmetric, nonrecursive Grassmann-type formula for the dimension of their sum. The formula expresses the total dimension loss through intersections of a distinguished subspace with partial sums of the remaining subspaces, with coefficients determined by the number of subspaces involved. We also show that the formula admits a Shapley-value interpretation for the associated representable polymatroid. Grouping the correction terms by the number of participating subspaces yields a nonnegative dimension-defect profile. We prove that this profile is monotone and identify its successive gaps with the discrete curvatures of the average-rank profile. These gaps give exact remainders in two-sided defect bounds. Equality in either bound holds precisely when the images of the subspaces in the quotient by their common intersection form an internal direct sum. We also give an exact geometric decomposition of the Kinser slack into nonnegative quotient dimensions and characterize equality. Averaging these slacks over permutations and contractions recovers every defect curvature except the final one; additional representability constraints remain in the individual ordered slacks. Weighted and dual formulas accompany the expansion, and entropy analogues express the defect levels and curvatures as averages of mutual and conditional mutual information, respectively.

For any Galois field $\mathbb{F}_q$ with $q$ elements and any positive integers $m$ and $n$, the directed graph $D(q;m,n)$ has vertex set $\mathbb{F}_q\times\mathbb{F}_q$, and there is an arc from vertex $(x_1,x_2)$ to vertex $(y_1,y_2)$ if and only if $x_2+y_2=x_1^my_1^n$. It was conjectured in earlier work that two digraphs $D(q;m_1,n_1)$ and $D(q;m_2,n_2)$ are isomorphic if and only if there exists an integer $k$ relatively prime to $q-1$ such that $m_2\equiv km_1$ and $n_2 \equiv kn_1$, where both congruences are modulo $q-1$. We prove this conjecture over prime fields and construct an infinite family of counterexamples over extension fields.

We establish staircase phase transitions for the largest eigenvalue of heavy-tailed sample correlation matrices formed from a $p_n \times n$ data matrix with i.i.d. real entries of mean zero and unit variance, allowing an infinite fourth moment. In the proportional regime $p_n / n \to \phi \in (0, \infty)$, the first-order asymptotics depend jointly on the aspect ratio and the entry tail. The transitions are driven by collisions of large entries in distinct rows of a common column. The first collision order capable of producing a separated upper outlier is $k_* (\phi) = \lfloor \sqrt{\phi} \rfloor + 2$, yielding the critical tail exponent $\alpha_* (\phi) = 2 + 2 / k_* (\phi)$. This exponent decreases in steps as $\phi$ increases, creating a staircase boundary between convergence to the upper Marčenko–Pastur edge and successive outlier levels. At exact critical tail scales, the point process of eigenvalues above the upper edge or the preceding deterministic level converges to a Poisson point process. The resulting nondegenerate limiting laws for the largest eigenvalue connect adjacent phases and have a positive atom at this baseline. If every fixed collision order is supercritical, the largest eigenvalue diverges in probability despite finite entry variance.

We prove that for any $p>4$ we can find a Schatten class Hankel operator $\Ha_\phi$ defined on the Hardy space of the infinite torus $H^2(\mathbb{T}^\infty)$ that belongs to the Schatten class $S^p$ but does not admit a bounded symbol, extending the previously known range $p>(1-\log\pi /\log 4)^{-1}=5.7388...$.

We design the first polynomial-time algorithms for approximately counting and almost uniformly sampling common bases of two matroids given by their independence oracles. Moreover, our algorithms generalize far beyond this to Hadamard products of two probability measures on the Boolean cube satisfying a simple nonnegative curvature condition. These algorithmic primitives have myriad applications in statistical physics, polyhedral combinatorics, the study of quantum many-body systems, and beyond. Our approach has two key ingredients. $\bullet$ We relax the intersection by imposing an overlap penalty on the product measure formed by the two input measures. We prove, via an integrated Bochner-type method, that this "$\textit{soft intersection}$" satisfies a Poincare inequality uniformly over all external fields. $\bullet$ We solve a dual maximum entropy convex program to compute external fields under which the hard constraint is satisfied with high probability under the soft intersection measure. We bound this success probability directly using the uniform Poincare inequality and smallness of the gradient norm. $\textbf{AI Disclosure}$ GPT-5.6 Sol Ultra and GPT-6 Astra Ultra were heavily used to develop the ideas in this paper. A more complete discussion is included in the acknowledgments.

The no-cloning theorem is a cornerstone result in quantum mechanics that forbids copying of general quantum information. More quantitatively, even given $N$ copies of an unknown state, any channel producing an $(N+1)$-copy state incurs a nonzero trace-distance error on some inputs. Turning this around, one can ask about the sample complexity of approximate $N\rightarrow N+1$ cloning: for a desired small error $\epsilon$, what $N$ suffices? For arbitrary pure states on a Hilbert space $H$, the answer is $N\simeq \dim H/\epsilon$ - astronomically large for most $H$ of interest. What if the input is promised to lie in a structured family? We examine a range of families fundamental to many-body physics and quantum information: $n$-mode fermionic Gaussian and Slater states, $n$-mode bosonic Gaussian states, and qudit phase states. For fermions, we construct optimal cloning channels, reducing the complexity from exponential to polynomial in $n$. For bosonic Gaussian states, we construct an explicit cloner that certifies polynomial complexity, without any constraint on the energy of the state; this channel, however, is not optimal in general. The fermionic and bosonic results follow from a single representation-theoretic framework, which generalizes previous work by Werner and by Chiribella and Yang; it also yields the cloners' unitary implementation. Furthermore, we explain why for many families the cloning complexity is linear in the dimension of the family manifold, via a controlled saddle-point evaluation of the frame potential at high $N$. The same calculation accounts for the analogous scaling of tomography complexity. The above framework does not cover phase states; for these we give an independent construction of a channel with optimal cloning fidelity. Our work complements recent results on cloning stabilizer states.