Arrays of gate-defined semiconductor quantum dots are among the leading candidates for building scalable quantum processors. High-fidelity initialization, control, and readout of spin qubit registers require exquisite and targeted control over key Hamiltonian parameters that define the electrostatic environment. However, due to the tight gate pitch, capacitive crosstalk between gates hinders independent tuning of chemical potentials and interdot couplings. While virtual gates offer a practical solution, determining all the required cross-capacitance matrices accurately and efficiently in large quantum dot registers is an open challenge. Here, we establish a modular automated virtualization system (MAViS)—a general and modular framework for autonomously constructing a complete stack of multilayer virtual gates in real time. Our method employs machine learning techniques to rapidly extract features from two-dimensional charge stability diagrams. We then utilize computer vision and regression models to self-consistently determine all relative capacitive couplings necessary for virtualizing plunger and barrier gates in both low- and high-tunnel-coupling regimes. Using MAViS, we successfully demonstrate accurate virtualization of a dense two-dimensional array comprising ten quantum dots defined in a high-quality $\mathrm{Ge}/\mathrm{SiGe}$ heterostructure. Our work offers an elegant and practical solution for the efficient control of large-scale semiconductor quantum dot systems.

The insulating magnetic pyrochlore ${\mathrm{Ce}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7$ has gained attention as a quantum spin-ice candidate with dipole-octupole character that arises from the crystal-electric-field ground-state doublet for the ${\mathrm{Ce}}^{3+}$ Kramers ion. This dipole-octupole character permits both spin-ice phases based on magnetic dipoles and those based on more-exotic octupoles. This work reports low-temperature neutron diffraction measurements on single-crystal ${\mathrm{Ce}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7$ with $Q$ coverage both at low $Q$, where the magnetic form factor for dipoles is near maximal, and at high $Q$, covering the region where the magnetic form factor for ${\mathrm{Ce}}^{3+}$ octupoles is near maximal. This study was motivated by recent powder neutron diffraction studies of other Ce-based dipole-octupole pyrochlores, ${\mathrm{Ce}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$ and ${\mathrm{Ce}}_2{\mathrm{Hf}}_2{\mathrm{O}}_7$, which each showed temperature-dependent diffuse diffraction at high $Q$, interpreted as arising from octupolar correlations. Our measurements use an optimized single-crystal diffuse scattering instrument that allows us to screen against strong Bragg scattering from ${\mathrm{Ce}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7$. The temperature-difference neutron diffraction reveals a low-$Q$ peak consistent with dipolar spin-ice correlations reported in previous work, and an alternation between positive and negative net intensity at higher $Q$. These features are consistent with our numerical-linked-cluster calculations using pseudospin interaction parameters previously reported for ${\mathrm{Ce}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7$, ${\mathrm{Ce}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$, and ${\mathrm{Ce}}_2{\mathrm{Hf}}_2{\mathrm{O}}_7$. Importantly, neither the measured data nor any of the NLC calculations show evidence for increased scattering at high $Q$ resulting from octupolar correlations. We conclude that at the lowest attainable temperature for our measurements ($T=0.05\mathrm{K}$), scattering from octupolar correlations in ${\mathrm{Ce}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7$ is not present in the neutron diffraction signal on the level of our observation threshold of around 0.1% of the low-$Q$ dipole scattering. We compare these results to those obtained earlier on powder ${\mathrm{Ce}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$ and ${\mathrm{Ce}}_2{\mathrm{Hf}}_2{\mathrm{O}}_7$, and to low-energy inelastic neutron scattering from single-crystal ${\mathrm{Ce}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7$.

An emerging application of resonant inelastic x-ray scattering (RIXS) is the study of lattice excitations and electron-phonon ($e$-ph) interactions in quantum materials. Despite the growing importance of this area of research, the community lacks a complete understanding of how the RIXS process excites the lattice and how these excitations encode information about the $e$-ph interactions. Here, we present a detailed study of the RIXS spectra of the Hubbard-Holstein model defined on extended one-dimensional lattices. Using the density matrix renormalization group method, we compute the RIXS response while treating the electron mobility, many-body interactions, and core-hole interactions on an equal footing. The predicted spectra exhibit notable differences from those obtained using the commonly adopted Lang-Firsov models, with important implications for analyzing past and future experiments. Our results provide a deeper understanding of how RIXS probes $e$-ph interactions and set the stage for a more realistic analysis of future experiments.

The superconducting state of the heavy-fermion metal ${\mathrm{UTe}}_2$ has attracted considerable interest because of evidence of spin-triplet Cooper pairing and nontrivial topology. Progress on these questions requires identifying the presence or absence of nodes in the superconducting gap function and their dimension. In this article, we report a comprehensive study of the influence of disorder on the thermal transport in the superconducting state of ${\mathrm{UTe}}_2$. Through detailed measurements of the magnetic-field dependence of the thermal conductivity in the zero-temperature limit, we obtain clear evidence of the presence of point nodes in the superconducting gap for all samples with transition temperatures ranging from 1.6 to 2.1 K obtained by different synthesis methods, including a refined self-flux method. This robustness implies the presence of symmetry-imposed nodes throughout the range studied, further confirmed via disorder-dependent calculations of the thermal transport in a model with a single pair of nodes. In addition to capturing the temperature dependence of the thermal conductivity up to $T_c$, this model provides some information about the locations of the nodes, suggesting a $B_{1u}$ or $B_{2u}$ symmetry for the superconducting order parameter. Additionally, comparing the new, ultrahigh conductivity samples to older samples reveals a crossover between a low-field and a high-field regime at a single value of the magnetic field in all samples. In the high-field regime, the thermal conductivity at different disorder levels differs from each other by a simple offset, suggesting that some simple principle determines the physics of the mixed state, a fact which may illuminate trends observed in other clean nodal superconductors.

Song and Bernevig (SB) have recently proposed a topological heavy-fermion description of the physics of magic angle twisted bilayer graphene (MATBG), involving the hybridization of flat-band electrons with a relativistic conduction sea. Here, we explore the consequences of this model, seeking a synthesis of understanding drawn from heavy-fermion physics and MATBG experiments. Our work identifies a key discrepancy between measured and calculated on-site Coulomb interactions, implicating renormalization effects that are not contained in the current model. With these considerations in mind, we consider a SB model with a single, renormalized on-site interaction between the $f$ electrons, containing a phenomenological heavy-fermion binding potential on the moiré $AA$ sites. This feature allows the simplified model to capture the periodic reset of the chemical potential with filling and the observed stability of local moment behavior. We argue that a two-stage Kondo effect will develop in MATBG as a consequence of the relativistic conduction band: Kondo I occurs at high temperatures, establishing a coherent hybridization at the $\mathrm{\Gamma }$ points and a non-Fermi liquid of incoherent fermions at the moiré $K$ points; at much lower temperatures, Kondo II leads to a Fermi liquid in the flat band. Utilizing an auxiliary-rotor approach, we formulate a mean-field treatment of MATBG that captures this physics, describing the evolution of the normal state across a full range of filling factors. By contrasting the relative timescales of phonons and valence fluctuations in bulk heavy-fermion materials with that of MATBG, we are led to propose a valley-polaron origin to the Coulomb renormalization and the heavy-fermion binding potential identified from experiment. We also discuss the possibility that the two-fluid, non-Fermi liquid physics of the relativistic Kondo lattice is responsible for the strange-metal physics observed in MATBG.

Recent breakthrough experiments have demonstrated how it is now possible to explore the dynamics of quantum Hall states interacting with quantum electromagnetic cavity fields. While the impact of strongly coupled nonlocal cavity modes on integer quantum Hall physics has been recently addressed, the effects on fractional quantum Hall (FQH) liquids—and, more generally, fractionalized states of matter—remain largely unexplored. In this work, we develop a theoretical framework for the understanding of FQH states coupled to quantum light. In particular, combining analytical arguments with tensor network simulations, we study the dynamics of a $\nu =1/3$ Laughlin state in a single-mode cavity with finite electric field gradients. We find that the topological signatures of the FQH state remain robust against the nonlocal cavity vacuum fluctuations, as indicated by the endurance of the quantized Hall resistivity. The entanglement spectra, however, carry direct fingerprints of light-matter entanglement and topology, revealing peculiar polaritonic replicas of the U(1) counting. As a further response to cavity fluctuations, we also find a squeezed FQH geometry, encoded in long-wavelength correlations. By exploring the low-energy excited spectrum inside the FQH phase, we identify a new neutral quasiparticle, the graviton polariton, arising from the hybridization between quadrupolar FQH collective excitations (known as gravitons) and light. Pushing the light-matter interaction to ultrastrong-coupling regimes, we find other two important effects, a cavity vacuum-induced Stark shift for charged quasiparticles and a potential instability toward a density modulated stripe phase, competing against the phase separation driven by the Stark shift. Finally, we discuss the experimental implications of our findings and possible extension of our results to more complex scenarios.

Understanding nanoscale electronic and thermal transport of two-dimensional (2D) electron systems in the quantum Hall regime, particularly in the bulk insulating state, poses considerable challenges. One of the primary difficulties arises from the presence of chiral edge channels, whose transport behavior obscures the investigation of the insulating bulk. Using near-field optical and photocurrent nanoscopy, we probe real-space variations of the optical and thermal dynamics of graphene in the quantum Hall regime without relying on complex sample or electrode geometries. Near the charge-neutrality point, we detect strong optical and photothermal signals from resonant inter-Landau-level (LL) magnetoexciton excitations between the zeroth and $\pm 1$ st LLs, which gradually weaken with increasing doping due to Pauli blocking. Interestingly, at higher doping levels and full-integer LL fillings, photothermal signals reappear across the entire sample over an approximately $10\text{−}\mathrm{\mu }\mathrm{m}$ scale, indicating unexpectedly long cooling lengths and nonlocal photothermal heating through the insulating bulk. This observation suggests that thermal conductivity persists for the localized states even as electronic transport is suppressed—a clear violation of the Wiedemann-Franz law. Our experiments provide novel insights into nanoscale thermal and electronic transport in incompressible 2D gases, highlighting the roles of magnetoexcitons and chiral edge states in the thermo-optoelectric dynamics of the Dirac quantum Hall state.

Competing interactions in frustrated magnets can give rise to highly degenerate ground states from which correlated liquidlike states of matter often emerge. The scaling of this degeneracy influences the ultimate ground state, with extensive degeneracies potentially yielding quantum spin liquids, while subextensive or smaller degeneracies yield static orders. A long-standing problem is to understand how ordered states precipitate from this degenerate manifold and what echoes of the degeneracy survive ordering. Here, we use neutron scattering to experimentally demonstrate a new “nodal-line” spin liquid, where spins collectively fluctuate within a subextensive manifold spanning one-dimensional lines in reciprocal space. Realized in the spin-orbit-coupled, face-centered-cubic iridate ${\mathrm{K}}_2{\mathrm{IrCl}}_6$, we show that the subextensive degeneracy is robust, but remains susceptible to fluctuations or longer-range interactions which cooperate to select a magnetic order at low temperatures. Proximity to the nodal-line spin liquid influences the ordered state, enhancing the effects of quantum fluctuations that in turn act to stabilize the sublattice magnetization through the self-consistent opening of a large spin-wave gap. Our results demonstrate how quantum fluctuations can act counterintuitively in frustrated materials: Even in a case where fluctuations are ineffective at selecting an ordered state from a degenerate manifold, at the brink of the nodal spin liquid, they can act to protect the ordered state and dictate its low-energy physics.

Monitored quantum systems undergo measurement-induced phase transitions (MiPTs) stemming from the interplay between measurements and unitary dynamics. When the detector readout is postselected to match a given value, the dynamics is generated by a non-Hermitian Hamiltonian with MiPTs characterized by different universal features. Here, we derive a partial postselected stochastic Schrödinger equation based on a microscopic description of continuous weak measurement. This formalism connects the monitored and postselected dynamics to a broader family of stochastic evolution. We apply the formalism to a chain of free fermions subject to partial postselected monitoring of local fermion parities. Within a two-replica approach, we obtain an effective bosonized Hamiltonian in the strong postselected limit. Using a renormalization group analysis, we find that the universality of the non-Hermitian MiPT is stable against a finite (weak) amount of stochasticity. We further show that the passage to the monitored universality occurs abruptly at finite partial postselection, which we confirm from the numerical finite size scaling of the MiPT. Our approach establishes a way to study MiPTs for arbitrary subsets of quantum trajectories and provides a potential route to tackle the experimental postselected problem.

Quantum critical phenomena are widely studied across various materials families, from high-temperature superconductors to magnetic insulators. They occur when a thermodynamic phase transition is suppressed to zero temperature as a function of some tuning parameter such as pressure or magnetic field. This generally yields a point of instability—a so-called quantum critical point—at which the phase transition is driven exclusively by quantum fluctuations. Here, we show that the heavy fermion metamagnet ${\mathrm{UTe}}_2$ possesses a quantum phase transition at extreme magnetic field strengths of over 70 T. Rather than terminating at one singular point, we find that the phase boundary is sensitive to magnetic field components in each of the three Cartesian axes of magnetic field space. This results in the transition surface being bounded by a continuous ring of quantum critical points, the locus of which forms an extended line of quantum criticality—a novel form of quantum critical phase boundary. Within this quantum critical line sits a magnetic field-induced superconducting state in a toroidal shape, which persists to fields over 70 T. We model our data by a phenomenological free energy expansion and show how a quantum critical line—rather than a more conventional singular point of instability—anchors the remarkable high magnetic field phase landscape of ${\mathrm{UTe}}_2$.

The symmetry of superconducting and normal states is at the core of superconductivity research. Emergent electronic nematicity, which spontaneously breaks the rotational symmetry, has been found in the normal state of various types of unconventional superconductors. Here, we exploit the angle-resolved resistivity method to systematically measure the nematicity of the interface superconducting ${\mathrm{LaAlO}}_3/{\mathrm{KTaO}}_3(111)$ ($\mathrm{LAO}/\mathrm{KTO}$). Compared to the normal state, electronic nematicity is enhanced substantially by superconducting fluctuations around the superconducting temperature $T_{\mathrm{c}}$. More importantly, $T_{\mathrm{c}}$ is also anisotropic in plane and angle dependent. The nematicity consists of a dominant ${\mathrm{C}}_2$ component and a ${\mathrm{C}}_4$ component, which can be explained by the presence of nematic domains. After the superconductivity is suppressed by a magnetic field, the uncovered quantum metal state manifests significant nematicity that is contributed by residual nematic superconducting fluctuations. A coherent picture of nematic interface superconductivity can be retrieved from the measured nematicity phase diagram that is crucial for the understanding of quantum metal state, electronic nematicity, and interface superconductivity.

A general framework for analyzing the topology of quantum channels of single-particle systems is developed to find a class of genuinely dynamical topological phases that can be realized by means of discrete quantum feedback control. We provide a symmetry classification of quantum channels by identifying ten symmetry classes of discrete quantum feedback control with projective measurements. We construct various types of topological feedback control by using topological Maxwell demons that achieve robust feedback-controlled chiral or helical transport against noise and decoherence. Topological feedback control thus offers a versatile tool for creating and controlling nonequilibrium topological phases in open quantum systems that are distinct from non-Hermitian and Lindbladian systems and should provide a guiding principle for topology-based design of quantum feedback control.

Solid-state quantum registers consisting of optically active electron spins with nearby nuclear spins are promising building blocks for future quantum technologies. For electron spin-1 registers, dynamical decoupling (DD) quantum gates have been developed that enable the precise control of multiple nuclear spin qubits. However, for the important class of electron spin-$1/2$ systems, this control method suffers from intrinsic selectivity limitations, resulting in reduced nuclear spin gate fidelities. Here, we demonstrate improved control of single nuclear spins by an electron spin-$1/2$ using dynamically decoupled radio-frequency (DDRF) gates. We make use of the electron spin-$1/2$ of a diamond tin-vacancy center, showing high-fidelity single-qubit gates, single-shot readout, and spin coherence beyond a millisecond. The DD control is used as a benchmark to observe and control a single ${}^{1}3\mathrm{C}$ nuclear spin. Using the DDRF control method, we demonstrate improved control on that spin. In addition, we find and control an additional nuclear spin that is insensitive to the DD control method. Using these DDRF gates, we show entanglement between the electron and the nuclear spin with 72(3)% state fidelity. Our extensive simulations indicate that DDRF gate fidelities well in excess are feasible. Finally, we employ time-resolved photon detection during readout to quantify the hyperfine coupling for the electron’s optically excited state. Our work provides key insights into the challenges and opportunities for nuclear spin control in electron spin-$1/2$ systems, opening the door to multiqubit experiments on these promising qubit platforms.

Although spin and charge stripes in high-$T_c$ cuprates have been extensively studied, the exact range of carrier concentration over which they form a static order remains uncertain, complicating efforts to understand their significance. The problem is challenging due to the combined effects of quenched disorder and competition with superconductivity—both significant in cuprates—which add to the inherent difficulty of determining phase boundaries. In ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$ (LSCO) and in zero external magnetic field, static spin stripes are confined to a doping range well below $p^{*}$, the pseudogap boundary at zero temperature. However, when high fields suppress the competing effect of superconductivity, spin-stripe order is found to extend up to $p^{*}$. Here, we investigate ${\mathrm{La}}_{1.8-x}{\mathrm{Eu}}_{0.2}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$ (Eu-LSCO) using ${}^{139}\mathrm{La}$ nuclear magnetic resonance and observe field-dependent spin fluctuations suggesting a similar competition between superconductivity and spin order as in LSCO. Nevertheless, we find that static spin stripes are present practically up to $p^{*}$ irrespective of field strength: The stronger stripe order in Eu-LSCO prevents superconductivity from enforcing a nonmagnetic ground state, except very close to $p^{*}$. Thus, spin-stripe order is consistently bounded by $p^{*}$ in both LSCO and Eu-LSCO, despite their differing balances between stripe order and superconductivity. This indicates that the canonical stripe order, where spins and charges are intertwined in a static pattern, is fundamentally tied to the pseudogap phase, though the exact nature of this connection has yet to be elucidated. Any stripe order beyond the pseudogap endpoint must then be of a different nature: Either spin and charge orders remain intertwined, but both fluctuating, or only spin order fluctuates while charge order remains static. The presence of spin-stripe order up to $p^{*}$ and the pervasive, slow, and field-dependent spin-stripe fluctuations, as well as the electronic inhomogeneity documented in this work, must all be carefully considered in discussions of Fermi surface transformations, putative quantum criticality, and strange metal behavior.

The discovery of superconductivity in pressurized bilayer and trilayer nickelates has generated significant interest. However, their superconducting properties are often dependent on sample quality and pressure conditions, complicating the interpretation of the underlying physics. Finding new systems with optimized bulk superconducting properties is therefore important for advancing our understanding of these materials. Unlike cuprates, where trilayer compounds typically exhibit the highest transition temperature ($T_c$), the bilayer nickelate ${\mathrm{La}}_3{\mathrm{Ni}}_2{\mathrm{O}}_7$ has thus far outperformed the trilayer ${\mathrm{La}}_4{\mathrm{Ni}}_3{\mathrm{O}}_{10}$ in reported $T_c$. Whether the trilayer nickelates have achieved the optimal $T_c$ remains unclear, with various scenarios suggesting different possibilities. Here, we report the discovery of bulk superconductivity in pressurized ${\mathrm{Pr}}_4{\mathrm{Ni}}_3{\mathrm{O}}_{10}$ single crystals, achieving a maximum onset $T_c$ of 40.5 K at 80.1 GPa, significantly exceeding the 30 K observed in ${\mathrm{La}}_4{\mathrm{Ni}}_3{\mathrm{O}}_{10}$. The bulk nature of superconductivity is confirmed by zero resistance and a strong diamagnetic response below $T_c$, with a superconducting volume fraction exceeding 80%. These findings establish trilayer nickelates as genuine bulk high-temperature superconductors, provide new insights into the mechanisms driving superconductivity, and point to a promising route toward further enhancing superconducting properties in nickelates.

We use nonlinear pump-probe spectroscopy to study optical excitations in a charge-tunable ${\mathrm{MoSe}}_2/{\mathrm{WS}}_2$ moiré heterostructure. An intense red-detuned laser pulse creates a photonic dressing of the material by introducing a large virtual population of excitons or exciton polarons in a deep moiré potential. By measuring the resulting ac Stark effect with a weak resonant laser pulse, we gain access to the nature and mutual interactions of the elementary optical excitations. At charge neutrality, our measurements reveal that different exciton resonances, associated with confinement of their center-of-mass motion in the moiré potential, have a significant spatial overlap. The resulting short-range interactions manifest themselves as a density-dependent blueshift for same-valley excitons and bound biexciton states for opposite-valley excitons. The attractive polaron resonance that appears upon injection of electrons into the heterostructure shows a contrasting behavior: Here, we observe an electron-density-independent light shift and a clear pump-power-dependent saturation. These features are equivalent to that of an ensemble of independent two-level emitters and indicate a breakdown of the Fermi-polaron picture for optical excitations of electrons subject to a strong moiré potential. Our work establishes an experimental approach to elucidate the elementary optical excitations of semiconductor moiré heterostructures, providing a solid ground for the spectroscopy of correlated electronic and excitonic states in such materials.

Electron-boson coupling in unconventional superconductors is one of the key parameters in understanding the superconducting pairing symmetry. Here, we report definitive photoemission evidence of electron-spin fluctuation coupling in the iron-based superconductor ${\mathrm{CaKFe}}_4{\mathrm{As}}_4$, obtained via high-resolution ARPES. Our study identifies a distinct kink structure on the $\alpha$ band, observable only in the superconducting phase and closely linked with the superconductivity, indicative of strong electron-boson interactions. Notably, this kink structure corresponds to two distinct bosonic modes at 11 meV and 13 meV, aligning with spin resonance modes previously observed in inelastic neutron-scattering experiments. This alignment underscores the significant role of antiferromagnetic fluctuations in the pairing mechanism of this superconductor. Furthermore, the unique momentum-dependent and orbital-selective properties of the coupling revealed by ARPES provide profound insights into the pairing symmetry, suggesting predominantly $s_{\pm }$-wave pairing facilitated by spin fluctuations. Our findings not only highlight the pivotal role of spin resonance in the superconductivity of ${\mathrm{CaKFe}}_4{\mathrm{As}}_4$ but also enhance our understanding of the electron-spin fluctuation interactions in unconventional superconductors.

Landau’s Fermi-liquid (FL) theory has been successful at the phenomenological description of the normal phase of many different Fermi systems. Using a dilute atomic Fermi fluid with tunable interactions, we investigate the microscopic basis of Landau’s theory with a system describable from first principles. We study transport properties of an interacting Fermi gas by measuring its density response to a periodic external perturbation. In an ideal Fermi gas, we measure for the first time the celebrated Lindhard function. As the system is brought from the collisionless to the hydrodynamic regime, we observe the emergence of sound and find that the experimental observations are quantitatively understood with a first-principle transport equation for the FL. When the system is more strongly interacting, we find deviations from such predictions. Finally, we measure the momentum-space shape of the quasiparticle excitations and see how it evolves from the collisionless to the collisional regime. Our study establishes this system as a clean platform for studying Landau’s theory of the FL and paves the way for extending the theory to more exotic conditions, such as nonlinear dynamics and FLs with strong correlations in versatile settings.

The novel concept of entanglement renormalization and its corresponding tensor network renormalization technique have been highly successful in developing a controlled real-space renormalization group (RG) scheme. Numerically approximate fixed-point (FP) tensors are widely used to extract the conformal data of the underlying conformal field theory (CFT) describing critical phenomena. In this paper, we present an explicit analytical construction of the FP tensor for 2D rational CFT. We define it as a correlation function between the “boundary-changing operators” (BCO) on triangles. Our construction fully captures all the real-space RG conditions. We also provide concrete examples, such as Ising, Yang-Lee, and tricritical Ising models, to compute the scaling dimensions explicitly based on the corresponding FP tensor. The BCO descendants turn out to be an optimal basis such that truncation in bond dimensions naturally produces comparable accuracies with the leading existing FP algorithms. Interestingly, our construction of FP tensors is closely related to a strange correlator, where the holographic picture naturally emerges. Our results also open a new door toward understanding CFT in higher dimensions.

Quantum entanglement measures of many-body states have been increasingly useful to characterize phases of matter. Here, we explore a surprising connection between mixed-state entanglement and ’t Hooft anomaly. More specifically, we consider lattice systems in $d$ space dimensions with anomalous symmetry $G$ where the anomaly is characterized by an invariant in the group cohomology $H^{d+2}[G,U(1)]$. We show that any mixed state $\rho$ that is strongly symmetric under $G$, in the sense that $G\rho \propto \rho$ is necessarily ($d+2$)-nonseparable, i.e., is not the mixture of tensor products of $d+2$ states in the Hilbert space. Furthermore, such states cannot be prepared from any ($d+2$)-separable states using finite-depth local quantum channels, so the nonseparability is long-ranged in nature. We provide proof of these results in $d\le 1$ and plausibility arguments in $d>1$. The anomaly-nonseparability connection, thus, allows us to generate simple examples of mixed states with nontrivial long-ranged multipartite entanglement. In particular, in $d=1$ we find an example of intrinsically mixed quantum phase, in the sense that states in this phase cannot be two-way connected to any pure state through finite-depth local quantum channels. We also analyze a mixed anomaly involving both strong and weak symmetries, including systems constrained by the Lieb-Schultz-Mattis type of anomaly. We find that, while strong-weak mixed anomaly, in general, does not constrain quantum entanglement, it does constrain long-range correlations of mixed states in nontrivial ways. Namely, such states are not symmetrically invertible and not gapped Markovian, generalizing familiar properties of anomalous pure states.