Papers

In this work, we define entanglement-assisted quantum locally recoverable codes with availability, in which any set of up to $\delta-1$ erased qudits can be recovered from any one of $t$ local recovery sets, each of size at most $r+\delta-1$, with the recovery sets intersecting exactly in the erased coordinates, where $r$ is a (small) positive integer. We show that shared entanglement permits $t>1$, meaning that multiple local recovery sets can be available for the same set of up to $\delta-1$ erasures. We establish a Singleton-like bound for this family of codes and present random constructions based on classical linear codes with Vandermonde parity-check matrices. We also provide explicit constructions of entanglement-assisted quantum locally recoverable codes with availability from several classical code families and their folded versions, including Tamo-Barg codes, fiber-product codes, and algebraic-geometry codes such as one-point Hermitian and Suzuki codes.

The description of open quantum system dynamics beyond the perturbative treatment (usually associated with Markovian master equations) is a computationally challenging task due to the unfavorable exponential scaling of memory kernels. Developed over recent decades in the context of quantum information and condensed matter, tensor networks provide both a new formalism and a toolbox for overcoming previous computational bottlenecks. This framework enables the formulation of non-perturbative, numerically exact methods for describing the dynamics of open quantum systems to controllable numerical accuracy. In this review, we present these methods and discuss their commonalities and differences to paint a comprehensive view of the field.

A quantum state is called absolutely separable (resp. absolutely positive partial transpose (APPT)) if it remains separable (resp. positive partial transpose (PPT)) under any global unitary transformation. It is known that the set of all absolutely separable states is a subset of the set of all APPT states. Moreover, these two sets are identical for qubit-qudit systems. In this note, we give an upper bound for the purity of APPT bipartite states (and therefore for the purity of absolutely separable bipartite states). For qubit-qudit systems, this upper bound becomes the maximum purity of APPT (and absolutely separable) states.

We revisit the notion of “quantum information flow” introduced in Bob Coecke's early work and seek to give it an explicit string-diagrammatic formalization. Given a semantics preserving string-diagram rewriting sequence, we first distinguish apparent through-paths, which depend on the current graphical presentation, from genuine through paths, which can be compatibly inherited through successive rewrites to a terminal decoupled bare wire factor. We then formally define a “Coecke flow line” in terms of this compatible inheritance relation. In particular, we use the ZX calculus, i.e., the ZX string diagram rewriting system, to illustrate the resulting formalism. In the physical setting of quantum protocols, a Coecke flow can be interpreted as a constrained, quasi-local, line-like presentation of a target bare wire morphism factor within the protocol string diagram.

The barren plateau (BP) phenomenon poses a fundamental challenge to the trainability of variational quantum eigensolvers (VQEs) by causing exponentially vanishing gradients as the system size increases. While extensive studies have investigated the geometric origins of BP, its impact on the optimization dynamics and complexity of practical algorithms under finite-shot measurements remains poorly understood. In this paper, we develop a theoretical framework that characterizes how the BP affects the optimization dynamics of the Simultaneous Perturbation Stochastic Approximation (SPSA) algorithm and quantifies the resulting iteration complexity and measurement budget. We derive non-asymptotic bias and variance characterizations of the SPSA gradient estimator, introduce a signal-to-noise ratio analysis to quantify gradient reliability, and establish convergence guarantees for SPSA under finite-shot measurements. Our results show that the exponentially decaying gradient energy associated with BP leads to an exponential increase in the number of iterations required to achieve a fixed relative optimization accuracy, which in turn results in an exponential increase in the total measurement budget.

Carleman lifting converts nonlinear polynomial dynamics into finite linear systems, but the resulting truncations are generally nonunitary and need not correspond to physical quantum evolution. We prove that a finite linear endpoint admits an autonomous Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) realization on vacuum coherences if and only if it is invertible and power bounded. The construction is explicit and realizes the nonunitary map directly as open-system dynamics, with no endpoint postselection and with one encoding and one decoding over repeated timesteps. We apply the result to the complete D2Q9 multiple-relaxation-time lattice Boltzmann (LB) timestep by compiling collision and periodic streaming into a single Carleman endpoint. The resulting GKSL evolution reproduces the classical Carleman trajectory over multiple timesteps, while the remaining discrepancy from nonlinear LB dynamics is the expected Carleman truncation error. The result establishes a general criterion for autonomous open-quantum realization of finite nonunitary dynamics, with Carleman–LB dynamics as a concrete example.

Detection-efficiency mismatch is a common problem in realistic quantum key distribution (QKD) systems. The existing security proofs for the case of the passive basis choice provide a nonzero secret key rate only for a small detection-efficiency mismatch. Unfortunately, in realistic setups, the detection-efficiency mismatch can be significant. Here we present a more precise estimation of the secret key rate for the BB84 protocol with the passive basis choice, which accounts for the detection-efficiency mismatch between four threshold detectors as well as an adaptation of the decoy-state method. The suggested approach is used to estimate the secret key rate in a QKD experiment between the Micius satellite and the Zvenigorod ground station.

Open boundaries host symmetry-protected Majorana modes, yet their imprint on the stabilizer Rényi entropy is obscured by a nonuniversal volume law. At Rényi index $\alpha=1/2$, we isolate a bulk-subtracted boundary response in free-fermion BDI chains using exact finite-open-chain relations, large-$L$ Pfaffian evaluations, and direct positive-weight checks. Across a mass inversion, this response approaches $|\Delta\omega|\ln 2$, where $\Delta\omega$ is the change in winding number. The response survives primitive bulk deformations, the tested local boundary perturbations, and broken bulk duality. Thus, for the families studied here, the boundary response counts the Majorana channels that change the fermionic SPT index.

Quantum computing is a promising way to study problems that are difficult for classical methods, but current quantum hardware still faces limits in scale, noise, and fidelity. Running quantum algorithms on physical machines can also be costly. Quantum circuit simulators therefore remain important because they let researchers design and test algorithms on classical computers before using quantum hardware. Most high-performance simulators provide GPU backends, while few offer native support for NPUs. This gap limits the computing platforms available for quantum-algorithm research. We developed Aicir to provide a full-stack quantum circuit simulator with a native Huawei Ascend NPU backend. Aicir connects circuit construction, several state representations, measurement, differentiation, variational algorithms, quantum machine learning, and quantum architecture search through one programming model. It also supports noise simulation, tensor-network and matrix-product-state engines, and distributed state simulation. On the NPU, paired real tensors, fixed-rank gate views, and hardware-specific formulas keep the tested simulation paths on the device. The same representation lets Aicir partition a state across $2^{p}$ NPUs while retaining reverse-mode differentiation. We validated native execution with CPU fallback disabled and checked distributed communication and gradients on 2, 4, and 8 NPUs. For the tested fused layered circuits, Aicir's CPU runtime is within $0.97$–$1.28\times$ that of Qiskit Aer and $0.76$–$1.10\times$ that of Cirq. These results place its CPU execution in the same range as established simulators for this workload, while the NPU tests establish correct native execution rather than CPU-to-NPU speedup.

Fault-tolerant quantum computation architectures are frequently bottlenecked by the overhead of producing high-fidelity magic states. In this work, we use algebraic geometric techniques to construct codes over binary extension fields $\mathbb{F}_{2^s}$, thus discovering new protocols for the distillation of qubit magic states, where our focus is on the regime of practical qubit-based quantum computing architectures. To do this, we show that multi-qubit gates of interest such as $\text{CS}$, $\text{CCZ}$, and $\text{TOF}\# = \text{CCZ}_{123}\text{CCZ}_{345}$, can be packaged into simple gates over the larger fields, and we derive simple algebraic conditions in the extension fields allowing the distillation of these gates. Because they are derived from Galois qudits, the corresponding qubits codes naturally handle the correlated errors present on such multi-qubit states. Moreover, the protocols we discover are extremely compact; for example, we show that 4 $\text{CS}$ states can be distilled to 1 $\text{CS}$ state at distance 2, using only 4 logical qubits. For a case study, we consider the distillation of $\text{CS}$ and $\text{CCZ}$ states from injected $\text{T}$ and $\text{CS}$ states. When optimized for magic state production per unit time, or logical spacetime volume, we find that our protocols outperform the state-of-the-art in almost every situation, both at input error rates $10^{-3}$ (direct injection), and $10^{-6}$ (allowing some cultivation pre-injection).

We study the emergence of complexity in finite-size quantum systems as their dynamics transition from closed and coherent evolution to interacting and effectively open behavior. Using a state-based geometric framework, we represent mixed quantum states as probability measures on complex projective Hilbert space. This representation allows us to track how interactions reshape the underlying pure-state geometry. We introduce two complementary diagnostics: a distinguishability measure, based on the Wasserstein distance between probability-measure representations of mixed states, that quantifies sensitivity to initial states, and a state-space coverage index that measures long-time exploration of the subsystem state space. These diagnostics provide a geometric perspective on the emergence and evolution of quantum dynamical complexity. When applied to the quantum kicked top, both diagnostics generally increase with interaction strength. Their dependence on environment size is structured by parity symmetry, with integer-spin systems often exhibiting greater sensitivity and state-space coverage than half-integer-spin systems. These results highlight finite-size quantum effects and provide a geometric approach to quantifying dynamical complexity deep in the quantum regime

We investigate the dynamics of entanglement in a generalized version of the kicked Ising chain, extending the model from the standard qubit case (local dimension $q=2$) to higher local dimensions ($q > 2$). We identify the existence of ”dual-unitary” points where the model's space-time duality allows for exact analytical solutions. Our analysis reveals that while a few unique dual-unitary points exist analytically for systems with local dimensions $q=3$ and $q=4$, such points do not exist for $q \ge 5$ due to the lack of a unique kicking strength that satisfies the required matrix element conditions. Utilizing the transfer matrix method and a replica trick specifically adapted for higher dimensions, we derive exact expressions for the growth of entanglement entropy in the $q=3$ (kicked Potts-type) model starting from a class of solvable initial states. Our results demonstrate that at the dual-unitary point, both Rényi and von Neumann entanglement entropies grow linearly with time until reaching a maximum value determined by the subsystem size.

We investigate the role of symmetry-protected dark states and metastable-like frozen states in the autonomous charging dynamics of open quantum batteries described by a transverse-field Ising model. By comparing local and collective dissipation over a range of system sizes, temperatures, and magnetic phases, we demonstrate that collective dissipation generates symmetry-protected dark states together with a much larger set of frozen (metastable) states, forming an extended protected Hilbert space. We derive the multiplicity of the collective dark sector analytically, showing that it follows the Catalan sequence for even system sizes, while such states are absent for odd sizes. Our results show that collective dissipation can enhance ergotropy and charging power, with its advantage depending on temperature, magnetic phase, and system size. While the number of dark and frozen states is identical in the ferromagnetic and antiferromagnetic phases, the achievable ergotropy differs substantially because of the different spectral locations of these protected states. In particular, the antiferromagnetic configuration exhibits considerably larger extractable work owing to the favorable positioning of the protected subspaces within the many-body energy spectrum. Finally, we analyze the active Hilbert-space fraction and show that metastable protection provides an effective mechanism for suppressing dissipative losses while preserving efficient charging pathways. These results establish the dark-state and frozen-state sectors as key resources for optimizing the performance of open quantum batteries through engineered dissipation.

Quantum algorithms for nonlinear partial differential equations remain challenging because nonlinear dynamics are not directly amenable to unitary quantum simulation. Building on the level-set formulation, we construct a quantum algorithm and provide an explicit gate-level implementation for solving scalar conservation laws. The nonlinear equation is first lifted to a linear Liouville equation, discretized by finite differences, and then embedded into a unitary evolution through Schrödingerisation. We further develop quantum procedures for estimating relevant observables from the evolved state. Error bounds and gate-complexity estimates are established for the complete algorithm. The resulting complexity comparison demonstrates a quantum advantage for observable estimation in sufficiently high spatial dimensions, under standard assumptions on state preparation and oracle access. Finally, numerical experiments validate the accuracy, multidimensional applicability, and predicted scaling of the proposed method.

Practical quantum key distribution (QKD) receivers may leak information about their measurement outcomes and settings to the channel (e.g., through detector backflashes or back-reflected Trojan-horse light) that could compromise the protocol's security. Here we present a simple finite-key security proof based on phase-error estimation for prepare-and-measure QKD in the presence of either a priori information leakage about the basis choices and/or a posteriori information leakage about the measurement outcomes. The proof requires only a bound on the distinguishability of the side-channel states. Furthermore, the analysis is modular and compatible with existing security proofs that address detector and source imperfections, making it applicable to a wide range of practical QKD implementations.

We study effective spin interactions arising from quantum electrodynamics (QED) scattering between localized fermionic spins. By integrating out photon and mediator fields, the dynamics reduce to an effective spin Hamiltonian. For two qubits in the nonrelativistic regime, the resulting interaction takes a tensor dipolar form with an asymptotic decay proportional to \(R^{-3}\). We obtain analytical expressions for the entanglement negativity, highlighting its dependence on coupling strength and spatial configuration. We then examine a setup in which two bath spins interact via a sequential exchange with an intermediate fermionic mediator. At the perturbative order considered, the mediator remains unentangled and induces an effective bath–bath interaction with stronger spatial suppression than in the photon-mediated case. Extending the construction to an \(N\)-spin setting yields an effective interaction network mediated by virtual exchange processes, which can support the generation of multipartite entanglement across the system.

Correlations that do not admit a deterministic explanation are a central feature of quantum theory and a key resource for quantum information processing. Identifying and certifying such correlations, however, remains a fundamental challenge. In this work, we advance on this problem by considering deterministic correlations in the prepare-and-measure scenario consistent with a wide range of communication restrictions. To certify correlations incompatible with such deterministic models, we ask whether they admit a scenario in which measurement outcomes can be perfectly predicted by an adversary equipped with classical side-information. We show the usefulness of this approach and propose semidefinite programming relaxations tailored to three representative communication restrictions: fixed ensemble, bounded overlaps and restricted observables.

Fluid dynamics workloads are dominated by repeated solves of large, structured linear systems, motivating the search for quantum acceleration. The Variational Quantum Linear Solver (VQLS) is a leading near-term candidate, but practical deployment on hybrid quantum–high–performance computing (HPC) systems faces three persistent challenges: (i) the linear-combination-of-unitaries (LCU) encoding of the system matrix explodes in memory and runtime as the problem size grows, (ii) ansatz selection is largely empirical, with no clear link between standard circuit metrics and solver convergence, and (iii) end-to-end VQLS pipelines have rarely been exercised on production HPC hardware at non-trivial qubit counts. This work addresses these challenges through three contributions. First, we benchmark four matrix-encoding strategies—naive LCU, PennyLane-integrated, Fast Walsh–Hadamard Transform (FWHT)-based parallel Pauli decomposition, and an singular value decomposition (SVD)-based two-term LCU—and show that the FWHT approach reduces peak memory by up to $1298\times$ on an $11\times 11$ Hele–Shaw grid, while the SVD-based coherent VQLS delivers over $10{,}000\times$ per-iteration speedup over standard Pauli-based VQLS at 8 qubits. Second, we evaluate 11 ansatz families with gradient-free and gradient-based optimizers on canonical Hele–Shaw flow, and find that expressibility and entanglement metrics correlate only weakly with VQLS convergence, motivating problem-aware ansatz design. Third, we deploy the full workflow on the OLCF Frontier supercomputer and successfully simulate a 15-qubit tridiagonal Toeplitz system on a single node. Together, these results establish a practical baseline for VQLS in hybrid quantum–HPC computation fluid dynamic (CFD) workflows and identify the remaining bottlenecks for larger problems.

A key challenge when designing hybrid quantum-classical workflows is the identification of quantum candidates, that is, determining for which specific problems quantum advantages could potentially be achieved. When choosing between several candidates, it is crucial to consider the characteristics specific to the problem, as these can fundamentally determine how successful quantum or classical approaches will be. This implies that specialized expertise is needed to use hybrid quantum-classical workflows successfully. To address this challenge, we propose a domain-specific language (DSL) to express best-practices in solution strategies using a universal representation that is easy to use and share. This DSL provides a flexible approach to design hybrid quantum-classical workflows and to automate decisions for a wide range of problems, supporting decisions down to problem-specific details while remaining technically independent. Furthermore, we propose a framework that is built around our DSL that enables the execution of defined workflows using the ProvideQ toolbox as an orchestration layer. All contributions from this publication are open source.

In the Noisy Intermediate-Scale Quantum (NISQ) era, limited qubit availability and hardware noise constrain the practical deployment of quantum machine learning (QML). Existing quantum neural network (QNN) and quantum convolutional neural network (QCNN) architectures often require increasing quantum resources as the input dimension grows, limiting scalability on near-term devices. We propose QSVM-RQNN, a low-qubit framework integrating centroid-based Quantum Support Vector Machine (QSVM) similarity learning with Recurrent Quantum Neural Networks (RQNNs) for fault classification. The framework reduces the feature space using principal component analysis (PCA), partitions the reduced representation into sequential timesteps, and processes them using a compact three-qubit recurrent quantum architecture with shared parameters. Two complementary variants are developed: QSVM-RQNN-V1 performs class-conditioned joint quantum encoding of input and centroid segments, whereas QSVM-RQNN-V2 performs recurrent learning over timestep-wise quantum similarity representations. Experimental evaluation on multiple fault diagnosis datasets shows competitive and, in several cases, state-of-the-art performance compared with QSVM, QNN, QCNN, QSVM-QNN, QSVM-QCNN, and RQNN models. The proposed architectures provide favorable performance-efficiency trade-offs, improved recall, and enhanced fault detection on highly imbalanced datasets. These results demonstrate that integrating centroid-based quantum similarity learning with low-qubit recurrent representation learning provides an effective and scalable approach to condition monitoring and fault classification on resource-constrained NISQ devices.

Magnetic order with vanishing net magnetization can produce non-relativistic spin-split bands, broadly categorized into even-parity altermagnets and odd-parity \(p\)-wave magnets. Here, we introduce a spin-bond theory that unifies these seemingly distinct phenomena into a single algebraic framework. We demonstrate that non-relativistic spin textures are fundamentally governed by two components of the electronic bond: unitary spin phases and Hermitian spin amplitudes. The unitary sector generates odd-parity p-wave and emergent spin-orbit-like textures, while the Hermitian sector generates even-parity spin fields, including the uniform \(\Gamma\)-split and bond-structured altermagnetic limits. Beyond unifying known phases, our theory uncovers a mixed non-commuting regime that emerges when the unitary and Hermitian sectors fail to commute, revealing an underlying non-commuting spin-bond structure. This regime generates a non-coplanar spin texture characterized by an even-in-momentum transverse spin polarization, providing a direct spectroscopic fingerprint for spin- and angle-resolved photoemission spectroscopy. Furthermore, we establish that this synthetic spin-orbit coupling can be dynamically tuned by geometrically controlling the non-commutation of the bond sectors. By providing a microscopic foundation for such tuning, our theory paves the way for advanced applications, including field-free spin qubits.

Projective measurements are foundational for quantum physics in general and quantum information tasks in particular. However, their direct implementation is often not warranted. Here, we investigate under what conditions continuous monitoring that manifests in quantum jump trajectories realises projective measurements over time. Considering finite-dimensional quantum systems and single, diagonalisable jump operators, we show if and how the detected quantum jump statistics force, and allow inferring, the convergence of individual quantum trajectories towards eigenstates of an observable in the long-time limit. We identify a necessary non-degeneracy condition that is related to the presence of a strong symmetry with non-degenerate symmetry sectors. We derive analytical expressions for the rate of convergence and the time-error relationship in finite-time measurements. Our results provide a transparent framework for understanding the emergence of projective measurements from continuous monitoring, with direct implications for the optimisation of quantum measurement protocols in experiments.

Diffusion-based quantum state tomography (QST) has shown promising results, but all existing methods implicitly adopt a single parameterization (typically Cholesky) without systematic evaluation. We present the first design space study of density matrix parameterizations for diffusion QST, introducing a geometric framework based on the Jacobian Gram matrix $\mathbf{J}^\top\mathbf{J}$. Our calibration of seven parameterizations at 2- and 3-qubit scales, validated by end-to-end training, reveals that geometric conditioning alone does not predict end-to-end performance: at 3-qubit scale, Hermitian direct ($\kappa = 2.0\times$) performs worse than Cholesky ($\kappa = 27\times$) at all shot levels—a $13.5\times$ isotropy advantage that translates into a fidelity disadvantage of up to $+0.51$. The 2-qubit ranking (Hermitian $>$ Bloch) reverses at 3 qubits (Bloch 0.907 vs.\ Hermitian 0.394). We provide a geometric explanation: unbounded parameterizations suffer projection-induced information loss because the PSD constraint couples diagonal and off-diagonal coordinates in ways the unconstrained model cannot respect, whereas the Bloch representation places the maximally mixed state at the center of the valid region, minimizing projection loss.

In this work, we study the emergence of relativistic effects in a composite quantum clock within the Page-Wootters relational formulation of quantum mechanics. We consider a system with internal and center-of-mass degrees of freedom and analyze the conditioned evolution of the center-of-mass relative to the internal system treated as an internal clock. We show that the internal sector exhibits an effective time dilation. Retaining the full mass-energy structure of the composite particle, a back-reaction of the internal energy on the center-of-mass sector appears through an operator-valued normalization factor. As a direct consequence of the back-reaction, the conditioned center-of-mass dynamics is governed by a Schroodinger equation that is non-local in the clock time, with an effective temporal non-locality of the order of the Compton time of the composite system. Applying the formalism to a center-of-mass momentum superposition, we find that the interferometric visibility acquires a correction that is quadratic in the internal energy, and therefore depends on the absolute distribution of clock energies rather than only on the energy gaps that control the standard time-dilation dephasing. Although this correction is parametrically small, it is a generic feature of relativistic composite clocks and can, in principle, be isolated through a differential visibility measurement comparing clocks with equal transition frequency but different mean internal energies.

A well-balanced decoder has been central to the development of modern fault-tolerant quantum computing. However, the inherent topologies of quantum error correcting codes can limit the performance of many well-studied decoding algorithms. In this work, we introduce Noise Assisted Ensemble Decoding (NAED), a highly accurate decoding framework with a significant advantage in real-time speed. NAED constructs an ensemble of Tanner forests, obtained as acyclic subgraphs of the original Tanner graph, and performs exact inference on each Tanner forest using a lightweight dynamic programming algorithm. The forest construction is guided by synthetic soft information derived jointly from the measured syndrome and channel statistics, with controlled noise perturbations generating diverse yet informative decoding matrix column orderings for the Tanner forest construction across the ensemble. Our benchmark results show that the proposed synthetic soft information-driven construction and inference on the Tanner forests can achieve improved or comparable decoding performances to the state-of-the-art decoding solutions, such as BP+OSD$0$, while also providing orders-of-magnitude improvements in per-round decoding speed under circuit-level noise.

Automatic uncomputation aims to provide programming-language-level support to facilitate the correct and safe use of ancilla qubits in quantum computing, but efforts have only been made for clean ancillas, leaving dirty ancillas unexplored. We present a unified formalization of the uncomputation of both clean and dirty ancillas. For the first time, we prove that checking the existence of uncomputation is coNP-hard. We introduce two complementary synthesis-oriented existence checking methods: a syntax-directed static reasoning system and a rewrite-based normalization procedure (\RwUnx), together forming a top-down pipeline. We implement \RwUnx in Qiskit. Compared with the state-of-the-art \Reqompx, \RwUnx handles practical benchmarks with complex dependencies where \Reqompx does not, and achieves higher success rates in random-circuit experiments.

Continuous monitoring can convert a dissipative channel into a resource for accelerating otherwise slow multiphoton transitions. We consider a three-level atom in a $\Lambda$ configuration and condition the evolution on the absence of photon emission through an auxiliary monitored decay channel. The resulting no-jump dynamics is governed by a non-Hermitian Rabi-type Hamiltonian. Using Floquet theory and Brillouin–Wigner projection-operator perturbation method, we derive effective two-state descriptions of odd-multiphoton resonances in the semiclassical and quantum Rabi models. The effective population-transfer rate, defined as the inverse of the time required for the first complete transfer between the atomic states, is maximized at an exceptional point, where its enhancement factor approaches $\pi/2$, corresponding to an approximately $57\%$ increase over the Hermitian value. Numerical results for three- and five-photon resonances closely reproduce the analytical transition times and yield non-negligible postselection probabilities. By contrast, the complete unconditioned dissipative evolution does not exhibit the same population-transfer enhancement. These results demonstrate a speed–success trade-off for measurement-conditioned multiphoton state transfer.

Coordination is a fundamental primitive in communication and information theory, in which distributed systems must collectively generate correlated behavior rather than merely exchange messages. In quantum networks, the nature of entanglement, quantum measurements, and nonclassical correlations introduces coordination possibilities unavailable classically. This article reviews recent advances in coordination over classical and quantum communication networks, focusing on empirical and strong coordination in multi-user settings. We consider the implications of coordination for nonlocal games, showing how it provides a natural framework for understanding the correlations that enable spatially separated players to improve their probability of winning. We present a unified framework for coordination using classical or quantum communication and pre-shared correlation resources. The review covers simulation of both entanglement and separable correlations across a variety of network architectures, including two-node, cascade, broadcast, and multiple-access networks. We study the operational differences between empirical and strong coordination, and the tradeoffs between communication and correlation resources, such as pre-shared randomness and entanglement. Coordination plays a major role in device-independent quantum key distribution (DI-QKD) schemes, in which parties can generate a secret key even if the devices used in the process have been prepared by an adversary. Furthermore, we examine the role of coordination in quantum repeaters, where distributed entanglement serves as a resource for long-distance quantum communication. The review highlights connections between coordination theory and applications such as distributed quantum systems, quantum internet architectures, and future communication networks.

Compound beams composed of simple experimental blocks (detected in simultaneous detection windows) that form specific quantum-correlated structures are suggested for simulating the properties of different quantum operations used for creating highly nonclassical and entangled multi-mode states needed in quantum communication, metrology, and information protocols. Qualitative and quantitative comparison of multi-photon addition and subtraction in compound multi-mode thermal as well as sub-Poissonian beams and multi-mode twin beams with their intensities extending over two orders in magnitude is provided. Adding and subtracting up to twenty photocounts, optimal conditions for the generation of experimental nonclassical states are identified. In general, photon addition is identified as advantageous over photon subtraction for the multi-mode thermal and sub-Poissonian beams: It induces (enhances) the nonclassicality in the former (latter) state. Contrary to this, photon subtraction outperforms photon addition in the multi-mode twin beams. Moreover, exploiting temporal photon-pair correlations in compound twin beams when post-selecting, nearly ideal experimental photon(s) addition is demonstrated. The results apply for the beams with mean photon number per mode much less than one.

We study constructive discrete multimarginal optimal transport (MOT) among $m$ distributions on $n$ points, where the cost tensor $C$ has $N=n^m$ entries. For additive accuracy $\varepsilon$, let $\kappa=\max\{1,(\max C-\min C)/\varepsilon\}$. Classically, we give two algorithms that return exactly feasible additive-$\varepsilon$ couplings. A deterministic box–simplex method with rounding runs in $O(m^2N\kappa\log N)=\widetilde O(m^3N\kappa)$ time, while an exact reduction to positive packing followed by rank-one completion runs in randomized time $\widetilde O(m^2N\kappa)$. At fixed $\kappa$, both match the $\Omega(N)$ cost of writing a dense coupling, up to factors in $m$ and logarithms. Quantumly, in the general entry-access model, tensor Sinkhorn followed by sparse recovery returns an exactly feasible additive-$\varepsilon$ coupling as a classical list of $\widetilde O(m^2n\kappa^2)$ atoms, using $\widetilde O(m^4\sqrt{Nn}\,\kappa^3)$ coherent cost queries without materializing the tensor. Finally, for fixed $m$ and constant normalized accuracy, we prove sparse-output lower bounds of $\widetilde\Omega(N)$ randomized classical queries, even with unrestricted output size, and $\widetilde\Omega(\sqrt{Nn})$ quantum queries for outputs with at most $n\operatorname{polylog}(n)$ atoms. Hence, for fixed $m$ and normalized accuracy between $(\log n)^{-O(1)}$ and a sufficiently small constant, explicit sparse MOT construction has query complexity $\widetilde\Theta(N)$ classically and $\widetilde\Theta(\sqrt{Nn})$ quantumly.