We show that the communication cost of quantum broadcast channel simulation under free entanglement assistance between the sender and the receivers is asymptotically characterized by an efficiently computable single-letter formula in terms of the channel's multipartite mutual information. Our core contribution is a new one-shot achievability result for multipartite quantum state splitting via multipartite convex splitting. As part of this, we face a general instance of the quantum joint typicality problem with arbitrarily overlapping marginals. The crucial technical ingredient to sidestep this difficulty is a conceptually novel multipartite mean-zero decomposition lemma, together with employing recently introduced complex interpolation techniques for sandwiched R\'enyi divergences. Moreover, we establish an exponential convergence of the simulation error when the communication costs are within the interior of the capacity region. As the costs approach the boundary of the capacity region moderately quickly, we show that the error still vanishes asymptotically.
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The objective of this study is to address the difficulty of simplifying the geometric model in which a differential problem is formulated, also called defeaturing, while simultaneously ensuring that the accuracy of the solution is maintained under control. This enables faster and more efficient simulations, without sacrificing accuracy. More precisely, we consider an isogeometric discretisation of an elliptic model problem defined on a two-dimensional hierarchical B-spline computational domain with a complex boundary. Starting with an oversimplification of the geometry, we build a goal-oriented adaptive strategy that adaptively reintroduces continuous geometrical features in regions where the analysis suggests a large impact on the quantity of interest. This strategy is driven by an a posteriori estimator of the defeaturing error based on first-order shape sensitivity analysis, and it profits from the local refinement properties of hierarchical B-splines. The adaptive algorithm is described together with a procedure to generate (partially) simplified hierarchical B-spline geometrical domains. Numerical experiments are presented to illustrate the proposed strategy and its limitations.
We study channel simulation under common randomness-assistance in the finite-blocklength regime and identify the smooth channel max-information as a linear program one-shot converse on the minimal simulation cost for fixed error tolerance. We show that this one-shot converse can be achieved exactly using no-signaling assisted codes, and approximately achieved using common randomness-assisted codes. Our one-shot converse thus takes on an analogous role to the celebrated meta-converse in the complementary problem of channel coding, and find tight relations between these two bounds. We asymptotically expand our bounds on the simulation cost for discrete memoryless channels, leading to the second-order as well as the moderate deviation rate expansion, which can be expressed in terms of the channel capacity and channel dispersion known from noisy channel coding. Our techniques extend to discrete memoryless broadcast channels. In stark contrast to the elusive broadcast channel capacity problem, we show that the reverse problem of broadcast channel simulation under common randomness-assistance allows for an efficiently computable single-letter characterization of the asymptotic rate region in terms of the broadcast channel's multi-partite mutual information. Finally, we present a Blahut-Arimoto type algorithm to compute the rate region efficiently.
We consider the point-to-point lossy coding for computing and channel coding problems with two-sided information. We first unify these problems by considering a new generalized problem. Then we develop graph-based characterizations and derive interesting reductions through explicit graph operations, which reduce the number of decision variables. After that, we design alternating optimization algorithms for the unified problems, so that numerical computations for both the source and channel problems are covered. With the help of extra root-finding techniques, proper multiplier update strategies are developed. Thus our algorithms can compute the problems for a given distortion or cost constraint and the convergence can be proved. Also, extra heuristic deflation techniques are introduced which largely reduce the computational time. Numerical results show the accuracy and efficiency of our algorithms.
Quantifying entanglement is an important task by which the resourcefulness of a quantum state can be measured. Here we develop a quantum algorithm that tests for and quantifies the separability of a general bipartite state, by making use of the quantum steering effect, the latter originally discovered by Schr\"odinger. Our separability test consists of a distributed quantum computation involving two parties: a computationally limited client, who prepares a purification of the state of interest, and a computationally unbounded server, who tries to steer the reduced systems to a probabilistic ensemble of pure product states. To design a practical algorithm, we replace the role of the server by a combination of parameterized unitary circuits and classical optimization techniques to perform the necessary computation. The result is a variational quantum steering algorithm (VQSA), which is a modified separability test that is better suited for the capabilities of quantum computers available today. We then simulate our VQSA on noisy quantum simulators and find favorable convergence properties on the examples tested. We also develop semidefinite programs, executable on classical computers, that benchmark the results obtained from our VQSA. Our findings here thus provide a meaningful connection between steering, entanglement, quantum algorithms, and quantum computational complexity theory. They also demonstrate the value of a parameterized mid-circuit measurement in a VQSA and represent a first-of-its-kind application for a distributed VQA.
In this work, we prove rigorous error estimates for a hybrid method introduced in [15] for solving the time-dependent radiation transport equation (RTE). The method relies on a splitting of the kinetic distribution function for the radiation into uncollided and collided components. A high-resolution method (in angle) is used to approximate the uncollided components and a low-resolution method is used to approximate the the collided component. After each time step, the kinetic distribution is reinitialized to be entirely uncollided. For this analysis, we consider a mono-energetic problem on a periodic domains, with constant material cross-sections of arbitrary size. To focus the analysis, we assume the uncollided equation is solved exactly and the collided part is approximated in angle via a spherical harmonic expansion ($\text{P}_N$ method). Using a non-standard set of semi-norms, we obtain estimates of the form $C(\varepsilon,\sigma,\Delta t)N^{-s}$ where $s\geq 1$ denotes the regularity of the solution in angle, $\varepsilon$ and $\sigma$ are scattering parameters, $\Delta t$ is the time-step before reinitialization, and $C$ is a complicated function of $\varepsilon$, $\sigma$, and $\Delta t$. These estimates involve analysis of the multiscale RTE that includes, but necessarily goes beyond, usual spectral analysis. We also compute error estimates for the monolithic $\text{P}_N$ method with the same resolution as the collided part in the hybrid. Our results highlight the benefits of the hybrid approach over the monolithic discretization in both highly scattering and streaming regimes.
Byzantine reliable broadcast is a primitive that allows a set of processes to agree on a message broadcast by a dedicated source process, even when some of them are malicious (Byzantine). It guarantees that no two correct processes deliver different messages, and if a message is delivered by a correct process, every correct process eventually delivers one. The primitive is known not to scale, as it requires $\Omega(n^2)$ message exchanges, where $n$ is the number of system members. The quadratic cost can be explained by the inherent need for every process to relay a message to every other process. In this paper, we explore ways to overcome this limitation, by casting the problem to the probabilistic setting. We propose a solution in which every broadcast message is validated by a small set of witnesses, which allows us to maintain low latency and small communication complexity. In order to tolerate a slow adaptive adversary, we dynamically select witnesses through a novel use of locality-preserving hash functions. Our simulations demonstrate significant scalability gains of our solution with respect to existing protocols.
ISAC is recognized as a promising technology for the next-generation wireless networks, which provides significant performance gains over individual S&C systems via the shared use of wireless resources. The characterization of the S&C performance tradeoff is at the core of the theoretical foundation of ISAC. In this paper, we consider a point-to-point ISAC model under vector Gaussian channels, and propose to use the CRB-rate region as a basic tool for depicting the fundamental S&C tradeoff. In particular, we consider the scenario where a unified ISAC waveform is emitted from a dual-functional ISAC Tx, which simultaneously performs S&C tasks with a communication Rx and a sensing Rx. In order to perform both S&C tasks, the ISAC waveform is required to be random to convey communication information, with realizations being perfectly known at both the ISAC Tx and the sensing Rx as a reference sensing signal as in typical radar systems. As the main contribution of this paper, we characterize the S&C performance at the two corner points of the CRB-rate region, namely, $P_{SC}$ indicating the max. achievable rate constrained by the min. CRB, and $P_{CS}$ indicating the min. achievable CRB constrained by the max. rate. In particular, we derive the high-SNR capacity at $P_{SC}$, and provide lower and upper bounds for the sensing CRB at $P_{CS}$. We show that these two points can be achieved by the conventional Gaussian signaling and a novel strategy relying on the uniform distribution over the Stiefel manifold, respectively. Based on the above-mentioned analysis, we provide an outer bound and various inner bounds for the achievable CRB-rate regions. Our main results reveal a two-fold tradeoff in ISAC systems, consisting of the subspace tradeoff (ST) and the deterministic-random tradeoff (DRT) that depend on the resource allocation and data modulation schemes employed for S&C, respectively.
Matrix factor model is drawing growing attention for simultaneous two-way dimension reduction of well-structured matrix-valued observations. This paper focuses on robust statistical inference for matrix factor model in the ``diverging dimension" regime. We derive the convergence rates of the robust estimators for loadings, factors and common components under finite second moment assumption of the idiosyncratic errors. In addition, the asymptotic distributions of the estimators are also derived under mild conditions. We propose a rank minimization and an eigenvalue-ratio method to estimate the pair of factor numbers consistently. Numerical studies confirm the iterative Huber regression algorithm is a practical and reliable approach for the estimation of matrix factor model, especially under the cases with heavy-tailed idiosyncratic errors . We illustrate the practical usefulness of the proposed methods by two real datasets, one on financial portfolios and one on the macroeconomic indices of China.
In many practical applications including remote sensing, multi-task learning, and multi-spectrum imaging, data are described as a set of matrices sharing a common column space. We consider the joint estimation of such matrices from their noisy linear measurements. We study a convex estimator regularized by a pair of matrix norms. The measurement model corresponds to block-wise sensing and the reconstruction is possible only when the total energy is well distributed over blocks. The first norm, which is the maximum-block-Frobenius norm, favors such a solution. This condition is analogous to the notion of low-spikiness in matrix completion or column-wise sensing. The second norm, which is a tensor norm on a pair of suitable Banach spaces, induces low-rankness in the solution together with the first norm. We demonstrate that the joint estimation provides a significant gain over the individual recovery of each matrix when the number of matrices sharing a column space and the ambient dimension of the shared column space are large relative to the number of columns in each matrix. The convex estimator is cast as a semidefinite program and an efficient ADMM algorithm is derived. The empirical behavior of the convex estimator is illustrated using Monte Carlo simulations and recovery performance is compared to existing methods in the literature.
Image segmentation is an important component of many image understanding systems. It aims to group pixels in a spatially and perceptually coherent manner. Typically, these algorithms have a collection of parameters that control the degree of over-segmentation produced. It still remains a challenge to properly select such parameters for human-like perceptual grouping. In this work, we exploit the diversity of segments produced by different choices of parameters. We scan the segmentation parameter space and generate a collection of image segmentation hypotheses (from highly over-segmented to under-segmented). These are fed into a cost minimization framework that produces the final segmentation by selecting segments that: (1) better describe the natural contours of the image, and (2) are more stable and persistent among all the segmentation hypotheses. We compare our algorithm's performance with state-of-the-art algorithms, showing that we can achieve improved results. We also show that our framework is robust to the choice of segmentation kernel that produces the initial set of hypotheses.
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