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In this work, we address the lossy quantum-classical source coding with the quantum side-information (QC-QSI) problem. The task is to compress the classical information about a quantum source, obtained after performing a measurement while incurring a bounded reconstruction error. Here, the decoder is allowed to use the side information to recover the classical data obtained from measurements on the source states. We introduce a new formulation based on a backward (posterior) channel, replacing the single-letter distortion observable with a single-letter posterior channel to capture reconstruction error. Unlike the rate-distortion framework, this formulation imposes a block error constraint. An analogous formulation is developed for lossy classical source coding with classical side information (C-CSI) problem. We derive an inner bound on the asymptotic performance limit in terms of single-letter quantum and classical mutual information quantities of the given posterior channel for QC-QSI and C-CSI cases, respectively. Furthermore, we establish a connection between rate-distortion and rate-channel theory, showing that a rate-channel compression protocol attains the optimal rate-distortion function for a specific distortion measure and level.

In this paper, we provide a lower and an upper bound for the strong converse exponent of the soft covering problem in the classical setting. This exponent characterizes the slowest achievable convergence speed of the total variation to one when a code with a rate below mutual information is applied to a discrete memoryless channel for synthesizing a product output distribution. We employ a type-based approach and additionally propose an equivalent form of our upper bound using the R\'enyi mutual information. Future works include tightening these two bounds to determine the exact bound of the strong converse exponent.

Federated learning (FL) has emerged as a promising framework for distributed machine learning. It enables collaborative learning among multiple clients, utilizing distributed data and computing resources. However, FL faces challenges in balancing privacy guarantees, communication efficiency, and overall model accuracy. In this work, we introduce CorBin-FL, a privacy mechanism that uses correlated binary stochastic quantization to achieve differential privacy while maintaining overall model accuracy. The approach uses secure multi-party computation techniques to enable clients to perform correlated quantization of their local model updates without compromising individual privacy. We provide theoretical analysis showing that CorBin-FL achieves parameter-level local differential privacy (PLDP), and that it asymptotically optimizes the privacy-utility trade-off between the mean square error utility measure and the PLDP privacy measure. We further propose AugCorBin-FL, an extension that, in addition to PLDP, achieves user-level and sample-level central differential privacy guarantees. For both mechanisms, we derive bounds on privacy parameters and mean squared error performance measures. Extensive experiments on MNIST and CIFAR10 datasets demonstrate that our mechanisms outperform existing differentially private FL mechanisms, including Gaussian and Laplacian mechanisms, in terms of model accuracy under equal PLDP privacy budgets.

Quantum machine learning through variational quantum algorithms (VQAs) has gained substantial attention in recent years. VQAs employ parameterized quantum circuits, which are typically optimized using gradient-based methods. However, these methods often exhibit sub-optimal convergence performance due to their dependence on Euclidean geometry. The quantum natural gradient descent (QNGD) optimization method, which considers the geometry of the quantum state space via a quantum information (Riemannian) metric tensor, provides a more effective optimization strategy. Despite its advantages, QNGD encounters notable challenges for learning from quantum data, including the no-cloning principle, which prohibits the replication of quantum data, state collapse, and the measurement postulate, which leads to the stochastic loss function. This paper introduces the quantum natural stochastic pairwise coordinate descent (2-QNSCD) optimization method. This method leverages the curved geometry of the quantum state space through a novel ensemble-based quantum information metric tensor, offering a more physically realizable optimization strategy for learning from quantum data. To improve computational efficiency and reduce sample complexity, we develop a highly sparse unbiased estimator of the novel metric tensor using a quantum circuit with gate complexity $\Theta(1)$ times that of the parameterized quantum circuit and single-shot quantum measurements. Our approach avoids the need for multiple copies of quantum data, thus adhering to the no-cloning principle. We provide a detailed theoretical foundation for our optimization method, along with an exponential convergence analysis. Additionally, we validate the utility of our method through a series of numerical experiments.

We propose quantum soft covering problems for fully quantum channels and classical-quantum (CQ) channels using relative entropy as a criterion of operator closeness. We prove covering lemmas by deriving one-shot bounds on the rates in terms of smooth min-entropies and smooth max-divergences, respectively. In the asymptotic regime, we show that for quantum channels, the rate infimum defined as the logarithm of the minimum rank of the input state is the coherent information between the reference and output state; for CQ channels, the rate infimum defined as the logarithm of the minimum number of input codewords is the Helovo information between the input and output state. Furthermore, we present a one-shot quantum decoupling theorem with relative entropy criterion. Our results based on the relative-entropy criterion are tighter than the corresponding results based on the trace norm considered in the literature due to the Pinsker inequality.

This work considers the non-interactive source simulation problem (NISS). In the standard NISS scenario, a pair of distributed agents, Alice and Bob, observe a distributed binary memoryless source $(X^d,Y^d)$ generated based on joint distribution $P_{X,Y}$. The agents wish to produce a pair of discrete random variables $(U_d,V_d)$ with joint distribution $P_{U_d,V_d}$, such that $P_{U_d,V_d}$ converges in total variation distance to a target distribution $Q_{U,V}$. Two variations of the standard NISS scenario are considered. In the first variation, in addition to $(X^d,Y^d)$ the agents have access to a shared Bell state. The agents each measure their respective state, using a measurement of their choice, and use its classical output along with $(X^d,Y^d)$ to simulate the target distribution. This scenario is called the entanglement-assisted NISS (EA-NISS). In the second variation, the agents have access to a classical common random bit $Z$, in addition to $(X^d,Y^d)$. This scenario is called the classical common randomness NISS (CR-NISS). It is shown that for binary-output NISS scenarios, the set of feasible distributions for EA-NISS and CR-NISS are equal with each other. Hence, there is not quantum advantage in these EA-NISS scenarios. For non-binary output NISS scenarios, it is shown through an example that there are distributions that are feasible in EA-NISS but not in CR-NISS. This shows that there is a quantum advantage in non-binary output EA-NISS.

We have constructed a dark energy cosmological model in the framework of generalized Brans Dicke theory (GBD) with a self-interacting potential. The source of dark energy is considered through a unified dark fluid (UDF) characterized by a linear equation of state (EoS). A time varying deceleration parameter simulating the cosmic transit behaviour has been introduced to get the dynamical behaviour of the model. The $H(z)$ data have been explored to constrain the model parameters and to study the dynamical aspects of the Brans-Dicke parameter and the scalar field., Comment: 13 pages, accepted version of Gravitation and Cosmology

We propose a quantum soft-covering problem for a given general quantum channel and one of its output states, which consists in finding the minimum rank of an input state needed to approximate the given channel output. We then prove a one-shot quantum covering lemma in terms of smooth min-entropies by leveraging decoupling techniques from quantum Shannon theory. This covering result is shown to be equivalent to a coding theorem for rate distortion under a posterior (reverse) channel distortion criterion by two of the present authors. Both one-shot results directly yield corollaries about the i.i.d. asymptotics, in terms of the coherent information of the channel. The power of our quantum covering lemma is demonstrated by two additional applications: first, we formulate a quantum channel resolvability problem, and provide one-shot as well as asymptotic upper and lower bounds. Secondly, we provide new upper bounds on the unrestricted and simultaneous identification capacities of quantum channels, in particular separating for the first time the simultaneous identification capacity from the unrestricted one, proving a long-standing conjecture of the last author., Comment: 30 pages, 3 figures; v2 fixes an error in Definition 36 and various typos and minor issues throughout; v3 fixes some further minor points and provides a proof of Lemma 11 due to F. Dupuis, it is the version accepted by IJQI (special issue in honour of Alexander S. Holevo)

The reliability function of a channel is the maximum achievable exponential rate of decay of the error probability as a function of the transmission rate. In this work, we derive bounds on the reliability function of discrete memoryless multiple-access channels (MAC) with noiseless feedback. We show that our bounds are tight for a variety of MACs, such as $m$-ary additive and two independent point-to-point channels. The bounds are expressed in terms of a new information measure called ``variable-length directed information". The upper bound is proved by analyzing stochastic processes defined based on the entropy of the message, given the past channel's outputs. Our method relies on tools from the theory of martingales, variable-length information measures, and a new technique called time pruning. We further propose a variable-length achievable scheme consisting of three phases: (i) data transmission, (ii) hybrid data-confirmation, and (iii) full confirmation. We show that two-phase-type schemes are strictly suboptimal in achieving the MAC's reliability function. Moreover, we study the shape of the lower-bound and show that it increases linearly with respect to a specific Euclidean distance measure defined between the transmission rate pair and the capacity boundary. As side results, we derive an upper bound on the capacity of MAC with noiseless feedback and study a new problem involving a hybrid of hypothesis testing and data transmission.

We establish a coding theorem for rate-limited quantum-classical optimal transport systems with limited classical common randomness. This theorem characterizes the rate region of measurement protocols on a product source state for faithful construction of a given destination state while maintaining the source-destination distortion below a prescribed threshold with respect to a general distortion observable. It also provides a solution to the problem of rate-limited optimal transport, which aims to find the optimal cost of transforming a source quantum state to a destination state via an entanglement-breaking channel with a limited communication rate. The coding theorem is further extended to cover Bosonic continuous-variable quantum systems. The analytical evaluation is performed for the case of a qubit measurement system with unlimited common randomness., Comment: 45 pages, 3 figures

We have studied a superbounce scenario in a set up of Brans-Dicke (BD) theory. The BD parameter is considered to be time dependent and is assumed to evolve with the Brans-Dicke scalar field. In the superbounce scenario, the model bounces at an epoch corresponding to a Big Crunch provided the ekpyrotic phase continues until that time. Within the given superbounce scenario, we investigate the evolution of the BD parameter for different equations of state. We chose an axially symmetric metric that has an axial symmetry along the x-axis. The metric is assumed to incorporate an anisotropic expansion effect. The effect of asymmetric expansion and the anisotropic parameter on the evolving and the non-evolving part of the BD parameter is investigated., Comment: 10 pages, 5 figures, accepted version of the journal Symmetry

In this paper, we leverage polar codes and the well-established channel polarization to design capacity-achieving codes with a certain constraint on the weights of all the columns in the generator matrix (GM) while having a low-complexity decoding algorithm. We first show that given a binary-input memoryless symmetric (BMS) channel $W$ and a constant $s \in (0, 1]$, there exists a polarization kernel such that the corresponding polar code is capacity-achieving with the \textit{rate of polarization} $s/2$, and the GM column weights being bounded from above by $N^s$. To improve the sparsity versus error rate trade-off, we devise a column-splitting algorithm and two coding schemes for BEC and then for general BMS channels. The \textit{polar-based} codes generated by the two schemes inherit several fundamental properties of polar codes with the original $2 \times 2$ kernel including the decay in error probability, decoding complexity, and the capacity-achieving property. Furthermore, they demonstrate the additional property that their GM column weights are bounded from above sublinearly in $N$, while the original polar codes have some column weights that are linear in $N$. In particular, for any BEC and $\beta <0.5$, the existence of a sequence of capacity-achieving polar-based codes where all the GM column weights are bounded from above by $N^\lambda$ with $\lambda \approx 0.585$, and with the error probability bounded by $O(2^{-N^{\beta}} )$ under a decoder with complexity $O(N\log N)$, is shown. The existence of similar capacity-achieving polar-based codes with the same decoding complexity is shown for any BMS channel and $\beta <0.5$ with $\lambda \approx 0.631$., Comment: 31 pages, single column. arXiv admin note: substantial text overlap with arXiv:2012.13977

Basic multimode impedance analysis grounded in the availability of nonequilibrium charge carriers and their retarded path towards equilibrium is used to access the inadequacy of equivalent circuits in nonlinear systems with inherent memory. On the basic grounds of generation and recombination (or trapping) of nonequilibrium carriers and their relaxation times, we show how seeming complexity of frequency-dependent impedance that matches a vast universe of experimental evidences can be reduced to simple combinations of basic microscopic ingredients. Counterintuitive features such as a negative capacitances or unexpected inductances become a metaphoric construction with poor physical meaning, pointing to the limitations and ambiguities of the symbolic nature of "equivalent" circuits. Our approach further provides a microscopic perspective that exposes the linkage of an apparent flux with an apparent inductance dismissing any magnetic essence., Comment: 9 pages, 7 figures

We consider the lossy quantum source coding problem where the task is to compress a given quantum source below its von Neumann entropy. Inspired by the duality connections between the rate-distortion and channel coding problems in the classical setting, we propose a new formulation for the lossy quantum source coding problem. This formulation differs from the existing quantum rate-distortion theory in two aspects. Firstly, we require that the reconstruction of the compressed quantum source fulfill a global error constraint as opposed to the sample-wise local error criterion used in the standard rate-distortion setting. Secondly, instead of a distortion observable, we employ the notion of a backward quantum channel, which we refer to as a "posterior reference map", to measure the reconstruction error. Using these, we characterize the asymptotic performance limit of the lossy quantum source coding problem in terms of single-letter coherent information of the given posterior reference map. We demonstrate a protocol to encode (at the specified rate) and decode, with the reconstruction satisfying the provided global error criterion, and therefore achieving the asymptotic performance limit. The protocol is constructed by decomposing coherent information as a difference of two Holevo information quantities, inspired from prior works in quantum communication problems. To further support the findings, we develop analogous formulations for the quantum-classical and classical variants and express the asymptotic performance limit in terms of single-letter mutual information quantities with respect to appropriately defined channels analogous to posterior reference maps. We also provide various examples for the three formulations, and shed light on their connection to the standard rate-distortion formulation wherever possible.

Motivated by quantum simulation, we consider lattice Hamiltonians for Yang-Mills gauge theories with finite gauge group, for example a finite subgroup of a compact Lie group. We show that the electric Hamiltonian admits an interpretation as a certain natural, non-unique Laplacian operator on the finite Abelian or non-Abelian group, and derive some consequences from this fact. Independently of the chosen Hamiltonian, we provide a full explicit description of the physical, gauge-invariant Hilbert space for pure gauge theories and derive a simple formula to compute its dimension. We illustrate the use of the gauge-invariant basis to diagonalize a dihedral gauge theory on a small periodic lattice., Comment: 28 pages, 10 figures