Quantum burst error correction codes (QBECCs) are of great importance to deal with the memory effect in quantum channels. As the most important family of QBECCs, quantum cyclic codes (QCCs) play a vital role in the correction of burst errors. In this work, we characterize the burst error correction ability of QCCs constructed from the Calderbank-Shor-Steane (CSS) and the Hermitian constructions. We determine the burst error correction limit of QCCs and quantum Reed-Solomon codes with algorithms in polynomial-time complexities. As a result, lots of QBECCs saturating the quantum Reiger bound are obtained. We show that quantum Reed-Solomon codes have better burst error correction abilities than the previous results. At last, we give the quantum error-trapping decoder (QETD) of QCCs for decoding burst errors. The decoder runs in linear time and can decode both degenerate and nondegenerate burst errors. What’s more, the numerical results show that QETD can decode much more degenerate burst errors than the nondegenerate ones.

Quantum-centric supercomputing presents a compelling framework for large-scale hybrid quantum-classical tasks. Although quantum machine learning (QML) offers theoretical benefits in various applications, challenges such as large-size data encoding in the input stage and the reliance on quantum resources in the inference stage limit its practicality for tasks like fine-tuning large language models (LLMs). Quantum parameter generation, a novel approach of QML, addresses these limitations by using quantum neural networks (QNNs) to generate classical model weights (parameters) exclusively during training, thereby decoupling inference from quantum hardware. In this work, we introduce Quantum Parameter Adaptation (QPA) in the framework of quantum parameter generation, which integrates QNNs with a classical multi-layer perceptron mapping model to generate parameters for fine-tuning methods. Using Gemma-2 and GPT-2 as case studies, QPA demonstrates significant parameter reduction for parameter-efficient fine-tuning methods, such as Low-Rank Adaptation (LoRA), while maintaining comparable or improved performance in text generation tasks. Specifically, QPA reduces the number of parameters to $52.06\%$ of the original LoRA for GPT-2 with a slight performance gain of $0.75\%$, and to $16.84\%$ for Gemma-2, with a marginal performance improvement of $0.07\%$. These results highlight QPA’s ability to achieve efficient parameter reduction without sacrificing performance in the quantum parameter generation framework. This work showcases the potential of quantum-enhanced parameter reduction, offering a scalable quantum-classical solution for fine-tuning LLMs while preserving the feasibility of inference on classical hardware.

Optimizing routing in Wireless Sensor Networks (WSNs) is pivotal for minimizing energy consumption and extending network lifetime. This paper introduces a resource-efficient compilation method for distributed quantum circuits tailored to address large-scale WSN routing problems. Leveraging a hybrid classical-quantum framework, we employ spectral clustering for network partitioning and the Quantum Approximate Optimization Algorithm (QAOA) for optimizing routing within manageable subgraphs. We formulate the routing problem as a Quadratic Unconstrained Binary Optimization (QUBO) problem, providing comprehensive mathematical formulations and complexity analyses. Comparative evaluations against traditional classical algorithms demonstrate significant energy savings and enhanced scalability. Our approach underscores the potential of integrating quantum computing techniques into wireless communication networks, offering a scalable and efficient solution for future network optimization challenges.

Abstract We present a verifier of quantum programs called AutoQ 2.0. Quantum programs extend quantum circuits (the domain of AutoQ 1.0) by classical control flow constructs, which enable users to describe advanced quantum algorithms in a formal and precise manner. The extension is highly non-trivial, as we needed to tackle both theoretical challenges (such as the treatment of measurement, the normalization problem, and lifting techniques for verification of classical programs with loops to the quantum world), and engineering issues (such as extending the input format with a support for specifying loop invariants). We have successfully used AutoQ 2.0 to verify two types of advanced quantum programs that cannot be expressed using only quantum circuits: the repeat-until-success (RUS) algorithm and the weak-measurement-based version of Grover’s search algorithm. AutoQ 2.0 can efficiently verify all our benchmarks: all RUS algorithms were verified instantly and, for the weak-measurement-based version of Grover’s search, we were able to handle the case of 100 qubits in $$\sim $$ ∼ 20 minutes.

Distributed source simulation is the task where two (or more) parties share some correlated randomness and use local operations and no communication to convert this into some target correlation. Wyner’s seminal result showed that asymptotically the rate of uniform shared randomness needed for this task is given by a mutual information induced measure, now referred to as Wyner’s common information. This asymptotic result was extended by Hayashi in the quantum setting to separable states, the largest class of states for which this task can be performed to vanishing error. In this work we characterize this task in a near-tight manner in the one-shot setting using the smooth entropy framework. We do this by introducing one-shot operational quantities and correlation measures that characterize them. We establish asymptotic equipartition properties for our correlation measures thereby recovering the previous vanishing-error asymptotic results. In doing so, we consider technical points in one-shot network information theory and provide methods for cardinality bounds in the smooth entropy calculus. We also introduce entangled state versions of the distributed source simulation task and determine bounds in this setting via quantum embezzling. This provides a strong characterization of this network task in the one-shot, quantum regime.

In this work, we continue the search for quantum locally testable codes (qLTCs) of new parameters by presenting three constructions that can make new qLTCs from old. The first analyses the soundness of a quantum code under Hastings’ weight reduction construction for qLDPC codes to give a weight reduction procedure for qLTCs. Secondly, we describe a novel ‘soundness amplification’ procedure for qLTCs which can increase the soundness of any qLTC to a constant while preserving its distance and dimension, with an impact only felt on its locality. Finally, we apply the AEL distance amplification construction to the case of qLTCs for the first time which can turn a high-distance qLTC into one with linear distance, at the expense of other parameters. These constructions can be used on as-yet undiscovered qLTCs to obtain new parameters, but we also find a number of present applications to prove the existence of codes in previously unknown parameter regimes. In particular, applications of these operations to the hypersphere product code and the hemicubic code yield many previously unknown parameters. In addition, applications of all three results are described to an upcoming work.

Recent advancements in understanding the impulse response of the first arrival position (FAP) channel in molecular communication (MC) have illuminated its Shannon capacity. While Lee et al. shed light on FAP channel capacities with vertical drifts, the zero-drift scenario remains a conundrum, primarily due to the challenges associated with the heavy-tailed Cauchy distributions whose first and second moments do not exist, rendering traditional mutual information constraints ineffective. This paper unveils a novel characterization of the zero-drift FAP channel capacity for both 2D and 3D. Interestingly, our results reveal a 3D FAP channel capacity that is double its 2D counterpart, hinting at a capacity increase with spatial dimension growth. Furthermore, our approach, which incorporates a modified logarithmic constraint and an output signal constraint, offers a simplified and more intuitive formula (similar to the well-known Gaussian case) for estimating FAP channel capacity.

Factorization Machine (FM) is the most commonly used model to build a recommendation system since it can incorporate side information to improve performance. However, producing item suggestions for a given user with a trained FM is time-consuming. To address this problem, we propose a quadratic unconstrained binary optimization (QUBO) scheme to combine with FM and apply quantum annealing (QA) computation. Compared to classical methods, this hybrid algorithm provides a fast sub-optimal sampling of good user suggestions. We then demonstrate the aforementioned computational behavior on current noisy intermediate-scale quantum (NISQ) hardware by experimenting with a real example on a D-Wave annealer.

This paper introduces a novel mathematical model for Molecular Communication (MC) systems, utilizing First Arrival Position (FAP) as a fundamental mode of information transmission. We address two critical challenges: the characterization of FAP density and the establishment of capacity bounds for channels with vertically-drifted FAP. Our method relate macroscopic Partial Differential Equation (PDE) models to microscopic Stochastic Differential Equation (SDE) models, resulting in a precise expression that links FAP density with elliptic-type Green’s function. This formula is distinguished by its wide applicability across any spatial dimensions, any drift directions, and various receiver geometries. We demonstrate the practicality of our model through case studies: 2D and 3D planar receivers. The accuracy of our formula is also validated by particle-based simulations. Advancing further, the explicit FAP density forms enable us to establish closed-form upper and lower bounds for the capacity of vertically-drifted FAP channels under a second-moment constraint, significantly advancing the understanding of FAP channels in MC systems.

Distributed source simulation is the task where two (or more) parties share some correlated randomness and use local operations and no communication to convert this into some target correlation. Wyner’s seminal result showed that asymptotically the rate of uniform shared randomness needed for this task is given by a mutual information induced measure, now referred to as Wyner’s common information. In this work we characterize the quantum version of this task in the one-shot setting using the smooth entropy framework and one-shot operational quantities. We further establish asymptotic equipartition properties for our correlation measures. We also introduce entanglement versions of the distributed source simulation task and determine bounds in this setting via quantum embezzling.

We construct a new explicit family of good quantum low-density parity-check codes which additionally have linear time decoders. Our codes are based on a three-term chain (2m× m)V →δ0 (2m)E →δ1 2F where V (X-checks) are the vertices, E (qubits) are the edges, and F (Z-checks) are the squares of a left-right Cayley complex, and where the maps are defined based on a pair of constant-size random codes CA,CB:2m→2Δ where Δ is the regularity of the underlying Cayley graphs.

Quantum computers are next-generation devices that hold promise to perform calculations beyond the reach of classical computers. A leading method towards achieving this goal is through quantum machine learning, especially quantum generative learning. Due to the intrinsic probabilistic nature of quantum mechanics, it is reasonable to postulate that quantum generative learning models (QGLMs) may surpass their classical counterparts. As such, QGLMs are receiving growing attention from the quantum physics and computer science communities, where various QGLMs that can be efficiently implemented on near-term quantum machines with potential computational advantages are proposed. In this paper, we review the current progress of QGLMs from the perspective of machine learning. Particularly, we interpret these QGLMs, covering quantum circuit Born machines, quantum generative adversarial networks, quantum Boltzmann machines, and quantum variational autoencoders, as the quantum extension of classical generative learning models. In this context, we explore their intrinsic relations and their fundamental differences. We further summarize the potential applications of QGLMs in both conventional machine learning tasks and quantum physics. Last, we discuss the challenges and further research directions for QGLMs.

In this paper, we utilize a concatenation scheme to construct new families of quantum error correction codes achieving the quantum Gilbert-Varshamov (GV) bound asymptotically. Weconcatenate alternant codes with any linear code achievingthe classical GV bound to construct Calderbank-Shor-Steane (CSS) codes. We show that the concatenated code can achieve the quantum GV bound asymptotically and can approach the Hashing bound for asymmetric Pauli channels. By combing Steane’s enlargement construction of CSS codes, we derive a family of enlarged stabilizer codes achieving the quantum GV bound for enlarged CSS codes asymptotically. Asapplications, we derive two families of fast encodable and decodable CSS codes with parameters$\mathscr {Q}_{1}=[[N,\Omega (\sqrt {N}),\Omega (\sqrt {N})]]$, and$\mathscr {Q}_{2}=[[N,\Omega (N/\log N),\Omega (N/\log N)/\Omega (\log N)]]$. We show that$\mathscr {Q}_{1}$can be encoded very efficiently by circuits of size$O(N)$and depth$O(\sqrt {N})$. For an input error syndrome,$\mathscr {Q}_{1}$can correct any adversarial error of weight up to half the minimum distance bound in$O(N)$time.$\mathscr {Q}_{1}$can also be decoded in parallel in$O(\sqrt {N})$time by using$O(\sqrt {N})$classical processors. For an input error syndrome, we proved that$\mathscr {Q}_{2}$can correct a linear number of${X}$-errors with high probability and an almost linear number of${Z}$-errors in$O(N)$time. Moreover,$\mathscr {Q}_{2}$can be decoded in parallel in$O(\log (N))$time by using$O(N)$classical processors.

A locally testable code (LTC) is an error correcting code with a property tester. The tester tests if a word is a codeword by reading constant random bits and rejects the word with probability proportional to the distance from the word to the closest codeword. An important open question until recently is whether there exist c3-LTCs which are LTCs with a constant rate, constant relative distance, and constant locality. In this work, we construct a new LTC family using 1-sided lossless expanders and balanced products.

Variational quantum circuits have been widely employed in quantum simulation and quantum machine learning in recent years. However, quantum circuits with random structures have poor trainability due to the exponentially vanishing gradient with respect to the circuit depth and the qubit number. This result leads to a general standpoint that deep quantum circuits would not be feasible for practical tasks. In this work, we propose an initialization strategy with theoretical guarantees for the vanishing gradient problem in general deep quantum circuits. Specifically, we prove that under proper Gaussian initialized parameters, the norm of the gradient decays at most polynomially when the qubit number and the circuit depth increase. Our theoretical results hold for both the local and the global observable cases, where the latter was believed to have vanishing gradients even for very shallow circuits. Experimental results verify our theoretical findings in quantum simulation and quantum chemistry.

In this paper, we establish an interesting duality between two different quantum information-processing tasks, namely, classical source coding with quantum side information, and channel coding over classical-quantum channels. The duality relates the optimal error exponents of these two tasks, generalizing the classical results of Ahlswede and Dueck [IEEE Trans. Inf. Theory, 28(3):430–443, 1982]. We establish duality both at the operational level and at the level of the entropic quantities characterizing these exponents. For the latter, the duality is given by an exact relation, whereas for the former, duality manifests itself in the following sense: an optimal coding strategy for one task can be used to construct an optimal coding strategy for the other task. Along the way, we derive a bound on the error exponent for classical-quantum channel coding with constant composition codes which might be of independent interest. Finally, we consider the task of variable-length classical compression with quantum side information, and a duality relation between this task and classical-quantum channel coding can also be established correspondingly. Furthermore, we study the strong converse of this task, and show that the strong converse property does not hold even in the i.i.d. scenario.

The eligibility of various advanced quantum algorithms will be questioned if they can not guarantee privacy. To fill this knowledge gap, here we devise an efficient quantum differentially private (QDP) Lasso estimator to solve sparse regression tasks. Concretely, given$N~d$-dimensional data points with$N\ll d$, we first prove that the optimal classical and quantum non-private Lasso requires$\Omega (N+d)$and$\Omega (\sqrt {N}+\sqrt {d})$runtime, respectively. We next prove that the runtime cost of QDP Lasso isdimension independent, i.e.,$O(N^{5/2})$, which implies that the QDP Lasso can be faster than both the optimal classical and quantum non-private Lasso. Last, we exhibit that the QDP Lasso attains a near-optimal utility bound$\tilde {O}(N^{-2/3})$with privacy guarantees and discuss the chance to realize it on near-term quantum chips with advantages.

We solve the entanglement-assisted (EA) classical capacity region of quantum multiple-access channels with an arbitrary number of senders, which is conjectured by Hsieh, Devetak and Winter. As an example, we consider the bosonic thermal-loss multiple-access channel and solve the rate region enabled by an entanglement source composed of sender-receiver pairwise two-mode squeezed vacuum states. The EA rate region is strictly larger than the capacity region without entanglement-assistance, therefore also larger than the Yen-Shapiro rate-region of Gaussian encoding or coherent-state encoding. When the senders have equal low brightness, we also numerically find that the two-mode squeezed vacuum source is optimal at a corner rate point. With two-mode squeezed vacuum states as the source and phase modulation as the encoding, we also design practical receiver protocols to realize the entanglement advantages. In the parameter region of a large noise background, the receivers can enable a simultaneous rate advantage of 82.0% for each sender with binary phase-shift keying. Due to teleportation and superdense coding, our results for EA classical communication can be directly extended to EA quantum communication at half of the rates.

In this paper, we present a new construction of asymmetric quantum codes (AQCs) by combining classical concatenated codes (CCs) with tensor product codes (TPCs), called asymmetric quantum concatenated and tensor product codes (AQCTPCs) which have the following three advantages. First, only the outer codes in AQCTPCs need to satisfy the orthogonal constraint in quantum codes, and any classical linear code can be used for the inner, which makes AQCTPCs very easy to construct. Second, most AQCTPCs are highly degenerate, which means they can correct many more errors than their classical TPC counterparts. Consequently, we construct several families of AQCs with better parameters than known results in the literature. Third, AQCTPCs can be efficiently decoded although they are degenerate, provided that the inner and outer codes are efficiently decodable. In particular, we significantly reduce the inner decoding complexity of TPCs from$\Omega (n_{2}a^{n_{1}})(a>1)$to$O(n_{2})$by considering error degeneracy, where$n_{1}$and$n_{2}$are the block length of the inner code and the outer code, respectively. Furthermore, we generalize our concatenation scheme by using the generalized CCs and TPCs correspondingly.

In this paper, we analyze classical data compression with quantum side information (also known as the classical-quantum Slepian–Wolf protocol) in the so-called large and moderate deviation regimes. In the non-asymptotic setting, the protocol involves compressing classical sequences of finite length$n$and decoding them with the assistance of quantum side information. In the large deviation regime, the compression rate is fixed, and we obtain bounds on the error exponent function, which characterizes the minimal probability of error as a function of the rate. Devetak and Winter showed that the asymptotic data compression limit for this protocol is given by a conditional entropy. For any protocol with a rate below this quantity, the probability of error converges to one asymptotically and its speed of convergence is given by the strong converse exponent function. We obtain finite blocklength bounds on this function, and determine exactly its asymptotic value. In the moderate deviation regime for the compression rate, the latter is no longer considered to be fixed. It is allowed to depend on the blocklength$n$, but assumed to decay slowly to the asymptotic data compression limit. Starting from a rate above this limit, we determine the speed of convergence of the error probability to zero and show that it is given in terms of the conditional information variance.

We consider state redistribution of a "hybrid" information source that has both classical and quantum components. The sender transmits classical and quantum information at the same time to the receiver, in the presence of classical and quantum side information both at the sender and at the decoder. The available resources are shared entanglement, and noiseless classical and quantum communication channels. We derive one-shot direct and converse bounds for these three resources, represented in terms of the smooth conditional entropies of the source state. The two bounds coincide in the asymptotic limit of infinitely many copies and vanishingly small error. Various coding theorems for two-party communication tasks are obtained by reduction from our results.

We consider a variation of the well-studied quantum state redistribution task, in which the starting state is known only to the receiver Bob and not to the sender Alice. We refer to this as quantum state redistribution with a one-sided promise. In addition, we consider communication from Alice to Bob over a noisy channel N, instead of the noiseless channel, as is usually considered in state redistribution. We take a natural approach towards the solution of this problem where we “embed” the promise as part of the state and then invoke known protocols for quantum state redistribution composed with known protocols for transfer of quantum information over noisy channels. Using our approach, we are able to reproduce the Alpha-bit capacities with or without entanglement assistance in Hayden and Penington, using known protocols for quantum state redistribution and quantum communication over noisy channels. Furthermore, we generalize the entanglement assisted classical Alpha-bit capacity, showing that any quantum state redistribution protocol can be used as a black box to simulate classical communication.

We study the communication capabilities of a quantum channel under the most general channel model known as the one-shot model. Unlike classical channels that can only be used to transmit classical information (bits), a quantum channel can be used for transmission of classical information, quantum information (qubits) and simultaneous transmission of classical and quantum information. In this work, we investigate the one-shot capabilities of a quantum channel for simultaneously transmitting bits and qubits. This problem was studied in the asymptotic regime for a memoryless channel where a regularized characterization of the capacity region was reported. It is known that the transmission of private classical information is closely related to the problem of quantum information transmission. We resort to this idea and find achievable and converse bounds on the simultaneous transmission of the public and private classical information. Then shifting the classical private rate to the quantum information rate leads to a rate region for simultaneous transmission of classical and quantum information. In the case of asymptotic i.i.d. setting, our one-shot result is evaluated to the known results in the literature. Our main tools used in the achievability proofs are position-based decoding and convex-split lemma.

The two-receiver broadcast channel with primary and third party receivers is studied. The sender wishes to reliably communicate a common (or public) message to both receivers as well as individualized and confidential messages to the primary receiver only. The third party receiver must be kept completely ignorant of the confidential message but there are no secrecy requirements associated to the individualized message. A trade-off arises between the rates of the three messages: when one of the rates is high, the other rates may need to back off to guarantee the reliable transmission of all three messages. In addition, the confidentiality requirement implies availability of local randomness at the transmitter in order to implement a stochastic encoding. This article studies the trade-off between the rates of the common, individualized and confidential messages as well as that of the local randomness in the one-shot regime of a quantum broadcast channel. We provide an achievability region, by proving a conditional version of the convex-split lemma combined with the position-based decoding, as well as a (weak) converse region. We study the asymptotic behaviour of our bounds and recover several well-known asymptotic results in the literature, including simultaneous transmission of classical and quantum information.

In this paper, we study tail inequalities of the largest eigenvalue of a matrix infinitely divisible (i.d.) series, which is a finite sum of fixed matrices weighted by i.d. random variables. We obtain several types of tail inequalities, including Bennett-type and Bernstein-type inequalities. This allows us to further bound the expectation of the spectral norm of a matrix i.d. series. Moreover, by developing a new lower-bound function for Q(s) = (s + 1) log(s + 1) - s that appears in the Bennett-type inequality, we derive a tighter tail inequality of the largest eigenvalue of the matrix i.d. series than the Bernstein-type inequality when the matrix dimension is high. The resulting lower-bound function is of independent interest and can improve any Bennett-type concentration inequality that involves the function Q(s). The class of i.d. probability distributions is large and includes Gaussian and Poisson distributions, among many others. Therefore, our results encompass the existing work on matrix Gaussian series as a special case. Lastly, we show that the tail inequalities of a matrix i.d. series have applications in several optimization problems including the chance constrained optimization problem and the quadratic optimization problem with orthogonality constraints. In addition, we also use the resulting tail bounds to show that random matrices constructed from i.d. random variables satisfy the restricted isometry property (RIP) when it acts as a measurement matrix in compressed sensing.

In this paper, we establish an interesting duality between two different quantum information-processing tasks, namely, classical source coding with quantum side information, and channel coding over classical-quantum channels. The duality relates the optimal error exponents of these two tasks, generalizing the classical results of Ahlswede and Dueck. We establish duality both at the operational level and at the level of the entropic quantities characterizing these exponents. For the latter, the duality is given by an exact relation, whereas for the former, duality manifests itself in the following sense: an optimal coding strategy for one task can be used to construct an optimal coding strategy for the other task. Along the way, we derive a bound on the error exponent for classical-quantum channel coding with constant composition codes which might be of independent interest.

The scaled Rényi information plays a significant role in evaluating the performance of information processing tasks by virtue of its connection to the error exponent analysis. In quantum information theory, there are three generalizations of the classical Rényi divergence-the Petz's, sandwiched, and log-Euclidean versions, that possess meaningful operational interpretation. The goal of this paper is thus to analyze fundamental properties of scaled Rényi information from a noncommutative measure-theoretic perspective. Firstly, we prove the uniform equicontinuity for all three quantum versions of Rényi information, hence it yields the joint continuity of these quantities in the orders and priors. Secondly, we establish the concavity in the region of s ∈ (-1, 0) for both Petz's and the sandwiched versions. This completes the open questions raised by Holevo, Mosonyi and Ogawa. For the applications, we show that the strong converse exponent in classical-quantum channel coding satisfies a minimax identity. The established concavity is further employed to prove an entropic duality between classical data compression with quantum side information and classical-quantum channel coding, and a Fenchel duality in joint source-channel coding with quantum side information.

The 2-receiver broadcast channel with primary and third-party receivers is studied. The messages are classified into public, private and confidential. The messages in the public class are messages intended for both receivers. The private messages are intended for the primary receiver with no secrecy requirements imposed upon them. And the confidential messages are aimed exclusively to the primary receiver such that they must not be accessible to the other receiver. The encoder performs the necessary encryption by virtue of local randomness whose rate is assumed to be limited. We find an achievability region on the trade-off between the rates of the three messages and the source of randomness in the one-shot regime of a quantum broadcast channel.

We study lower bounds on the optimal error probability in classical coding over classical-quantum channels at rates below the capacity, commonly termed quantum sphere-packing bounds. Winter and Dalai have derived such bounds for classical-quantum channels; however, the exponents in their bounds only coincide when the channel is classical. In this paper, we show that these two exponents admit a variational representation and are related by the Golden-Thompson inequality, reaffirming that Dalai's expression is stronger in general classical-quantum channels. Second, we establish a finite blocklength sphere-packing bound for classical-quantum channels, which significantly improves Dalai's prefactor from the order of subexponential to polynomial. Furthermore, the gap between the obtained error exponent for constant composition codes and the best known classical random coding exponent vanishes in the order of o(logn/n), indicating our sphere-packing bound is almost exact in the high rate regime. Finally, for a special class of symmetric classical-quantum channels, we can completely characterize its optimal error probability without the constant composition code assumption. The main technical contributions are two converse Hoeffding bounds for quantum hypothesis testing and the saddle-point properties of error exponent functions.

In this paper, we analyze classical data compression with quantum side information (also known as the classical-quantum Slepian- Wolf protocol) in the so-called large and moderate deviation regimes. In the non-asymptotic setting, the protocol involves compressing classical sequences of finite length n and decoding them with the assistance of quantum side information. In the large deviation regime, the compression rate is fixed, and we obtain bounds on the error exponent function, which characterizes the minimal probability of error as a function of the rate. Devetak and Winter showed that the asymptotic data compression limit for this protocol is given by a conditional entropy. For any protocol with a rate below this quantity, the probability of error converges to one asymptotically and its speed of convergence is given by the strong converse exponent function. We obtain finite blocklength bounds on this function, and determine exactly its asymptotic value, thus improving on previous results by Tomamichel. In the moderate deviation regime for the compression rate, the latter is no longer considered to be fixed. It is allowed to depend on the blocklength n, but assumed to decay slowly to the asymptotic data compression limit. Starting from a rate above this limit, we determine the speed of convergence of the error probability to zero and show that it is given in terms of the conditional information variance. Our results complement earlier results obtained by Tomamichel and Hayashi, in which they analyzed the so-called small deviation regime of this protocol.

In practical communication and computation systems, errors occur predominantly in adjacent positions rather than in a random manner. In this paper, we develop a stabilizer formalism for quantum burst error correction codes (QBECC) to combat such error patterns in the quantum regime. Our contributions are as follows. Firstly, we derive an upper bound for the correctable burst errors of QBECCs, the quantum Reiger bound (QRB). Secondly, we propose two constructions of QBECCs: one by heuristic computer search and the other by concatenating two quantum tensor product codes (QTPCs). We obtain several new QBECCs with better parameters than existing codes with the same coding length. Moreover, some of the constructed codes can saturate the quantum Reiger bounds. Finally, we perform numerical experiments for our constructed codes over Markovian correlated depolarizing quantum memory channels, and show that QBECCs indeed outperform standard QECCs in this scenario.

We aim to study the optimal rates of transmission of public and private classical information over a quantum channel in the most general channel model. To this end, we discuss a scenario in which a quantum channel is being used only once, i.e., one-shot regime is considered. A quantum channel can be used to send classical information (bits) either publicly or privately and for either case, one-shot bounds have been reported in the literature. This paper investigates the one-shot capacity capabilities of a quantum channel for simultaneous transmission of public and private information. We derive an achievable rate region in the form of a tradeoff between public and private rates. We also provide converse bounds assessing the tightness of our achievable rates. Our main tools used in the achievability proofs are position-based decoding and convex-split lemma.

In this paper, we investigate the additivity phenomenon in the quantum dynamic capacity region of a quantum channel for trading the resources of classical communication, quantum communication, and entanglement. Understanding such an additivity property is important if we want to optimally use a quantum channel for general communication purposes. However, in a lot of cases, the channel one will be using only has an additive single or double resource capacity region, and it is largely unknown if this could lead to a strictly superadditive double or triple resource capacity region, respectively. For example, if a channel has additive classical and quantum capacities, can the classical-quantum capacity region be strictly superadditive? In this paper, we answer such questions affirmatively. We give proof-of-principle requirements for these channels to exist. In most cases, we can provide an explicit construction of these quantum channels. The existence of these superadditive phenomena is surprising in contrast to the result that the additivity of both classical-entanglement and classical-quantum capacity regions imply the additivity of the triple resource capacity region for a given channel.

In this paper, we study the tradeoffs between the error probabilities of classical-quantum channels and the block-length n when the transmission rates approach the channel capacity at a rate lower than 1/√n, a research topic known as moderate deviation analysis. We show that the optimal error probability vanishes under this rate convergence. Our main technical contributions are a tight quantum sphere-packing bound, obtained via Chaganty and Sethuraman's concentration inequality in strong large deviation theory, and asymptotic expansions of error-exponent functions. Moderate deviation analysis for quantum hypothesis testing is also established. The converse directly follows from our channel coding result, while the achievability relies on a martingale inequality.

In this paper, we consider two extensions of the Gács-Körner common information to three variables, theconditional common information(cCI) and thecoarse-grained conditional common information(ccCI). Both quantities are shown to be useful technical tools in the study of classical and quantum resource transformations. In particular, the ccCI is shown to have an operational interpretation as the optimal rate of secret key extraction from an eavesdropped classical sourcepXYZwhen Alice (X) and Bob (Y) are unable to communicate but share common randomness with the eavesdropper Eve (Z). Moving to the quantum setting, we consider two different ways of generating a tripartite quantum state from classical correlationspXYZ: (1) coherent encodings Σxyz√(pxyz)|xyz> and (2) incoherent encodings Σxyzpxyz|xyz>xyz|. We study how well can Alice and Bob extract secret key from these quantum sources using quantum operations compared with the extraction of key from the underlying classical sourcespXYZusing classical operations. While the power of quantum mechanics increases Alice and Bob's ability to generate shared randomness, it also equips Eve with a greater arsenal of eavesdropping attacks. Therefore, it is not obvious who gains the greatest advantage for distilling secret key when replacing a classical source with a quantum one. We first demonstrate that the classical key rate ofpXYZis equivalent to the quantum key rate for an incoherent quantum encoding of the distribution. For coherent encodings, we next show that the classical and quantum rates are generally incomparable, and in fact, their difference can be arbitrarily large in either direction. Finally, we introduce a “zoo” of entangled tripartite states all characterized by the conditional common information of their encoded probability distributions. Remarkably, for these states almost all entanglement measures, such as Alice and Bob's entanglement cost, squashed entanglement, and relative entropy of entanglement, can be sharply bounded or even exactly expressed in terms of the conditional common information. In the latter case, we thus present a rare instance in which the various entropic entanglement measures of a quantum state can be explicitly calculated.

“To be considered for the 2017 IEEE Jack Keil Wolf ISIT Student Paper Award.” We show that the reliable communication through a classical-quantum channel is possible when the transmission rate approaches the channel capacity sufficiently slowly. This scenario exists between the non-vanishing error probability regime, where the rate tends to capacity with a fixed error, and the small error probability regime, where the error vanishes given a rate below capacity. The proof employs a sharp concentration bound in strong large deviation theory, and the asymptotic expansions of the error-exponent functions.

“To be considered for the 2017 IEEE Jack Keil Wolf ISIT Student Paper Award.” We study the asymptotic behavior of the type-I error in quantum hypothesis testing when the exponent of the type-II error approaches the quantum relative entropy sufficiently slowly. Our result shows that the moderate deviation principle holds for the testing problem if the quantum relative variance is positive. Our proof strategy employs strong large deviation theory and a martingale inequality.

“To be considered for the 2017 IEEE Jack Keil Wolf ISIT Student Paper Award.” We provide a sphere-packing lower bound for the optimal error probability in finite blocklengths when coding over a symmetric classical-quantum channel. Our result shows that the pre-factor can be significantly improved from the order of the subexponential to the polynomial, This established pre-factor is arguably optimal because it matches the best known random coding upper bound in the classical case. Our approaches rely on a sharp concentration inequality in strong large deviation theory and crucial properties of the error-exponent function.

We study lower bounds on the optimal error probability in channel coding at rates below capacity, commonly termed sphere-packing bounds. In this work, we establish a sphere-packing bound for classical-quantum channels, which significantly improves previous prefactor from the order of subexponential to polynomial. Furthermore, the gap between the obtained error exponent for constant composition codes and the best known classical random coding exponent vanishes in the order of o(log n/n), indicating our sphere-packing bound is almost exact in the high rate regime. The main technical contributions are two converse Hoeffding bounds for quantum hypothesis testing and the saddle-point properties of error exponent functions.

The weight generating functions associated with convolutional codes (CCs) are based on state space realizations or the weight adjacency matrices (WAMs). The MacWilliams identity for CCs on the WAMs was first established by Gluesing-Luerssen and Schneider in the case of minimal encoders, and generalized by Forney. We consider this problem in the viewpoint of constraint codes and obtain a simple and direct proof of this MacWilliams identity in the case of minimal encoders. For our purpose, we choose a different representation for the exact weight generating function (EWGF) of a block code, by defining it as a linear combination of orthonormal vectors in Dirac bra-ket notation. This representation provides great flexibility so that general split weight generating functions and their MacWilliams identities can be easily obtained from the MacWilliams identity for EWGFs. As a result, we also obtain the MacWilliams identity for the input-parity WAMs of a systematic convolutional code and its dual. Finally, paralleling the development of the classical case, we establish the MacWilliams identity for quantum CCs.

The auxiliary function of a classical channel appears in two fundamental quantities, the random coding exponent and the sphere-packing exponent, which yield upper and lower bounds on the error probability of decoding, respectively. A crucial property of the auxiliary function is its concavity, and this property consequently leads to several important results in finite blocklength analysis. In this paper, we prove that the auxiliary function of a classical-quantum channel also enjoys the same concavity property, extending an earlier partial result to its full generality. We also prove that the auxiliary function satisfies the data-processing inequality, among various other important properties. Furthermore, we show that the concavity property of the auxiliary function enables a geometric interpretation of the random coding exponent and the sphere-packing exponent of a classical-quantum channel. The key component in our proof is an important result from the theory of matrix geometric means.

In this paper, we consider the problem of generating arbitrary three-party correlations from a combination of public and secret correlations. Two parties-called Alice and Bob-share perfectly correlated bits that are secret from a collaborating third party, Charlie. At the same time, all three parties have access to a separate source of correlated bits, and their goal is to convert these two resources into multiple copies of some given tripartite distribution P(XYZ). We obtain a single-letter characterization of the tradeoff between public and private bits that are needed to achieve this task. The rate of private bits is shown to generalize Wyner's classic notion of common information held between a pair of random variables. The problem we consider can be contrasted fruitfully with the task of secrecy formation, in which P(XYZ) is generated using public communication and local randomness but with Charlie functioning as an adversary instead of a collaborator. We describe in detail the differences between the collaborative and adversarial scenarios.

Coded source compression, also known as source compression with helpers, has been a major variant of distributed source compression, but has hitherto received little attention in the quantum regime. This letter treats and solves the corresponding quantum coded source compression through an observation that connects coded source compression with channel simulation. First, we consider classical source coding with quantum side information, where the quantum side information is observed by a helper and sent to the decoder via a classical channel. We derive a single-letter characterization of the achievable rate region for this problem. The direct coding theorem of our result is proved via the measurement compression theory of Winter, a quantum-to-classical channel simulation. Our result reveals that a helper's scheme which separately conducts a measurement and a compression is suboptimal, and measurement compression seems necessary to achieve the optimal rate region. We then study coded source compression in the fully quantum regime, where two different scenarios are considered depending on the types of communication channels between the legitimate source and the receiver. We further allow entanglement assistance from the quantum helper in both scenarios. We characterize the involved quantum resources and derive single-letter expressions of the achievable rate region. The direct coding proofs are based on well-known quantum protocols, the quantum state merging protocol, and the fully quantum Slepian-Wolf protocol, together with the quantum reverse Shannon theorem.

We study classical source coding with quantum side-information where the quantum side-information is observed by a helper and sent to the decoder via a classical channel. We derive a single-letter characterization of the achievable rate region for this problem. The direct part of our result is proved via the measurement compression theory by Winter. Our result reveals that a helper's scheme that separately conducts a measurement and a compression is suboptimal, and the measurement compression is fundamentally needed to achieve the optimal rate region.

In this paper, we propose a definition of the dual code of a quantum convolutional code, with or without entanglement assistance. We then derive a MacWilliams identity for quantum convolutional codes. Along the way, we obtain a direct proof of the MacWilliams identity, first found by Gluesing-Luerssen and Schneider, in the setting of classical convolutional codes.

In this paper, we prove a MacWilliams identity for the weight adjacency matrices based on the constraint codes of a convolutional code (CC) and its dual. Our result improves upon a recent result by Gluesing-Luerssen and Schneider, where the requirement of a minimal encoder is assumed. We can also establish the MacWilliams identity for the input-parity weight adjacency matrices of a systematic CC and its dual. Most importantly, we show that a type of Hamming weight enumeration functions of all codewords of a CC can be derived from the weight adjacency matrix, which thus provides a connection between these two very different notions of weight enumeration functions in the convolutional code literature. Finally, the relations between various enumeration functions of a CC and its dual are summarized in a diagram. This explains why no MacWilliams identity exists for the free-distance enumerators.

We develop the theory of entanglement-assisted quantum error-correcting (EAQEC) codes, a generalization of the stabilizer formalism to the setting in which the sender and receiver have access to preshared entanglement. Conventional stabilizer codes are equivalent to self-orthogonal symplectic codes. In contrast, EAQEC codes do not require self-orthogonality, which greatly simplifies their construction. We show how any classical binary or quaternary block code can be made into an EAQEC code. We provide a table of best known EAQEC codes with code length up to 10. With the self-orthogonality constraint removed, we see that the distance of an EAQEC code can be better than any standard quantum error-correcting code with the same fixed net yield. In a quantum computation setting, EAQEC codes give rise to catalytic quantum codes, which assume a subset of the qubits are noiseless. We also give an alternative construction of EAQEC codes by making classical entanglement-assisted codes coherent.

In this paper, we consider the conditions under which a given ensemble of two-qubit states can be optimally distinguished by local operations and classical communication (LOCC). We begin by completing the perfect distinguishability problem of two-qubit ensembles-both for separable operations and LOCC-by providing necessary and sufficient conditions for the perfect discrimination of one pure and one mixed state. Then, for the well-known task of minimum error discrimination, it is shown that almost all two-qubit ensembles consisting of three pure states cannot be optimally discriminated using LOCC. This is surprising considering that any two pure states can be distinguished optimally by LOCC. Special attention is given to ensembles that lack entanglement, and we prove an easy sufficient condition for when a set of three product states cannot be optimally distinguished by LOCC, thus providing new examples of the phenomenon known as non-locality without entanglement. We next consider an example of N parties who each share the same state but who are ignorant of its identity. The state is drawn from the rotationally invariant trine ensemble, and we establish a tight connection between the N-copy ensemble and Shor's lifted single-copy ensemble. For any finite N, we prove that optimal identification of the states cannot be achieved by LOCC; however, as N→∞, LOCC can indeed discriminate the states optimally. This is the first result of its kind. Finally, we turn to the task of unambiguous discrimination and derive new lower bounds on the LOCC inconclusive probability for symmetric states. When applied to the double trine ensemble, this leads to a rather different distinguishability character than when the minimum error probability is considered.

An unexpected breakdown in the existing theory of quantum serial turbo coding is that a quantum convolutional encoder cannot simultaneously be recursive and non-catastrophic. These properties are essential for quantum turbo code families to have a minimum distance growing with blocklength and for their iterative decoding algorithm to converge, respectively. Here, we show that the entanglement-assisted paradigm simplifies the theory of quantum turbo codes, in the sense that an entanglement-assisted quantum (EAQ) convolutional encoder can possess both of the aforementioned desirable properties. We give several examples of EAQ convolutional encoders that are both recursive and non-catastrophic and detail their relevant parameters. We then modify the quantum turbo decoding algorithm of Poulin , in order to have the constituent decoders pass along only extrinsic information to each other rather than a posteriori probabilities as in the decoder of Poulin , and this leads to a significant improvement in the performance of unassisted quantum turbo codes. Other simulation results indicate that entanglement-assisted turbo codes can operate reliably in a noise regime 4.73 dB beyond that of standard quantum turbo codes, when used on a memoryless depolarizing channel. Furthermore, several of our quantum turbo codes are within 1 dB or less of their hashing limits, so that the performance of quantum turbo codes is now on par with that of classical turbo codes. Finally, we prove that entanglement is the resource that enables a convolutional encoder to be both non-catastrophic and recursive because an encoder acting on only information qubits, classical bits, gauge qubits, and ancilla qubits cannot simultaneously satisfy them.

We study entanglement-assisted quantum and classical communication over a single use of a quantum channel, which itself can correspond to a finite number of uses of a channel with arbitrarily correlated noise. We obtain characterizations of the corresponding one-shot capacities by establishing upper and lower bounds on them in terms of the difference of two smoothed entropic quantities. In the case of a memoryless channel, the upper and lower bounds converge to the known single-letter formulas for the corresponding capacities, in the limit of asymptotically many uses of it. Our results imply that the difference of two smoothed entropic quantities characterizing the one-shot entanglement-assisted capacities serves as a one-shot analog of the mutual information, since it reduces to the mutual information, between the output of the channel and a system purifying its input, in the asymptotic, memoryless scenario.

We derive quantum counterparts of two key theorems of classical information theory, namely, the rate-distortion theorem and the source-channel separation theorem. The rate-distortion theorem gives the ultimate limits on lossy data compression, and the source-channel separation theorem implies that a two-stage protocol consisting of compression and channel coding is optimal for transmitting a memoryless source over a memoryless channel. In spite of their importance in the classical domain, there has been surprisingly little work in these areas for quantum information theory. In this paper, we prove that the quantum rate-distortion function is given in terms of the regularized entanglement of purification. We also determine a single-letter expression for the entanglement-assisted quantum rate-distortion function, and we prove that it serves as a lower bound on the unassisted quantum rate-distortion function. This implies that the unassisted quantum rate-distortion function is nonnegative and generally not equal to the coherent information between the source and distorted output (in spite of Barnum's conjecture that the coherent information would be relevant here). Moreover, we prove several quantum source-channel separation theorems. The strongest of these are in the entanglement-assisted setting, in which we establish a necessary and sufficient condition for transmitting a memoryless source over a memoryless quantum channel up to a given distortion.

We use the smooth entropy approach to treat the problems of binary quantum hypothesis testing and the transmission of classical information through a quantum channel. We provide lower and upper bounds on the optimal type II error of quantum hypothesis testing in terms of the smooth max-relative entropy of the two states representing the two hypotheses. Then using a relative entropy version of the quantum asymptotic equipartition property (QAEP), we can recover the strong converse rate of the i.i.d. hypothesis testing problem in the asymptotics. On the other hand, combining Stein's lemma with our bounds, we obtain a stronger ( ε-independent) version of the relative entropy-QAEP. Similarly, we provide bounds on the one-shot ε-error classical capacity of a quantum channel in terms of a smooth max-relative entropy variant of its Holevo capacity. Using these bounds and the ε-independent version of the relative entropy-QAEP, we can recover both the Holevo- Schumacher- Westmoreland theorem about the optimal direct rate of a memoryless quantum channel with product state encoding, as well as its strong converse counterpart.

We extend quantum rate-distortion theory by considering auxiliary resources that might be available to a sender and receiver performing lossy quantum data compression. The first setting we consider is that of quantum rate-distortion coding with the help of a classical side channel. Our result here is that the regularized entanglement of formation characterizes the quantum rate-distortion function, extending earlier work of Devetak and Berger. We also combine this bound with the entanglement-assisted bound from our prior work to obtain the best known bounds on the quantum rate-distortion function for an isotropic qubit source. The second setting we consider is that of quantum rate-distortion coding with quantum side information (QSI) available to the receiver. In order to prove results in this setting, we first state and prove a quantum reverse Shannon theorem with QSI (for tensor-power states), which extends the known tensor-power quantum reverse Shannon theorem. The achievability part of this theorem relies on the quantum state redistribution protocol, while the converse relies on the fact that the protocol can cause only a negligible disturbance to the joint state of the reference and the receiver's QSI. This quantum reverse Shannon theorem with QSI naturally leads to quantum rate-distortion theorems with QSI, with or without entanglement assistance.

Contrary to the assumption that most quantum error-correcting codes (QECC) make, it is expected that phase errors are much more likely than bit errors in physical devices. By employing the entanglement-assisted stabilizer formalism, we develop a new kind of error-correcting protocol which can flexibly trade error correction abilities between the two types of errors, such that high error correction performance is achieved both in symmetric and in asymmetric situations. The characteristics of the QECCs can be optimized in an adaptive manner during information transmission. The proposed entanglement-assisted QECCs require only one ebit regardless of the degree of asymmetry at a given moment and can be decoded in polynomial time.

One of the unexpected breakdowns in the existing theory of quantum serial turbo coding is that a quantum convolutional encoder cannot simultaneously be recursive and non-catastrophic. These properties are essential for a quantum turbo code to have an unbounded minimum distance and for its iterative decoding algorithm to converge, respectively. Here, we show that the entanglement-assisted paradigm gives a theoretical and simulated “turbo boost” to these codes, in the sense that an entanglement-assisted quantum (EAQ) convolutional encoder can possess both of the aforementioned desirable properties, and simulation results indicate that entanglement-assisted turbo codes can operate reliably in a noise regime 5.5 dB beyond that of standard quantum turbo codes. Entanglement is the resource that enables a convolutional encoder to satisfy both properties because an encoder acting on only information qubits, classical bits, gauge qubits, and ancilla qubits cannot simultaneously satisfy them. Simulation results demonstrate that interleaved serial concatenation of EAQ convolutional encoders leads to a powerful code construction with excellent performance on a memoryless depolarizing channel.

Though the entanglement-assisted formalism provides a universal connection between a classical linear code and an entanglement-assisted quantum error-correcting code (EAQECC), the issue of maintaining large amount of pure maximally entangled states in constructing EAQECCs is a practical obstacle to its use. It is also conjectured that the power of entanglement-assisted formalism to convert those good classical codes comes from massive consumption of maximally entangled states. We show that the above conjecture is wrong by providing families of EAQECCs with an entanglement consumption rate that diminishes linearly as a function of the code length. Notably, two families of EAQECCs constructed in the paper require only one copy of maximally entangled state no matter how large the code length is. These families of EAQECCs that are constructed from classical finite geometric LDPC codes perform very well according to our numerical simulations. Our work indicates that EAQECCs are not only theoretically interesting, but also physically implementable. Finally, these high performance entanglement-assisted LDPC codes with low entanglement consumption rates allow one to construct high-performance standard QECCs with very similar parameters.

We introduce a new protocol, the channel-state coding protocol, to quantum Shannon theory. This protocol generates entanglement between a sender and receiver by coding for a noisy quantum channel with the aid of a noisy shared state. The mother and father protocols arise as special cases of the channel-state coding protocol, where the channel is noiseless or the state is a noiseless maximally entangled state, respectively. The channel-state coding protocol paves the way for formulating entanglement-assisted quantum error-correcting codes that are robust to noise in shared entanglement. Finally, the channel-state coding protocol leads to a Smith-Yard superactivation, where we can generate entanglement using a zero-capacity erasure channel and a non-distillable bound entangled state.

In this paper, we consider the problem of transmitting classical and quantum information reliably over an entanglement-assisted quantum (EAQ) channel. Our main result is a capacity theorem that gives a 3-D achievable rate region. Points in the region are rate triples, consisting of the classical communication rate, the quantum communication rate, and the entanglement consumption rate of a particular coding scheme. The crucial protocol in achieving the boundary points of the capacity region is a protocol that we name the classically enhanced father(CEF) protocol. The CEF protocol is more general than other protocols in the family tree of quantum Shannon theoretic protocols, in the sense that several previously known quantum protocols are now child protocols of it. The CEF protocol also shows an improvement over a timesharing strategy for the case of a qubit dephasing channel-this result justifies the need for simultaneous coding of classical and quantum information over an EAQ channel. Our capacity theorem is of a multiletter nature (requiring a limit over many uses of the channel), but it reduces to a single-letter characterization for at least three channels: the completely depolarizing channel, the quantum erasure channel, and the qubit dephasing channel.

In this paper, we give tradeoffs between classical communication, quantum communication, and entanglement for processing information in the Shannon-theoretic setting. We first prove a “unit-resource” capacity theorem that applies to the scenario where only the above three noiseless resources are available for consumption or generation. The optimal strategy mixes the three fundamental protocols of teleportation, superdense coding, and entanglement distribution. We then provide an achievable rate region and a matching multiletter converse for the “direct-static” capacity theorem. This theorem applies to the scenario where a large number of copies of a noisy bipartite state are available (in addition to consumption or generation of the above three noiseless resources). Our coding strategy involves a protocol that we name theclassically assisted state redistribution protocoland the three fundamental protocols. We finally provide an achievable rate region and a matching multiletter converse for the “direct-dynamic” capacity theorem. This theorem applies to the scenario where a large number of uses of a noisy quantum channel are available in addition to the consumption or generation of the three noiseless resources. Our coding strategy combines theclassically enhanced father protocolwith the three fundamental unit protocols.

We find a regularized formula for the entanglement-assisted (EA) capacity region for quantum multiple-access channels (QMAC). We illustrate the capacity region calculation with the example of the collective phase-flip channel which admits a single-letter characterization. On the way, we provide a first-principles proof of the EA coding theorem based on a packing argument. We observe that the Holevo-Schumacher-Westmoreland theorem may be obtained from a modification of our EA protocol. We remark on the existence of a family hierarchy of protocols for multiparty scenarios with a single receiver, in analogy to the two-party case. In this way, we relate several previous results regarding QMACs.

Entanglement-assisted quantum error-correcting codes (EAQECCs) make use of pre-existing entanglement between the sender and receiver to boost the rate of transmission. It is possible to construct an EAQECC from any classical linear code, unlike standard QECCs which can only be constructed from dual-containing codes. Operator quantum error-correcting codes (OQECCs) allow certain errors to be corrected (or prevented) passively, reducing the complexity of the correction procedure. We combine these two extensions of standard quantum error correction into a unified entanglement- assisted quantum error correction formalism. This new scheme, which we call entanglement-assisted operator quantum error correction (EAOQEC), is the most general and powerful quantum error-correcting technique known, retaining the advantages of both entanglement-assistance and passive correction. We present the formalism, show the considerable freedom in constructing EAOQECCs from classical codes, and demonstrate the construction with examples.

Both direct sequence and frequency hopping spread spectrum communication systems co-exist in the unlicensed band such as 2.4 GHz ISM band. We propose a novel structure to obtain the channel information so that more robust communication is possible. Both theoretical and numerical results are presented to demonstrate effectiveness.