arXiv subjects

Hao-Chung Cheng

Publications and source records attributed to Hao-Chung Cheng.

68 records · Page 4Linked to original sources

arXiv2021

On the Existence of the Augustin Mean

The existence of a unique Augustin mean and its invariance under the Augustin operator are established for arbitrary input distributions with finite Augustin information for channels with countably generated output $σ$-algebras. The existence is established by representing the conditional Rényi divergence as a lower semicontinuous and convex functional in an appropriately chosen uniformly convex space and then invoking the Banach--Saks property in conjunction with the lower semicontinuity and the convexity. A new family of operators is proposed to establish the invariance of the Augustin mean under the Augustin operator for orders greater than one. Some members of this new family strictly decrease the conditional Rényi divergence, when applied to the second argument of the divergence, unless the second argument is a fixed point of the Augustin operator.

cs.IT↗

arXiv2020

Refined Strong Converse for the Constant Composition Codes

A strong converse bound for constant composition codes of the form $P_{e}^{(n)} \geq 1- A n^{-0.5(1-E_{sc}'(R,W,p))} e^{-n E_{sc}(R,W,p)}$ is established using the Berry-Esseen theorem through the concepts of Augustin information and Augustin mean, where $A$ is a constant determined by the channel $W$, the composition $p$, and the rate $R$, i.e., $A$ does not depend on the block length $n$.

cs.IT↗

arXiv2019

Strong converse bounds in quantum network information theory: distributed hypothesis testing and source coding

We consider a distributed quantum hypothesis testing problem with communication constraints, in which the two hypotheses correspond to two different states of a bipartite quantum system, multiple identical copies of which are shared between Alice and Bob. They are allowed to perform local operations on their respective systems and send quantum information to Charlie at limited rates. By doing measurements on the systems that he receives, Charlie needs to infer which of the two different states the original bipartite state was in, that is, which of the two hypotheses is true. We prove that the Stein exponent for this problem is given by a regularized quantum relative entropy. The latter reduces to a single letter formula when the alternative hypothesis consists of the products of the marginals of the null hypothesis, and there is no rate constraint imposed on Bob. Our proof relies on certain properties of the so-called quantum information bottleneck function. The second part of this paper concerns the general problem of finding finite blocklength strong converse bounds in quantum network information theory. In the classical case, the analogue of this problem has been reformulated in terms of the so-called image size characterization problem. Here, we extend this problem to the classical-quantum setting and prove a second order strong converse bound for it. As a by-product, we obtain a similar bound for the Stein exponent for distributed hypothesis testing in the special case in which the bipartite system is a classical-quantum system, as well as for the task of quantum source coding with compressed classical side information. Our proofs use a recently developed tool from quantum functional inequalities, namely, the tensorization property of reverse hypercontractivity for the quantum depolarizing semigroup.

quant-ph↗

arXiv2019

Strong Converse for Classical-Quantum Degraded Broadcast Channels

We consider the transmission of classical information through a degraded broadcast channel, whose outputs are two quantum systems, with the state of one being a degraded version of the other. Yard et al. proved that the capacity region of such a channel is contained in a region characterized by certain entropic quantities. We prove that this region satisfies the strong converse property, that is, the maximal probability of error incurred in transmitting information at rates lying outside this region converges to one exponentially in the number of uses of the channel. In establishing this result, we prove a second-order Fano-type inequality, which might be of independent interest. A powerful analytical tool which we employ in our proofs is the tensorization property of the quantum reverse hypercontractivity for the quantum depolarizing semigroup.

quant-ph↗

arXiv2019

Matrix Poincaré, Φ-Sobolev inequalities, and quantum ensembles

Sobolev-type inequalities have been extensively studied in the frameworks of real-valued functions and non-commutative $\mathbb{L}_p$ spaces, and have proven useful in bounding the time evolution of classical/quantum Markov processes, among many other applications. In this paper, we consider yet another fundamental setting - matrix-valued functions - and prove new Sobolev-type inequalities for them. Our technical contributions are two-fold: (i) we establish a series of matrix Poincaré inequalities for separably convex functions and general functions with Gaussian unitary ensembles inputs; and (ii) we derive $Φ$-Sobolev inequalities for matrix-valued functions defined on Boolean hypercubes and for those with Gaussian distributions. Our results recover the corresponding classical inequalities (i.e.~real-valued functions) when the matrix has one dimension. Finally, as an application of our technical outcomes, we derive the upper bounds for a fundamental entropic quantity - the Holevo quantity - in quantum information science since classical-quantum channels are a special instance of matrix-valued functions. This is obtained through the equivalence between the constants in the strong data processing inequality and the $Φ$-Sobolev inequality.

quant-ph↗

arXiv2019

Quantum Sphere-Packing Bounds with Polynomial Prefactors

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 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 $o(\log n / n)$, 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.

quant-ph↗

arXiv2018

Duality between source coding with quantum side information and c-q channel coding

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 c-q 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 c-q channel coding with constant composition codes which might be of independent interest.

quant-ph↗

arXiv2018

Non-Asymptotic Classical Data Compression with Quantum Side Information

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. Our results complement earlier results obtained by Tomamichel and Hayashi, in which they analyzed the so-called small deviation regime of this protocol.

quant-ph↗

arXiv2017

Moderate Deviation Analysis for Classical-Quantum Channels and Quantum Hypothesis Testing

In this work, we study the tradeoffs between the error probabilities of classical-quantum channels and the blocklength $n$ when the transmission rates approach the channel capacity at a rate slower than $1/\sqrt{n}$, 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.

quant-ph↗

arXiv2017

Sphere-Packing Bound for Symmetric Classical-Quantum Channels

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. The established pre-factor is essentially 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.

quant-ph↗

arXiv2016

On the Concavity of Auxiliary Function in Classical-Quantum Channels

The auxiliary function of a classical channel appears in two fundamental quantities that upper and lower bound the error probability, respectively. A crucial property of the auxiliary function is its concavity, which leads to several important results in finite block length analysis. In this paper, we prove that the auxiliary function of a classical-quantum channel also enjoys the same concave property, extending an earlier partial result to its full generality. The key component in our proof is a beautiful result of geometric means of operators.

quant-ph↗

arXiv2016

Characterisations of Matrix and Operator-Valued $Φ$-Entropies, and Operator Efron-Stein Inequalities

We derive new characterisations of the matrix $\mathrmΦ$-entropy functionals introduced in [Electron.~J.~Probab., 19(20): 1--30, 2014]. Notably, all known equivalent characterisations of the classical $Φ$-entropies have their matrix correspondences. Next, we propose an operator-valued generalisation of the matrix $Φ$-entropy functionals, and prove their subadditivity under Löwner partial ordering. Our results demonstrate that the subadditivity of operator-valued $Φ$-entropies is equivalent to the convexity of various related functions. This result can be used to demonstrate an interesting result in quantum information theory: the matrix $Φ$-entropy of a quantum ensemble is monotone under unital quantum channels. Finally, we derive the operator Efron-Stein inequality to bound the operator-valued variance of a random matrix.

math-ph↗

arXiv2015

Exponential Decay of Matrix $Φ$-Entropies on Markov Semigroups with Applications to Dynamical Evolutions of Quantum Ensembles

In the study of Markovian processes, one of the principal achievements is the equivalence between the $Φ$-Sobolev inequalities and an exponential decrease of the $Φ$-entropies. In this work, we develop a framework of Markov semigroups on matrix-valued functions and generalize the above equivalence to the exponential decay of matrix $Φ$-entropies. This result also specializes to spectral gap inequalities and modified logarithmic Sobolev inequalities in the random matrix setting. To establish the main result, we define a non-commutative generalization of the carré du champ operator, and prove a de Bruijn's identity for matrix-valued functions. The proposed Markov semigroups acting on matrix-valued functions have immediate applications in the characterization of the dynamical evolution of quantum ensembles. We consider two special cases of quantum unital channels, namely, the depolarizing channel and the phase-damping channel. In the former, since there exists a unique equilibrium state, we show that the matrix $Φ$-entropy of the resulting quantum ensemble decays exponentially as time goes on. Consequently, we obtain a stronger notion of monotonicity of the Holevo quantity - the Holevo quantity of the quantum ensemble decays exponentially in time and the convergence rate is determined by the modified log-Sobolev inequalities. However, in the latter, the matrix $Φ$-entropy of the quantum ensemble that undergoes the phase-damping Markovian evolution generally will not decay exponentially. This is because there are multiple equilibrium states for such a channel. Finally, we also consider examples of statistical mixing of Markov semigroups on matrix-valued functions. We can explicitly calculate the convergence rate of a Markovian jump process defined on Boolean hypercubes, and provide upper bounds of the mixing time on these types of examples.

quant-ph↗

arXiv2015

The Learnability of Unknown Quantum Measurements

Quantum machine learning has received significant attention in recent years, and promising progress has been made in the development of quantum algorithms to speed up traditional machine learning tasks. In this work, however, we focus on investigating the information-theoretic upper bounds of sample complexity - how many training samples are sufficient to predict the future behaviour of an unknown target function. This kind of problem is, arguably, one of the most fundamental problems in statistical learning theory and the bounds for practical settings can be completely characterised by a simple measure of complexity. Our main result in the paper is that, for learning an unknown quantum measurement, the upper bound, given by the fat-shattering dimension, is linearly proportional to the dimension of the underlying Hilbert space. Learning an unknown quantum state becomes a dual problem to ours, and as a byproduct, we can recover Aaronson's famous result [Proc. R. Soc. A 463:3089-3144 (2007)] solely using a classical machine learning technique. In addition, other famous complexity measures like covering numbers and Rademacher complexities are derived explicitly. We are able to connect measures of sample complexity with various areas in quantum information science, e.g. quantum state/measurement tomography, quantum state discrimination and quantum random access codes, which may be of independent interest. Lastly, with the assistance of general Bloch-sphere representation, we show that learning quantum measurements/states can be mathematically formulated as a neural network. Consequently, classical ML algorithms can be applied to efficiently accomplish the two quantum learning tasks.

quant-ph↗