在香农理论（无论是经典的还是量子的）中，一个重要的基本量是给定信道 W 和速率 R 的最优误差指数：即常数 E(W,R)，它刻画了当使用固定速率 R 的越来越大的码字在给定信道 W 的越来越多的（无记忆）实例上进行通信时，译码误差的指数衰减规律。本文证明，Hayashi [CMP 333, 335 (2015)] 针对隐私放大中类似量给出的一个界，蕴含了对称经典-量子信道上通信误差指数的一个下界。所得下界在高于某一临界值的速率范围内与 Dalai [IEEE TIT 59, 8027 (2013)] 的球填充上界相匹配，并重现了对称信道上众所周知的经典结果。论证过程首先将隐私放大的误差指数与具有量子边信息的经典信息压缩的误差指数联系起来，由此得到一个与 Cheng 等人 [IEEE TIT 67, 902 (2021)] 的球填充上界相匹配的下界。反过来，Cheng 等人所发现的球填充界中的多项式前置因子可以转移到隐私放大问题中，从而至少在针对线性随机性提取器的情况下，加强了 Li、Yao 和 Hayashi [arXiv:2111.01075 [quant-ph]] 最近的一个结果。
英文摘要：
A fundamental quantity of interest in Shannon theory, classical or quantum, is the optimal error exponent of a given channel W and rate R: the constant E(W,R) which governs the exponential decay of decoding error when using ever larger codes of fixed rate R to communicate over ever more (memoryless) instances of a given channel W. Here I show that a bound by Hayashi [CMP 333, 335 (2015)] for an analogous quantity in privacy amplification implies a lower bound on the error exponent of communication over symmetric classical-quantum channels. The resulting bound matches Dalai's [IEEE TIT 59, 8027 (2013)] sphere-packing upper bound for rates above a critical value, and reproduces the well-known classical result for symmetric channels. The argument proceeds by first relating the error exponent of privacy amplification to that of compression of classical information with quantum side information, which gives a lower bound that matches the sphere-packing upper bound of Cheng et al. [IEEE TIT 67, 902 (2021)]. In turn, the polynomial prefactors to the sphere-packing bound found by Cheng et al. may be translated to the privacy amplification problem, sharpening a recent result by Li, Yao, and Hayashi [arXiv:2111.01075 [quant-ph]], at least for linear randomness extractors.
We study a protocol in which many parties use quantum communication to transfer a shared state to a receiver without communicating with each other. This protocol is a multiparty version of the fully quantum Slepian-Wolf protocol for two senders and arises through the repeated application of the two-sender protocol. We describe bounds on the achievable rate region for the distributed compression problem. The inner bound arises by expressing the achievable rate region for our protocol in terms of its vertices and extreme rays and, equivalently, in terms of facet inequalities. We also prove an ou…
We resume the investigation of the problem of independent local compression of correlated quantum sources, the classical case of which is covered by the celebrated Slepian-Wolf theorem. We focus specifically on classical-quantum (cq) sources, for which one edge of the rate region, corresponding to the compression of the classical part, using the quantum part as side information at the decoder, was previously determined by Devetak and Winter [Phys. Rev. A 68, 042301 (2003)]. Whereas the Devetak-Winter protocol attains a rate-sum equal to the von Neumann entropy of the joint source, here we show…
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.
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 work 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 achi…
We study source compression with a helper in the fully quantum regime, extending our earlier result on classical source compression with a quantum helper [arXiv:1501.04366, 2015]. We characterise the quantum resources involved in this problem, and derive a single-letter expression of the achievable rate region when entanglement assistance is available. The direct coding proof is based on a combination of two fundamental protocols, namely the quantum state merging protocol and the quantum reverse Shannon theorem (QRST). This result demonstrates an unexpected connection between distributed sourc…
In this paper, we provide a simple framework for deriving one-shot achievable bounds for some problems in quantum information theory. Our framework is based on the joint convexity of the exponential of the collision relative entropy, and is a (partial) quantum generalization of the technique of Yassaee et al. (2013) from classical information theory. Based on this framework, we derive one-shot achievable bounds for the problems of communication over classical-quantum channels, quantum hypothesis testing, and classical data compression with quantum side information. We argue that our one-shot a…
In blind compression of quantum states, a sender Alice is given a specimen of a quantum state $ρ$ drawn from a known ensemble (but without knowing what $ρ$ is), and she transmits sufficient quantum data to a receiver Bob so that he can decode a near perfect specimen of $ρ$. For many such states drawn iid from the ensemble, the asymptotically achievable rate is the number of qubits required to be transmitted per state. The Holevo information is a lower bound for the achievable rate, and is attained for pure state ensembles, or in the related scenario of entanglement-assisted visible compression…
In this paper we obtain a lower bound of exponent of average probability of error for classical quantum multiple access channel, which implies that for all rate pairs in the capacity region is achievable by a code with exponential probability of error. Thus we re-obtain the direct coding theorem.
A single quantum state can be shared by many distant parties. In this thesis, we try to characterize the information contents of such distributed states by defining the multiparty information and the multiparty squashed entanglement, two steps toward a general theory of multiparty quantum information. As a further step in that direction, we partially solve the multiparty distributed compression problem where multiple parties use quantum communication to faithfully transfer their shares of a state to a common receiver. We build a protocol for multiparty distributed compression based on the full…
On a class of memoryless quantum channels which includes the depolarizing channel, the highest fidelity of quantum error-correcting codes of length n and rate R is proven to be lower bounded by 1-exp[-nE(R)+o(n)] for some function E(R). The E(R) is positive below some threshold R', which implies R' is a lower bound on the quantum capacity.
We consider visible compression for discrete memoryless sources of mixed quantum states when only classical information can be sent from Alice to Bob. We assume that Bob knows the source statistics, and that Alice and Bob have identical random number generators. We put in an information theoretic framework some recent results on visible compression for sources of states with commuting density operators, and remove the commutativity requirement. We derive a general achievable compression rate, which is for the noncommutative case still higher than the known lower bound. We also present several …
This thesis addresses problems in the field of quantum information theory. The first part of the thesis is opened with concrete definitions of general quantum source models and their compression, and each subsequent chapter addresses the compression of a specific source model as a special case of the initially defined general models. First, we find the optimal compression rate of a general mixed state source which includes as special cases all the previously studied models such as Schumacher's pure and ensemble sources and other mixed state ensemble models. For an interpolation between the vis…
Based on the problem of quantum data compression in a lossless way, we present here an operational interpretation for the family of quantum Rényi entropies. In order to do this, we appeal to a very general quantum encoding scheme that satisfies a quantum version of the Kraft-McMillan inequality. Then, in the standard situation, where one is intended to minimize the usual average length of the quantum codewords, we recover the known results, namely that the von Neumann entropy of the source bounds the average length of the optimal codes. Otherwise, we show that by invoking an exponential averag…
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…
A major challenge of today's quantum communication systems lies in the transmission of quantum information with high rates over long distances in the presence of unavoidable losses. Thereby the achievable quantum communication rate is fundamentally limited by the amount of energy that can be transmitted per use of the channel. It is hence vital to develop quantum communication protocols that encode quantum information as energy efficiently as possible. To this aim we investigate continuous-variable quantum teleportation as a method of distributing quantum information. We explore the possibilit…
We consider the sequential quantum channel discrimination problem using adaptive and non-adaptive strategies. In this setting the number of uses of the underlying quantum channel is not fixed but a random variable that is either bounded in expectation or with high probability. We show that both types of error probabilities decrease to zero exponentially fast and, when using adaptive strategies, the rates are characterized by the measured relative entropy between two quantum channels, yielding a strictly larger region than that achievable by non-adaptive strategies. Allowing for quantum memory,…
In order to evaluate, validate, and refine the design of new quantum algorithms or quantum computers, researchers and developers need methods to assess their correctness and fidelity. This requires the capabilities of quantum circuit simulations. However, the number of quantum state amplitudes increases exponentially with the number of qubits, leading to the exponential growth of the memory requirement for the simulations. In this work, we present our memory-efficient quantum circuit simulation by using lossy data compression. Our empirical data shows that we reduce the memory requirement to 1…
We show that any language in nondeterministic time $\exp(\exp(\cdots \exp(n)))$, where the number of iterated exponentials is an arbitrary function $R(n)$, can be decided by a multiprover interactive proof system with a classical polynomial-time verifier and a constant number of quantum entangled provers, with completeness $1$ and soundness $1 - \exp(-C\exp(\cdots\exp(n)))$, where the number of iterated exponentials is $R(n)-1$ and $C>0$ is a universal constant. The result was previously known for $R=1$ and $R=2$; we obtain it for any time-constructible function $R$. The result is based on a c…
Classical simulation of quantum circuits is crucial for evaluating and validating the design of new quantum algorithms. However, the number of quantum state amplitudes increases exponentially with the number of qubits, leading to the exponential growth of the memory requirement for the simulations. In this paper, we present a new data reduction technique to reduce the memory requirement of quantum circuit simulations. We apply our amplitude-aware lossy compression technique to the quantum state amplitude vector to trade the computation time and fidelity for memory space. The experimental resul…
