Arbitrarily varying and compound classical-quantum channels and a note on quantum zero-error capacities
Igor Bjelaković, Holger Boche, Gisbert Janßen, Janis Nötzel
备注：
37 pages, 0 figures. Accepted for publication in the LNCS Volume in Memory of Rudolf Ahlswede. Includes a section on certain differences between classical and classical-quantum channels regarding their zero-error capacities
We consider compound as well as arbitrarily varying classical-quantum channel models. For classical-quantum compound channels, we give an elementary proof of the direct part of the coding theorem. A weak converse under average error criterion to this statement is also established. We use this result together with the robustification and elimination technique developed by Ahlswede in order to give an alternative proof of the direct part of the coding theorem for a finite classical-quantum arbitrarily varying channels with the criterion of success being average error probability. Moreover we provide a proof of the strong converse to the random coding capacity in this setting.The notion of symmetrizability for the maximal error probability is defined and it is shown to be both necessary and sufficient for the capacity for message transmission with maximal error probability criterion to equal zero. Finally, it is shown that the connection between zero-error capacity and certain arbitrarily varying channels is, just like in the case of quantum channels, only partially valid for classical-quantum channels.
We determine the capacity of the classical compound quantum wiretapper channel with channel state information at the transmitter. Moreover we derive a lower bound on the capacity of this channel without channel state information and determine the capacity of the classical quantum compound wiretap channel with channel state information at the transmitter.
We consider classical message transmission under entanglement assistance for compound memoryless and arbitrarily varying quantum channels. In both cases, we prove general coding theorems together with corresponding weak converse bounds. In this way, we obtain single-letter characterizations of the entanglement-assisted classical capacities for both channel models. Moreover, we show that the entanglement-assisted classical capacity does exhibit no strong converse property for some compound quantum channels for the average as well as the maximal error criterion. A strong converse to the entangle…
Compound channel models offer a simple and straightforward way of analyzing the stability of decoder design under model variations. With this work we provide a coding theorem for a large class of practically relevant compound channel models. We give explicit formulas for the cases of the Gaussian classical-quantum compound channels with unknown noise, unknown phase and unknown attenuation. We show analytically how the classical compound channel capacity formula motivates nontrivial choices of the displacement parameter of the Kennedy receiver. Our work demonstrates the value of the compound ch…
We determine the secrecy capacity of the compound channel with quantum wiretapper and channel state information at the transmitter. Moreover, we derive a lower bound on the secrecy capacity of this channel without channel state information and determine the secrecy capacity of the compound classical-quantum wiretap channel with channel state information at the transmitter. We use this result to derive a new proof for a lower bound on the entanglement generating capacity of compound quantum channel. We also derive a new proof for the entanglement generating capacity of compound quantum channel …
We study universal quantum codes for entanglement-assisted quantum communication over compound quantum channels. In this setting, sender and receiver do not know the specific channel that will be used for communication, but only know the set that the channel is selected from. We investigate different variations of the problem: uninformed users, informed receiver, informed sender, and feedback assistance. We derive single-letter formulas for all corresponding channel capacities. Our proofs are based on one-shot decoupling bounds and properties of smooth entropies.
We determine the capacity of compound classical-quantum channels. As a consequence we obtain the capacity formula for the averaged classical-quantum channels. The capacity result for compound channels demonstrates, as in the classical setting, the existence of reliable universal classical-quantum codes in scenarios where the only a priori information about the channel used for the transmission of information is that it belongs to a given set of memoryless classical-quantum channels. Our approach is based on the universal classical approximation of the quantum relative entropy which in turn rel…
We determine the secrecy capacities under common randomness assisted coding of arbitrarily varying classical-quantum wiretap channels.Furthermore, we determine the secrecy capacity of a mixed channel model which is compound from the sender to the legal receiver and varies arbitrarily from the sender to the eavesdropper. As an application we examine when the secrecy capacity is a continuous function of the system parameters and show that resources, i.e., having access to a perfect copy of the outcome of a random experiment. are helpful for channel stability.
We consider one-way quantum state merging and entanglement distillation under compound and arbitrarily varying source models. Regarding quantum compound sources, where the source is memoryless, but the source state an unknown member of a certain set of density matrices, we continue investigations begun in the work of Bjelaković et. al. [Universal quantum state merging, J. Math. Phys. 54, 032204 (2013)] and determine the classical as well as entanglement cost of state merging. We further investigate quantum state merging and entanglement distillation protocols for arbitrarily varying quantum so…
We initiate the study of passive environment-assisted communication via a quantum channel, modeled as a unitary interaction between the information carrying system and an environment. In this model, the environment is controlled by a benevolent helper who can set its initial state such as to assist sender and receiver of the communication link. (The case of a malicious environment, also known as jammer, or arbitrarily varying channel, is essentially well-understood and comprehensively reviewed.) Here, after setting out precise definitions, focussing on the problem of quantum communication, we …
We investigate entanglement transmission over an unknown channel in the presence of a third party (called the adversary), which is enabled to choose the channel from a given set of memoryless but non-stationary channels without informing the legitimate sender and receiver about the particular choice that he made. This channel model is called arbitrarily varying quantum channel (AVQC). We derive a quantum version of Ahlswede's dichotomy for classical arbitrarily varying channels. This includes a regularized formula for the common randomness-assisted capacity for entanglement transmission of an …
As our main result we show that, in order to achieve the randomness assisted message - and entanglement transmission capacities of a finite arbitrarily varying quantum channel it is not necessary that sender and receiver share (asymptotically perfect) common randomness. Rather, it is sufficient that they each have access to an unlimited amount of uses of one part of a correlated bipartite source. This access might be restricted to an arbitrary small (nonzero) fraction per channel use, without changing the main result. We investigate the notion of common randomness. It turns out that this is a …
We determine the secrecy capacities of AVQCs (arbitrarily varying quantum channels). Both secrecy capacity with average error probability and with maximal error probability are derived. Both derivations are based on one common code construction. The code we construct fulfills a stringent secrecy requirement, which is called the strong code concept. We determine when the secrecy capacity is a continuous function of the system parameters and completely characterize its discontinuity points both for average error criterion and for maximal error criterion. Furthermore, we prove the phenomenon "sup…
We consider energy-constrained infinite-dimensional quantum channels from a given system (satisfying a certain condition) to any other systems. We show that dealing with basic capacities of these channels we may assume (accepting arbitrarily small error $\varepsilon$) that all channels have the same finite-dimensional input space -- the subspace corresponding to the $m(\varepsilon)$ minimal eigenvalues of the input Hamiltonian. We also show that for the class of energy-limited channels (mapping energy-bounded states to energy-bounded states) the above result is valid with substantially smaller…
We consider secret-key distillation from tripartite compound classical-quantum-quantum (cqq) sources with free forward public communication under strong security criterion. We design protocols which are universally reliable and secure in this scenario. These are shown to achieve asymptotically optimal rates as long as a certain regularity condition is fulfilled by the the set of its generating density matrices. We derive a multi-letter formula which describes the optimal forward secret-key capacity for all compound cqq sources being regular in this sense. We also determine the forward secret-k…
In quantum mechanics, a fundamental law prevents quantum communications to simultaneously achieve high rates and long distances. This limitation is well known for point-to-point protocols, where two parties are directly connected by a quantum channel, but not yet fully understood in protocols with quantum repeaters. Here we solve this problem bounding the ultimate rates for transmitting quantum information, entanglement and secret keys via quantum repeaters. We derive single-letter upper bounds for the end-to-end capacities achievable by the most general (adaptive) protocols of quantum and pri…
We consider a model of communication via a fully quantum jammer channel with quantum jammer, quantum sender and quantum receiver, which we dub quantum arbitrarily varying channel (QAVC). Restricting to finite dimensional user and jammer systems, we show, using permutation symmetry and a de Finetti reduction, how the random coding capacity (classical and quantum) of the QAVC is reduced to the capacity of a naturally associated compound channel, which is obtained by restricting the jammer to i.i.d. input states. Furthermore, we demonstrate that the shared randomness required is at most logarithm…
This work is motivated by a quite general question: Under which circumstances are the capacities of information transmission systems continuous? The research is explicitly carried out on arbitrarily varying quantum channels (AVQCs). We give an explicit example that answers the recent question whether the transmission of messages over AVQCs can benefit from distribution of randomness between the legitimate sender and receiver in the affirmative. The specific class of channels introduced in that example is then extended to show that the deterministic capacity does have discontinuity points, whil…
Strong Converses for Quantum Channel Capacities from Blowing-Up Lemmata
School of Mathematics, Institute for Research in Fundamental Sciences (IPM)（基础科学研究所数学学院）；Centre for Quantum Technologies, National University of Singapore（新加坡国立大学量子技术中心）；Department of Electrical and Computer Engineering, National University of Singapore（新加坡国立大学电气与计算机工程系）
In this paper we propose a new model for arbitrarily varying classical-quantum channels. In this model a jammer has side information. We consider two scenarios. In the first scenario the jammer knows the channel input, while in the second scenario the jammer knows both the channel input and the message. The transmitter and receiver share a secret random key with a vanishing key rate. We determine the capacity for both average and maximum error criteria for both scenarios. We also establish the strong converse. We show that all these corresponding capacities are equal, which means that addition…
We consider quantum and private communications assisted by repeaters, from the basic scenario of a single repeater chain to the general case of an arbitrarily-complex quantum network, where systems may be routed through single or multiple paths. In this context, we investigate the ultimate rates at which two end-parties may transmit quantum information, distribute entanglement, or generate secret keys. These end-to-end capacities are defined by optimizing over the most general adaptive protocols that are allowed by quantum mechanics. Combining techniques from quantum information and classical …
An elementary introduction into algebraic approach to unified quantum information theory and operational approach to quantum entanglement as generalized encoding is given. After introducing compound quantum state and two types of informational divergences, namely, Araki-Umegaki (a-type) and of Belavkin-Staszewski (b-type) quantum relative entropic information, this paper treats two types of quantum mutual information via entanglement and defines two types of corresponding quantum channel capacities as the supremum via the generalized encodings. It proves the additivity property of quantum chan…
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 t…
We implement a mechanism - originally proposed as a model for the large memory storage capacity of black holes - in quantum neural networks and show that an exponentially increased capacity of pattern storage and recognition is achieved in certain critically excited states, without involvement of synaptic plasticity. We consider a simple network of N interconnected quantum neurons with weak excitatory synaptic connections. We show that for frozen synaptic weights there exist the critical states of enhanced memory storage capacity. These states are achieved thanks to the high excitation levels …
We introduce one-way LOCC protocols for quantum state merging for compound sources, which have asymptotically optimal entanglement as well as classical communication resource costs. For the arbitrarily varying quantum source (AVQS) model, we determine the one-way entanglement distillation capacity, where we utilize the robustification and elimination techniques, well-known from classical as well as quantum channel coding under assumption of arbitrarily varying noise. Investigating quantum state merging for AVQS, we demonstrate by example, that the usual robustification procedure leads to subop…
We derive universal codes for simultaneous transmission of classical messages and entanglement through quantum channels, possibly under attack of a malignant third party. These codes are robust to different kinds of channel uncertainty. To construct such universal codes, we invoke and generalize properties of random codes for classical and quantum message transmission through quantum channels. We show these codes to be optimal by giving a multi-letter characterization of regions corresponding to the capacity of compound quantum channels for simultaneously transmitting and generating entangleme…
We determine the optimal rates of universal quantum codes for entanglement transmission and generation under channel uncertainty. In the simplest scenario the sender and receiver are provided merely with the information that the channel they use belongs to a given set of channels, so that they are forced to use quantum codes that are reliable for the whole set of channels. This is precisely the quantum analog of the compound channel coding problem. We determine the entanglement transmission and entanglement-generating capacities of compound quantum channels and show that they are equal. Moreov…
We study the identification capacity of classical-quantum channels ("cq-channels"), under channel uncertainty and privacy constraints. To be precise, we consider first compound memoryless cq-channels and determine their identification capacity; then we add an eavesdropper, considering compound memoryless wiretap cqq-channels, and determine their secret identification capacity. In the first case (without privacy), we find the identification capacity always equal to the transmission capacity. In the second case, we find a dichotomy: either the secrecy capacity (also known as private capacity) of…
No information transmission through quantum channels above capacity
National Taiwan University（国立台湾大学）；National Center for Theoretical Sciences, Taiwan（台湾理论科学中心）；Hon Hai (Foxconn) Quantum Computing Center, Taiwan（台湾鸿海（富士康）量子计算中心）；National University of Singapore（新加坡国立大学）；Centre for Quantum Technologies, National University of Singapore（新加坡国立大学量子技术中心）
Transmitting data reliably over noisy communication channels is one of the most important applications of information theory, and well understood when the channel is accurately modelled by classical physics. However, when quantum effects are involved, we do not know how to compute channel capacities. The capacity to transmit quantum information is essential to quantum cryptography and computing, but the formula involves maximising the coherent information over arbitrarily many channel uses. This is because entanglement across channel uses can increase the coherent information, even from zero t…
We study how an arbitrary Gaussian state of two localized wave packets, prepared in an inertial frame of reference, is described by a pair of uniformly accelerated observers. We explicitly compute the resulting state for arbitrarily chosen proper accelerations of the observers and independently tuned distance between them. To do so, we introduce a generalized Rindler frame of reference and analytically derive the corresponding state transformation as a Gaussian channel. Our approach provides several new insights into the phenomenon of vacuum entanglement such as the highly non-trivial effect o…
In this article we show that the Holevo capacity of a classical quantum channel, can be reached not only by a POVM but by von Neumann measurement too. Furthermore we show two use of this fact 1) We can generalize the classical compound channel to quantum environment and with this new measure technique we can give an optimal coding/decoding algorithm to it. 2) We give an algorithm, with it a classical system can be decoded by quantum apparatus in linear time, which is a surprising result, because best known optimal classical decoding algorithm need n*log(n) time. This means that if a quantum ma…
Quantum information is the next frontier in information science, promising unconditionally secure communications, enhanced channel capacities, and computing capabilities far beyond their classical counterparts. And as quantum information processing devices continue to transition from the lab to the field, the demand for the foundational infrastructure connecting them with each other and their users---the quantum internet---will only increase. Due to the remarkable success of frequency multiplexing and control in the classical internet, quantum information encoding in optical frequency offers a…
RWTH Aachen University（亚琛工业大学）；Inria, ENS Lyon, UCBL, LIP（法国国家信息与自动化研究所、里昂高等师范学院、里昂第一大学、里昂信息处理实验室）；Technical University of Munich（慕尼黑工业大学）；Munich Center for Quantum Science and Technology (MCQST)（慕尼黑量子科学与工程中心）
