The minimum entropy output is computed for rotationally invariant quantum channels acting on spin-1/2 and spin-1 systems. For the case of two parallel such channels and initial entangled (singlet) state the entropy of the output is higher then the doubled minimal entropy output of the single channel. This gives a certain moral support to the additivity hypothesis. Another related simple function of the channel (minimum entropy gain) is shown to be additive in general.
We prove additivity of the minimum output entropy and the Holevo capacity for rotationally invariant quantum channels acting on spin-1/2 and spin-1 systems. The physical significance of these channels and their relations to other known channels is also discussed.
We reduce the number of open additivity problems in quantum information theory by showing that four of them are equivalent. We show that the conjectures of additivity of the minimum output entropy of a quantum channel, additivity of the Holevo expression for the classical capacity of a quantum channel, additivity of the entanglement of formation, and strong superadditivity of the entanglement of formation, are either all true or all false.
The purpose of these notes is to discuss the relation between the additivity questions regarding the quantities (Holevo) capacity of a quantum channel T and entanglement of formation of a given bipartite state. In particular, using the Stinespring dilation theorem, we give a formula for the channel capacity involving entanglement of formation. This can be used to show that additivity of the latter for some states can be inferred from the additivity of capacity for certain channels. We demonstrate this connection for a family of group--covariant channels, allowing us to calculate the entangleme…
The design of error-correcting codes used in modern communications relies on information theory to quantify the capacity of a noisy channel to send information [1]. This capacity can be expressed using the mutual information between input and output for a single use of the channel: although correlations between subsequent input bits are used to correct errors, they cannot increase the capacity. For quantum channels, it has been an open question whether entangled input states can increase the capacity to send classical information [2]. The additivity conjecture [3,4] states that entanglement do…
When can noiseless quantum information be sent across noisy quantum devices? And at what maximum rate? These questions lie at the heart of quantum technology, but remain unanswered because of non-additivity -- a fundamental synergy which allows quantum devices (aka quantum channels) to send more information than expected. Previously, non-additivity was known to occur in very noisy channels with coherent information much smaller than that of a perfect channel; but, our work shows non-additivity in a simple low-noise channel. Our results extend even further. We prove a general theorem concerning…
Additivity of quantum relative entropies as a single-copy criterion
School of Mathematics, Institute for Research in Fundamental Sciences (IPM)（基础科学研究所）；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, College of Design and Engineering, National University of Singapore（新加坡国立大学设计与工程学院电气与计算机工程系）
We make a number of simplifications in Gour and Friedland's proof of local additivity of minimum output entropy of a quantum channel. We follow them in reframing the question as one about entanglement entropy of bipartite states associated with a $d_B \times d_E $ matrix. We use a different approach to reduce the general case to that of a square positive definite matrix. We use the integral representation of the log to obtain expressions for the first and second derivatives of the entropy, and then exploit the modular operator and functional calculus to streamline the proof following their und…
In this article, we investigate the additivity phenomenon in the dynamic capacity of a quantum channel for trading classical communication, quantum communication and entanglement. Understanding such additivity property is important if we want to optimally use a quantum channel for general communication purpose. However, in a lot of cases, the channel one will be using only has an additive single or double resource capacity, and it is largely unknown if this could lead to an superadditive double or triple resource capacity. For example, if a channel has an additive classical and quantum capacit…
Information theory establishes the fundamental limits on data transmission, storage, and processing. Quantum information theory unites information theoretic ideas with an accurate quantum-mechanical description of reality to give a more accurate and complete theory with new and more powerful possibilities for information processing. The goal of both classical and quantum information theory is to quantify the optimal rates of interconversion of different resources. These rates are usually characterized in terms of entropies. However, nonadditivity of many entropic formulas often makes finding a…
Additivity and chain rules for quantum entropies via multi-index Schatten norms
Inria（法国国家信息与自动化研究所）；ENS Lyon（里昂高等师范学院）；UCBL（里昂第一大学）；LIP（计算机科学实验室）；Télécom Paris - LTCI（巴黎电信学院-通信与图像实验室）；Institut Polytechnique de Paris（巴黎理工学院）；Inria Paris（法国国家信息与自动化研究所巴黎分部）
It is known that the minimal output entropy is additive for any product of entanglement breaking (EB) channels. The same is true for the Renyi entropy, where additivity is equivalent to multiplicativity of the $1 \rightarrow q$ norm for all $q \ge 1$. In this paper we consider the related question of multiplicativity of the $2 \rightarrow q$ norm for entanglement breaking channels. We prove that multiplicativity holds in this case for certain classes of EB channels, including both the CQ and QC channels.
A problem in quantum information theory that has received considerable attention in recent years is the question of multiplicativity of the so-called maximal output purity (MOP) of a quantum channel. This quantity is defined as the maximum value of the purity one can get at the output of a channel by varying over all physical input states, when purity is measured by the Schatten $q$-norm, and is denoted by $ν_q$. The multiplicativity problem is the question whether two channels used in parallel have a combined $ν_q$ that is the product of the $ν_q$ of the two channels. A positive answer would …
Non-Multiplicativity of the Holevo Barycenter of Quantum Channels
Universität zu Köln（科隆大学）；University of Camerino（卡梅里诺大学）；Istituto Nazionale di Fisica Nucleare, Sezione di Perugia（意大利国家核物理研究所佩鲁吉亚分部）；Universitat Autònoma de Barcelona（巴塞罗那自治大学）
In this paper we give a self contained introduction to the conceptional and mathematical foundations of quantum information theory. In the first part we introduce the basic notions like entanglement, channels, teleportation etc. and their mathematical description. The second part is focused on a presentation of the quantitative aspects of the theory. Topics discussed in this context include: entanglement measures, channel capacities, relations between both, additivity and continuity properties and asymptotic rates of quantum operations. Finally we give an overview on some recent developments a…
The class QMA(k), introduced by Kobayashi et al., consists of all languages that can be verified using k unentangled quantum proofs. Many of the simplest questions about this class have remained embarrassingly open: for example, can we give any evidence that k quantum proofs are more powerful than one? Does QMA(k)=QMA(2) for k>=2? Can QMA(k) protocols be amplified to exponentially small error? In this paper, we make progress on all of the above questions. First, we give a protocol by which a verifier can be convinced that a 3SAT formula of size n is satisfiable, with constant soundness, given …
A quantum communication channel can be put to many uses: it can transmit classical information, private classical information, or quantum information. It can be used alone, with shared entanglement, or together with other channels. For each of these settings there is a capacity that quantifies a channel's potential for communication. In this short review, I summarize what is known about the various capacities of a quantum channel, including a discussion of the relevant additivity questions. I also give some indication of potentially interesting directions for future research.
We consider the image of some classes of bipartite quantum states under a tensor product of random quantum channels. Depending on natural assumptions that we make on the states, the eigenvalues of their outputs have new properties which we describe. Our motivation is provided by the additivity questions in quantum information theory, and we build on the idea that a Bell state sent through a product of conjugated random channels has at least one large eigenvalue. We generalize this setting in two directions. First, we investigate general entangled pure inputs and show that that Bell states give…
The quantum capacity of degradable quantum channels has been proven to be additive. On the other hand, there is no general rule for the behavior of quantum capacity for non-degradable quantum channels. We introduce the set of partially degradable (PD) quantum channels to answer the question of additivity of quantum capacity for a well-separable subset of non-degradable channels. A quantum channel is partially degradable if the channel output can be used to simulate the degraded environment state. PD channels could exist both in the degradable, non-degradable and conjugate degradable family. We…
Entangled inputs can enhance the capacity of quantum channels, this being one of the consequences of the celebrated result showing the non-additivity of several quantities relevant for quantum information science. In this work, we answer the converse question (whether entangled inputs can ever render noisy quantum channels have maximum capacity) to the negative: No sophisticated entangled input of any quantum channel can ever enhance the capacity to the maximum possible value; a result that holds true for all channels both for the classical as well as the quantum capacity. This result can henc…
Perimeter Institute for Theoretical Physics（佩里尔研究所）；Institute for Quantum Computing, University of Waterloo（滑铁卢大学量子计算研究所）；Dahlem Center for Complex Quantum Systems, Freie Universität Berlin（柏林自由大学达勒姆复杂量子系统中心）；Department of Applied Mathematics, University of Waterloo（滑铁卢大学应用数学系）；Department of Combinatorics and Optimization, University of Waterloo（滑铁卢大学组合优化系）
