Quantum channels showing superadditivity in capacity

Quantum superadditivity in linear optics networks: sending bits via multiple access Gaussian channels

We study classical capacity regions of quantum Gaussian multiple access channels (MAC). In classical variants of such channels, whilst some capacity superadditivity-type effects such as the so called {\it water filling effect} may be achieved, a fundamental classical additivity law can still be identified, {\it viz.} adding resources to one sender is never advantageous to other senders in sending their respective information to the receiver. Here, we show that quantum resources allows violation of this law, by providing two illustrative schemes of experimentally feasible Gaussian MACs.

Superadditivity of communication capacity using entangled inputs

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…

Superadditivity of the Classical Capacity with Limited Entanglement Assistance

Finding the optimal encoding strategies can be challenging for communication using quantum channels, as classical and quantum capacities may be superadditive. Entanglement assistance can often simplify this task, as the entanglement-assisted classical capacity for any channel is additive, making entanglement across channel uses unnecessary. If the entanglement assistance is limited, the picture is much more unclear. Suppose the classical capacity is superadditive, then the classical capacity with limited entanglement assistance could retain superadditivity by continuity arguments. If the class…

Dephrasure channel and superadditivity of coherent information

The quantum capacity of a quantum channel captures its capability for noiseless quantum communication. It lies at the heart of quantum information theory. Unfortunately, our poor understanding of nonadditivity of coherent information makes it hard to understand the quantum capacity of all but very special channels. In this paper, we consider the dephrasure channel, which is the concatenation of a dephasing channel and an erasure channel. This very simple channel displays remarkably rich and exotic properties: we find nonadditivity of coherent information at the two-letter level, a big gap betw…

Schemes of transmission of classical information via quantum channels with many senders: discrete and continuous variables cases

Superadditivity effects in the classical capacity of discrete multi-access channels (MACs) and continuous variable (CV) Gaussian MACs are analysed. New examples of the manifestation of superadditivity in the discrete case are provided including, in particular, a channel which is fully symmetric with respect to all senders. Furthermore, we consider a class of channels for which {\it input entanglement across more than two copies of the channels is necessary} to saturate the asymptotic rate of transmission from one of the senders to the receiver. The 5-input entanglement of Shor error correction…

Multipartite entanglement to boost superadditivity of coherent information in quantum communication lines with polarization dependent losses

Coherent information quantifies the achievable rate of the reliable quantum information transmission through a communication channel. Use of the correlated quantum states instead of the factorized ones may result in an increase in the coherent information, a phenomenon known as superadditivity. However, even for simple physical models of channels it is rather difficult to detect the superadditivity and find the advantageous multipartite states. Here we consider the case of polarization dependent losses and propose some physically motivated multipartite entangled states which outperform all fac…

Superadditivity in trade-off capacities of quantum channels

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…

Superadditivity of private information for any number of uses of the channel

The quantum capacity of a quantum channel is always smaller than the capacity of the channel for private communication. However, both quantities are given by the infinite regularization of respectively the coherent and the private information. Here, we construct a family of channels for which the private and coherent information can remain strictly superadditive for unbounded number of uses. We prove this by showing that the coherent information is strictly larger than the private information of a smaller number of uses of the channel. This implies that even though the quantum capacity is uppe…

Superadditivity of channel capacity through quantum fields

Given that any communication is communication through quantum fields, we here study the scenario where a sender, Alice, causes information-carrying disturbances in a quantum field. We track the exact spread of these disturbances in space and time by using the technique of quantum information capsules (QIC). We find that the channel capacity between Alice and a receiver, Bob, is enhanced by Bob placing detectors not only inside but in addition also outside the causal future of Alice's encoding operation. Intuitively, this type of superadditivity arises because the field outside the causal futur…

量子容量在 qudit 去极化信道中的超可加性效应随维度增加而减小

The superadditivity effects of quantum capacity decrease with the dimension for qudit depolarizing channels

本文研究 qudit 去极化信道的量子容量超可加性，证明其增益随系统维度增加而减小，并在 d→∞ 时容量等于相干信息，同时得到私密经典容量的类似结论。

Computable limits of optical multiple-access communications

Communication rates over quantum channels can be boosted by entanglement, via superadditivity phenomena or entanglement assistance. Superadditivity refers to the capacity improvement from entangling inputs across multiple channel uses. Nevertheless, when unlimited entanglement assistance is available, the entanglement between channel uses becomes unnecessary -- the entanglement-assisted (EA) capacity of a single-sender and single-receiver channel is additive. We generalize the additivity of EA capacity to general multiple-access channels (MACs) for the total communication rate. Furthermore, fo…

General immunity and superadditivity of two-way Gaussian quantum cryptography

We consider two-way continuous-variable quantum key distribution, studying its security against general eavesdropping strategies. Assuming the asymptotic limit of many signals exchanged, we prove that two-way Gaussian protocols are immune to coherent attacks. More precisely we show the general superadditivity of the two-way security thresholds, which are proven to be higher than the corresponding one-way counterparts in all cases. We perform the security analysis first reducing the general eavesdropping to a two-mode coherent Gaussian attack, and then showing that the superadditivity is achiev…

Experimental Proposal for Achieving Superadditive Communication Capacities with a Binary Quantum Alphabet

We demonstrate superadditivity in the communication capacity of a binary alphabet consisting of two nonorthogonal quantum states. For this scheme, collective decoding is performed two transmissions at a time. This improves upon the previous schemes of Sasaki et al. [Phys. Rev. A 58, 146 (1998)] where superadditivity was not achieved until a decoding of three or more transmissions at a time. This places superadditivity within the regime of a near-term laboratory demonstration. We propose an experimental test based upon an alphabet of low photon-number coherent states where the signal decoding i…

Heralded Channel Holevo Superadditivity Bounds from Entanglement Monogamy

We show that for a particular class of quantum channels, which we call heralded channels, monogamy of squashed entanglement limits the superadditivity of Holevo capacity. Heralded channels provide a means to understand the quantum erasure channel composed with an arbitrary other quantum channel, as well as common situations in experimental quantum information that involve frequent loss of qubits or failure of trials. We also show how entanglement monogamy applies to non-classicality in quantum games, and we consider how faithful, monogamous entanglement measures may bound other entanglement-de…

广义直和信道中的极端容量

Extreme Capacities in Generalized Direct Sum Channels

本文构建了广义直和（GDS）信道框架，通过在Kraus算子中引入直和结构来研究量子信道容量，揭示了量子容量趋近零时无界超加性，以及私有与经典容量任意大导致的容量差距。

Superadditivity of Quantum Channel Coding Rate with Finite Blocklength Joint Measurements

The maximum rate at which classical information can be reliably transmitted per use of a quantum channel strictly increases in general with $N$, the number of channel outputs that are detected jointly by the quantum joint-detection receiver (JDR). This phenomenon is known as superadditivity of the maximum achievable information rate over a quantum channel. We study this phenomenon for a pure-state classical-quantum (cq) channel and provide a lower bound on $C_N/N$, the maximum information rate when the JDR is restricted to making joint measurements over no more than $N$ quantum channel outputs…

纠缠辅助通信的超可加性

Superadditivity for Entanglement-Assisted Communication

研究纠缠辅助通信中，虽跨信道使用相关性不提升最终速率，但能增强可靠性。通过证明Petz-Rényi信道信息对特定α严格超可加，实现纠缠辅助随机编码错误指数多副本增强，且无需传输系统间纠缠。

Additivity of the Classical Capacity of Entanglement-Breaking Quantum Channels

We show that for the tensor product of an entanglement-breaking quantum channel with an arbitrary quantum channel, both the minimum entropy of an output of the channel and the Holevo-Schumacher-Westmoreland capacity are additive. In addition, for the tensor product of two arbitrary quantum channels, we give a bound involving entanglement of formation for the amount of subadditivity (for minimum entropy output) or superadditivity (for channel capacity) that can occur.

关于相干信息的小扰动

On small perturbations of coherent information

本文提出了一种分析量子信道相干信息对输入状态小扰动行为的微扰方法，推导出三个判断输入状态是否产生次优相干信息的准则，并展示了其在单次量子容量超加性和私有容量与量子容量间隙检测中的应用。

How Deep the Theory of Quantum Communications Goes: Superadditivity, Superactivation and Causal Activation

In the theory of quantum communications, a deeper structure has been recently unveiled, showing that the capacity does not completely characterize the channel ability to transmit information due to phenomena -- namely, superadditivity, superactivation and causal activation -- with no counterpart in the classical world. Although how deep goes this structure is yet to be fully uncovered, it is crucial for the communication engineering community to own the implications of these phenomena for understanding and deriving the fundamental limits of communications. Hence, the aim of this treatise is to…

Subadditivity of the minimum output entropy and superactivation of the classical capacity of quantum multiple access channels

We study subadditivity of the minimum output entropy ($H_{min}$) of quantum multiple access channels (MACs). We provide an example of violation of the additivity theorem for $H_{min}$ known in classical information theory. Our result is based on a fundamental property of $MACs$ i.e. independence of each sender. The channels used in the example can be constructed explicitly. On the basis of subadditivity of $H_{min}$ we also provide an example of extremal superadditivity (super activation) of the classical capacity region of MACs.

标记扩展与量子信道容量的数值模拟：连接理论与计算

Flagged Extensions and Numerical Simulations for Quantum Channel Capacity: Bridging Theory and Computation

本文通过标记扩展技术和数值模拟研究量子信道容量，验证了幅度退相干信道的退化性及去极化信道的多字母超加性。

On the additivity conjecture for channels with arbitrary constraints

Recently Shor proved equivalence of several open (sub)additivity problems related to the Holevo capacity and the entanglement of formation [15]. In our previous note [6] equivalence of these to the additivity of the Holevo capacity for channels with arbitrary linear constraints was shown. This note is the development of the previous one in the direction of channels with general constraints. Introducing input constraints provides greater flexibility in the treatment of the additivity conjecture. The Holevo capacity of arbitrarily constrained channel is considered and the characteristic property…

直接信道与互补信道容量的同时超加性

Simultaneous superadditivity of the direct and complementary channel capacities

本文挑战了纠缠单配性直觉，证明了量子信道及其互补信道的相干信息与私密信息在任意多次使用下可同时具有超加性，并研究了最大与总私密信息量在不同信道使用次数下与相干信息的不等式交替关系。

完全关联多能级振幅阻尼信道的信息容量分析

Information capacity analysis of fully correlated multi-level amplitude damping channels

本文研究连续使用存在关联的三能级振幅阻尼信道，通过分析可退化性给出单次经典容量上界，并计算了普通及纠缠辅助场景下的量子与经典容量。

Bounding quantum capacities via partial orders and complementarity

Quantum capacities are fundamental quantities that are notoriously hard to compute and can exhibit surprising properties such as superadditivity. Thus, a vast amount of literature is devoted to finding tight and computable bounds on these capacities. We add a new viewpoint by giving operationally motivated bounds on several capacities, including the quantum capacity and private capacity of a quantum channel and the one-way distillable entanglement and private key of a quantum state. These bounds are generally phrased in terms of capacity quantities involving the complementary channel or state.…

Entanglement can completely defeat quantum noise

We describe two quantum channels that individually cannot send any information, even classical, without some chance of decoding error. But together a single use of each channel can send quantum information perfectly reliably. This proves that the zero-error classical capacity exhibits superactivation, the extreme form of the superadditivity phenomenon in which entangled inputs allow communication over zero capacity channels. But our result is stronger still, as it even allows zero-error quantum communication when the two channels are combined. Thus our result shows a new remarkable way in whic…

Nonadditivity of quantum and classical capacities for entanglement breaking multiple-access channels and butterfly network

We analyze quantum network primitives which are entanglement-breaking. We show superadditivity of quantum and classical capacity regions for quantum multiple-access channel and quantum butterfly network. Since the effects are especially visible at high noise they suggest that quantum information effects may be particularly helpful in the case of the networks with occasional high noise rates. To our knowledge the present effects provide the first qualitative borderline between superadditivities of bipartite and multipartite systems.

Remarks on additivity of the Holevo channel capacity and of the entanglement of formation

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…

Methods for Estimating Capacities and Rates of Gaussian Quantum Channels

Optimization methods aimed at estimating the capacities of a general Gaussian channel are developed. Specifically evaluation of classical capacity as maximum of the Holevo information is pursued over all possible Gaussian encodings for the lossy bosonic channel, but extension to other capacities and other Gaussian channels seems feasible. Solutions for both memoryless and memory channels are presented. It is first dealt with single use (single-mode) channel where the capacity dependence from channel's parameters is analyzed providing a full classification of the possible cases. Then it is deal…

基于量子信道-态对偶的一般信道容量

General channel capacities from quantum channel-state duality

本研究基于量子信道-态对偶，证明了将Choi态作为信息源时的量子信道容量基本定理，丰富了信道容量集合，并为理解超加性现象与发现新量子纠错码提供了新机遇。

On environment-assisted capacities of quantum channels

Following initial work by Gregoratti and Werner [J. Mod. Optics 50, 913-933, 2003 and quant-ph/0403092] and Hayden and King [quant-ph/0409026], we study the problem of the capacity of a quantum channel assisted by a "friendly (channel) environment" that can locally measure and communicate classical messages to the receiver. Previous work [quant-ph/0505038] has yielded a capacity formula for the quantum capacity under this kind of help from the environment. Here we study the problem of the environment-assisted classical capacity, which exhibits a somewhat richer structure (at least, it seems to…

振幅阻尼信道的经典容量与纠缠代价

Classical Capacity and Entanglement Cost of the Amplitude Damping Channel

本文针对具有纯输出的量子比特到量子比特信道，消除了经典容量与纠缠代价的正则化，证明振幅阻尼信道的经典容量等于单次Holevo信息，纠缠代价为$h_2((1+\sqrt p)/2)$，并给出支撑准则与有限块亏缺结果。

Minimal Entropy of States Emerging from Noisy Quantum Channels

In this paper, we consider the minimal entropy of qubit states transmitted through two uses of a noisy quantum channel, which is modeled by the action of a completely positive trace-preserving (or stochastic) map. We provide strong support for the conjecture that this minimal entropy is additive, namely that the minimum entropy can be achieved when product states are transmitted. Explicitly, we prove that for a tensor product of two unital stochastic maps on qubit states, using an entanglement that involves only states which emerge with minimal entropy cannot decrease the entropy below the min…

Approximate unitary $n^{2/3}$-designs give rise to quantum channels with super additive classical Holevo capacity

In a breakthrough, Hastings' showed that there exist quantum channels whose classical capacity is superadditive i.e. more classical information can be transmitted by quantum encoding strategies entangled across multiple channel uses as compared to unentangled quantum encoding strategies. Hastings' proof used Haar random unitaries to exhibit superadditivity. In this paper we show that a unitary chosen uniformly at random from an approximate $n^{2/3}$-design gives rise to a quantum channel with superadditive classical Holevo capacity, where $n$ is the dimension of the unitary exhibiting the Stin…

Equivalence of Additivity Questions in Quantum Information Theory

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.

纠缠辅助传输容量是识别任务的强逆界

The entanglement-assisted transmission capacity is a strong converse bound for identification

该研究针对量子信道识别任务，证明纠缠辅助传输容量是其强逆界，低噪声信道中二者相等，一般信道中存在严格不等的转置退极化信道，还得到识别容量严格超加性的首个实例。

通过隐私实现量子容量放大

Quantum capacity amplification via privacy

本文通过私有信道研究量子容量放大，提出基于Holevo信息的充分条件，给出具体阈值，并构造量子容量消失但私有容量无界的信道，证明容量放大与反降解信道集合的度量分离。

量子通信中噪声是资源上下文的

Noise is resource-contextual in quantum communication

研究量子信道信息传输能力的上下文依赖性，提出单参数信道族和状态族，展示容量与可蒸馏纠缠的相反变化趋势，证明量子通信中噪声是上下文相关的。

The classical capacity achievable by a quantum channel assisted by limited entanglement

We give the trade-off curve showing the capacity of a quantum channel as a function of the amount of entanglement used by the sender and receiver for transmitting information. The endpoints of this curve are given by the Holevo-Schumacher-Westmoreland capacity formula and the entanglement-assisted capacity, which is the maximum over all input density matrices of the quantum mutual information. The proof we give is based on the Holevo-Schumacher-Westmoreland formula, and also gives a new and simpler proof for the entanglement-assisted capacity formula.

Local Additivity Revisited

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…

Activation of zero-error classical capacity in low-dimensional quantum systems

Channel capacities of quantum channels can be nonadditive even if one of two quantum channels has no channel capacity. We call this phenomenon \emph{activation} of the channel capacity. In this paper, we show that when we use a quantum channel on a qubit system, only a noiseless qubit channel can generate the activation of the zero-error classical capacity. In particular, we show that the zero-error classical capacity of two quantum channels on qubit systems cannot be activated. Furthermore, we present a class of examples showing the activation of the zero-error classical capacity in low-dimen…

Heat Capacity and Entanglement Measure in a simple two-qubit model

A simple two-qubit model showing Quantum Phase Transitions as a consequence of ground state level crossings is studied in detail. Using the Concurrence of the system as an entanglement measure and heat capacity as a marker of thermodynamical properties, an analytical expression giving the latter in terms of the former is obtained. A protocol allowing an experimental measure of entanglement is then presented and compared with a related proposal recently reported by Wieśniak, Vedral and Brukner

Experimental witnessing of the quantum channel capacity in the presence of correlated noise

We present an experimental method to detect lower bounds to the quantum capacity of two-qubit communication channels. We consider an implementation with polarisation degrees of freedom of two photons and report on the efficiency of such a method in the presence of correlated noise for varying values of the correlation strength. The procedure is based on the generation of separable states of two qubits and local measurements at the output. We also compare the performance of the correlated two-qubit channel with the single-qubit channels corresponding to the partial trace on each of the subsyste…

Error Thresholds for Arbitrary Pauli Noise

The error threshold of a one-parameter family of quantum channels is defined as the largest noise level such that the quantum capacity of the channel remains positive. This in turn guarantees the existence of a quantum error correction code for noise modeled by that channel. Discretizing the single-qubit errors leads to the important family of Pauli quantum channels; curiously, multipartite entangled states can increase the threshold of these channels beyond the so-called hashing bound, an effect termed superadditivity of coherent information. In this work, we divide the simplex of Pauli chann…

在量子计算机上估计量子相对熵

Estimating quantum relative entropies on quantum computers

本研究提出首个可直接估计两个未知量子态量子相对熵与Petz Renyi散度的量子算法，电路规模小、适用于分布式场景，验证了无贫瘠高原现象，并发现量子比特信道相干信息严格超可加性新实例。

量子容量在简单非退化量子信道中的可加性

Additivity of quantum capacities in simple non-degradable quantum channels

本文提出两类非退化量子信道，其相干信息保持可加性，从而简化量子容量计算，并分析结构约束对强、弱可加性的影响。

Quantum Entanglement Capacity with Classical Feedback

For any quantum discrete memoryless channel, we define a quantity called quantum entanglement capacity with classical feedback ($E_B$), and we show that this quantity lies between two other well-studied quantities. These two quantities - namely the quantum capacity assisted by two-way classical communication ($Q_2$) and the quantum capacity with classical feedback ($Q_B$) - are widely conjectured to be different: there exists quantum discrete memoryless channel for which $Q_2>Q_B$. We then present a general scheme to convert any quantum error-correcting codes into adaptive protocols for this n…

一般物理理论中的单次与渐近经典容量

One-shot and asymptotic classical capacity in general physical theories

本文在一般物理理论中研究单次经典容量，利用假设检验相对熵给出上下界，并证明其与经典容量的渐近等价性。

Reliably distinguishing states in qutrit channels using one-way LOCC

We present numerical evidence showing that any three-dimensional subspace of C^3 \otimes C^n has an orthonormal basis which can be reliably distinguished using one-way LOCC, where a measurement is made first on the 3-dimensional part and the result used to select an optimal measurement on the n-dimensional part. This conjecture has implications for the LOCC-assisted capacity of certain quantum channels, where coordinated measurements are made on the system and environment. By measuring first in the environment, the conjecture would imply that the environment-assisted classical capacity of any …

量子通信的几何优化

Geometric optimization for quantum communication

本文开发了黎曼优化方法，通过在流形上参数化状态和信道扩展来建立量子容量和可蒸馏纠缠的双侧界限，成功解锁超加性增益并取得了新的最先进结果。

玻色子–自旋模型中的 Holevo 界与客观性

Holevo bound and objectivity in the boson-spin model

本文将量子信道理论用于开放量子系统的玻色子–自旋模型，借助无反冲近似与 Floquet 理论计算 Holevo 量，揭示容量的短时二次增长、渐近行为及参数依赖。

Quantum Capacities of Bosonic Channels

We investigate the capacity of bosonic quantum channels for the transmission of quantum information. Achievable rates are determined from measurable moments of the channel by showing that every channel can asymptotically simulate a Gaussian channel which is characterized by second moments of the initial channel. We calculate the quantum capacity for a class of Gaussian channels, including channels describing optical fibers with photon losses, by proving that Gaussian encodings are optimal. Along the way we provide a complete characterization of degradable Gaussian channels and those arising fr…

Capacities of linear quantum optical systems

A wide variety of communication channels employ the quantized electromagnetic field to convey information. Their communication capacity crucially depends on losses associated to spatial characteristics of the channel such as diffraction and antenna design. Here we focus on the communication via a finite pupil, showing that diffraction is formally described as a memory channel. By exploiting this equivalence we then compute the communication capacity of an optical refocusing system, modeled as a converging lens. Even though loss of information originates from the finite pupil of the lens, we sh…