A. S. Holevo
In this note we introduce purification for a pair $(ρ,Φ),$ where $ρ$ is a quantum state and $Φ$ is a channel, which allows in particular a natural extension of the properties of related information quantities (such as mutual and coherent informations) to the channels with arbitrary input and output spaces.
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This work provides a parametrization of the set of purifications of a 2x2 mixed state, and joint purifications (if any) of a given pair of mixed 2x2 states. The former is parametrized by SO(3, R), while the latter is parametrized by SO(2,R), except when the 2x2 state is pure. Using this parametrization some quantum information measures are obtained. In the process it is shown how to solve one variation of the classical Procustes problem. The appendix is devoted to the first step in extending this work to higher dimensions.
Experimental Quantum Channel Purification
本研究针对量子网络信道噪声问题，提出利用光子空间与偏振特性、基于双Fredkin门的实验性量子信道纯化方案，实现多类型多水平噪声抑制，在纠缠分发场景下抗噪保纠缠性能优于传统方法。
A geometric Fano--Procrustes framework for purification-based distances and quantum channels analysis
本文提出了一种基于净化的几何框架，用于分析混合的双量子比特态和量子通道，通过Fano表示法将净化过程参数化为Bloch矢量、辅助系统Bloch矢量和相关矩阵，并将净化优化问题转化为在SO(3)李群上的正交Procrustes问题，从而得到最大净化重叠和最优旋转，定义了净化偏移角Θ，提供几何信息。
Capacity Enhancement of n-GHZ State Super-dense Coding Channels by Purification and Quantum Neural Network
提出基于n-GHZ态的超密编码协议，结合纠缠纯化与量子神经网络提升信道容量，实验验证二者叠加增强效果显著。
We present a protocol based on continuous-variable quantum teleportation and Gaussian post- selection that can be used to correct errors introduced by a lossy channel. We first show that the global transformation enacted by the protocol is equivalent to an effective system composed of a noiseless amplification (or attenuation), and an effective quantum channel, which can in theory have no loss and an amount of thermal noise arbitrarily small, hence tending to an identity channel. An application of our protocol is the probabilistic purification of quantum non-Gaussian states using only Gaussian…
We give a proof that entanglement purification, even with noisy apparatus, is sufficient to disentangle an eavesdropper (Eve) from the communication channel. Our proof applies to all possible attacks (individual and coherent). Due to the quantum nature of the entanglement purification protocol, it is also possible to use the obtained quantum channel for secure transmission of quantum information.
It has been recently shown that there exist universal fundamental limits to the accuracy and efficiency of the transformation from noisy resource states to pure ones (e.g.,~distillation) in any well-behaved quantum resource theory [Fang/Liu, Phys. Rev. Lett. 125, 060405 (2020)]. Here, we develop a novel and powerful method for analyzing the limitations on quantum resource purification, which not only leads to improved bounds that rule out exact purification for a broader range of noisy states and are tight in certain cases, but also enable us to establish a robust no-purification theory for qu…
We study the use of entanglement purification for quantum communication over long distances. For distances much longer than the coherence length of a corresponding noisy quantum channel, the fidelity of transmission is usually so low that standard purification methods are not applicable. It is however possible to divide the channel into shorter segments that are purified separately and then connected by the method of entanglement swapping. This method can be much more efficient than schemes based on quantum error correction, as it makes explicit use of two-way classical communication. An impor…
Purification enables an unbounded query separation in sequential quantum channel discrimination
本文证明在顺序量子信道鉴别中，利用纯化资源（即使环境参考系未知）能严格提升成功概率，优于裸访问和平行策略，并给出具体信道实例验证该优势。
Asymptotically optimal purification of noisy unitary channels in any dimension
该研究解决任意维度下噪声幺正信道的渐近最优纯化问题，推导最优保真度，给出SU(d)协变并行策略，明确查询复杂度并分析其扩展性，还证明噪声幺正共轭对偶任务的最优保真度与纯化任务的主导阶结果一致。
The essence of quantum error-correction is to use redundant Hilbert space to identify and correct errors, and the channel fidelity of the quantum channel does not affect which errors can be identified and corrected. Based on this, it is found that quantum error-correction can be used to classify the independent quantum channel into 5 types, and 4 of the 5 types can be purified. It is found in quantum error-correction, the decoherence of quantum state may be related to the degree of identification for the state under quantum noise, and the results of this work confirmed that the degree of purit…
We give a proof that entanglement purification, even with noisy apparatus, is sufficient to disentangle an eavesdropper (Eve) from the communication channel. In the security regime, the purification process factorises the overall initial state into a tensor-product state of Alice and Bob, on one side, and Eve on the other side, thus establishing a completely private, albeit noisy, quantum communication channel between Alice and Bob. The security regime is found to coincide for all practical purposes with the purification regime of a two-way recurrence protocol. This makes two-way entanglement …
Virtual Channel Purification
本文提出虚拟信道纯化（VCP）协议，结合旗标容错与虚拟态纯化思想，在无需目标态和噪声模型知识的情况下实现强误差抑制，并衍生出虚拟误差校正（VEC）协议，以低采样成本实现QEC级保护，适用于量子网络任务。
Recently, Briegel et al. present a quantum repeaters protocol using nest entanglement purification for long distance quantum communication(Physical Review Letters \textbf{81},5932). In this paper we present a modified scheme for constructing an optimal quantum repeater using doubly entangled photon pairs that overcomes the limitations of quantum communication via noisy channels. Based on the imperfect quantum operations and classical communications, the polarization entangled quantum channels can be built with a better efficiency.
Scaling-optimal purification of noisy qubit unitary channels
针对带噪声的量子比特酉通道，提出基于纠缠辅助量子纠错码的U(2)-协变并行协议，实现噪声强度O(1/n)抑制，并证明在低噪声区域渐近最优。
We investigate general probabilistic theories in which every mixed state has a purification, unique up to reversible channels on the purifying system. We show that the purification principle is equivalent to the existence of a reversible realization of every physical process, namely that every physical process can be regarded as arising from a reversible interaction of the system with an environment, which is eventually discarded. From the purification principle we also construct an isomorphism between transformations and bipartite states that possesses all structural properties of the Choi-Ja…
We present a protocol in which two or more parties can share multipartite entanglement over noisy quantum channels. The protocol is based on the entanglement purification presented by Shor and Preskill [Phys. Rev. Lett. 85, 441 (2000)] and the quantum teleportation via an isotropic state. We show that a nearly perfect purification implies a nearly perfect sharing of multipartite entanglement between two parties so that the protocol can assure a faithful sharing of multipartite entanglement with Shor and Preskill's proof on the entanglement purification.
We present an improved protocol for entanglement purification of bipartite mixed states. The protocol requires two-way classical communication and hence implies an improved lower bound on the quantum capacity with two-way classical communication of the quantum depolarizing channel.
Error-mitigated quantum metrology via enhanced virtual purification
本文提出增强虚拟通道净化方法，通过提升虚拟态净化效果，提高量子计量的精度和鲁棒性，适用于单参数和多参数任务，并拓展至更广泛的量子信息领域。
We investigate the probabilistic cloning and purification of quantum states. The performance of these probabilistic operations is quantified by the average fidelity between the ideal and actual output states. We provide a simple formula for the maximal achievable average fidelity and we explictly show how to construct a probabilistic operation that achieves this fidelity. We illustrate our method on several examples such as the phase covariant cloning of qubits, cloning of coherent states, and purification of qubits transmitted via depolarizing channel and amplitude damping channel. Our exampl…
Quantum Purification for Amplitude Damping Noise
本文提出一种针对振幅阻尼噪声的量子纯化新方法，仅需少量辅助比特和克利福德门，即可显著提升态或信道的保真度，且成功概率较高，为实际量子系统提供了可扩展的噪声缓解方案。
Carrier-Assisted Entanglement Purification
提出一种利用单拷贝纠缠态量子存储器和传输单量子比特的载子辅助纠缠提纯协议，通过分析噪声信道条件并采用多载子方案，使固定点保真度趋近于1，显著降低实验开销。
Two separated observers, by applying local operations to a supply of not-too-impure entangled states ({\em e.g.} singlets shared through a noisy channel), can prepare a smaller number of entangled pairs of arbitrarily high purity ({\em e.g.} near-perfect singlets). These can then be used to faithfully teleport unknown quantum states from one observer to the other, thereby achieving faithful transfrom one observer to the other, thereby achieving faithful transmission of quantum information through a noisy channel. We give upper and lower bounds on the yield $D(M)$ of pure singlets ($\ket{Ψ^-}$)…
Hyperentanglement is a promising resource in quantum information processing, especially for increasing the channel capacity of long-distance quantum communication. Hyperentanglement purification is an important method to obtain high-fidelity nonlocal hyperentangled states from mixed hyperentangled states in a long-distance quantum communication process with noisy channels. Here, we present a two-step hyperentanglement purification protocol for nonlocal mixed hyperentangled states with polarization bit-flip errors and spatial-mode phase-flip errors, resorting to polarization-spatial phase-check…
A random purification channel for arbitrary symmetries with applications to fermions and bosons
该研究提出了一种适用于任意对称性的随机纯化通道，实现了费米子和玻色子高斯态的最优标定和性质测试。
We study the stability under quantum noise effects of the quantum privacy amplification protocol for the purification of entanglement in quantum cryptography. We assume that the E91 protocol is used by two communicating parties (Alice and Bob) and that the eavesdropper Eve uses the isotropic Bužek-Hillery quantum copying machine to extract information. Entanglement purification is then operated by Alice and Bob by means of the quantum privacy amplification protocol and we present a systematic numerical study of the impact of all possible single-qubit noise channels on this protocol. We find th…
We analyze the estimation of a finite ensemble of quantum bits which have been sent through a depolarizing channel. Instead of using the depolarized qubits directly, we first apply a purification step and show that this improves the fidelity of subsequent quantum estimation. Even though we lose some qubits of our finite ensemble the information is concentrated in the remaining purified ones.
Quantum communication is a secure way to transfer quantum information and to communicate with legitimate parties over distant places in a network. Although communication over a long distance has already been attained, technical problem arises due to unavoidable loss of information through the transmission channel. Quantum repeaters can extend the distance scale using entanglement swapping and purification scheme. Here we demonstrate the working of a quantum repeater by the above two processes. We use IBM's real quantum processor `ibmqx4' to create two pair of entangled qubits and design an equ…
We present an approach to purification and entanglement routing on complex quantum network architectures, that is, how a quantum network equipped with imperfect channel fidelities and limited memory storage time can distribute entanglement between users. We explore how network parameters influence the performance of path-finding algorithms necessary for optimizing routing and, in particular, we explore the interplay between the bandwidth of a quantum channels and the choice of purification protocol. Finally, we demonstrate multi-path routing on various network topologies with resource constrai…
Chip-to-chip hyperentanglement distribution and entanglement purification using silicon integrated photonics
该研究首次在集成硅芯片间实现偏振-空间超纠缠分发，并通过消耗空间自由度完成纠缠纯化，弥补了片上量子中继器的关键环节。
Random purification channel for passive Gaussian bosons
本研究提出一种高斯随机纯化通道，用于被动高斯玻色子态的纯化，具有两倍平均光子数的特性。
Random purification channel made simple
提出随机纯化通道的简单构造，证明其可纯化非独立同分布态，并用于简化量子散度的Uhlmann定理证明。
Security of One-Way Entanglement Purification with Quantum Sampling Against a Restricted Adversary
针对受限为单量子比特泡利门的对手，提出基于量子采样的单向纠缠纯化协议，利用与纠错码的等价性估计汉明重量并纠错，实现安全纯化。
We discuss the problem of transfering a qubit from Alice to Bob using a noisy quantum channel and only finite resources. As the basic protocol for the transfer we apply quantum teleportation. It turns out that for a certain quality of the channel direct teleportation combined with qubit purification is superior to entanglement purification of the channel. If, however, the quality of the channel is rather low one should simply apply an estimation-preparation scheme.
We present a family of entanglement purification protocols that generalize four previous methods, namely the recurrence method, the modified recurrence method, and the two methods proposed by Maneva-Smolin and Leung-Shor. We will show that this family of protocols have improved yields over a wide range of initial fidelities F, and hence imply new lower bounds on the quantum capacity assisted by two-way classical communication of the quantum depolarizing channel. In particular, the yields of these protocols are higher than the yield of universal hashing for F less than 0.993 and as F goes to 1.
Symmetric channel verification for purifying noisy quantum channels
本文提出利用量子信道固有对称性的对称信道验证（SCV）协议，以检测并纠正破缺对称噪声，并给出仅需单量子比特辅助比特的硬件高效实现。
Probabilistic and approximate universal quantum purification machines
研究如何通过概率精确和确定性近似方法将任意量子态和通道提升为纯化和斯蒂尼尔松扩张。证明纯化两个不同秩输入的概率非零即可排除线性正映射，揭示量子理论的根本障碍。近似设置中分析不同策略的性能，发现纯输出策略在大环境维度最优，非纯输出策略在小环境维度更优。
Preservability of Measurement Incompatibility: Purification, Activation, and a No-Go Theorem
本文在测量不相容性可保持性的资源理论中，引入基于鲁棒性的度量，证明预过滤无法从不兼容湮灭信道激活可保持性（不可行定理），但后过滤可随机激活，为量子信息处理提供实用框架。
Observation of residual entanglement in chip-based entanglement purification
利用硅基芯片实现基于超纠缠的单拷贝纠缠纯化方案，实验观测到残余纠缠，并证明在比特翻转噪声下总能获得适合进一步纯化的残余纠缠。
Advances in quantum entanglement purification
本文综述量子纠缠纯化的理论、协议类型与线性光学实验进展，并展望其在长距离量子通信和量子网络中的应用前景。
Capacity-Achieving Entanglement Purification Protocol for Pauli Dephasing Channel
本文提出了一种实现Pauli退相位通道容量的双向纠缠纯化协议，通过可扩展的电路实现渐进式保真度提升，有效抑制退相干误差。
The distribution of entangled states between distant locations will be essential for the future large scale realization of quantum communication schemes such as quantum cryptography and quantum teleportation. Because of the unavoidable noise in the quantum communication channel, the entanglement between two particles is more and more degraded the further they propagate. Entanglement purification is thus essential to distill highly entangled states from less entangled ones. Existing general purification protocols are based on the quantum controlled-NOT (CNOT) or similar quantum logic operations…
We introduce the distribution of a secret multipartite entangled state in a real-world scenario as a quantum primitive. We show that in the presence of noisy quantum channels (and noisy control operations) any state chosen from the set of two-colorable graph states (CSS codewords) can be created with high fidelity while it remains unknown to all parties. This is accomplished by either blind multipartite entanglement purification, which we introduce in this paper, or by multipartite entanglement purification of enlarged states, which offers advantages over an alternative scheme based on standar…
Design of an entanglement purification protocol selection module
本文通过密度矩阵仿真研究现实误差下多种纠缠纯化协议的性能，并提出一个结合网络层规格与物理层能力的协议选择模块，在基准测试中识别最优协议的成功率接近90%。
Purification and correction of quantum channels by commutation-derived quantum filters
本文提出基于交换子导出的量子滤波器，结合QEM与QEC思想，通过辅助量子比特显式校正Clifford电路噪声，并实现指数级误差降低，为可扩展量子误差抑制提供新工具。
Streaming quantum state purification
提出一种基于交换测试的流式量子态纯化协议，适用于任意维度量子比特，样本复杂度在最终误差参数上渐近最优，且仅需少量量子存储器。
Entanglement purification takes a number of noisy EPR pairs and processes them to produce a smaller number of more reliable pairs. If this is done with only a forward classical side channel, the procedure is equivalent to using a quantum error-correcting code (QECC). We instead investigate entanglement purification protocols with two-way classical side channels (2-EPPs) for finite block sizes. In particular, we consider the analog of the minimum distance problem for QECCs, and show that 2-EPPs can exceed the quantum Hamming bound and the quantum Singleton bound. We also show that 2-EPPs can ac…
Random dimension reduction and learning symmetric properties of quantum states
提出随机降维方法，在保持等距张量积作用下的不变性质的同时降低多个量子态的维度，从而改进距离、保真度和相对熵的估计上界，并给出高效的量子电路实现。
High-fidelity quantum entanglement is a key resource for quantum communication and distributed quantum computing, enabling quantum state teleportation, dense coding, and quantum encryption. Any sources of decoherence in the communication channel however degrade entanglement fidelity, thereby increasing the error rates of entangled state protocols. Entanglement purification provides a method to alleviate these non-idealities, by distilling impure states into higher-fidelity entangled states. Here we demonstrate the entanglement purification of Bell pairs shared between two remote superconductin…
Additivity Results for the Rényi-2 Entanglement of Purification
本文将Rényi-2纠缠纯化熵转化为受限的最小输出Rényi熵问题，并证明了在特定条件下其可加性。
Integrated Generation and Purification of Entangled Coherent States for Non-Gaussian Teleportation
本文提出了一种集成光子方案，通过光子子traction和催化生成高保真度的非高斯纠缠相干态，实现了高保真度的量子 teleportation，超越了经典极限。
We investigate measurement-based quantum communication with noisy resource states that are generated by entanglement purification. We consider the transmission of encoded information via noisy quantum channels using a measurement-based implementation of encoding, error correction and decoding. We show that such an approach offers advantages over direct transmission, gate-based error correction and measurement-based schemes with direct generation of resource states. We analyze the noise structure of resource states generated by entanglement purification and show that a local error model, i.e. n…
Recently, two-photon six-qubit hyperentangled states were produced in experiment and they can improve the channel capacity of quantum communication largely. Here we present a scheme for the hyperentanglement purification of nonlocal two-photon systems in three degrees of freedom (DOFs), including the polarization, the first-longitudinal-momentum, and the second longitudinal momentum DOFs. Our hyperentanglement purification protocol (hyper-EPP) is constructed with two steps resorting to parity-check quantum nondemolition measurement on the three DOFs and SWAP gates, respectively. With these two…
High-quality long-distance entanglement is essential for both quantum communication and scalable quantum networks. Entanglement purification is to distill high-quality entanglement from low-quality entanglement in a noisy environment and it plays a key role in quantum repeaters. The previous significant entanglement purification experiments require two pairs of low-quality entangled states and were demonstrated in table-top. Here we propose and report a high-efficiency and long-distance entanglement purification using only one pair of hyperentangled states. We also demonstrate its practical ap…
We address security of the quantum key distribution scheme based on the noisy modulation of coherent states and investigate how it is robust against noise in the modulation regardless of the particular technical implementation. As the trusted preparation noise is shown to be security breaking even for purely lossy channels, we reveal the essential difference between two types of trusted noise, namely sender-side preparation noise and receiver-side detection noise, the latter being security preserving. We consider the method of sender-side state purification to compensate the preparation noise …
Design and Optimization of Adaptive Diversity Schemes in Quantum MIMO Channels
针对量子MIMO信道中噪声、串扰等导致的传输可靠性问题，提出基于发射端通用非对称克隆和接收端概率纯化的自适应分集策略，推导解码器最优保真度表达式并优化纯化映射，显著提升串扰主导环境下的保真度。
We study the preparation and distribution of high-fidelity multi-party entangled states via noisy channels and operations. In the particular case of GHZ and cluster states, we study different strategies using bipartite or multipartite purification protocols. The most efficient strategy depends on the target fidelity one wishes to achieve and on the quality of transmission channel and local operations. We show the existence of a crossing point beyond which the strategy making use of the purification of the state as a whole is more efficient than a strategy in which pairs are purified before the…
Optimistic Entanglement Purification in Quantum Networks
本文提出量子网络单链路中的乐观纠缠纯化方法，通过延迟预示和纯化所需的经典通信来缩短 EPR 对存储时间，在低初始保真度和短相干时间场景下相比传统方案可同时提高保真度与速率。
No Universal Purification in Quantum Mechanics
本文证明量子力学的线性和正性对纯化施加了普遍限制，排除了未知量子态或信道的精确通用纯化，并给出了近似纯化的样本复杂度下界，揭示了纯化与量子学习之间的深层联系。
Establishing end-to-end quantum connections requires quantified link characteristics, and operations need to coordinate decision-making between nodes across a network. We introduce the RuleSet-based communication protocol for supporting quantum operations over distant nodes to minimize classical packet transmissions for guaranteeing synchronicity. RuleSets are distributed to nodes along a path at connection set up time, and hold lists of operations that need to be performed in real time. We simulate the RuleSet-based quantum link bootstrapping protocol, which consists of recurrent purification…
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