John Watrous
It is proved that every doubly stochastic quantum channel that is properly averaged with the completely depolarizing channel can be written as a convex combination of unitary channels. As a consequence, we find that the collection of channels expressible as convex combinations of unitary channels has non-zero Borel measure within the space of doubly stochastic channels.
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Experimental demonstration of genuine quantum information transmission through completely depolarizing channels in a superposition of cyclic orders
本研究通过可编程硅基光子芯片实验演示了利用循环序叠加的不定因果序，在完全退极化信道中实现真正量子信息传输，保真度显著超过经典阈值，为高噪声环境下的量子通信提供了新途径。
If two identical copies of a completely depolarizing channel are put into a superposition of their possible causal orders, they can transmit non-zero classical information. Here, we study how well we can transmit classical information with $N$ depolarizing channels put in superposition of $M$ causal orders via quantum SWITCH. We calculate Holevo quantity if the superposition uses only cyclic permutations of channels and find that it increases with $M$ and it is independent of $N$. For a qubit it never reaches $1$ if we are increasing $M$. On the other hand, the classical capacity decreases wit…
Guaranteeing Completely Positive Quantum Evolution
本文研究将非完全正（NCP）映射通过非对称退极化复合转化为完全正（CP）映射，证明无需完全退极化即可实现，并给出扰动度量，表明非对称退极化优于结构物理近似，能更好保持原映射结构。
The qubit depolarizing channel with noise parameter $η$ transmits an input qubit perfectly with probability $1-η$, and outputs the completely mixed state with probability $η$. We show that its complementary channel has positive quantum capacity for all $η>0$. Thus, we find that there exists a single parameter family of channels having the peculiar property of having positive quantum capacity even when the outputs of these channels approach a fixed state independent of the input. Comparisons with other related channels, and implications on the difficulty of studying the quantum capacity of the …
Depolarizing maps acting on an N dimensional system are completely positive maps resulting into compression of the Bloch ball along the different polarization directions. In the qubit case these maps are a convex sum of four extremal maps and form a simplex in the space of compression coefficients along the three polarization directions. We calculate the compression domain for three and four level systems. For a three level system the region has curved surfaces, but it is a simplex for a four level system. We conjecture that it is a simplex in the case of 2^n level systems.
We introduce a method to construct non-Markovian variants of completely positive (CP) dynamical maps, particularly, qubit Pauli channels. We identify non-Markovianity with the breakdown in CP-divisibility of the map, i.e., appearance of a not-completely-positive (NCP) intermediate map. In particular, we consider the case of non-Markovian dephasing in detail. The eigenvalues of the Choi matrix of the intermediate map crossover at a point which corresponds to a singularity in the canonical decoherence rate of the corresponding master equation, and thus to a momentary non-invertibility of the map…
Strong converse bounds on the classical identification capacity of the qubit depolarizing channel
本文研究了量子退相干信道的经典识别容量，推导出强反向界限，该界限在噪声趋于1时趋于零，解决了完全噪声极限下的正确行为问题，并证明在同时经典识别约束下，识别容量等于信道的经典容量。
The notion of potential output purity of a completely positive map is introduced as a generalization of the regularized output purity. An upper bound is derived for this quantity, and for several classes of maps (including CQ, QC and Hadamard channels) it is shown that potential purity does not exceed the standard output purity. As an application the potential purity is used to bound the logarithmic Sobolev constant of a product of depolarizing channel semigroups.
Construction of channels which in every dimension anti-degrade the depolarizing channel
本文研究了在x≥1/2时去极化通道的反退化性质，并证明其容量为零，同时给出了互补通道的闭式容量计算。
We investigate the speed of disentanglement in the multiqubit systems under the local depolarizing channel, in which each qubit is independently coupled to the environment. We focus on the bipartition entanglement between one qubit and the remaining qubits constituting the system, which is measured by the negativity. For the two-qubit system, the speed for the pure state completely depends on its entanglement. The upper and lower bounds of the speed for arbitrary two-qubit states, and the necessary conditions for a state achieving them, are obtained. For the three-qubit system, we study the sp…
Quantum coherence is a prime resource in quantum computing and quantum communication. Quantum coherence of an arbitrary qubit state can be created at a remote location using maximally entangled state, local operation and classical communication. However, if there is a noisy channel acting on one side of the shared resource, then, it is not possible to create perfect quantum coherence remotely. Here, we present a method for the creation of quantum coherence at a remote location via the use of entangled state and indefinite causal order. We show this specifically for the superposition of two com…
We propose an effective Hamiltonian approach to investigate decoherence of a quantum system in a non-Markovian reservoir, naturally imposing the complete positivity on the reduced dynamics of the system. The formalism is based on the notion of an effective reservoir, i.e., certain collective degrees of freedom in the reservoir that are responsible for the decoherence. As examples for completely positive decoherence, we present three typical decoherence processes for a qubit such as dephasing, depolarizing, and amplitude-damping. The effects of the non-Markovian decoherence are compared to the …
We study the convergence of states under continuous-time depolarizing channels with full rank fixed points in terms of the relative entropy. The optimal exponent of an upper bound on the relative entropy in this case is given by the log-Sobolev-1 constant. Our main result is the computation of this constant. As an application we use the log-Sobolev-1 constant of the depolarizing channels to improve the concavity inequality of the von-Neumann entropy. This result is compared to similar bounds obtained recently by Kim et al. and we show a version of Pinsker's inequality, which is optimal and tig…
Exactness of the doubly nonnegative relaxation for qubit-output quantum channels
研究证明了所有输出为qubit的CPDNN量子通道实际上都是CPCP，解决了关于是否存在非CPCP的CPDNN通道的问题。
We introduce a 3-parameter class of maps acting on a bipartite system that are a natural generalisation of the depolarizing channel (and include it as a special case). Then, we find the exact regions of the parameter space that alternatively determine a positive, completely positive, entanglement-breaking or entanglement-annihilating map. This model displays a much richer behaviour than the one shown by a simple depolarizing channel, yet it stays exactly solvable. As an example of this richness, PPT but not entanglement-breaking maps are found. A simple example of a positive yet indecomposable…
Quantum error-correcting codes so far proposed have not worked in the presence of noise which introduces more than one bit of entropy per qubit sent through a quantum channel, nor can any code which identifies the complete error syndrome. We describe a code which does not find the complete error syndrome and can be used for reliable transmission of quantum information through channels which add more than one bit of entropy per transmitted bit. In the case of the depolarizing channel our code can be used in a channel of fidelity .8096. The best existing code worked only down to .8107.
Reciprocal pairs of quantum channels are defined as completely positive transformations which admit a rigid, distance-preserving, yet not completely-positive transformation that allows to reproduce the outcome of one from the corresponding outcome of the other. From a classical perspective these transmission lines should exhibit the same communication efficiency. This is no longer the case in the quantum setting: explicit asymmetric behaviours are reported studying the classical communication capacities of reciprocal pairs of depolarizing and Weyl-covariant channels.
We demonstrate how insights gained from reformulating the problem of quantum teleportation into one of reversing quantum operations, and designing optimum completely positive maps for teleportation, can enable one to explore optimal approximate reversal of quantum operations on a single qubit. In particular, we show that the optimal approximate reversal of a generalized depolarizing channel can be achieved using only unitary transformations. We also show that for a quantum channel, which reveals some information about the input state, extremal completely positive maps and not unitary transform…
The depolarization channel is usually modelled as a quantum operation that destroys all input information, replacing it by a completely chaotic state. For qubits this has a quite intuitive interpretation as a shrinking of the Bloch sphere. We propose a way to deal with depolarizing dynamics (in the Markov approximation) for systems with arbitrary symmetries.
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。
Additivity of the Holevo capacity is proved for product channels, under the condition that one of the channels is in a certain class of unital qubit channels, with the other completely arbitrary. This qubit class includes the depolarizing channel. As a byproduct this proves that the Holevo bound is the ultimate information capacity of such qubit channels (assuming no prior entanglement between sender and receiver). Additivity of minimal entropy and multiplicativity of p-norms are also proved under the same assumptions.
General Communication Enhancement via the Quantum Switch
本文定义了一个与量子开关相关的可计算量，并猜想其大于零是通信增强的充要条件，通过推导泡利信道经典容量的解析表达式验证了该猜想，并构造协议增强了BB84信道的私密容量。
Birkhoff's Theorem states that doubly stochastic matrices are convex combinations of permutation matrices. Quantum mechanically these matrices are doubly stochastic channels, i.e. they are completely positive maps preserving both the trace and the identity. We expect these channels to be convex combinations of unitary channels and yet it is known that some channels cannot be written that way. Recent work has suggested that $n$ copies of a single channel might approximate a mixture (convex combination) of unitaries. In this paper we show that $n(n+1)/2$ copies of a symmetric unital quantum chan…
In this work, we study two different approaches to defining the entropy of a quantum channel. One of these is based on the von Neumann entropy of the corresponding Choi-Jamiołkowski state. The second one is based on the relative entropy of the output of the extended channel relative to the output of the extended completely depolarizing channel. This entropy then needs to be optimized over all possible input states. Our results first show that the former entropy provides an upper bound on the latter. Next, we show that for unital qubit channels, this bound is saturated. Finally, we conjecture a…
Dephasing decoherence induced by interaction of one qubit with a quantum bath can be simulated classically by random unitary evolution without the need for a bath and this random unitary evolution is equivalent to the quantum case. For a general dephasing model and a single qubit system, we explicitly construct the noise functional and completely specify the random unitary evolution. To demonstrate the technique, we applied our results to three paradigmatic models: spin-boson, central spin, and quantum impurity. For multiple qubits, we identify a class of generalized quantum dephasing models t…
We investigate quantum channels, which after a finite number $k$ of repeated applications erase all input information, i.e., channels whose $k$-th power (but no smaller power) is a completely depolarizing channel. We show that on a system with Hilbert space dimension $d$, the order is bounded by $k\leq d^2-1$, and give an explicit construction scheme for such channels. We also consider strictly forgetful memory channels, i.e., channels with an additional input and output in every step, which after exactly $k$ steps retain no information about the initial memory state. We establish an explicit …
The set of doubly-stochastic quantum channels and its subset of mixtures of unitaries are investigated. We provide a detailed analysis of their structure together with computable criteria for the separation of the two sets. When applied to O(d)-covariant channels this leads to a complete characterization and reveals a remarkable feature: instances of channels which are not in the convex hull of unitaries can return to it when either taking finitely many copies of them or supplementing with a completely depolarizing channel. In these scenarios this implies that a channel whose noise initially r…
In this work, we study the transmission of classical information through three completely depolarizing channels in superposition of different causal orders. We thus introduce the quantum 3-switch as a resource for quantum communications. We perform a new kind of quantum control that was not accessible to the previously treated two-channel case. The fine and full quantum control achieved using selected combinations of causal orders let us uncover new features: non monotonous behavior on the transmission of information with respect to the number of causal orders involved, and different values of…
In quantum Shannon theory, the way information is encoded and decoded takes advantage of the laws of quantum mechanics, while the way communication channels are interlinked is assumed to be classical. In this Letter we relax the assumption that quantum channels are combined classically, showing that a quantum communication network where quantum channels are combined in a superposition of different orders can achieve tasks that are impossible in conventional quantum Shannon theory. In particular, we show that two identical copies of a completely depolarizing channel become able to transmit info…
Quantum operations represented by completely positive maps encompass many of the physical processes and have been very powerful in describing quantum computation and information processing tasks. We introduce the notion of relative phase change for a quantum system undergoing quantum operation. We find that the relative phase shift of a system not only depends on the state of the system, but also depends on the initial state of the ancilla with which it might have interacted in the past. The relative phase change during a sequence of quantum operations is shown to be non-additive in nature. Th…
The highest information rate at which quantum error-correction schemes work reliably on a channel, which is called the quantum capacity, is proven to be lower bounded by the limit of the quantity termed coherent information maximized over the set of input density operators which are proportional to the projections onto the code spaces of symplectic stabilizer codes. Quantum channels to be considered are those subject to independent errors and modeled as tensor products of copies of a completely positive linear map on a Hilbert space of finite dimension, and the codes that are proven to have th…
Simulation and analysis of quantum phase estimation algorithm in the presence of incoherent quantum noise channels
本文通过将非相干噪声建模为保迹完全正量子信道，仿真分析了退极化、相位翻转、比特翻转及比特-相位翻转等噪声对量子相位估计算法的影响，发现本征值标准差与单比特错误概率呈指数依赖，而在小错误概率下随量子比特数仅线性增长。
We investigate the completely positive semidefinite cone $\mathcal{CS}_+^n$, a new matrix cone consisting of all $n\times n$ matrices that admit a Gram representation by positive semidefinite matrices (of any size). In particular we study relationships between this cone and the completely positive and doubly nonnegative cones, and between its dual cone and trace positive non-commutative polynomials. We use this new cone to model quantum analogues of the classical independence and chromatic graph parameters $α(G)$ and $χ(G)$, which are roughly obtained by allowing variables to be positive semid…
Quantum Channel Masking
本文提出量子通道掩码的概念，研究了量子通道层面的动态扩展，证明了qubit通道在特定条件下可被身份掩码，揭示了通道噪声通过广播映射的局部化特性。
In quantum Shannon theory, transmission of information is enhanced by quantum features. Up to very recently, the trajectories of transmission remained fully classical. Recently, a new paradigm was proposed by playing quantum tricks on two completely depolarizing quantum channels i.e. using coherent control in space or time of the two quantum channels. We extend here this control to the transmission of information through a network of an arbitrary number $N$ of channels with arbitrary individual capacity i.e. information preservation characteristics in the case of indefinite causal order. We pr…
A simple criterion for local equality between the constrained Holevo capacity and the quantum mutual information of a quantum channel is obtained. It implies that the set of all states for which this equality holds is determined by the kernel of the channel (as a linear map). Applications to Bosonic Gaussian channels are considered. It is shown that for a Gaussian channel having no completely depolarizing components the above characteristics may coincide only at non-Gaussian mixed states and a criterion of existence of such states is given. All the obtained results may be reformulated as condi…
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。
Quantum Skyrmions in general quantum channels: topological noise rejection and the discretization of quantum information
本研究利用双光子纠缠纯态拓扑实现量子信息离散化，构建噪声模型并通过理论与数值验证其抗噪性，为噪声环境下量子信息分发提供新框架。
Transport and Channel Normal Forms for Clifford Interactions
本文研究 Clifford 相互作用对量子传输的影响，分类局部等价类，揭示正卷积导致纠缠破坏而其他相互作用可实现无噪声传输，并利用环境 Clifford 传输约化信道以获得私有容量上界。
A phase transition in the exactness of the NPA hierarchy at the critical doubly-tilted CHSH functional
研究双倾斜CHSH泛函在临界线处NPA层次精确性的相变，通过量化对称切片上的机制，在超临界侧证明层次精确性，亚临界侧验证前四个层次，确定极限障碍为Motzkin多项式，借助定理给出相位边界几何解读并记录勘误。
An $n\times n$ matrix $X$ is called completely positive semidefinite (cpsd) if there exist $d\times d$ Hermitian positive semidefinite matrices $\{P_i\}_{i=1}^n$ (for some $d\ge 1$) such that $X_{ij}= {\rm Tr}(P_iP_j),$ for all $i,j \in \{ 1, \ldots, n \}$. The cpsd-rank of a cpsd matrix is the smallest $d\ge 1$ for which such a representation is possible. In this work we initiate the study of the cpsd-rank which we motivate twofold. First, the cpsd-rank is a natural non-commutative analogue of the completely positive rank of a completely positive matrix. Second, we show that the cpsd-rank is …
Quantum Nonlocality and Device-Independent Randomness are Robust to Noisy Signaling Channels
本文研究了在噪声信号通道下量子非局域性和设备无关随机性的鲁棒性，通过分析局部多面体的顶点和面，确定了能够认证非信号量子相关性的贝尔不等式，并展示了这些不等式在对抗去极化噪声时比Clauser-Horne-Shimony-Holt不等式更鲁棒。
In quantum error correction, the description of noise channel cannot be completely accurate, and fluctuation always appears in noise channel. It is found that when fluctuation of physical noise channel is considered, the average effective channel is dependent only on the average of physical noise channel, and the average of physical noise channel here plays the role of the independent error model in the previous works. Now, one may conclude that in the independent error model, the results in previous works are also valid for average channel where fluctuation exists. In some typical cases, our …
Entanglement Fidelity in Standard Quantum Channels
本文推导了标准量子噪声模型下的纠缠保真度闭式表达式，分析了通道和源参数对纠缠保持的影响，并提供了性能比较和状态排名的方法。
Quantum channels can be described via a unitary coupling of system and environment, followed by a trace over the environment state space. Taking the trace instead over the system state space produces a different mapping which we call the conjugate channel. We explore the properties of conjugate channels and describe several different methods of construction. In general, conjugate channels map M_d to M_d' with d < d', and different constructions may differ by conjugation with a partial isometry. We show that a channel and its conjugate have the same minimal output entropy and maximal output p-n…
The method of complementary channel for analysis of reversibility (sufficiency) of a quantum channel with respect to families of input states (pure states for the most part) are considered and applied to Bosonic linear (quasi-free) channels, in particular, to Bosonic Gaussian channels. The obtained reversibility conditions for Bosonic linear channels have clear physical interpretation and their sufficiency is also shown by explicit construction of reversing channels. The method of complementary channel gives possibility to prove necessity of these conditions and to describe all reversed famili…
Quantum Channel Certification with Incoherent Strategies
研究量子信道认证问题，针对酉信道和完全退极化信道两种极端情形，提出了最优非相干策略，并分别给出了自适应与非自适应设定下黑盒使用次数的上下界。
We consider the problem of transmitting classical and quantum information reliably over an entanglement-assisted quantum channel. Our main result is a capacity theorem that gives a three-dimensional achievable rate region. Points in the region are rate triples, consisting of the classical communication rate, the quantum communication rate, and the entanglement consumption rate of a particular coding scheme. The crucial protocol in achieving the boundary points of the capacity region is a protocol that we name the classically-enhanced father protocol. The classically-enhanced father protocol is…
We analyze the set ${\cal A}_N^Q$ of mixed unitary channels represented in the Weyl basis and accessible by a Lindblad semigroup acting on an $N$-level quantum system. General necessary and sufficient conditions for a mixed Weyl quantum channel of an arbitrary dimension to be accessible by a semigroup are established. The set ${\cal A}_N^Q$ is shown to be log--convex and star-shaped with respect to the completely depolarizing channel. A decoherence supermap acting in the space of Lindblad operators transforms them into the space of Kolmogorov generators of classical semigroups. We show that fo…
To achieve universal quantum computation via general fault-tolerant schemes, stabilizer operations must be supplemented with other non-stabilizer quantum resources. Motivated by this necessity, we develop a resource theory for magic quantum channels to characterize and quantify the quantum "magic" or non-stabilizerness of noisy quantum circuits. For qudit quantum computing with odd dimension $d$, it is known that quantum states with non-negative Wigner function can be efficiently simulated classically. First, inspired by this observation, we introduce a resource theory based on completely posi…
Quantum information theory and Fourier multipliers on quantum groups
本文利用局部紧量子群上的傅里叶乘子理论，计算了矩阵代数上大类量子信道的最小输出熵和完全有界最小熵的精确值，并给出经典容量上界，同时揭示了其与子因子平面代数、单位量子比特信道及纠缠破坏性质的联系。
The von Neumann entropy of a quantum state is a central concept in physics and information theory, having a number of compelling physical interpretations. There is a certain perspective that the most fundamental notion in quantum mechanics is that of a quantum channel, as quantum states, unitary evolutions, measurements, and discarding of quantum systems can each be regarded as certain kinds of quantum channels. Thus, an important goal is to define a consistent and meaningful notion of the entropy of a quantum channel. Motivated by the fact that the entropy of a state $ρ$ can be formulated as …
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