M. A. Nielsen
This note quantifies the continuity properties of entanglement: how much does entanglement vary if we change the entangled quantum state just a little? This question is studied for the pure state entanglement of a bipartite system and for the entanglement of formation of a bipartite system in a mixed state.
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We show that an entanglement measure called relative entropy of entanglement satisfies a strong continuity condition. If two states are close to each other then so are their entanglements per particle pair in this measure. It follows in particular, that the measure is appropriate for the description of entanglement manipulations in the limit of an infinite number of pairs of particles.
We show that, in any open set of distillable states, all asymptotic entanglement measures $E(ρ)$ are continuous as a function of (a single copy of) $ρ$, even though they quantify the entanglement properties of $ρ^{\otimes N}$ is the large $N$ limit.
We introduce a complex purity density and its associated current for pure states of continuous variable systems. The scheme is constructed by analogy with the notions of probability density and probability current. Taking advantage of the formal continuity equations obtained this way we can introduce an entanglement subdynamics. We suggest the use of the dimensionality of this subdynamics as a potential measure of the complexity of entanglement. The scheme also provides insights into the relation between the global and local aspects of quantum correlations.
We propose experiments on quantum entanglement for investigating the Einstein Podolsky Rosen (EPR) problem with the polarization directions of photons. These experiments are performed to investigate whether the defined polarization directions in an entangled state are teleported between entangled photons. EPR-type sequential experiments are performed using a twin-photon beam and two pairs of linear polarization analyzers under the cross-Nicol condition (i.e., orthogonal to each other). If the third filter whose polarization angle is 45 degrees is set between the first cross-Nicol filters, the …
We consider universal methods for obtaining (uniform) continuity bounds for characteristics of multipartite quantum systems. We pay a special attention to infinite-dimensional multipartite quantum systems under the energy constraints. By these methods we obtain continuity bounds for several important characteristics of a multipartite quantum state: the quantum (conditional) mutual information, the squashed entanglement, the c-squashed entanglement and the conditional entanglement of mutual information. The continuity bounds for the multipartite quantum mutual information are asymptotically tig…
We study rates asymptotic of transformations between entangled states by local operations and classical communication and a sublinear amount of quantum communication. It is known that additive asymptotically continuous entanglement measures provide upper bounds on the rates that are achievable with asymptotically vanishing error. We show that for transformations between pure states, the optimal rate between any pair of states can be characterized as the infimum of such upper bounds provided by fully additive asymptotically continuous entanglement measures.
In the first part of this thesis, we present a general technique for establishing local and uniform continuity bounds for Schur concave functions. Our technique uses a particular relationship between majorization and the trace distance between quantum states. Namely, the majorization pre-order attains a minimum over $ε$-balls in this distance. By tracing the path of the majorization-minimizer as a function of the distance $ε$, we obtain the path of "majorization flow". This yields a new proof of the Audenaert-Fannes continuity bound for the von Neumann entropy in a universal framework which ex…
We analyse two possible definitions of the squashed entanglement in an infinite-dimensional bipartite system: direct translation of the finite-dimensional definition and its universal extension. It is shown that the both definitions produce the same lower semicontinuous entanglement measure possessing all basis properties of the squashed entanglement on the set of states having at least one finite marginal entropy. Is also shown that the second definition gives an adequate extension of this measure to the set of all states of infinite-dimensional bipartite system. A general condition relating …
Continuity of the relative entropy of resource
该研究建立资源相对熵局部连续性判据，推导其基本推论，并证明多体纠缠相对熵的局部连续性可由量子互信息的局部连续性得出。
Entanglement Measure Based on Optimal Entanglement Witness
本文提出了一种基于最优纠缠 witness 的新纠缠度量，证明了其满足零纠缠、凸性、连续性及 LOCC 下非增加等必要性质，并给出了任意两体混合态下界的数学表达式，进一步改进了 2⊗2 系统的下界并进行了数值模拟。
The possibility of realizing lattice analogs of fractional quantum Hall (FQH) states, so-called fractional Chern insulators (FCIs), in nearly flat topological (Chern) bands has attracted a lot of recent interest. Here, we make the connection between Abelian as well as non-Abelian FQH states and FCIs more precise. Using a gauge-fixed version of Qi's Wannier basis representation of a Chern band, we demonstrate that the interpolation between several FCI states, obtained by short-range lattice interactions in a spin-orbit-coupled kagome lattice model, and the corresponding continuum FQH states is …
We present a bouquet of continuity bounds for quantum entropies, falling broadly into two classes: First, a tight analysis of the Alicki-Fannes continuity bounds for the conditional von Neumann entropy, reaching almost the best possible form that depends only on the system dimension and the trace distance of the states. Almost the same proof can be used to derive similar continuity bounds for the relative entropy distance from a convex set of states or positive operators. As applications we give new proofs, with tighter bounds, of the asymptotic continuity of the relative entropy of entangleme…
Purity based continuity bounds for quantum information measures
本文基于量子态纯度差推导了多种量子信息测度（如von Neumann熵、条件熵、量子互信息等）的连续性界，并探讨了其在压缩纠缠和量子Markov链中的应用。
General Continuity Bounds for Quantum Relative Entropies
本文提出一种通用方法，为量子相对熵导出的熵量建立连续性界，并用于推导新的熵不确定性关系。
Continuity of entropies via integral representations
本文利用Frenkel的量子相对熵积分表示推导出维度无关的半连续性关系，并据此解决了多个量子信息连续性猜想，改进了量子容量与纠缠成本估计。
Attainability and lower semi-continuity of the relative entropy of entanglement, and variations on the theme
本文证明了纠缠相对熵在无限维下也具有可达性与下半连续性，并推导出其对偶变分表达式，该结果适用于多种资源量化指标，关键在于自由状态生成锥的弱*闭性，同时给出了能量约束下的连续性估计。
The Renyi entropy plays an essential role in quantum information theory. We study the continuity estimation of the Renyi entropy. An inequality relating the Renyi entropy difference of two quantum states to their trace norm distance is derived. This inequality is shown to be tight in the sense that equality can be attained for every prescribed value of the trace norm distance. It includes the sharp Fannes inequality for von Neumann entropy as a special case.
We provide an upper bound on the quasi-relative entropy in terms of the trace distance. The bound is derived for two cases: 1) any operator monotone decreasing function and full rank mixed qubit or classical states; 2) a large class of operator monotone decreasing function and any mixed qubit or classical states. Moreover, we derive an upper bound for the Umegaki and Tsallis relative entropies in the case of any finite-dimensional states. The bound for the relative entropy improves the known bounds for some states in any dimensions larger than four. The bound for the Tsallis entropy improves t…
We study the reduced dynamics of open quantum spin chains of arbitrary length $N$ with nearest neighbour $XX$ interactions, immersed within an external constant magnetic field along the $z$ direction, whose end spins are weakly coupled to heat baths at different temperatures, via energy preserving couplings. We find the analytic expression of the unique stationary state of the master equation obtained in the so-called global approach based on the spectralization of the full chain Hamiltonian. Hinging upon the explicit stationary state, we reveal the presence of sink and source terms in the spi…
We derive a Lipschitz continuity bound for quantum-classical conditional entropies with respect to angular distance, with a Lipschitz constant that is independent of the dimension of the conditioning system. This bound is sharper in some situations than previous continuity bounds, which were either based on trace distance (where Lipschitz continuity is not possible), or based on angular distance but did not include a conditioning system. However, we find that the bound does not directly generalize to fully quantum conditional entropies. To investigate possible counterexamples in that setting, …
In this short note, I show how a recent result of Alhejji and Smith [arXiv:1909.00787] regarding an optimal uniform continuity bound for classical conditional entropy leads to an optimal uniform continuity bound for quantum conditional entropy of classical--quantum states. The bound is optimal in the sense that there always exists a pair of classical--quantum states saturating the bound, and so no further improvements are possible. An immediate application is a uniform continuity bound for entanglement of formation that improves upon the one previously given by Winter in [arXiv:1507.07775]. Tw…
Continuity of robustness measures in quantum resource theories
本文研究量子资源理论中鲁棒性度量的连续性，证明其连续性取决于自由态集合形状，星凸性在许多情况下足以保证Lipschitz连续性，并给出非连续度量的例子，最后以隐形传态能力和量子失谐的鲁棒性为例展示应用。
We prove a tight uniform continuity bound for the conditional Shannon entropy of discrete finitely supported random variables in terms of total variation distance.
We introduce a notion of majorization flow, and demonstrate it to be a powerful tool for deriving simple and universal proofs of continuity bounds for entropic functions relevant in information theory. In particular, for the case of the alpha-Rényi entropy, whose connections to thermodynamics are discussed in this article, majorization flow yields a Lipschitz continuity bound for the case alpha > 1, thus resolving an open problem and providing a substantial improvement over previously known bounds.
By extending the concept of energy-constrained diamond norms, we obtain continuity bounds on the dynamics of both closed and open quantum systems in infinite-dimensions, which are stronger than previously known bounds. We extensively discuss applications of our theory to quantum speed limits, attenuator and amplifier channels, the quantum Boltzmann equation, and quantum Brownian motion. Next, we obtain explicit log-Lipschitz continuity bounds for entropies of infinite-dimensional quantum systems, and classical capacities of infinite-dimensional quantum channels under energy-constraints. These …
We prove a simple uniform continuity bound for the sandwiched Rényi conditional entropy for $α\in [1/2, 1) \cup (1, \infty]$, which is independent of the dimension of the conditioning system.
Continuity bounds on observational entropy and measured relative entropies
本文推导出一般 POVM 测量下观测熵的渐近连续性界，并扩展至测量相对熵与条件观测熵，揭示了其连续性但证明无法具象为渐近界。
Generalized entropies and relative entropies are the subject of active research. Similar to the standard relative entropy, the relative $q$-entropy is generally unbounded for $q>1$. Upper bounds on the quantum relative $q$-entropy in terms of norm distances between its arguments are obtained in finite-dimensional context. These bounds characterize a continuity property in the sense of Fannes.
We show how to use properties of the quantum conditional mutual information to obtain continuity bounds for information characteristics of quantum channels depending on their input dimension. First we prove tight estimates for variation of the output Holevo quantity with respect to simultaneous variations of a channel and of an input ensemble. Then we obtain tight continuity bounds for output conditional mutual information for a single channel and for $n$ copies of a channel. As a result tight and close-to-tight continuity bounds for basic capacities of quantum channels depending on the input …
Close-to-optimal continuity bound for the von Neumann entropy and other quasi-classical applications of the Alicki-Fannes-Winter technique
本文提出 Alicki-Fannes-Winter 技术的准经典版本，用于推导多种约束下量子与经典熵量的连续性及半连续性界，并在单模振荡器情形获得接近最优的 von Neumann 熵界。
We consider a very natural generalization of quantum theory by letting the dimension of the Bloch ball be not necessarily three. We analyze bipartite state spaces where each of the components has a d-dimensional Euclidean ball as state space. In addition to this we impose two very natural assumptions: the continuity and reversibility of dynamics, and the possibility of characterizing bipartite states by local measurements. We classify all these bipartite state spaces and prove that, except for the quantum two-qubit state space, none of them contains entangled states. Equivalently, in any of th…
The quantum relative entropy is frequently used as a distance, or distinguishability measure between two quantum states. In this paper we study the relation between this measure and a number of other measures used for that purpose, including the trace norm distance. More precisely, we derive lower and upper bounds on the relative entropy in terms of various distance measures for the difference of the states based on unitarily invariant norms. The upper bounds can be considered as statements of continuity of the relative entropy distance in the sense of Fannes. We employ methods from optimisati…
We start with Fannes' type and Winter's type tight continuity bounds for the quantum conditional mutual information and their specifications for states of special types. Then we analyse continuity of the Holevo quantity with respect to nonequivalent metrics on the set of discrete ensembles of quantum states. We show that the Holevo quantity is continuous on the set of all ensembles of m states with respect to all the metrics if either m or the dimension of underlying Hilbert space is finite and obtain Fannes' type tight continuity bounds for the Holevo quantity in this case. In general case co…
The tight, in a sense, lower estimates of diamond-norm distance from a given quantum channel to the sets of degradable, antidegradable and entanglement-breaking channels are obtained via the tight continuity bounds for quantum mutual information and for relative entropy of entanglement in finite-dimensional case. As an auxiliary result there are established lower bounds of trace-norm distance from a given bipartite state to the set of all separable states.
We prove a tight uniform continuity bound for a family of entropies which includes the von Neumann entropy, the Tsallis entropy and the $α$-Rényi entropy, $S_α$, for $α\in (0,1)$. We establish necessary and sufficient conditions for equality in the continuity bound and prove that these conditions are the same for every member of the family. Our result builds on recent work in which we constructed a state which was majorized by every state in a neighbourhood ($\varepsilon$-ball) of a given state, and thus was the minimal state in majorization order in the $\varepsilon$-ball. This minimal state …
Continuity of quantum entropic quantities via almost convexity
本文提出ALAFF方法，基于条件熵连续性证明，推导多种量子熵量的连续性界，得到新结果并证明相关熵的几乎凹性，可应用于量子信息领域。
We adapt modulus of continuity estimates to the study of spectra of combinatorial graph Laplacians, as well as the Dirichlet spectra of certain weighted Laplacians. The latter case is equivalent to stoquastic Hamiltonians and is of current interest in both condensed matter physics and quantum computing. In particular, we introduce a new technique which bounds the spectral gap of such Laplacians (Hamiltonians) by studying the limiting behavior of the oscillations of their eigenvectors when introduced into the heat equation. Our approach is based on recent advances in the PDE literature, which i…
We obtain continuity bounds for basic information characteristics of quantum channels depending on their input dimension (if it is finite) and on the input energy bound (if the input dimension is infinite). We pay a special attention to the case of a multi-mode quantum oscillator as an input system. First we prove continuity bounds for the output conditional mutual information for a single channel and for $n$ copies of a channel. Then we obtain estimates for variation of the output Holevo quantity with respect to simultaneous variations of a channel and of an input ensemble. As a result tight …
We are interested in the properties and relations of entanglement measures. Especially, we focus on the squashed entanglement and relative entropy of entanglement, as well as their analogues and variants. Our first result is a monogamy-like inequality involving the relative entropy of entanglement and its one-way LOCC variant. The proof is accomplished by exploring the properties of relative entropy in the context of hypothesis testing via one-way LOCC operations, and by making use of an argument resembling that by Piani on the faithfulness of regularized relative entropy of entanglement. Foll…
We prove a tight uniform continuity bound for Arimoto's version of the conditional $α$-Rényi entropy, for the range $α\in [0, 1)$. This definition of the conditional Rényi entropy is the most natural one among the multiple forms which exist in the literature, since it satisfies two desirable properties of a conditional entropy, namely, the fact that conditioning reduces entropy, and that the associated reduction in uncertainty cannot exceed the information gained by conditioning. Furthermore, it has found interesting applications in various information theoretic tasks such as guessing with sid…
Continuity bounds for quantum entropies arising from a fundamental entropic inequality
该研究针对两个量子态的冯·诺依曼熵差建立紧上界，提出蕴含并细化Audenaert-Fannes不等式的新型熵不等式，据此得到量子条件熵和双变量相对熵的连续性界，且不等式在无限维中有效。
Optimal continuity bound for the von Neumann entropy under energy constraints
该研究为满足吉布斯假设的任意哈密顿量能量约束下的冯·诺依曼熵构造了全局最优连续性界，解决了此前仅对接近状态对已知的问题，并改进了相关半连续性界与Fano型不等式。
From Classical to Quantum: Uniform Continuity Bounds on Entropies in Infinite Dimensions
本文证明了无限维经典与量子系统熵的一致连续性界，利用Fano型不等式和极大耦合获得Shannon与von Neumann熵的紧致估计，并提出基于算子Hölder连续性的新近似方案处理Rényi等熵，解决了有限维近似性质的开放问题。
A complete characterization of eventually entanglement breaking unital quantum channels
本文通过特征向量交换性质完整刻画了最终纠缠破坏的酉信道，并利用矩阵积分与等角紧框架理论刻画了其极端点。
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