Sergio Albeverio, Shao-Ming Fei
The quantum entanglements are studied in terms of the invariants under local unitary transformations. A generalized formula of concurrence for $N$-dimensional quantum systems is presented. This generalized concurrence has potential applications in studying separability and calculating entanglement of formation for high dimensional mixed quantum states.
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An alternative approach to "entanglement, nonlocality, and superluminal signaling" is advocated. It is inspired on Faraday's way of viewing physics.
These are notes on some entanglement properties of quantum field theory, aiming to make accessible a variety of ideas that are known in the literature. The main goal is to explain how to deal with entanglement when -- as in quantum field theory -- it is a property of the algebra of observables and not just of the states.
Quantum and classical models for delayed choice entanglement swapping by postselection of measurements are discussed.
In quant-ph/0205017 v1 Chen, Wu and Yang formulated a necessary separability criterion based on a realignment method for matrices. This note is to point out that this criterion is identical to the necessary cross norm criterion previously put forward by this author.
We discuss a kind of generalized concurrence for a class of high dimensional quantum pure states such that the entanglement of formation is a monotonically increasing convex function of the generalized concurrence. An analytical expression of the entanglement of formation for a class of high dimensional quantum mixed states is obtained.
We study the Bell nonlocality of high dimensional quantum systems based on quantum entanglement. A quantitative relationship between the maximal expectation value B of Bell operators and the quantum entanglement concurrence C is obtained for even dimension pure states, with the upper and lower bounds of B governed by C.
We study quantum states for which the PPT criterion is both sufficient and necessary for separability. We present a class of 3x3 bipartite mixed states and show that these states are separable if and only if they are PPT.
A note on entanglement detection via the generalized realignment moments
本文提出基于广义重排矩的两个分离性判据，通过引入额外参数提升判据的灵活性和强度，并通过实例验证其在纠缠检测中的有效性。
A note on the holographic time-like entanglement entropy in Lifshitz theory
本文在三维Lifshitz时空中检验多种全息类时间纠缠熵提案，发现结果一致，且TEE的实部与虚部均依赖各向异性参数。
A note on the lower bounds of genuine multipartite entanglement concurrence
本文研究真正多体纠缠并发度的下界，通过推导其与二分及多体纠缠并发度的关系，提出更优下界，并探讨其与其他纠缠度量的联系。
Using the information content of correlations between multipartite systems, together with the notion of partitioning, we show that some general results about the evolution of correlations in quantum systems can be derived with only elementary methods. In particular, we show that for 2 quantum systems A and B, each comprised of a number of sub-systems, in which a partition of A interacts unitarily with a partition of B, then the total correlation can only increase (or remain unchanged) and is given simply by the sum of the initial correlation and the correlation that develops as a result of the…
We study generalized concurrences as a tool to detect the entanglement of bipartite quantum systems. By considering the case of 2 X 4 states of rank 2, we prove that generalized concurrences do not, in general, give a necessary and sufficient condition of separability. We identify a set of entangled states which are undetected by this method.
Lecture Notes: Introduction to random unitary circuits and the measurement-induced entanglement phase transition
本讲义以轻松对话形式介绍随机幺正电路中的纠缠动力学与测量诱导纠缠相变，重点阐述最小割概念及其统计力学映射。
This is an expanded version of the short report arXiv:1401.0539, where we stud- ied the (Renyi) entanglement entropies for the excited state defined by acting a given local operator on the ground state. We introduced the (Renyi) entanglement entropies of given local operators which measure the degrees of freedom of local operators and characterize them in conformal field theories from the viewpoint of quantum entanglement. In present paper, we explain how to compute them in free massless scalar field theories and we also investigate their time evolution. The results are interpreted in terms of…
We elaborate on the extended Hilbert space factorization of Chern Simons theory and show how this arises naturally from a proper regularization of the entangling surface in the Euclidean path integral. The regularization amounts to stretching the entangling surface into a co-dimension one surface which hosts edge modes of the Chern Simons theory when quantized on a spatial subregion. The factorized state is a regularized Ishibashi state and reproduces the well known topological entanglement entropies. We illustrate how the same factorization arises from the glueing of two spatial subregions vi…
We show that two related measures of k-coherence, called the standard and generalized robustness of k-coherence, are equal to each other when restricted to pure states. As a direct application of the result, we establish an equivalence between two analogous measures of Schmidt rank k-entanglement for all pure states. This answers conjectures raised in the literature regarding the evaluation of the quantifiers, and facilitates an efficient quantification of pure-state resources by introducing computable closed-form expressions for the two measures.
Quantum entropy inequalities are studied. Some quantum entropy inequalities are obtained by several methods. For entanglement breaking channel, we show that the entanglement-assisted classical capacity is upper bounded by $\log d$. A relationship between entanglement-assisted and one-shot unassisted capacities is obtained. This relationship shows the entanglement-assisted channel capacity is upper bounded by the sum of $\log d$ and the one-shot unassisted classical capacity.
We show using a numerical approach that gives necessary and sufficient conditions for the existence of local realism, that the bound entangled state presented in Bennett et. al. Phys. Rev. Lett. 82, 5385 (1999) admits a local and realistic description. We also find the lowest possible amount of some appropriate entangled state that must be ad-mixed to the bound entangled state so that the resulting density operator has no local and realistic description and as such can be useful in quantum communication and quantum computation.
In this paper, we find the invariant for $n$-qubits and propose the residual entanglement for $n$-qubits by means of the invariant. Thus, we establish a relation between SLOCC entanglement and the residual entanglement. The invariant and the residual entanglement can be used for SLOCC entanglement classification for $n$-qubits.
Chien-Shiung Wu as the experimental pioneer in quantum entanglement: a 2022 note
本文回顾了吴健雄在电子-正电子湮灭产生纠缠光子方面的早期开创性实验工作，该文基于作者在纪念吴健雄诞辰110周年国际研讨会上的演讲。
In this Brief Report we show the relation between three-qubit entanglement invariants and two-qubit concurrence with the help of projective measurements. How to use these invariants to represent the entanglement property of three-qubit pure states is also investigated.
We construct entanglement monotones for multi-qubit states based on Plücker coordinate equations of Grassmann variety, which are central notion in geometric invariant theory. As an illustrative example, we in details investigate entanglement monotones of a three-qubit state.
We calculate the relative entropy of entanglement for rotationally invariant states of spin-1/2 and arbitrary spin-$j$ particles or of spin-1 particle and spin-$j$ particle with integer $j$. A lower bound of relative entropy of entanglement and an upper bound of distillable entanglement are presented for rotationally invariant states of spin-1 particle and spin-$j$ particle with half-integer $j$.
We develop a simple method for constructing polynomial invariants of degree 4 for even-$n$ qubits and give explicit expressions for these polynomial invariants. We demonstrate the invariance of the polynomials under stochastic local operations and classical communication and exemplify the use of the invariance in classifying entangled states. The absolute values of these polynomial invariants are entanglement monotones, thereby allowing entanglement measures to be built. Finally, we discuss the properties of these entanglement measures.
We analyze the entanglement of SU(2)-invariant density matrices of two spins $\vec S_{1}$, $\vec S_{2}$ using the Peres-Horodecki criterion. Such density matrices arise from thermal equilibrium states of isotropic spin systems. The partial transpose of such a state has the same multiplet structure and degeneracies as the original matrix with eigenvalue of largest multiplicity being non-negative. The case $S_{1}=S$, $S_{2}=1/2$ can be solved completely and is discussed in detail with respect to isotropic Heisenberg spin models. Moreover, in this case the Peres-Horodecki ciriterion turns out to …
Pairwise entanglement properties of a symmetric multi-qubit system are analyzed through a complete set of two-qubit local invariants. Collective features of entanglement, such as spin squeezing, are expressed in terms of invariants and a classifcation scheme for pairwise entanglement is proposed. The invariant criteria given here are shown to be related to the recently proposed (Phys. Rev. Lett. 95, 120502 (2005)) generalized spin squeezing inequalities for pairwise entanglement in symmetric multi-qubit states.
We investigate entanglement dynamics and properties of double Jaynes-Cummings model. In particular we study the dynamics of double Jaynes-Cummings model based on geometric invariants for four qubits. We show that these geometric invariants provide us with useful information about different types of entanglement during the quantum evolution of the system.
In this work we consider the permutational properties of multipartite entanglement monotones. Based on the fact that genuine multipartite entanglement is a property of the entire multi-qubit system, we argue that ideal definitions for its characterizing quantities must be permutation-invariant. Using this criterion, we examine the three 4-qubit entanglement monotones introduced by Osterloh and Siewert [Phys. Rev. A. 72, 012337]. By expressing them in terms of quantities whose permutational properties can be easily derived, we find that one of these monotones is not permutation-invariant. We pr…
Entanglement properties of random multipartite quantum states which are invariant under global SU($d$) action are investigated. The random states live in the tensor power of an irreducible representation of SU($d$). We calculate and analyze the expectation and fluctuation of the second-order Rényi entanglement measure of the random invariant and near-invariant states in high dimension, and reveal the phenomenon of concentration of measure the random states exhibit. We show that with high probability a random SU($d$)-invariant state is close to being maximally entangled with respect to any bipa…
We consider the problem of characterizing genuine multiparticle entanglement for permutationally invariant states using the approach of PPT mixtures. We show that the evaluation of this necessary biseparability criterion scales polynomially with the number of particles. In practice, it can be evaluated easily up to ten qubits and improves existing criteria significantly. Finally, we show that our approach solves the problem of characterizing genuine multiparticle entanglement for permutationally invariant three-qubit states.
We investigate the relation between local unitary symmetries and entanglement invariants of multi-qubit systems. The Hilbert space of such systems can be stratified in terms of states with different types of symmetry. We review the connection between this stratification and the ring of entanglement invariants and the corresponding geometric description in terms of algebraic varieties. On a stratum of a non-trivial symmetry group the invariants of the symmetry preserving operations gives a sufficient description of entanglement. Finding these invariants is often a simpler problem than finding t…
Quantum entanglements are of fundamental importance in quantum physics ranging from the quantum information processing to the physics of black hole. Here, we show that the quantum entanglement is not invariant in special relativity. This suggests that nearly all aspects of quantum information processing would be affected significantly when relativistic effects are considered because present schemes are based on the general assumption that entanglement is invariant. There should be additional protocols to compensate the variances of entanglement in in quantum information processing. Furthermore…
The concept of the Lorentz-invariant mass of a group of particles is shown to be applicable to biphoton states formed in the process of spontaneous parametric down conversion. The conditions are found when the Lorentz-invariant mass is related directly with (proportional to) the Schmidt parameter $K\gg 1$ determining a high degree of entanglement of a biphoton state with respect to transverse wave vectors of emitted photons.
We compare the polynomial invariants for four qubits introduced by Luque and Thibon, PRA {\bf 67}, 042303 (2003), with optimized Bell inequalities and a combination of two qubit concurrences. It is shown for various parameter dependent states from different SLOCC classes that it is possible to measure a genuine 4-qubit entanglement with these polynomials.
We study entanglement dynamics between four qubits interacting through two isolated Jaynes-Cummings hamiltonians, via the entanglement measure based on the wedge product. We compare the results with similar results obtained using bipartite concurrence resulting in what is referred to as "entanglement sudden death". We find a natural entanglement invariant under evolution demonstrating that entanglement sudden death is caused by ignoring (tracing over) some of the system's degrees of freedom that become entangled through the interaction.
We investigate the dynamics of entanglement and nonlocality for multipartite quantum systems under collective dephasing. Using an exact and computable measure for genuine entanglement, we demonstrate the possibility of a non trivial phenomenon of time-invariant entanglement for multipartite quantum systems. We find that for four qubits, there exist quantum states, which are changing continously nevertheless their genuine entanglement remains constant. Based on our numerical results, we conjecture that there is no evidence of time-invariant entanglement for quantum states of three qubits. We po…
These are lecture notes from the 44th IFF Spring School "Quantum Information Processing" in Juelich, discussing applications of entanglement theory in condensed matter. The focus of the notes is on tensor network states, in particular Matrix Product States (MPS) and Projected Entangled Pair States (PEPS), which provide an efficient description of many-body states based on their entanglement structure; both numerical and analytical aspects of tensor networks are being covered. The notes close with a brief introduction to quantum complexity, which allows to assess the limitations to our ability …
Entanglement detection via third-order local invariants from randomized measurements
本研究通过随机测量计算三阶局部不变量并推导可分性判据，实现任意维数多体态的纠缠检测，以Werner态为例，将检测阈值从二阶的p>1/2改进至三阶的p>1/∛10。
There has been spectacular progress in the field of quantum information in recent decades. The development of this field highlights the importance of the role of entanglement in quantum computing, quantum teleportation and quantum cryptography. These notes serve to provide a gentle introduction to the entanglement of bipartite states. In these notes, we introduce Bell's theorem in the form derived by Clauser, Horne, Shimony and Holt. We discuss the Schmidt decomposition and the Peres-Horodecki criterion in the entanglement of pure and mixed bipartite states. Finally, we describe a teleportatio…
It has been suggested in arXiv:2004.00613 that in Dp-brane holography, entanglement in the target space of the D-brane Yang-Mills theory provides a precise notion of bulk entanglement in the gravity dual. We expand on this discussion by providing a gauge invariant characterization of operator sub-algebras corresponding to such entanglement. This is achieved by finding a projection operator which imposes a constraint characterizing the target space region of interest. By considering probe branes in the Coulomb branch we provide motivation for why the operator sub-algebras we consider are approp…
The entanglement between spins of a pair of particles may change because the spin and momentum become mixed when viewed by a moving observer [R.M. Gingrich and C. Adami, Phys. Rev. Lett. 89, 270402 (2002)]. In this paper, it is shown that, if the momenta are appropriately entangled, the entanglement between the spins of the Bell states can remain maximal when viewed by any moving observer. Further, we suggest a relativistic-invariant protocol for quantum communication, with which the non-relativistic quantum information theory could be invariantly applied to relativistic situations.
We present local invariants of multi-partite pure or mixed states, which can be easily calculated and have a straight-forward physical meaning. As an application, we derive a new entanglement criterion for arbitrary mixed states of $n$ parties. The new criterion is weaker than the partial transposition criterion but offers advantages for the study of multipartite systems. A straightforward generalization of these invariants allows for the construction of a complete set of observable polynomial invariants.
The main objective of the paper is to unveil an adequate mathematics hidden behind entanglement, that is Geometric Invariant Theory. More specifically relation between these two subjects can be described by the following theses. (i) Total variance of completely entangled state is maximal. (ii) This distinguishes the state as a minimal vector in its orbit under action of complexified dynamic group. (iii) An ultimate aim of Geometric Invariant Theory is a description of complex orbits and their minimal vectors. It suggests that noncompletely entangled states are just GIT semistable vectors.
The purpose of this short article is to build on the work of Ghirardi, Marinatto and Weber (Ghirardi, Marinatto & Weber 2002; Ghirardi & Marinatto 2003, 2004, 2005) and Ladyman, Linnebo and Bigaj (2013), in supporting a redefinition of entanglement for "indistinguishable" systems, particularly fermions. According to the proposal, non-separability of the joint state is insufficient for entanglement. The re-definition is justified by its physical significance, as enshrined in three biconditionals whose analogues hold of "distinguishable" systems.
In this note a new method for computing the entanglement entropy of a CFT holographically is explored. It consists of finding a bulk background with a boundary metric that has the conical singularities needed to compute the entanglement entropy in the usual QFT definition. An explicit calculation is presented for d=2.
Kempe-Invariant Ordering of Three-Qubit Pairwise Entanglement under Local Depolarization
该研究推导了三量子比特系统中成对平方形成纠缠之和的紧全局边界，将三量子比特态映射为三维空间中的点，可解释动力学演化轨迹，并研究了马尔可夫噪声下成对纠缠的动力学行为。
In a recent paper [Phys. Rev. A 76, 032304(2007)], Li et al. proposed the definition of the residual entanglement for n qubits by means of the Stochastic local operations and classical communication. Here we argue that their definition is not suitable for the case of odd-n qubits.
We have reviewed the comment in [3], posted on arXiv.org concerning our recent work in [1]. We reply to the comment in this paper.
This short note describes a method to tackle the (bipartite) quantum separability problem. The method can be used for solving the separability problem in an experimental setting as well as in the purely mathematical setting. The idea is to invoke the following characterization of entangled states: A state is entangled if and only if there exists an entanglement witness that detects it. The method is basically a search for an entanglement witness that detects the given state.
Genuine Tripartite Entanglement Selects Gauge-Invariant Theories
该研究将极值纠缠的方法扩展至三方系统，发现真正三方纠缠可唯一识别胶子与引力子散射中对应规范不变、微分同胚不变的理论。
A scheme is proposed for the fast generation of three-dimensional entanglement between two atoms trapped in two cavities connected by a fiber via invariant-based shortcut to adiabatic passage. With the help of quantum Zeno dynamics, the technique of invariant-based shortcut to adiabatic passage is applied for the generation of two-atom three-dimensional entanglement. The numerical simulation results show that, within a short time, the scheme has a high fidelity and is robust against the decoherence caused by the atomic spontaneous emission, photon leakage, and the variations in the parameters …
We show how to simplify the computation of the entanglement of formation and the relative entropy of entanglement for states, which are invariant under a group of local symmetries. For several examples of groups we characterize the state spaces, which are invariant under these groups. For specific examples we calculate the entanglement measures. In particular, we derive an explicit formula for the entanglement of formation for UU-invariant states, and we find a counterexample to the additivity conjecture for the relative entropy of entanglement.
We describe a direct method to determine the negativity of an arbitrary two-qubit state in experiments. The method is derived by analyzing the relation between the purity, negativity, and a universal entanglement witness for two-qubit entanglement. We show how the negativity of a two-qubit state can be calculated from just three experimentally accessible moments of the partially transposed density matrix of a two-photon state. Moreover, we show that the negativity can be given as a function of only six invariants, which are linear combinations of nine invariants from the complete set of 21 fun…
We examine whether it is possible for one-dimensional translationally-invariant Hamiltonians to have ground states with a high degree of entanglement. We present a family of translationally invariant Hamiltonians {H_n} for the infinite chain. The spectral gap of H_n is Omega(1/poly(n)). Moreover, for any state in the ground space of H_n and any m, there are regions of size m with entanglement entropy Omega(min{m,n}). A similar construction yields translationally-invariant Hamiltonians for finite chains that have unique ground states exhibiting high entanglement. The area law proven by Hastings…
We investigate the dynamics of quantum entanglement and more general quantum correlations quantified respectively via negativity and local quantum uncertainty for two qubit systems undergoing Markovian collective dephasing. Focusing on a two-parameter family of initial two-qubit density matrices, we study the relation of the emergence of the curious phenomenon of time-invariant entanglement and the dynamical behavior of local quantum uncertainty. Developing an illustrative geometric approach, we demonstrate the existence of distinct regions of quantum entanglement for the considered initial st…
We point out that a geometric measure of quantum entanglement is related to the matrix permanent when restricted to permutation invariant states. This connection allows us to interpret the permanent as an angle between vectors. By employing a recently introduced permanent inequality by Carlen, Loss and Lieb, we can prove explicit formulas of the geometric measure for permutation invariant basis states in a simple way.
We consider two spacelike separated Dirac particles and construct five invariants under the spinor representations of the local proper orthochronous Lorentz groups. All of the constructed Lorentz invariants are identically zero for product states. The behaviour of the Lorentz invariants under local unitary evolutions that act unitarily on any subspace with fixed particle momenta is studied. All of the Lorentz invariants have invariant absolute values on such subspaces if the evolutions are generated by local zero-mass Dirac Hamiltonians. Some of them also for the case of nonzero-mass. Therefor…
In recent years there has been a great deal of focus on a globe-spanning quantum network, including linked satellites for applications ranging from quantum key distribution to distributed sensors and clocks. In many of these schemes, relativistic transformations may have deleterious effects on the purity of the distributed entangled pairs. This becomes particularly important for the application of distributed clocks. In this paper, we have developed a Lorentz invariant entanglement distribution protocol that completely removes the effects due to the relative motions of the satellites.
A note on lower bounds of concurrence for arbitrary dimensional bipartite quantum states
该研究针对量子纠缠量化问题，基于两体量子态关联矩阵提出concurrence与2-concurrence的解析下界，相比现有下界对大纯度态的纠缠估计更优。
We show that a positive homogeneous function that is invariant under determinant-1 stochastic local operations and classical communication (SLOCC) transformations defines an N-qubit entanglement monotone if and only if the homogeneous degree is not larger than four. In particular this implies that any power larger than one of the well-known N-tangle (N > 2) is not an entanglement monotone anymore. We then describe a common basis and formalism for the N-tangle and other known invariant polynomials of degree four. This allows us to elucidate the relation of the four-qubit invariants defined by L…
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