A. A. Klyachko, A. S. Shumovsky
A definition of entanglement in terms of local measurements is discussed. Viz, the maximum entanglement corresponds to the states that cause the highest level of quantum fluctuations in all local measurements determined by the dynamic symmetry group of the system. A number of examples illustrating this definition is considered.
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It is shown that spontaneous symmetry breaking does not modify the ground-state entanglement of two spins, as defined by the concurrence, in the XXZ- and the transverse field Ising-chain. Correlation function inequalities, valid in any dimensions for these models, are presented outlining the regimes where entanglement is unaffected by spontaneous symmetry breaking.
In this lectures I explain a connection between geometric invariant theory and entanglement, and give a number of examples how this approach works.
We investigate the relationship between multipartite entanglement and symmetry, focusing on permutation symmetric states. We use the Majorana representation, where these states correspond to points on a sphere. Symmetry of the representation under rotation is equivalent to symmetry of the states under products of local unitaries. The geometric measure of entanglement is thus phrased entirely as a geometric optimisation, and a condition for the equivalence of entanglement measures written in terms of point symmetries. Finally we see that different symmetries of the states correspond to differen…
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 study the separability of permutationally symmetric quantum states. We show that for bipartite symmetric systems most of the relevant entanglement criteria coincide. However, we provide a method to generate examples of bound entangled states in symmetric systems, for the bipartite and the multipartite case. These states shed some new light on the nature of bound entanglement.
We study the temporal evolution of entanglement pertaining to two qubits interacting with a thermal bath. In particular we consider the simplest nontrivial spin bath models where symmetry breaking occurs and treat them by mean field approximation. We analytically find decoherence free entangled states as well as entangled states with an exponential decay of the quantum correlation at finite temperature.
Ichikawa et al. [Phys. Rev. A 78, 052105 (2008)] showed that exchange symmetry gives rise to simple characterization of whether multipartite pure quantum states being either globally entangled or fully separable. In this Brief Report, we provide a simple alternative approach and some extension to their conclusions.
A relationship between a recently introduced multipartite entanglement measure, state mixedness, and spin-flip symmetry is established for any finite number of qubits. It is also shown that, within those classes of states invariant under the spin-flip transformation, there is a complementarity relation between multipartite entanglement and mixedness. A number of example classes of multiple-qubit systems are studied in light of this relationship.
We study finite-dimensional product Hilbert spaces, coupled spin systems, entanglement and energy level crossing. The Hamilton operators are based on the Pauli group. We show that swapping the interacting term can lead from unentangled eigenstates to entangled eigenstates and from an energy spectrum with energy level crossing to avoided energy level crossing.
Recently, a new type of symmetry for three-qubit quantum states was introduced, the so-called Greenberger-Horne-Zeilinger (GHZ) symmetry. It includes the operations which leave the three-qubit standard GHZ state unchanged. This symmetry is powerful as it yields families of mixed states that are, on the one hand, complex enough from the physics point of view and, on the other hand, simple enough mathematically so that their properties can be characterized analytically. We show that by using the properties of GHZ-symmetric states it is straightforward to derive optimal witnesses for three-qubit …
So-called direct measurements of entanglement are collective measurements on multiple copies of a (bipartite or multipartite) quantum system that directly provide one a value for some entanglement measure, such as the concurrence for bipartite states. Multiple copies are needed since the entanglement of a mixed state is not a linear function of the density matrix. Unfortunately, so far all experimental implementations of direct measurements made unverified assumptions about the form of the states, and, therefore, do not qualify as entanglement verification tests. I discuss how a direct measure…
We study the significance of Photon/Phonons bunching and antibunching on the dynamics of the quantum entanglement in the presence of coupled PT-symmetry systems with balanced/unbalanced gain and loss. We suggest a hybrid electromechanical system to realize a strong and tunable coupling between a Coplanar-Waveguide (CPW) microwave cavity and a nanomechanical resonator (NAMR) via a superconducting Transmon qubit. The hybrid electromechanical system consists of a non-hermitian Hamiltonian with balanced/unbalanced gain and loss. The interplay between the quantum entanglement and the -symmetry syst…
Entanglement in bipartite systems with symmetry: coupled chaotic kicked Bose-Hubbard systems
本文以耦合混沌踢玻色-哈伯德系统为研究对象，结合对称性分辨纠缠与量子混沌微扰理论，揭示了二分多体系统中由对称性破缺引发的普适纠缠相变及本征态局域化特性。
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…
Bath engineering, which utilizes coupling to lossy modes in a quantum system to generate non-trivial steady states, is a tantalizing alternative to gate- and measurement-based quantum science. Here, we demonstrate dissipative stabilization of entanglement between two superconducting transmon qubits in a symmetry-selective manner. We utilize the engineered symmetries of the dissipative environment to stabilize a target Bell state; we further demonstrate suppression of the Bell state of opposite symmetry due to parity selection rules. This implementation is resource-efficient, achieves a steady-…
Entanglement properties of a basic set of eight entangled three particle pure states possessing certain permutation symmetries are studied. They fall into four sets of two entangled states, differing in their patterns of robustness to entanglement when one of the states is lost. These features are related to the permutation symmetries of the spin states of the three particles and their corresponding marginal two-particle states. It is interesting to note that the eight entangled three-qubit states discussed here are eigenstates of a three-spin Heisenberg Hamiltonian.
We investigate the entanglement properties of multi-mode Gaussian states, which have some symmetry with respect to the ordering of the modes. We show how the symmetry constraints the entanglement between two modes of the system. In particular, we determine the maximal entanglement of formation that can be achieved in symmetric graphs like chains, 2d and 3d lattices, mean field models and the platonic solids. The maximal entanglement is always attained for the ground state of a particular quadratic Hamiltonian. The latter thus yields the maximal entanglement among all quadratic Hamiltonians hav…
Squeezing and Entanglement Dynamics in Phase-Sensitive Non-Hermitian Systems
本文构建相位敏感非厄米波混频模型，揭示其同时具备单模APT与双模正交PT对称性，并发现APT相变伴随量子纠缠从无到有。
A simple scheme is presented for achieving effectively maximal pure-state entanglement between non-interacting atoms through purely collective decay and controlled symmetry breaking. The scheme requires no measurements or feedback or even knowledge of the initial states of the atoms. It relies on breaking the symmetry of the system Hamiltonian to ensure the existence of a unique attractive steady state and minimal control to achieve almost perfect overlap of this steady state with the maximally entangled singlet state. We demonstrate how our scheme can be implemented for two qubits encoded in …
Within a special multi-coin quantum walk scheme we analyze the effect of the entanglement of the initial coin state. For states with a special entanglement structure it is shown that this entanglement can be meausured with the mean value of the walk, which depends on the i-concurrence of the initial coin state. Further on the entanglement evolution is investigated and it is shown that the symmetry of the probability distribution is reflected by the symmetry of the entanglement distribution.
Symmetry-resolved entanglement is a useful tool for characterizing symmetry-protected topological states. In two dimensions, their entanglement spectra are described by conformal field theories but the symmetry resolution is largely unexplored. However, addressing this problem numerically requires system sizes beyond the reach of exact diagonalization. Here, we develop tensor network methods that can access much larger systems and determine universal and nonuniversal features in their entanglement. Specifically, we construct one-dimensional matrix product operators that encapsulate all the ent…
Symmetry protected topological phases (SPTs) have universal degeneracies in the entanglement spectrum in one dimension (1D). Here, we formulate this phenomenon in the framework of symmetry-resolved entanglement (SRE) using cohomology theory. We develop a general approach to compute entanglement measures of SPTs in any dimension and specifically SRE via a discrete path integral on multi-sheet Riemann surfaces with generalized defects. The resulting path integral is expressed in terms of group cocycles describing the topological actions of SPTs. Their cohomology classification allows to identify…
Revealing high-dimensional entanglement through symmetry
提出一种基于粒子交换对称性的线性光学方案，利用两个二分对称性测量即可认证时间仓编码态的高维纠缠并给出维数下界，实验证明即使在远低于探测器时间分辨率的尺度下仍有效。
Entanglement of multipartite systems is studied based on exchange symmetry under the permutation group S_N. With the observation that symmetric property under the exchange of two constituent states and their separability are intimately linked, we show that anti-symmetric (fermionic) states are necessarily globally entangled, while symmetric (bosonic) states are either globally entangled or fully separable and possess essentially identical states in all the constituent systems. It is also shown that there cannot exist a fully separable state which is orthogonal to all symmetric states, and that…
Efficient certification of time-reversal symmetry requires entanglement
本文提出利用纠缠高效认证时间反演对称性，证明最大纠缠探针和SWAP测量可将查询复杂度降至常数，建立了纠缠与时间反演对称性的定量联系。
In this work we offer an approach to protect the entanglement based on the anti-symmetric property of the hamiltonian. Our main objective is to protect the entanglement of a given initial three-qubit state which is governed by hamiltonian of a three-spin Ising chain in site-dependent transverse fields. We show that according to anti-symmetric property of the hamiltonian with respect to some operators mimicking the time reversal operator, the dynamics of the system can be effectively reversed. It equips us to control the dynamics of the system. The control procedure is implemented as a sequence…
We study two-spin entanglement and order parameter fluctuations as a function of the system size in the XY model in a transverse field and in the isotropic XXX model. Both models are characterized by the occurrence of ground state degeneracy also when systems of finite size are considered. This is always true for the XXX model, but only at the factorizing field for the XY model. We study the size dependence of symmetric states, which, in the presence of degeneracy, can be expanded as a linear combination of broken symmetry states. We show that, while the XY model looses its quantum superpositi…
We propose a method for the realization of the two-qubit quantum Fourier transform (QFT) using a Hamiltonian which possesses the circulant symmetry. Importantly, the eigenvectors of the circulant matrices are the Fourier modes and do not depend on the magnitude of the Hamiltonian elements as long as the circulant symmetry is preserved. The QFT implementation relies on the adiabatic transition from each of the spin product states to the respective quantum Fourier superposition states. We show that in ion traps one can obtain a Hamiltonian with the circulant symmetry by tuning the spin-spin inte…
Permutation symmetries of multipartite quantum states are defined only when the constituent subsystems are of equal dimensions. In this work we extend this notion of permutation symmetry to heterogeneous systems, that is, systems composed of subsystems having unequal dimensions. Given a tensor product space of $k$ subsystems (of arbitrary dimensions) and a permutation operation $σ$ over $k$ symbols, these states are such that they have identical decompositions (up to an overall phase) in the given tensor product space and the tensor product space obtained by the permuting the subsystems by $σ$…
Here we evaluate the many-body entanglement properties of a generalized SU(n) valence bond solid state on a chain. Our results follow from a derivation of the transfer matrix of the system which, in combination with symmetry properties, allows for a new, elegant and straightforward evaluation of different entanglement measures. In particular, the geometric entanglement per block, correlation length, von Neumann and Rényi entropies of a block, localizable entanglement and entanglement length are obtained in a very simple way. All our results are in agreement with previous derivations for the SU…
We investigate the change of entanglement of photons due to propagation. We find that post-selected entanglement in general varies by propagation and, as a consequence, states with maximum bi- and tri-partite entanglement can be generated from propagation of unentangled photons. We generalize the results to n photons and show that entangled states with permutation symmetry can be generated from propagation of unentangled states. Generation of n-photon GHZ states is discussed as an example of a class of states with the desired symmetry.
The paper contains a brief review of an approach to quantum entanglement based on analysis of dynamic symmetry of systems and quantum uncertainties, accompanying the measurement of mean value of certain basic observables. The latter are defined in terms of the orthogonal basis of Lie algebra, corresponding to the dynamic symmetry group. We discuss the relativity of entanglement with respect to the choice of basic observables and a way of stabilization of robust entanglement in physical systems.
Symmetry Resolved Multipartite Entanglement Entropy
本文对可分辨粒子系统的全局纠缠$Q$进行对称性解析，发现Haar随机态系综满足等分且其首阶行为和有限尺寸修正呈幂律缩放，并探讨了该结果的理论意义与实验验证方法。
We study the effect of symmetry breaking in a quantum phase transition on pairwise entanglement in spin-1/2 models. We give a set of conditions on correlation functions a model has to meet in order to keep the pairwise entanglement unchanged by a parity symmetry breaking. It turns out that all mean-field solvable models do meet this requirement, whereas the presence of strong correlations leads to a violation of this condition. This results in an order-induced enhancement of entanglement, and we report on two examples where this takes place.
Symmetry-Resolved Entanglement: General considerations, calculation from correlation functions, and bounds for symmetry-protected topological phases
本文研究具有粒子数守恒系统的对称性分辨von-Neumann纠缠熵，提出从关联函数计算纠缠分量的方法，引入majorization推导手性和Cn对称拓扑相中粒子数与构型熵的下界，并改进了部分已知的纠缠熵下界。
Symmetry is an important property of quantum mechanical systems which may dramatically influence their behavior in and out of equilibrium. In this paper, we study the effect of symmetry on tripartite entanglement properties of typical states in symmetric sectors of Hilbert space. In particular, we consider Abelian symmetries and derive an explicit expression for the logarithmic entanglement negativity of systems with $\mathbb{Z}_N$ and $U(1)$ symmetry groups. To this end, we develop a diagrammatic method to incorporate partial transpose within the random matrix theory of symmetric states and f…
We analyze the bipartite and multipartite entanglement for the ground state of the one-dimensional XY model in a transverse magnetic field in the thermodynamical limit. We explicitly take into account the spontaneous symmetry breaking in order to explore the relation between entanglement and quantum phase transitions. As a result we show that while both bipartite and multipartite entanglement can be enhanced by spontaneous symmetry breaking deep into the ferromagnetic phase, only the latter is affected by it in the vicinity of the critical point. This result adds to the evidence that multipart…
Entanglement properties are routinely used to characterize phases of quantum matter in theoretical computations. For example the spectrum of the reduced density matrix, or so-called "entanglement spectrum", has become a widely used diagnostic for universal topological properties of quantum phases. However, while being convenient to calculate theoretically, it is notoriously hard to measure in experiments. Here we use the IBM quantum computer to make the first ever measurement of the entanglement spectrum of a symmetry-protected topological state. We are able to distinguish its entanglement spe…
When a quantum pure state is drawn uniformly at random from a Hilbert space, the state is typically highly entangled. This property of a random state is known as generic entanglement of quantum states and has been long investigated from many perspectives, ranging from the black hole science to quantum information science. In this paper, we address the question of how symmetry of quantum states changes the properties of generic entanglement. More specifically, we study bipartite entanglement entropy of a quantum state that is drawn uniformly at random from an invariant subspace of a given symme…
Symmetry enforces entanglement at high temperatures
该研究针对高温下多体量子系统的纠缠突然死亡问题，证明具有在位阿贝尔对称性的强/弱对称热态在任意高温下仍能维持非零纠缠，费米子吉布斯态也存在持续纠缠，揭示了全局对称性对热平衡下量子纠缠的保护作用。
We calculate the hybrid entanglement entropy between coin and walker degrees of freedom in a non-unitary quantum walk. The model possesses a joint parity and time-reversal symmetry or PT-symmetry and supports topological phases when this symmetry is unbroken by its eigenstates. An asymptotic analysis at long times reveals that the quantum walk can indefinitely sustain hybrid entanglement in the unbroken symmetry phase even when gain and loss mechanisms are present. However, when the gain-loss strength is too large, the PT-symmetry of the model is spontaneously broken and entanglement vanishes.…
We investigate the global-symmetry projections applied to the tensor network states from the view point of the entanglement entropy and the mutual information. The projections to the translational invariant space and to the total-$S^z$-zero space give logarithmically increasing mutual information with respect to the system size. In the anti-ferromagnetic $S=1/2$ Heisenberg chain and lattice, the optimized energies become accurate numerically by using variational states of the projected tensor network states, because the projections reflecting symmetries of the ground states generate quantum en…
Total and Symmetry resolved Entanglement spectra in some Fermionic CFTs from the BCFT approach
本文使用BCFT方法研究了某些2维费米子CFT中单个区间的全纠缠谱与对称性分辨纠缠谱，发现模不变Dirac费米子的全纠缠谱与对偶半径处紧致玻色子结果匹配，且U(1)对称性的特征loglog(ℓ/ε)项在对称性分辨纠缠谱中与M成正比。
Crossing Symmetry and Entanglement
本文针对含SU(N)全局对称性的定域量子场论2→2散射过程，将散射振幅重构为量子操作，证明量子门重耦合关系源于振幅交叉性质，得出此类相互作用场论必在至少一个散射通道生成纠缠的结论。
We observe that the many-body eigenstates of any quadratic, fermionic Hamiltonian with sublattice symmetry have quantized entanglement entropies between the sublattices: the entanglement comes in multiple singlets. Moreover, such systems always have a ground state that is maximally entangled between the two sublattices. In fact we also show that under the same assumptions there always exists a (potentially distinct) basis of energy eigenstates that do not conserve the particle number in which each energy eigenstate is maximally entangled between the sublattices. No additional properties, such …
We study the entanglement structure of symmetry-protected topological (SPT) phases from an operational point of view by considering entanglement distillation in the presence of symmetries. We demonstrate that non-trivial SPT phases in one-dimension necessarily contain some entanglement which is inaccessible if the symmetry is enforced. More precisely, we consider the setting of local operations and classical communication (LOCC) where the local operations commute with a global onsite symmetry group $G$, which we call $G$-LOCC, and we define the inaccessible entanglement $E_{inacc}$ as the enta…
Symmetry classification of typical quantum entanglement
本文基于时间反演、电荷共轭和手征变换的十重对称类，对自由费米子典型量子纠缠进行全面分类，发现平均纠缠熵的体积律贡献不受对称性影响，而常数项具有十重普适值，并阐明了对称性与纠缠的相互作用及对称性增强的量子混沌表征。
One-dimensional valence bond solid (VBS) states represent the simplest symmetry protected topological phases. We show that their ground state entanglement spectrum contains both topological and non-topological structures. For the $SO(3)$ symmetric VBS states with odd-integer spins, the two-fold topological degeneracy is associated with an underlying $Z_{2}\times Z_{2}$ symmetry that protects the corresponding topological phase. In general, for the $SO(2S+1)$ symmetric VBS states with integer spins $S$, the corresponding protecting symmetry is identified as the $(Z_{2}\times Z_{2})^{S}$ symmetr…
Entanglement Resolution with Respect to Conformal Symmetry
本研究在共形场论框架下针对共形族解析纠缠，揭示领头阶与量子维数相关、全阶依赖零向量的规律，给出均分判据，证明拓扑纠缠熵可对称分辨Affleck-Ludwig边界熵，并分析构型与涨落熵。
A universal formula for the entanglement asymmetry of matrix product states
本文提出了一种适用于矩阵乘积态的纠缠不对称性通用公式，揭示了其与对称破缺模式的关系，为研究量子相变提供了新工具。
The excess entanglement resulting from exciting a finite number of quasiparticles above the ground state of a free integrable quantum field theory has been investigated quite extensively in the literature. It has been found that it takes a very simple form, depending only on the number of excitations and their statistics. There is now mounting evidence that such formulae also apply to interacting and even higher-dimensional quantum theories. In this paper we study the entanglement content of such zero-density excited states focusing on the symmetry resolved entanglement, that is on 1+1D quantu…
Symmetry enforced entanglement in maximally mixed states
本文研究强对称保迹量子信道的最大混合稳态，证明不变扇区 MMIS 的生成纠缠与蒸馏纠缠可精确计算，并揭示 Abelian 对称性下可分、非 Abelian 对称性下纠缠且连续情形随 N 对数增长。
We present an example where Spontaneous Symmetry Breaking may effect not only the behavior of the entanglement at Quantum Phase Transitions, but also the origin of the non-analyticity. In particular, in the XXZ model, we study the non analyticities in the concurrence between two spins, which was claimed to be accidental, since it had its origin in the optimization involved in the concurrence definition. We show that when one takes in account the effect of the Spontaneous Symmetry Breaking, even tough the values of the entanglement measure does not change, the origin the the non-analytical beha…
Entanglement asymmetry as a probe of symmetry breaking
本文引入纠缠不对称性这一子系统层面的对称性破缺度量，通过自旋链量子淬火发现对称性初始破坏越严重恢复越快，揭示了广泛的量子 Mpemba 效应。
Entanglement-enabled symmetry-breaking orders
本文提出纠缠使能的对称性破缺序，它无法由张量积态实现，并给出判据及一维精确可解模型实例，其基态具有离散对称性破缺序和无能隙边缘模，并推广至连续对称性破缺情形。
We use an entanglement measure that respects the superselection of particle number to study the non-local properties of symmetry-protected topological edge states. Considering half-filled M-leg Su-Schrieffer-Heeger (SSH) ladders as an example, we show that the topological properties and the operational entanglement extractable from the boundaries are intimately connected. Topological phases with at least two filled edge states have the potential to realize genuine, non-bipartite, many-body entanglement which can be transferred to a quantum register. The entanglement is extractable when the fil…
It is well known that the matrix product state (MPS) description of a gapped ground state with a global on-site symmetry can exhibit "symmetry fractionalization". Namely, even though the symmetry acts as a linear representation on the physical degrees of freedom, the MPS matrices---which act on some virtual degrees of freedom---can transform under a projective representation. This was instrumental in classifying gapped symmetry protected phases that manifest in one dimensional quantum many-body systems. Here we consider the multi-scale entanglement renormalization ansatz (MERA) description of …
Entanglement asymmetry and symmetry defects in boundary conformal field theory
本文利用二维边界共形场论，通过纠缠不对称性研究边界条件对全局对称性破缺的影响，计算了领头阶及次领头阶修正，并扩展镜像方法处理含可逆对称性缺陷的动力学淬火情形。
We discover a simple factorization law describing how multipartite entanglement of a composite quantum system evolves when one of the subsystems undergoes an arbitrary physical process. This multipartite entanglement decay is determined uniquely by a single factor we call the entanglement resilient factor (ERF). Since the ERF is a function of the quantum channel alone, we find that multipartite entanglement evolves in exactly the same way as bipartite (two qudits) entanglement. For the two qubits case, our factorization law reduces to the main result of Nature Physics 4, 99 (2008). In addition…
We investigate the symmetry resolution of entanglement in the presence of long-range couplings. To this end, we study the symmetry-resolved entanglement entropy in the ground state of a fermionic chain that has dimerised long-range hoppings with power-like decaying amplitude -- a long-range generalisation of the Su-Schrieffer-Heeger model. This is a system that preserves the number of particles. The entropy of each symmetry sector is calculated via the charged moments of the reduced density matrix. We exploit some recent results on block Toeplitz determinants generated by a discontinuous symbo…
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