Rob Clifton, Brian Hepburn, Christian Wuthrich
In support of a recent conjecture by Nielsen (1999), we prove that the phenomena of 'incomparable entanglement'--whereby, neither member of a pair of pure entangled states can be transformed into the other via local operations and classical communication (LOCC)--is a generic feature when the states at issue live in an infinite-dimensional Hilbert space.
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Incomparability of pure bipartite entangled states under deterministic LOCC is a very strange phenomena. We find two possible ways of getting our desired pure entangled state which is incomparable with the given input state, by collective LOCC with certainty. The first one is by providing some pure entanglement through the lower dimensional maximally-entangled states or using further less amount of entanglement and the next one is by collective operation on two pairs which are individually incomparable. It is quite surprising that we are able to achieve maximally entangled states of any Schmid…
The work is intended to represent some interesting and apparently peculiar features of entangled system in both pure as well as mixed states level. In the pure state level, we are largely concerned about the existence and characteristics of bipartite incomparable pair of states. We have found some schemes for local transformation of one pure bipartite state to another, incomparable in nature, by exploiting free unpreserved entanglement. Also, we observed that different incomparable pair of states in $3\times 3$ may co-operate with each other to cross the barrier of incomparability. Our next go…
For states in infinite dimensional Hilbert spaces entanglement quantities like the entanglement of distillation can become infinite. This leads naturally to the question, whether one system in such an infinitely entangled state can serve as a resource for tasks like the teleportation of arbitrarily many qubits. We show that appropriate states cannot be obtained by density operators in an infinite dimensional Hilbert space. However, using techniques for the description of infinitely many degrees of freedom from field theory and statistical mechanics, such states can nevertheless be constructed …
Embezzlement as a "Self-Test" for Infinite Copies of Entangled States
该研究通过证明催化剂态必须包含无限多个目标纠缠态的副本，揭示了纠缠贪污现象背后的算子代数结构，并为无限维量子系统中的态认证提供了新工具。
An example is given of an interaction that produces an infinite amount of entanglement in an infinitely short time, but only a finite amount in longer times. The interaction arises from a standard Kerr nonlinearity and a 50/50 beamsplitter, and the initial state is a coherent state. For certain finite interaction times multi-dimensional generalizations of entangled coherent states are generated, for which we construct a teleportation protocol. Similarities between probabilistic teleportation and unambiguous state discrimination are pointed out.
By introducing the concept of $ε$-convertibility, we extend Nielsen's and Vidal's theorems to the entanglement transformation of infinite-dimensional systems. Using an infinite-dimensional version of Vidal's theorem we derive a new stochastic-LOCC (SLOCC) monotone which can be considered as an extension of the Schmidt rank. We show that states with polynomially-damped Schmidt coefficients belong to a higher rank of entanglement class in terms of SLOCC convertibility. For the case of Hilbert spaces of countable, but infinite dimensionality, we show that there are actually an uncountable number …
By using the thermo entangled state representation we solve the master equation for a dissipative cavity with Kerr medium to obtain density operators' infinite operator-sum representation}$ρ(t) =\sum_{m,n,l=0}^{\infty}M_{m,n,l}ρ_{0}\mathcal{M}_{m,n,l}^{\dagger}.$ It is noticeable that}$M_{m,n,l}$ is not hermite conjugate to $\mathcal{M}_{m,n,l}^{\dagger}$, nevertheless the normalization}$\sum_{m,n,l=0}^{\infty}\mathcal{M}_{nm,,l}^{\dagger}M_{m,n,l}=1$ still holds}, i.e., they are trace-preserving in a general sense. This example may stimulate further studying if general superoperator theory ne…
Decoherence-free interaction and maximally entangled state generation in giant-atom semi-infinite waveguide systems
本文研究了在半无限波导系统中，通过编织和嵌套配置实现巨原子的去相干自由相互作用，并生成最大纠缠态。
We show that the global infinitesimal change in the multi-particle pure product state gives rise to an entangled state. This suggests that even if there is no interaction present between the subsystems, i.e., at each time instant the state is non-entangled, the tangent vector is typically entangled. Since the tangent space vectors tell the state-space vectors how to change this implies that quantum entanglement is necessary for motion or change in general. This is truly a `hidden power' of quantum entanglement. During quantum computation even though at each time instant the state is not entang…
We demonstrate how to map out the phase diagram of a two dimensional quantum many body system with no prior physical knowledge by applying deep \textit{anomaly detection} to ground states from infinite projected entangled pair state simulations. As a benchmark, the phase diagram of the 2D frustrated bilayer Heisenberg model is analyzed, which exhibits a second-order and two first-order quantum phase transitions. We show that in order to get a good qualitative picture of the transition lines, it suffices to use data from the cost-efficient simple update optimization. Results are further improve…
We obtain the necessary and sufficient separability and distillability conditions of mixtures of a maximally entangled state and the completely separable state in relativistic setting. In an inertial frame we study the entanglement under Wigner rotations induced by Lorentz transformations. We also investigate the mixed state entanglement of scalar and Dirac fields as seen by two relatively accelerated observers. For scalar field we show that in infinite acceleration limit the state has no longer distillable entanglement. For dirac field the entanglement in the infinite acceleration limit is fi…
Local unitary equivalence of absolutely maximally entangled states constructed from orthogonal arrays
本研究利用不变量分析由正交阵列构造的绝对最大纠缠态的局部酉等价性，证明存在无限多个局部酉不等价的三方和五方AME态。
It is shown that accelerated projective measurements on the vacuum of a free Dirac spinor field results in an entangled state for an inertial observer. The physical mechanism at work is the Davies-Unruh effect. The produced state is always entangled and its entanglement increases as a function of the acceleration, reaching maximal entanglement in the asymptotic limit of infinite acceleration where Bell states are produced.
We present calculations of the time-evolution of the driven-dissipative XYZ model using the infinite Projected Entangled Pair Operator (iPEPO) method, introduced by [A. Kshetrimayum, H. Weimer and R. Orús, Nat. Commun. 8, 1291 (2017)]. We explore the conditions under which this approach reaches a steady state. In particular, we study the conditions where apparently converged calculations may become unstable with increasing bond dimension of the tensor-network ansatz. We discuss how more reliable results could be obtained.
Consider an infinite collection of qubits arranged in a line, such that every pair of nearest neighbors is entangled: an "entangled chain." In this paper we consider entangled chains with translational invariance and ask how large one can make the nearest neighbor entanglement. We find that it is possible to achieve an entanglement of formation equal to 0.285 ebits between each pair of nearest neighbors, and that this is the best one can do under certain assumptions about the state of the chain.
We present a scheme to perform an iterative variational optimization with infinite projected entangled-pair states (iPEPS), a tensor network ansatz for a two-dimensional wave function in the thermodynamic limit, to compute the ground state of a local Hamiltonian. The method is based on a systematic summation of Hamiltonian contributions using the corner transfer-matrix method. Benchmark results for challenging problems are presented, including the 2D Heisenberg model, the Shastry-Sutherland model, and the t-J model, which show that the variational scheme yields considerably more accurate resul…
We generalize the method introduced in Phys. Rev. B 101, 041108 (2020) of extracting information about topological order from the ground state of a strongly correlated two-dimensional system represented by an infinite projected entangled pair state (iPEPS) to non-Abelian topological order. When wrapped on a torus the unique iPEPS becomes a superposition of degenerate and locally indistinguishable ground states. We find numerically symmetries of the iPEPS, represented by infinite matrix product operators (MPO), and their fusion rules. The rules tell us how to combine the symmetries into project…
The infinite Projected Entangled-Pair State (iPEPS) algorithm is one of the most efficient techniques for studying the ground-state properties of two-dimensional quantum lattice Hamiltonians in the thermodynamic limit. Here, we show how the algorithm can be adapted to explore nearest-neighbor local Hamiltonians on the ruby and triangle-honeycomb lattices, using the Corner Transfer Matrix (CTM) renormalization group for 2D tensor network contraction. Additionally, we show how the CTM method can be used to calculate the ground state fidelity per lattice site and the boundary density operator and…
We present a much simplified version of the CGLMP inequality for the 2 x 2 x d Bell scenario. Numerical maximization of the violation of this inequality over all states and measurements suggests that the optimal state is far from maximally entangled, while the best measurements are the same as conjectured best measurements for the maximally entangled state. For very large values of d the inequality seems to reach its minimal value given by the probability constraints. This gives numerical evidence for a tight quantum Bell inequality (or generalized Csirelson inequality) for the 2 x 2 x inf sce…
We present a method of extracting information about the topological order from the ground state of a strongly correlated two-dimensional system computed with the infinite projected entangled pair state (iPEPS). For topologically ordered systems, the iPEPS wrapped on a torus becomes a superposition of degenerate, locally indistinguishable ground states. Projectors in the form of infinite matrix product operators (iMPO) onto states with well-defined anyon flux are used to compute topological $S$ and $T$ matrices (encoding mutual- and self-statistics of emergent anyons). The algorithm is shown to…
The quantum teleportation plays an important role in quantum information process, in this sense, the quantum entanglement properties involving an infinite chain structure is quite remarkable because real materials could be well represented by an infinite chain. We study the teleportation of an entangled state through a couple of quantum channels, composed by Heisenberg dimers in an infinite Ising-Heisenberg diamond chain, the couple of chains are considered sufciently far away from each other to be ignored the any interaction between them . To teleporting a couple of qubits through the quantum…
We propose an improved approach to carry out the imaginary time evolution of infinite projected entangled-pair states (iPEPS), especially for systems with criticality. A cyclic optimal truncation is introduced to update the tensors along a closed loop, aiming to remove the redundant internal correlations. We demonstrate the algorithm by considering an elaborate evolution based on simple update on a small plaquette. This scheme can also be applied to a full update strategy. We demonstrate their performances on simulating the ground states of the spin-$1/2$ anti-ferromagnetic Heisenberg model an…
We address the question: Given an arbitrary initial state and a general physical interaction what is the minimum time for reaching a target entangled state? We show that the minimum time is inversely proportional to the quantum mechanical uncertainty in the non-local Hamiltonian. We find that the presence of initial entanglement helps to minimize the waiting time. Furthermore, we find that in a bi-local rotating frame the entangling capability is actually a geometric quantity. We give an universal bound for the time average of entanglement rate for general quantum systems. The time average of …
We investigate entanglement measures in the infinite-dimensional regime. First, we discuss the peculiarities that may occur if the Hilbert space of a bi-partite system is infinite-dimensional, most notably the fact that the set of states with infinite entropy of entanglement is trace-norm dense in state space, implying that in any neighborhood of every product state there lies an arbitrarily strongly entangled state. The starting point for a clarification of this counterintuitive property is the observation that if one imposes the natural and physically reasonable constraint that the mean ener…
We present an approach to identify topological order based on unbiased infinite projected entangled-pair states (iPEPS) simulations, i.e. where we do not impose a virtual symmetry on the tensors during the optimization of the tensor network ansatz. As an example we consider the ground state of the toric code model in a magnetic field exhibiting $Z_2$ topological order. The optimization is done by an efficient energy minimization approach based on a summation of tensor environments to compute the gradient. We show that the optimized tensors, when brought into the right gauge, are approximately …
We study second order finite temperature phase transitions of the 2D quantum Ising and interacting honeycomb fermions models using infinite projected entangled pair states (iPEPS). We obtain an iPEPS thermal state representation by Variational Tensor Network Renormalization (VTNR). We find that at the critical temperature $T_c$ the iPEPS correlation length is finite for the computationally accessible values of the iPEPS bond dimension $D$. Motivated by this observation we investigate the application of Finite Correlation Length Scaling (FCLS), which has been previously used for iPEPS simulatio…
Simulating fermionic fractional Chern insulators with infinite projected entangled-pair states
本文通过变分优化U(1)对称的费米子无限投影纠缠对态，研究分数量子陈绝缘体相，发现临界键维数并利用体观测量和边缘纠缠谱表征该相。
A typical quantum state obeying the area law for entanglement on an infinite 2D lattice can be represented by a tensor network ansatz -- known as an infinite projected entangled pair state (iPEPS) -- with a finite bond dimension $D$. Its real/imaginary time evolution can be split into small time steps. An application of a time step generates a new iPEPS with a bond dimension $k$ times the original one. The new iPEPS does not make optimal use of its enlarged bond dimension $kD$, hence in principle it can be represented accurately by a more compact ansatz, favourably with the original $D$. In th…
Spectral functions with infinite projected entangled-pair states
本文扩展无限投影纠缠对态方法，通过实时演化与角转移矩阵重正化群计算非等时关联函数，实现谱函数的高效求解，并在二维横向场伊辛模型中验证了其准确性。
Completely determining the relationship between quantum correlation sets is a long-standing open problem, known as Tsirelson's problem. Following recent progress by Slofstra [arXiv:1606.03140 (2016), arXiv:1703.08618 (2017)] only two instances of the problem remain open. One of them is the question of whether the set of finite-dimensional quantum correlations is strictly contained in the set of infinite-dimensional ones (i.e. whether $\mathcal C_{q} \neq \mathcal C_{qs}$). The usual formulation of the question assumes finite question and answer sets. In this work, we show that, when one allows…
In a recent work by Maitra et al. (Phys. Rev. A, 2017), it was shown that the existing Quantum Private Query (QPQ) protocols fail to maintain the database security if the entangled states shared between Alice and Bob are not of a certain form. So it is necessary to certify the states a priori. In this regard, the local CHSH test was proposed. However, the proposed scheme works perfectly for the asymptotic case when we have infinite number of qubits. In this brief report, we upgrade the protocol for finite number of qubits and connect the sample size to the success probability of CHSH test. We …
Time evolution of an infinite 2D many body quantum lattice system can be described by the Suzuki-Trotter decomposition applied to the infinite projected entangled pair state (iPEPS). Each Trotter gate increases the bond dimension of the tensor network, $D$, that has to be truncated back in a way that minimizes a suitable error measure. This paper goes beyond simplified error measures -- like the one used in the full update (FU), the simple update (SU), and their intermediate neighborhood tensor update (NTU) -- and directly maximizes an overlap between the exact iPEPS with the increased bond di…
High-dimensional maximally entangled photon pairs in parametric down-conversion
针对无限OAM基矢高维子空间中难以直接生成最大纠缠态的问题，提出同时利用泵浦光束空间分布和晶体非线性轮廓的方法，无需额外空间滤波即可生成最大纠缠态，并以d=3和d=5为例进行了演示。
We study the spin-$1$ Heisenberg model on the square lattice with the antiferromagnetic nearest-neighbor $J_1$ and the next-nearest-neighbor $J_2$ couplings by using the infinite projected entangled-pair state (iPEPS) ansatz and density matrix renormalization group (DMRG) calculation. The iPEPS simulation, which studies the model directly in the thermodynamic limit, finds a crossing of the ground state from the Néel magnetic state to the stripe magnetic state at $J_2/J_1 \simeq 0.549$, showing a direct phase transition. In the finite-size DMRG calculation on the cylinder geometry up to the cyl…
KARIPAP: Quantum-Inspired Tensor Network Compression of Large Language Models Using Infinite Projected Entangled Pair States and Tensor Renormalization Group
提出量子启发张量网络压缩方法KARIPAP，利用iPEPS和TRG捕获层间多方向纠缠，在LLaMA-2 7B上实现93%内存和70%参数缩减，仅损失2-3%准确率。
We address the question, does a system A being entangled with another system B, put any constraints on the Heisenberg uncertainty relation (or the Schrodinger-Robertson inequality)? We find that the equality of the uncertainty relation cannot be reached for any two noncommuting observables, for finite dimensional Hilbert spaces if the Schmidt rank of the entangled state is maximal. One consequence is that the lower bound of the uncertainty relation can never be attained for any two observables for qubits, if the state is entangled. For infinite-dimensional Hilbert space too, we show that there…
It is shown that, every entangled state in an infinite-dimensional composite system has a simple entanglement witness of the form $αI+T$ with $α$ a nonnegative number and $T$ a finite rank self-adjoint operator. We also provide two methods of constructing entanglement witness and apply them to obtain some entangled states that cannot be detected by the PPT criterion and the realignment criterion.
The simple update (SU) and full update (FU) are the two paradigmatic time evolution algorithms for a tensor network known as the infinite projected entangled pair state (iPEPS). They differ by an error measure that is either, respectively, local or takes into account full infinite tensor environment. In this paper we test an intermediate neighborhood tensor update (NTU) accounting for the nearest neighbor environment. This small environment can be contracted exactly in a parallelizable way. It provides an error measure that is Hermitian and non-negative down to machine precision. In the 2D qua…
A pure state decoheres into a mixed state as it entangles with an environment. When an entangled two-mode system is embedded in a thermal environment, however, each mode may not be entangled with its environment by their simple linear interaction. We consider an exactly solvable model to study the dynamics of a total system, which is composed of an entangled two-mode system and a thermal environment, and also an array of infinite beam splitters. It is shown that many-body entanglement of the system and the environment plays a crucial role in the process of disentangling the system.
An introduction to infinite projected entangled-pair state methods for variational ground state simulations using automatic differentiation
本文综述基于自动微分的变分 iPEPS 框架，介绍其原理、可适配二维晶格的通用基础及基准结果，以提升基态与激发态模拟的灵活性和精度。
Variationally optimizing infinite projected entangled-pair states at large bond dimensions: A split corner transfer matrix renormalization group approach
针对大键维度PEPS变分优化中双层CTMRG收缩成本高的问题，提出保持层分离并引入新环境张量的split-CTMRG算法，在保持精度的同时显著加速。
We show that all entangled Gaussian states of two infinite dimensional systems can be distilled to maximally entangled states in finite dimensions. The distillation protocol involves local squeezing operations, local homodyne measurements with ancilla systems prepared in coherent states as well as local joint measurements of the total number operator of two copies of the state.
An infinite projected entangled pair state (iPEPS) is a tensor network ansatz to represent a quantum state on an infinite 2D lattice whose accuracy is controlled by the bond dimension $D$. Its real, Lindbladian or imaginary time evolution can be split into small time steps. Every time step generates a new iPEPS with an enlarged bond dimension $D' > D$, which is approximated by an iPEPS with the original $D$. In Phys. Rev. B 98, 045110 (2018) an algorithm was introduced to optimize the approximate iPEPS by maximizing directly its fidelity to the one with the enlarged bond dimension $D'$. In thi…
We derive sufficient conditions for infinite-dimensional systems whose entanglement is not completely lost in a finite time during its decoherence by a passive interaction with local vacuum environments. The sufficient conditions allow us to clarify a class of bipartite entangled states which preserve their entanglement or, in other words, are tolerant against decoherence in a vacuum. We also discuss such a class for entangled qubits.
Probing universal imaginary-time relaxation critical dynamics with infinite projected entangled pair states
利用无限投影纠缠对态（iPEPS）全更新策略，研究二维横向场伊辛模型的虚时弛豫临界动力学，通过磁化强度的幂律标度精确确定临界点和临界指数，并验证了临界初始滑移指数。
Absolutely maximally entangled state equivalence and the construction of infinite quantum solutions to the problem of 36 officers of Euler
本文证明四量子三能级AME态在局部幺正等价下仅有一类，而更大局部维度下等价类无穷多，并为欧拉36军官问题构造并证明了无穷多个不等价的量子解。
By establishing CHSH operators and CHSH-type inequalities, we show that any entangled pure state in infinite-dimensional systems is entangled in a $2\otimes2$ subspace. We find that, for infinite-dimensional systems, the corresponding properties are similar to that of the two-qubit case: (i) The CHSH-type inequalities provide a sufficient and necessary condition for separability of pure states; (ii) The CHSH operators satisfy the Cirel'son inequalities; (iii) Any state which violates one of these Bell inequalities is distillable.
Quantum multipartite entangled states play significant roles in quantum information processing. By using difference schemes and orthogonal partitions, we construct a series of infinite classes of irredundant mixed orthogonal arrays (IrMOAs) and thus provide positive answers to two open problems. The first is the extension of the method for constructing homogeneous systems from orthogonal arrays (OAs) to heterogeneous multipartite systems with different individual levels. The second is the existence of $k$-uniform states in heterogeneous quantum systems. We present explicit constructions of two…
In a recent article [Phys. Rev. A 54, 1793 (1996)] Krenn and Zeilinger investigated the conditional two-particle correlations for the subensemble of data obtained by selecting the results of the spin measurements by two observers 1 and 2 with respect to the result found in the corresponding measurement by a third observer. In this paper we write out explicitly the condition required in order for the selected results of observers 1 and 2 to violate Bell's inequality for general measurement directions. It is shown that there are infinitely many sets of directions giving the maximum level of viol…
We develop an improved variant of $U(1)$-symmetric infinite projected entangled-pair state (iPEPS) ansatz to investigate the ground state phase diagram of the spin-$1/2$ square $J_{1}-J_{2}$ Heisenberg model. In order to improve the accuracy of the ansatz, we discuss a simple strategy to select automatically relevant symmetric sectors and also introduce an optimization method to treat second-neighbor interactions more efficiently. We show that variational ground-state energies of the model obtained by the $U(1)$-symmetric iPEPS ansatz (for a fixed bond dimension $D$) set a better upper bound, …
The entangled quantum state of a photon pair propagating through atmospheric turbulence suffers decay of entanglement due to the scintillation it experiences. Here we investigate the robustness against this decay for different qutrit states. We use an infinitesimal-propagation equation to obtain the density matrix as a function of the propagation distance and we use the tangle to quantify the entanglement between a pair of qutrits. The evolution of various initial states as they propagate through turbulence is considered. Using optimization of the initial parameters, we obtain expressions for …
An experimental verification of the maximally entangled state ensures that the constructed state is close to the maximally entangled state, but it does not guarantee that the state is exactly the same as the maximally entangled state. Further, the entanglement structure is not uniquely determined in general probabilistic theories even if we impose that the local subsystems are fully equal to quantum systems. Therefore, the existence of the maximally entangled state depends on whether the standard entanglement structure is valid. To examine this issue, we introduce pseudo standard entanglement …
We introduce several classes of quantum combinatorial designs, namely quantum Latin squares, cubes, hypercubes and a notion of orthogonality between them. A further introduced notion, quantum orthogonal arrays, generalizes all previous classes of designs. We show that mutually orthogonal quantum Latin arrangements can be entangled in the same way than quantum states are entangled. Furthermore, we show that such designs naturally define a remarkable class of genuinely multipartite highly entangled states called $k$-uniform, i.e. multipartite pure states such that every reduction to $k$ parties …
The problem of conditions on the initial correlations between the system and the environment that lead to completely positive (CP) or not-completely positive (NCP) maps has been studied by various authors. Two lines of study may be discerned: one concerned with families of initial correlations that induce CP dynamics under the application of an arbitrary joint unitary on the system and environment; the other concerned with specific initial states that may be highly entangled. Here we study the latter problem, and highlight the interplay between the initial correlations and the unitary applied.…
Recently, several hybrid approaches to quantum information emerged which utilize both continuous- and discrete-variable methods and resources at the same time. In this work, we investigate the bipartite hybrid entanglement between a finite-dimensional, discrete-variable quantum system and an infinite-dimensional, continuous-variable quantum system. A classification scheme is presented leading to a distinction between pure hybrid entangled states, mixed hybrid entangled states (those effectively supported by an overall finite-dimensional Hilbert space), and so-called truly hybrid entangled stat…
We consider an infinite spin chain as a bipartite system consisting of the left and right half-chain and analyze entanglement properties of pure states with respect to this splitting. In this context we show that the amount of entanglement contained in a given state is deeply related to the von Neumann type of the observable algebras associated to the half-chains. Only the type I case belongs to the usual entanglement theory which deals with density operators on tensor product Hilbert spaces, and only in this situation separable normal states exist. In all other cases the corresponding state i…
Fractional quantum Hall states with variational Projected Entangled-Pair States: a study of the bosonic Harper-Hofstadter model
本文针对玻色型Harper-Hofstadter模型，利用变分无限投影纠缠对态（iPEPS）避免虚假有限尺寸效应，识别分数量子霍尔态并表征其体关联指数衰减与手性边缘模特性。
We discuss the notion of bound entanglement (BE) for continuous variables (CV). We show that the set of non--distillable states (NDS) for CV is nowhere dense in the set of all states, i.e., the states of infinite--dimensional bipartite systems are generically distillable. This automatically implies that the sets of separable states, entangled states with positive partial transpose, and bound entangled states are also nowhere dense in the set of all states. All these properties significantly distinguish quantum CV systems from the spin like ones. The aspects of the definition of BE for CV is al…
Van Dam and Hayden introduced a concept commonly referred to as embezzlement, where, for any entangled quantum state $ϕ$, there is an entangled catalyst state $ψ$, from which a high fidelity approximation of $ϕ\otimes ψ$ can be produced using only local operations. We investigate a version of this where the embezzlement is perfect (i.e., the fidelity is 1). We prove that perfect embezzlement is impossible in a tensor product framework, even with infinite-dimensional Hilbert spaces and infinite entanglement entropy. Then we prove that perfect embezzlement is possible in a commuting operator fra…
The question under what conditions different witnesses may detect some entangled states simultaneously is answered for both finite- and infinite-dimensional bipartite systems. Finite many different witnesses can detect some entangled states simultaneously if and only if their convex combinations are still witnesses; they can not detect any entangled state simultaneously if and only if the set of their convex combinations contains a positive operator. For two witnesses $W_1$ and $W_2$, some more can be said: (1) $W_1$ and $W_2$ can detect the same set of entangled states if and only if they are…
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