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 measurement can be turned into a quantitative entanglement verification test by exploiting a recent theorem by Renner (R. Renner, Nature Physics 3, 645 (2007)).
We describe and examine entanglement between different degrees of freedom in multiphoton states based on the permutation properties. From the state description, the entanglement comes from the permutation asymmetry. According to the different permutation properties, the multiphoton states can be divided into several parts. It will help to deal with the multiphoton interference, which can be used as the measurement of the entanglement.
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…
With an easily applicable criterion based on permutation symmetries of (identically prepared) replicas of quantum states we identify distinct entanglement classes in high-dimensional multi- partite systems. The different symmetry properties of inequivalent states provide a rather intuitive picture of the otherwise very abstract classification of many-body entangled states.
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.
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 present new algorithms for mixed-state multi-copy entanglement distillation for pairs of qubits. Our algorithms perform significantly better than the best known algorithms. Better algorithms can be derived that are tuned for specific initial states. The new algorithms are based on a characterization of the group of all locally realizable permutations of the 4^n possible tensor products of n Bell states.
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.
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…
Multipartite entanglement plays an essential role in both quantum information science and many-body physics. Due to the exponentially large dimension and complex geometric structure of the state space, the detection of entanglement in many-body systems is extremely challenging in reality. Conventional means, like entanglement witness and entropy criterion, either highly depend on the prior knowledge of the studied systems or the detection capability is relatively weak. In this work, we propose a framework for designing multipartite entanglement criteria based on permutation moments, which have…
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 $σ$…
Let $n\geq 2$ and $Φ_{n,t,π}: M_n({\mathbb C}) \rightarrow M_n({\mathbb C})$ be a linear map defined by $Φ_{n,t,π}(A)=(n-t)\sum_{i=1}^nE_{ii}AE_{ii}+t\sum_{i=1}^nE_{i,π(i)}AE_{i,π(i)}^†-A$, where $0\leq t\leq n$, $E_{ij}$s are the matrix units and $π$ is a non-identity permutation of $(1,2,\cdots,n)$. Denote by $\{{ F}_s: s=1,2\ldots, k\}$ the set of all minimal cycles of $π$ and $l(π)=\max\{\# { F}_s: s=1,2,\ldots,k\}$ the length of $π$. It is shown that the Hermitian matrix $W_{n,t,π}$ induced by $Φ_{n,t,π}$ is an indecomposable entanglement witness if and only if $π^2\not={\rm id}$ (the ide…
Many-body states that are invariant under particle relabelling, the permutation symmetric states, occur naturally when the system dynamics is described by symmetric processes or collective spin operators. We derive expressions for the reduced density matrix for arbitrary subsystem decomposition for these states and study properties of permutation symmetric states and their subsystems when the joint system is picked randomly and uniformly. Thus defining a new random matrix ensemble, we find the average linear entropy and von Neumann entropy which implies that random permutation symmetric states…
It is known that any bipartite unitary operator of Schmidt rank three is equivalent to a controlled unitary under local unitaries. We propose a standard form of such operators. Using the form we improve the upper bound for the entanglement cost to implement such operators under local operations and classical communications (LOCC), and provide a corresponding protocol. A part of our protocol is based on a recursive-control protocol which is helpful for implementing other unitary operators. We show that any bipartite permutation unitary of Schmidt rank three can be implemented using LOCC and two…
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 provide a simplified characterization of entanglement in physical systems which are symmetric under the action of subgroups of the symmetric group acting on the party labels. Sets of entanglements are inherently equal, lying in the same orbit under the group action, which we demonstrate for cyclic, dihedral, and polyhedral groups. We then introduce new, generalized relationships for the maxima of those entanglement by exploiting the normalizer and normal subgroups of the physical symmetry group.
The study of entanglement properties of multi-qubit states that are invariant under permutations of qubits is motivated by potential applications in quantum computing, quantum communication, and quantum metrology. In this work, we generalize the notions of symmetrization, Dicke states, and the Majorana representation to the alternating, cyclic, and dihedral subgroups of the full group of permutations. We use these tools to characterize states that are invariant under these subgroups and analyze their entanglement properties.
Using the SU($N$) representation of the group theory, we derive the general form of the spin swapping operator for the quantum Heisenberg spin-$s$ systems. We further prove that such a spin swapping operator is equal to the spin singlet pairing operator under the partial transposition. For SU(2) invariant states, it is shown that the expectation value of the spin swapping operator and its generalizations, the permutations, can be used as an entanglement witness, especially, for the formulation of observable conditions of entanglement.
A novel use of Majorana geometric representation brings out distinct entanglement families of permutation symmetric states of qubits. The paradigmatic W and GHZ (Greenberger-Horne-Zeilinger) states of three qubits respectively contain two and three independent Majorana spinors. Another unique state with three distinct Majorana spinors -- constructed through a permutation symmetric superposition of two up qubits and one down qubit (W state) and its obverse state Wbar) exhibits genuine three-party entanglement, which is robust under loss of a qubit. While the GHZ state has irreducible correlatio…
Recently, several aspects of controlled quantum communication (e.g., bidirectional controlled state teleportation, controlled quantum secure direct communication, controlled quantum dialogue, etc.) have been studied using $n$-qubit ($n\geq3$) entanglement. Specially, a large number of schemes for bidirectional controlled state teleportation are proposed using $m$-qubit entanglement ($m\in\{5,6,7\}$). Here, we propose a set of protocols to illustrate that it is possible to realize all these tasks related to controlled quantum communication using only Bell states and permutation of particles (Po…
In this paper we discuss the properties of the reduced density matrix of quantum many body systems with permutational symmetry and present basic quantification of the entanglement in terms of the von Neumann (VNE), Renyi and Tsallis entropies. In particular, we show, on the specific example of the spin $1/2$ Heisenberg model, how the RDM acquires a block diagonal form with respect to the quantum number $k$ fixing the polarization in the subsystem conservation of $S_{z}$ and with respect to the irreducible representations of the $\mathbf{S_{n}}$ group. Analytical expression for the RDM elements…
We analyse entanglement classes for permutation-symmetric states for n qudits (i.e. d-level systems), with respect to local unitary operations (LU-equivalence) and stochastic local operations and classical communication (SLOCC equivalence). In both cases, we show that the search can be restricted to operations where the same local operation acts on all qudits, and we provide an explicit construction for it. Stabilizers of states in the form of one-particle operations preserving permutation symmetry are shown to provide a coarse-grained classification of entanglement classes. We prove that the …
We give a capacity formula for the classical communication over a noisy quantum channel, when local operations and global permutations allowed in the encoding and bipartite states preshared between the sender and the receiver. The two endpoints of this formula are the Holevo capacity (without entanglement assistance) and the entanglement-assisted capacity (with unlimited entanglement assistance). What's more, we show that the capacity satisfies the strong converse property and thus the formula serves as a sharp dividing line between achievable and unachievable rates of communication. We prove …
We investigate the von Neumann entanglement entropy as function of the size of a subsystem for permutation invariant ground states in models with finite number of states per site, e.g., in quantum spin models. We demonstrate that the entanglement entropy of $n$ sites in a system of length $L$ generically grows as $σ\log_{2}[2πen(L-n)/L]+C$, where $σ$ is the on-site spin and $C$ is a function depending only on magnetization.
We explore the connection between quantum entanglement and the exchange symmetry of the states of N identical particles. Each particle has n-levels. The N particles span the nN dimensional Hilbert space. We shall call the general state of the particle as a qunit. The direct product of the N qunit space is given a decomposition in terms of states with definite permutation symmetry that are found to have a measure of entanglement which is related to the representation of the permutation group. The maximally entangled states are generated from the linear combinations of fully correlated but unent…
From Permutation Symmetry to Communication Bounds and Additivity
Perimeter Institute for Theoretical Physics（Perimeter理论物理研究所）；University of Waterloo（滑铁卢大学）；Institute for Quantum Computing, University of Waterloo（滑铁卢大学量子计算研究所）
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.
In this paper we study the non-local properties of permutation symmetric states of n-qubits. We extend the bipartite Hardy paradox and the associated CH-inequality to n-party permutation symmetric states to show that all symmetric states exhibit non-locality. Natural extensions of both the paradoxes and the inequalities are developed which relate different entanglement classes to different non-local features. We define inequalities which are violated by all states of one entanglement class, whereas there are states outside that class which do not violate.
We show that violation of variance based local sum uncertainty relation (LSUR) for angular momentum operators of a bipartite system, proposed by Hofmann and Takeuchi~[Phys.Rev.A {\bf 68}, 032103 (2003)], reflects entanglement in the equal bipartitions of an $N$-qubit symmetric state with even qubits. We establish the one-to-one connection with the violation of LSUR with negativity of covariance matrix [Phys. Lett. A, {\bf 364}, 203 (2007)] of the two-qubit reduced system of a permutation symmetric $N$-qubit state.
As early as 1932, Majorana had proposed that a pure permutation symmetric state of N spin- 1 2 particles can be represented by N spinors, which correspond geometrically to N points on the Bloch sphere. Several decades after its conception, the Majorana representation has recently attracted a great deal of attention in connection with multiparticle entanglement. A novel use of this represen- tation led to the classification of entanglement families of permutation symmetric qubits - based on the number of distinct spinors and their arrangement in constituting the multiqubit state. An elegant app…
We provide a group-theoretical classification of the entangled states of N identical particles. The connection between quantum entanglement and the exchange symmetry of the states of N identical particles is made explicit using the duality between the permutation group and the simple unitary group. Each particle has n-levels and spans the n-dimensional Hilbert space. We shall call the general state of the particle as a qunit. The direct product of the N qunit space is given a decomposition in terms of states with definite permutation symmetry. The nature of fundamental entanglement of a state …
Entanglement in multipartite quantum systems is much more elusive than its bipartite counterpart. In recent past the usefulness of multipartite entangled states in several information theoretic tasks have been demonstrated. Being a resource, the detection of multipartite entanglement is an imperative necessity. Among the different classes of multipartite entangled states the Dicke state has found importance in several tasks due to its permutation symmetric nature. In this work we propose a simple and elegant way of detecting n qubit Dicke states using permutation symmetric Bell operators. We c…
In this article I expound an understanding of the quantum mechanics of so-called "indistinguishable" systems in which permutation invariance is taken as a symmetry of a special kind, namely the result of representational redundancy. This understanding has heterodox consequences for the understanding of the states of constituent systems in an assembly and for the notion of entanglement. It corrects widespread misconceptions about the inter-theoretic relations between quantum mechanics and both classical particle mechanics and quantum field theory. The most striking of the heterodox consequences…
The quantification of quantum entanglement is a central issue in quantum information theory. Recently, Gao \emph{et al}. ( \href{http://dx.doi.org/10.1103/PhysRevLett.112.180501}{Phys. Rev. Lett. \textbf{112}, 180501 (2014)}) pointed out that the maximum of entanglement measure of the permutational invariant part of $ρ$ ought to be a lower bound on entanglement measure of the original state $ρ$, and proposed that this argument can be used as an additional requirement for (multipartite) entanglement measures. Whether any individual proposed entanglement measure satisfies the requirement still h…
We study the von Neumann and Rényi bipartite entanglement entropies in the thermodynamic limit of many-body quantum states with spin-s sites, that possess full symmetry under exchange of sites. It turns out that there is essentially a one-to-one correspondence between such thermodynamic states and probability measures on CP^{2s}. Let a measure be supported on a set of possibly fractal real dimension d with respect to the Study-Fubini metric of CP^{2s}. Let m be the number of sites in a subsystem of the bipartition. We give evidence that in the limit where m goes to infinity, the entanglement e…
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…
Quantities invariant under local unitary transformations are of natural interest in the study of entanglement. This paper deduces and studies a particularly simple quantity that is constructed from a combination of two standard permutations of the density matrix, namely realignment and partial transpose. This bipartite quantity, denoted here as $R_{12}$, vanishes on large classes of separable states including classical-quantum correlated states, while being maximum for only maximally entangled states. It is shown to be naturally related to the 3-tangle in three qubit states via their two-qubit…
It is a recent observation that entanglement classification for qubits is closely related to local $SL(2,\CC)$-invariants including the invariance under qubit permutations, which has been termed $SL^*$ invariance. In order to single out the $SL^*$ invariants, we analyze the $SL(2,\CC)$-invariants of four resp. five qubits and decompose them into irreducible modules for the symmetric group $S_4$ resp. $S_5$ of qubit permutations. A classifying set of measures of genuine multipartite entanglement is given by the ideal of the algebra of $SL^*$-invariants vanishing on arbitrary product states. We …
Measurement-based quantum correlation mimics several characteristics of multipartite quantum correlations and at the same time, it reduces the parent system to a smaller subsystem. On the other hand, genuine multipartite entanglement measures can capture certain features of a multisite composite system that are inaccessible via bipartite quantum correlation quantifiers. We merge these two concepts by introducing localizable genuine multimode entanglement for continuous variable systems, both for Gaussian and non-Gaussian multimode parent states. We report a compact form of localizable generali…
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 introduce a family of highly symmetric bipartite quantum states in arbitrary dimensions. It consists of all states that are invariant under local phase rotations and local cyclic permutations of the basis. We solve the separability problem for a subspace of these states and show that a sizable part of the family is bound entangled. We also calculate some of the Schmidt numbers for the family in $d = 3$, thereby characterizing the dimensionality of entanglement. Our results allow us to estimate entanglement properties of arbitrary states, as general states can be symmetrized to the considere…
Spin squeezing inequalities (SSI) represent a major tool to probe quantum entanglement among a collection of few-level atoms, and are based on collective spin measurements and their fluctuations. Yet, for atomic ensembles of spin-$j$ atoms and ultracold spinor gases, many experiments can image the populations in all Zeeman sublevels $s=-j, -j+1, \dots, j$, potentially revealing finer features of quantum entanglement not captured by SSI. Here we present a systematic approach which exploits Zeeman-sublevel population measurements in order to construct novel entanglement criteria, and illustrate …
We demonstrate the existence of general constraints on distributed quantum correlations, which impose a trade-off on bipartite and multipartite entanglement at once. For all N-mode Gaussian states under permutation invariance, we establish exactly a monogamy inequality, stronger than the traditional one, that by recursion defines a proper measure of genuine N-partite entanglement. Strong monogamy holds as well for subsystems of arbitrary size, and the emerging multipartite entanglement measure is found to be scale invariant. We unveil its operational connection with the optimal fidelity of con…
Nonlocal unitary operations can create quantum entanglement between distributed particles, and the quantification of created entanglement is a hard problem. It corresponds to the concepts of entangling and assisted entangling power when the input states are, respectively, product and arbitrary pure states. We analytically derive them for Schmidt-rank-two bipartite unitary and some complex bipartite permutation unitaries. In particular, the entangling power of permutation unitary of Schmidt rank three can take only one of two values: $\log_2 9 - 16/9$ or $\log_2 3$ ebits. The entangling power, …
We study separability properties in a 5-dimensional set of states of quantum systems composed of three subsystems of equal but arbitrary finite Hilbert space dimension d. These are the states, which can be written as linear combinations of permutation operators, or, equivalently, commute with unitaries of the form UxUxU. We compute explicitly the following subsets and their extreme points: (1) triseparable states, which are convex combinations of triple tensor products, (2) biseparable states, which are separable for a twofold partition of the system, and (3) states with positive partial trans…
A general method in constructing a complete set of wave functions for multipartite identical qubits is presented based on the irreducible representations of the permutation group and the nth rank tensors. Particular examples for n =2, 3, and 4 are derived and the entanglement behavior for each state is examined from several criteria. It is found that the states so constructed are all bound entangled states. For the case of even n, all the states are found to have maximum "n-tangle". The symmetry in spin space is found to increase the n-tangle in general. The "n-tangle" for n = 4 is found not a…
We analyze a class of entangled states for bipartite $d \otimes d$ systems, with $d$ non-prime. The entanglement of such states is revealed by the construction of canonically associated entanglement witnesses. The structure of the states is very simple and similar to the one of isotropic states: they are a mixture of a separable and a pure entangled state whose supports are orthogonal. Despite such simple structure, in an opportune interval of the mixing parameter their entanglement is not revealed by partial transposition nor by the realignment criterion, i.e. by any permutational criterion i…
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…
