We propose a method to generate entanglement measures systematically by using the irreducible decomposition of some copies of a state under the local unitary (LU) transformations. It is applicable to general multipartite systems. We show that there are entanglement monotones corresponding to singlet representations of the LU group. They can be evaluated efficiently in an algebraic way, and experimentally measurable by local projective measurements of the copies of the state. Non-singlet representations are also shown to be useful to classify entanglement. Our method reproduces many well-known measures in a unified way, and produces also a lot of new ones.
Using very general arguments, we prove that any entanglement measures based on distance must be maximal on pure states. Furthermore, we show that Bures measure of entanglement and geometric measure of entanglement satisfy the monogamy inequality on all pure multiqubit states. Finally, using the power of Bures measure of entanglement and geometric measure of entanglement, we present a class of tight monogamy relations for pure states of multiqubit systems.
In this work we study the entanglement of pure fourpartite of qubit states. The analysis is realized through the comparison between two different entanglement measures: the Groverian entanglement measure and the residual entanglement measured with negativities. Furthemore, we discuss some applications of four-way entangled fourpartite states.
Entanglement of formation is an important measure of quantum entanglement. We present an experimental way to measure the entanglement of formation for arbitrary dimensional pure states. The measurement only evolves local quantum mechanical observables.
We examine the application of Schmidt-mode analysis to pure state entanglement. Several examples permitting exact analytic calculation of Schmidt eigenvalues and eigenfunctions are included, as well as evaluation of the associated degree of entanglement.
Entanglement is one of the important resources in quantum tasks. Recently, Yang $et$ $al.$ [arXiv:2205.08801] proposed an entanglement polygon inequalities (EPI) in terms of some entanglement measures for $n$-qudit pure states. Here we continue to consider the entanglement polygon inequalities. Specifially, we show that the EPI is valid for $n$-qudit pure states in terms of geometric entanglement measure (GEM), then we study the residual entanglement in terms of GEM for pure states in three-qubit systems. At last, we present counterexamples showing that the EPI is invalid for higher dimensiona…
An entanglement bound based on local measurements is introduced for multipartite pure states. It is the upper bound of the geometric measure and the relative entropy of entanglement. It is the lower bound of minimal measurement entropy. For pure bipartite states, the bound is equal to the entanglement entropy. The bound is applied to pure tripartite qubit states and the exact tripartite relative entropy of entanglement is obtained for a wide class of states.
A necessary and sufficient entanglement criterion based on variances of Mermin-Klyshko's Bell operators is proved for multiqubit pure states. Contrary to Bell's inequalities, entangled pure states strictly satisfy a quadratic inequality but product ones can attain the equality under some local unitary transformations, which can be obtained by solving a quadratic maximum problem. This presents a characterization of multiqubit pure-state entanglement.
We construct a measure of entanglement for general pure multipartite states based on Segre variety. We also construct a class of entanglement monotones based on the Plücker coordinate equations of the Grassmann variety. Moreover, we discuss and compare these measures of entanglement.
The problem of the experimental determination of the amount of entanglement of a bipartite pure state is addressed. We show that measuring a single observable does not suffice to determine the entanglement of a given unknown pure state of two particles. Possible minimal local measuring strategies are discussed and a comparison is made on the basis of their best achievable precision.
We propose an explicit formula for an entanglement measure of pure multipartite quantum states, then study a general pure tripartite state in detail, and at end we give some simple but illustrative examples on four-qubits and m-qubits states.
A complete analysis of entangled triqubit pure states is carried out based on a new simple entanglement measure. An analysis of all possible extremally entangled pure triqubit states with up to eight terms is shown to reduce, with the help of SLOCC transformations, to three distinct types. The analysis presented are most helpful for finding different entanglement types in multipartite pure state systems.
We define an entanglement measure, called the partial tangle, which represents the residual two-qubit entanglement of a three-qubit pure state. By its explicit calculations for three-qubit pure states, we show that the partial tangle is closely related to the faithfulness of a teleportation scheme over a three-qubit pure state.
A simple entanglement measure for multipartite pure states is formulated based on the partial entropy of a series of reduced density matrices. Use of the proposed new measure to distinguish disentangled, partially entangled, and maximally entangled multipartite pure states is illustrated.
We prove the strong super-additivity of the entanglement of formation for stabilizer pure states, and the set of mixed states which minimize their average entropy of entanglement as a mixture of stabilizer pure states sharing the same stabilizer group up to phases. The implications of the result on the additivity of the Holevo capacity of a quantum channel transmitting stabilizer states with Pauli noise is discussed.
Monogamy of bipartite correlations leads, for arbitrary pure multi-qubit states, to simple conditions able to indicate various types of multipartite entanglement by being capable to exclude the possibility of k-separability.
The entanglement of general pure Gaussian two-mode states is examined in terms of the coefficients of the quadrature components of the wavefunction. The entanglement criterion and the entanglement of formation are directly evaluated as a function of these coefficients, without the need for deriving local unitary transformations. These reproduce the results of other methods for the special case of symmetric pure states which employ a relation between squeezed states and Einstein-Podolsky-Rosen correlations. The modification of the quadrature coefficients and the corresponding entanglement due t…
We present a general theoretical framework for both deterministic and probabilistic entanglement transformations of bipartite pure states achieved via local operations and classical communication. This framework unifies and greatly simplifies previous works. A necessary condition for ``pure contraction'' transformations is given. Finally, constructive protocols to achieve both probabilistic and deterministic entanglement transformations are presented.
We find a minimal set of necessary and sufficient conditions for the existence of a local procedure that converts a finite pure state into one of a set of possible final states. This result provides a powerful method for obtaining optimal local entanglement manipulation protocols for pure initial states. As an example, we determine analytically the optimal distillable entanglement for arbitrary finite pure states. We also construct an explicit protocol achieving this bound.
To quantify the entanglement is one of the most important topics in quantum entanglement theory. In [arXiv: 2006.12408], the authors proposed a method to build a measure from the orginal domain to a larger one. Here we apply that method to build an entanglement measure from measures for pure states. First, we present conditions when the entanglement measure is an entanglement monotone and convex, we also present an interpretation of the smoothed one-shot entanglement cost under the method here. At last, we present a difference between the local operation and classical communication (LOCC) and …
Various parameterizations for the orbits under local unitary transformations of three-qubit pure states are analyzed. The interconvertibility, symmetry properties, parameter ranges, calculability and behavior under measurement are looked at. It is shown that the entanglement monotones of any multipartite pure state uniquely determine the orbit of that state under local unitary transformations. It follows that there must be an entanglement monotone for three-qubit pure states which depends on the Kempe invariant defined in [Phys. Rev. A 60, 910 (1999)]. A form for such an entanglement monotone …
The geometric measure of entanglement for a symmetric pure state with nonnegative amplitudes has attracted much attention. On the other hand, the spectral theory of nonnegative tensors (hypermatrices) has been developed rapidly. In this paper, we show how the spectral theory of nonnegative tensors can be applied to the study of the geometric measure of entanglement for a pure state with nonnegative amplitudes. Especially, an elimination method for computing the geometric measure of entanglement for symmetric pure multipartite qubit or qutrit states with nonnegative amplitudes is given. For sym…
We construct a class of algebraic invariants for N-qubit pure states based on bipartite decompositions of the system. We show that they are entanglement monotones, and that they differ from the well know linear entropies of the sub-systems. They therefore capture new information on the non-local properties of multipartite systems.
A new criterion necessary and sufficient for the separability of pure bipartite systems for arbitrary finite dimensions is demonstrated; and the corresponding finer quantitative measures or characterizations of entanglement (beyond mere separability or non-separability determination) are discussed. Based on this criterion, we proved that the well-known Peres-Horodecki positivity-of-partial-transform criterion is also necessary and sufficient for separability in the case of pure bipartite systems. The maximum value of entanglement, and the corresponding maximally-entangled states are also worke…
We investigate the patterns in distributions of localizable entanglement over a pair of qubits for random multi-qubit pure states. We observe that the mean of localizable entanglement increases gradually with increasing the number of qubits of random pure states while the standard deviation of the distribution decreases. The effects on the distributions, when the random pure multi-qubit states are subjected to local as well as global noisy channels, are also investigated. Unlike the noiseless scenario, the average value of the localizable entanglement remains almost constant with the increase …
Complete complementarity relations, as e.g. $P(ρ_{A})^{2} + C(ρ_{A})^{2} + E(|Ψ\rangle_{AB})^{2}=1$, constrain the local predictability, $P$, and local coherence, $C$, and the entanglement, $E$, of bipartite pure states. For pairs of qubits prepared initially in a particular class of partially entangled pure states with null local coherence, these relations were used in Ref. [Phys. Lett. A, 451, 128414 (2022)] to provide an operational connection between local predictability of the pre-measurement states with the probability of the maximally entangled components of the states after the Bell-ba…
We address the question of quantifying entanglement in pure graph states. Evaluation of multipartite entanglement measures is extremely hard for most pure quantum states. In this paper we demonstrate how solving one problem in graph theory, namely the identification of maximum independent set, allows us to evaluate three multipartite entanglement measures for pure graph states. We construct the minimal linear decomposition into product states for a large group of pure graph states, allowing us to evaluate the Schmidt measure. Furthermore we show that computation of distance-like measures such …
We present an entanglement purification protocol for a mixture of a pure entangled state and a pure product state, which are orthogonal to each other. The protocol is combination of bisection method and one-way hashing protocol. We give recursive formula for the rate of the protocol for different states, i.e. the number of maximally entangled two-qubit pairs obtained with the protocol per a single copy of the initial state. We also calculate numerically the rate for some states.
Entanglement of pure states of bipartite quantum systems has been shown to have a unique measure in terms of the von Neumann entropy of the reduced states of either of its subsystems. The measure is established under entanglement manipulation of an asymptotically large number of copies of the states. In this paper, different asymptotic measures of entanglement assigned to arbitrary sequences of bipartite pure states are shown to coincide only when the sequence is information stable, in terms of the quantum spectral information rates of the sequence of subsystem states. Additional bounds on the…
We propose an entanglement tensor to compute the entanglement of a general pure multipartite quantum state. We compare the ensuing tensor with the concurrence for bipartite state and apply the tensor measure to some interesting examples of entangled three-qubit and four-qubit states. It is shown that in defining the degree of entanglement of a multi-partite state, one needs to make assumptions about the willingness of the parties to cooperate. We also discuss the degree of entanglement of the multi-qubit $\ket{W_{M}}$-states.
I show that two distant parties can transform pure entangled states to arbitrary pure states by stochastic local operations and classical communication (SLOCC) at the single copy level, if they share bound entangled states. This is the effect of bound entanglement since this entanglement processing is impossible by SLOCC alone. Similar effect of bound entanglement occurs in three qubits where two incomparable entangled states of GHZ and W can be inter-converted. In general multipartite settings composed by $N$ distant parties, all $N$-partite pure entangled states are inter-convertible by SLOC…
We show that not all 4-party pure states are GHZ reducible (i.e., can be generated reversibly from a combination of 2-, 3- and 4-party maximally entangled states by local quantum operations and classical communication asymptotically) through an example, we also present some properties of the relative entropy of entanglement for those 3-party pure states that are GHZ reducible, and then we relate these properties to the additivity of the relative entropy of entanglement.
We investigate how much amount of Greenberger-Horne-Zeilinger (GHZ) entanglement is required in order to prepare a given multipartite state by local operations and classical communication (LOCC).We present a LOCC procedure that asymptotically converts GHZ states into an arbitrary multipartite pure state, whose conversion rate is given by the multipartite discord of the state. This reveals that the GHZ-entanglement cost of preparing a pure state is not higher than the multipartite discord of the state. It also provides an operational interpretation of multipartite discord for pure states, namel…
It has been observed that the reduced density matrices of bipartite qudit pure states possess a Gram matrix structure. This observation has opened a possibility of analysing the entanglement in such systems from the purely geometrical point of view. In particular, a new quantitative measure of an entanglement of the geometrical nature, has been proposed. Using the invented Gram matrix approach, a version of a non-linear purification of mixed states describing the system analysed has been presented.
We use concurrence as an entanglement measure and experimentally demonstrate the entanglement classification of arbitrary three-qubit pure states on a nuclear magnetic resonance (NMR) quantum information processor. Computing the concurrence experimentally under three different bipartitions, for an arbitrary three-qubit pure state, reveals the entanglement class of the state. The experiment involves measuring the expectation values of Pauli operators. This was achieved by mapping the desired expectation values onto the local $z$ magnetization of a single qubit. We tested the entanglement classi…
We present optimal measuring strategies for the estimation of the entanglement of unknown two-qubit pure states and of the degree of mixing of unknown single-qubit mixed states, of which N identical copies are available. The most general measuring strategies are considered in both situations, to conclude in the first case that a local, although collective, measurement suffices to estimate entanglement, a non-local property, optimally.
The concept of entanglement splitting is introduced by asking whether it is possible for a party possessing half of a pure bipartite quantum state to transfer some of his entanglement with the other party to a third party. We describe the unitary local transformation for symmetric and isotropic splitting of a singlet into two branches that leads to the highest entanglement of the output. The capacity of the resulting quantum channels is discussed. Using the same transformation for less than maximally entangled pure states, the entanglement of the resulting states is found. We discuss whether t…
The problem of the reliable transfer of entanglement from one pure bipartite quantum state to another using local operations is analyzed. It is shown that in the case of qubits the amount that can be transferred is restricted to the difference between the entanglement of the two states. In the presence of a catalytic state the range of the transferrable amount broadens to a certain degree.
Based on the quantitative complementarity relations, we analyze thoroughly the properties of multipartite quantum correlations and entanglement in four-qubit pure states. We find that, unlike the three-qubit case, the single residual correlation, the genuine three- and four-qubit correlations are not suited to quantify entanglement. More interestingly, from our qualitative and numerical analysis, it is conjectured that the sum of all the residual correlations may constitute a good measure for the total multipartite entanglement in the system.
Quantum entanglement of pure states is usually quantified via the entanglement entropy, the von Neumann entropy of the reduced state. Entanglement entropy is closely related to entanglement distillation, a process for converting quantum states into singlets, which can then be used for various quantum technological tasks. The relation between entanglement entropy and entanglement distillation has been known only for the asymptotic setting, and the meaning of entanglement entropy in the single-copy regime has so far remained open. Here we close this gap by considering entanglement catalysis. We …
The Groverian entanglement measure of pure quantum states of $n$ qubits is generalized to the case in which the qubits are divided into any $m \le n$ parties and the entanglement between these parties is evaluated. To demonstrate this measure we apply it to general states of three qubits and to symmetric states with any number of qubits such as the Greenberg-Horne-Zeiliner state and the W state.
Based on the complementarity relation between entanglement of a composite system and the purity of a subsystem, we propose a simple method to measure the amount of entanglement. The method can be applied to a bipartite system in a pure state of any arbitrary dimension. It requires only single qudit rotations and straightforward probability measurements performed on one of the subsystems, and can thus be easily implemented experimentally using linear optical devices.
We calculate the effect of polarization-dependent scattering by disorder on the degree of polarization-entanglement of two beams of radiation. Multi-mode detection converts an initially pure state into a mixed state with respect to the polarization degrees of freedom. The degree of entanglement decays exponentially with the number of detected modes if the scattering mixes the polarization directions and algebraically if it does not.
We study non-local properties of generic three-qubit pure states. First, we obtain the distributions of both the coefficients and the only phase in the five-term decomposition of Acín et al. for an ensemble of random pure states generated by the Haar measure on U(8). Furthermore, we analyze the probability distributions of two sets of polynomial invariants. One of these sets allows us to classify three-qubit pure states into four classes. Entanglement in each class is characterized using the minimal Rényi-Ingarden-Urbanik entropy. Besides, the fidelity of a three-qubit random state with the cl…
We present a mapping which associates pure N-qubit states with a polynomial. The roots of the polynomial characterize the state completely. Using the properties of the polynomial we construct a way to determine the separability and the number of unentangled qubits of pure N-qubit states.
We analyze multipartite entanglement between atomic ensembles within quantum matter-light interfaces. In our proposal, a polarized light beam crosses sequentially several polarized atomic ensembles impinging on each of them at a given angle α_i. These angles are crucial parameters for shaping the entanglement since they are directly connected to the appropriate combinations of the collective atomic spins that are squeezed. We exploit such scheme to go beyond the pure state paradigm proposing realistic experimental settings to address multipartite mixed state entanglement in continuous variable…
A bipartite state which is secretly chosen from a finite set of known entangled pure states cannot be immediately useful in standard quantum information processing tasks. To effectively make use of the entanglement contained in this unknown state, we introduce a new way to locally manipulate the original quantum system: either identify the state successfully or distill some pure entanglement. Remarkably, if many copies are available, we show that any finite set of entangled pure states, whatever orthogonal or not, can be locally distinguished in this way, which further implies that pure entang…
We present a multipartite entanglement measure for $N$-qudit pure states, using the norm of the correlation tensor which occurs in the Bloch representation of the state. We compute this measure for important class of $N$-qutrit pure states, namely general GHZ states. We prove that this measure possesses almost all the properties expected of a good entanglement measure, including monotonicity. Finally, we extend this measure to $N$-qudit mixed states via convex roof construction and establish its various properties, including its monotonicity.
We show that for a finite-dimensional Hilbert space, there exist observables that induce a tensor product structure such that the entanglement properties of any pure state can be tailored. In particular, we provide an explicit, finite method for constructing observables in an unstructured d-dimensional system so that an arbitrary known pure state has any Schmidt decomposition with respect to an induced bipartite tensor product structure. In effect, this article demonstrates that in a finite-dimensional Hilbert space, entanglement properties can always be shifted from the state to the observabl…
The quantification and classification of quantum entanglement is a very important and still open question of quantum information theory. In this paper, we describe an entanglement measure for multipartite pure states (the minimum of Shannon's entropy of orthogonal measurements). This measure is additive, monotone under LOCC, and coincides with the reduced von Neumann entropy on bipartite states. A method for numerical calculation of this measure by genetic algorithms is also presented. Moreover, the minimization of entropy technique is extended to fermionic states.
In this note we generalize Nielsen's marjoization criterion for the convertibility of bipartite pure states [Phys. Rev. Lett \textbf{83}, 436(1999)] to a special class of multipartite pure states which have generalized Schmidt decompositions.
We quantify the geometric measure of entanglement in terms of mean values of observables of entangled system. For pure states we find the relation of geometric measure of entanglement with the mean value of spin one-half for the system composed of spin and arbitrary quantum system. The geometric measure of entanglement for mixed states of rank-2 is studied as well. We find the explicit expression for geometric entanglement and the relation of entanglement in this case with the values of spin correlations. These results allow to find experimentally the value of entanglement by measuring a value…
We show that for tripartite quantum pure states of qubits, all the kinds of entanglement in terms of SLOCC classification are experimentally measurable by simple projective measurements, provided that four copies of the composite quantum system are available. In particular, the entanglement of reduced density matrices, even though they are mixed states, can be exactly determined in experiment. Concurrence of assistance is also shown to be measurable by introducing an interesting equations with explicit physical meanings.
We present a geometric approach to the characterization of separability and entanglement in pure Gaussian states of an arbitrary number of modes. The analysis is performed adapting to continuous variables a formalism based on single subsystem unitary transformations that has been recently introduced to characterize separability and entanglement in pure states of qubits and qutrits [arXiv:0706.1561]. In analogy with the finite-dimensional case, we demonstrate that the $1 \times M$ bipartite entanglement of a multimode pure Gaussian state can be quantified by the minimum squared Euclidean distan…
We propose a qubit-environment entanglement measure which is tailored for evolutions that lead to pure dephasing of the qubit, such as are abundant in solid state scenarios. The measure can be calculated directly form the density matrix without minimization of any kind. In fact it does not require the knowledge of the full density matrix, and it is enough to know the initial qubit state and the states of the environment conditional on qubit pointer states. This yields a computational advantage over standard entanglement measures, which becomes large when there are no correlations between envir…
Entanglement of any pure state of an N times N bi-partite quantum system may be characterized by the vector of coefficients arising by its Schmidt decomposition. We analyze various measures of entanglement derived from the generalized entropies of the vector of Schmidt coefficients. For N >= 3 they generate different ordering in the set of pure states and for some states their ordering depends on the measure of entanglement used. This odd-looking property is acceptable, since these incomparable states cannot be transformed to each other with unit efficiency by any local operation. In analogy t…
We analyze approximate transformations of pure entangled quantum states by local operations and classical communication, finding explicit conversion strategies which optimize the fidelity of transformation. These results allow us to determine the most faithful teleportation strategy via an initially shared partially entangled pure state. They also show that procedures for entanglement manipulation such as entanglement catalysis [Jonathan and Plenio, Phys. Rev. Lett. 83, 3566 (1999)] are robust against perturbation of the states involved, and motivate the notion of non-local fidelity, which qua…
The entanglement quantification and classification of multipartite quantum states are two important research fields in quantum information. In this work, we study the entanglement of arbitrary-dimensional multipartite pure states by looking at the averaged partial entropies of various bipartite partitions of the system, namely, the so-called Manhattan distance ($l_1$ norm) of averaged partial entropies (MAPE), and it is proved to be an entanglement measure for pure states. We connected the MAPE with the coefficient matrices, which are important tools in entanglement classification and reexpres…
An optimal local conversion strategy between any two pure states of a bipartite system is presented. It is optimal in that the probability of success is the largest achievable if the parties which share the system, and which can communicate classically, are only allowed to act locally on it. The study of optimal local conversions sheds some light on the entanglement of a single copy of a pure state. We propose a quantification of such an entanglement by means of a finite minimal set of new measures from which the optimal probability of conversion follows.
