The majorization relation has been shown to be useful in classifying which transformations of jointly held quantum states are possible using local operations and classical communication. In some cases, a direct transformation between two states is not possible, but it becomes possible in the presence of another state (known as a catalyst); this situation is described mathematically by the trumping relation, an extension of majorization. The structure of the trumping relation is not nearly as well understood as that of majorization. We give an introduction to this subject and derive some new results. Most notably, we show that the dimension of the required catalyst is in general unbounded; there is no integer $k$ such that it suffices to consider catalysts of dimension $k$ or less in determining which states can be catalyzed into a given state. We also show that almost all bipartite entangled states are potentially useful as catalysts.
Entanglement catalysis allows one to convert certain entangled states into others by the temporary involvement of another entangled state (so-called catalyst), where after the conversion the catalyst is returned to the same state. For bipartite pure entangled states that can be transformed in this way with unit probability, the respective Schmidt coefficients are said to satisfy the trumping relation, a mathematical relation which is an extension of the majorization relation. This article provides all necessary and sufficient conditions for the trumping relation in terms of the Schmidt coeffic…
We prove that the bipartite entangled state of rank three is distillable. So there is no rank three bipartite bound entangled state. By using this fact, We present some families of rank four states that are distillable. We also analyze the relation between the low rank state and the Werner state.
We show that a bipartite state on a tensor product of two matrix algebras is almost surely entangled if its rank is not greater than that of one of its reduced density matrices.
We give conditions under which general bipartite entangled nonorthogonal states become maximally entangled states. By the conditions we construct a large class of entangled nonorthogonal states with exact one ebit of entanglement in both bipartite and multipartite systems. One remarkable property is that the amount of entanglement in this class of states is independent on the parameters involved in the states. Finally we discuss how to generate the bipartite maximally entangled nonorthogonal states.
A complete analysis of entangled bipartite qutrit pure states is carried out based on a simple entanglement measure. An analysis of all possible extremally entangled pure bipartite qutrit states is shown to reduce, with the help of SLOCC transformations, to three distinct types. The analysis and the results should be helpful for finding different entanglement types in multipartite pure state systems.
It is an interesting problem to construct genuine tripartite entangled states based on the collective use of two bipartite entangled states. We consider the case that the states are two-qubit Werner states, we construct the interval of parameter of Werner states such that the tripartite state is genuine entangled. Further, we present the way of detecting the tripartite genuine entanglement using current techniques in experiments. We also investigate the lower bound of genuine multipartite entanglement concurrence.
We derive Bohm's trajectories from Bell's beables for arbitrary bipartite systems composed by dissipative noninteracting harmonic oscillators at finite temperature. As an application of our result, we calculate the Bohmian trajectories of particles described by a generalized Werner state, comparing the trajectories when the sate is either separable or entangled. We show that qualitative differences appear in the trajectories for entangled states as compared with those for separable states.
We develop a systematic method to construct the Bell states of a qubit bipartite system while taking $SU(2)$ group as the basis group. An alternative formulation of fidelity, called $SU(2)$ fidelity, is proposed which gives the Bell-CHSH inequality. Taking analogy from the octet theory governed by the $SU(3)$ group, it is shown that the method can be easily generalized to derive all possible entangled states and the $SU(3)$ fidelity of the higher dimensional bipartite qutrit system. The properties of these states are studied which reveal the existence of some inequalities for the qutrit bipart…
A class of quantum protocols to teleport bipartite (entangled) states of two qubits is suggested. Our schemes require a single entangled pair shared by the two parties and the transmission of three bits of classical information, as well as a two-qubit gate with an additional qubit at the receiver' location. Noisy quantum channels are considered and the effects on both the teleportation fidelity and the entanglement of the replica are evaluated.
Construction of genuinely entangled multipartite subspaces with certain characteristics has become a relevant task in various branches of quantum information. Here we show that such subspaces can be obtained from an arbitrary collection of bipartite entangled subspaces under joining of their adjacent subsystems. In addition, it is shown that direct sums of such constructions under certain conditions are genuinely entangled. These facts are then used in detecting entanglement of tensor products of mixed states and constructing subspaces that are distillable across every bipartite cut, where for…
Recently the explicit applicability of bound entanglement in quantum cryptography has been shown. In this paper some of recent results respecting this topic are reviewed. In particular relevant notions and definitions are reminded. The new construction of bound entangled states containing secure correlations is presented. It provides low dimensional 6\otimes6 bound entangled states with nonzero distillable key.
We propose a particular encoding for bipartite entangled states derived from multipartite cluster-type entangled coherent states (CTECSs). We investigate the effects of amplitude damping on the entanglement content of this bipartite state, as well as its usefulness as a quantum channel for teleportation. We find interesting relationships among the amplitude of the coherent states constituting the CTECSs, the number of subsystems forming the logical qubits (redundancy), and the extent to which amplitude damping affects the entanglement of the channel. For instance, in the sense of sudden death …
We examine the existence and structure of particular sets of mutually unbiased bases (MUBs) in bipartite qudit systems. In contrast to well-known power-of-prime MUB constructions, we restrict ourselves to using maximally entangled stabilizer states as MUB vectors. Consequently, these bipartite entangled stabilizer MUBs (BES MUBs) provide no local information, but are sufficient and minimal for decomposing a wide variety of interesting operators including (mixtures of) Jamiolkowski states, entanglement witnesses and more. The problem of finding such BES MUBs can be mapped, in a natural way, to …
We prove a powerful theorem for tripartite remote entanglement distribution protocols that establishes an upper bound on the amount of entanglement of formation that can be created between two single-qubit nodes of a quantum network. Our theorem also provides an operational interpretation of concurrence as a type of entanglement capacity.
The positivity of the partial transpose is in general only a necessary condition for separability. There exist quantum states that are not separable, but nevertheless are positive under partial transpose. States of this type are known as bound entangled states meaning that these states are entangled but they do not allow distillation of pure entanglement by means of local operations and classical communication (LOCC). We present a parametrization of a class of $2\times 2$ bound entangled Gaussian states for bipartite continuous-variable quantum systems with two modes on each side. We propose a…
Quantum correlations in compound systems are of great importance, and they are fundamental resource for the development of quantum computation protocols and quantum information. In this work we construct bipartite pure coherent states using the $f$-deformed oscillator formalism and the Barut-Girardello deformation function. Also we construct the extension of these systems to mixed states using Werner like states. For the states constructed, we study the dependence of the entanglement and quantum discord upon the deformation and mixed parameters.
We study the explicit relation between violation of Bell inequalities and bipartite distillability of multi-qubit states. It has been shown that even though for $N\ge 8$ there exist $N$-qubit bound entangled states which violates a Bell inequality [Phys. Rev. Lett. {\bf 87}, 230402 (2001)], for all the states violating the inequality there exists at least one splitting of the parties into two groups such that pure-state entanglement can be distilled [Phys. Rev. Lett. {\bf 88}, 027901 (2002)]. We here prove that for all $N$-qubit states violating the inequality the number of distillable biparti…
We explore the relation between the rank of a bipartite density matrix and the existence of bound entanglement. We show a relation between the rank, marginal ranks, and distillability of a mixed state and use this to prove that any rank n bound entangled state must have support on no more than an n \times n Hilbert space. A direct consequence of this result is that there are no bipartite bound entangled states of rank two. We also show that a separability condition in terms of a quantum entropy inequality is associated with the above results. We explore the idea of how many pure states are nee…
Assuming the validity of a conjecture in quant-ph/9910026 and quant-ph/9910022 we show that the distillable entanglement for two bipartite states, each of which individually has zero distillable entanglement, can be nonzero. We show that this also implies that the distillable entanglement is not a convex function. Our example consists of the tensor product of a bound entangled state based on an unextendible product basis with a Werner state which lies in the class of conjectured undistillable states.
The idea of self-testing is to render guarantees concerning the inner workings of a device based on the measurement statistics. It is one of the most formidable quantum certification and benchmarking schemes. Recently it was shown by Coladangelo et. al. (Nat Commun 8, 15485 (2017)) that all pure bipartite entangled states can be self tested in the device independent scenario by employing subspace methods introduced by Yang et. al. (Phys. Rev. A 87, 050102(R)). Here, we have adapted their method to show that any bipartite pure entangled state can be certified in the semi-device independent scen…
Quantum entanglement in multipartite systems cannot be shared freely. In order to illuminate basic rules of entanglement sharing between qubits we introduce a concept of an entangled structure (graph) such that each qubit of a multipartite system is associated with a point (vertex) while a bi-partite entanglement between two specific qubits is represented by a connection (edge) between these points. We prove that any such entangled structure can be associated with a pure state of a multi-qubit system. Moreover, we show that a pure state corresponding to a given entangled structure is a superpo…
We derive lower limits on the inefficiency and classical communication costs of dilution between two-term bipartite pure states that are partially entangled. We first calculate explicit relations between the allowable error and classical communication costs of entanglement dilution using the protocol of Lo-Popescu and then consider a two-stage dilution from singlets with this protocol followed by some unknown protocol for conversion between partially entangled states. Applying the lower bounds on classical communication and inefficiency of Harrow-Lo and Hayden-Winter to this two-stage protocol…
We present the full experimental reconstruction of Gaussian entangled states generated by a type--II optical parametric oscillator (OPO) below threshold. Our scheme provides the entire covariance matrix using a single homodyne detector and allows for the complete characterization of bipartite Gaussian states, including the evaluation of purity, entanglement and nonclassical photon correlations, without a priori assumptions on the state under investigation. Our results show that single homodyne schemes are convenient and robust setups for the full characterization of OPO signals and represent a…
Quantum coherence and quantum entanglement are two strong pillars in quantum information theory. We study here for the possibility of any connection between these two important aspects of quantum mechanics while studying the entanglement detection problem for the detection of bipartite higher dimensional entangled states and multipartite entangled states. To achieve our goal, we derive the lower bound $L$ of $l_{1}$ norm of coherence of bipartite qubit-qudit system using the criterion that detect entanglement. Furthermore, we deduce the upper bound $U$ of $l_{1}$ norm of coherence of separable…
The determination of genuine entanglement is a central problem in quantum information processing. We investigate the tripartite state as the tensor product of two bipartite entangled states by merging two systems. We show that the tripartite state is a genuinely entangled state when the range of both bipartite states are entanglement-breaking subspaces. We further investigate the tripartite state when one of the two bipartite states has rank two. Our results provide the latest progress on a conjecture proposed in the paper [Yi Shen $\textit{et al}$, J. Phys. A 53, 125302 (2020)]. We apply our …
Bipartite quantum states are classified into three categories: separable states, bound entangled states, and free entangled states. It is of great importance to characterize these families of states for the development of quantum information science. In this paper, I show that the separable states and the bound entangled states have a common spectral property. More precisely, I prove that for undistillable -- separable and bound entangled -- states, the eigenvalue vector of the global system is majorized by that of the local system. This result constitutes a new sufficient condition for distil…
Device-independent self-testing allows to uniquely characterize the quantum state shared by untrusted parties (up to local isometries) by simply inspecting their correlations, and requiring only minimal assumptions, namely a no-signaling constraint on the untrusted parties and the validity of quantum mechanics. The device-independent approach exploits the fact that certain non-local correlations can be uniquely achieved by measurements on a particular quantum state. We can think of these correlations as a "classical fingerprint" of the self-tested quantum state. In this work, we answer affirma…
We show that a violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality can be demonstrated in a certain kind of Bell experiment for all bipartite entangled states. Our protocol allows local filtering measurements and involves shared ancilla states that do not themselves violate CHSH. Our result follows from two main steps. We first provide a simple characterization of the states that violate the CHSH-inequality after local filtering operations in terms of witness-like operators. Second, we prove that for each entangled state $σ$, there exists another state $ρ$ not violating CHSH, such th…
The ability to teleport entanglement through maximally entangled mixed states as defined by concurrence and linear entropy is studied. We show how the teleported entanglement depends on the quality of the quantum channel used, as defined through its entanglement and mixedness, as well as the form of the target state to be teleported. We present new results based on the fidelity of the teleported state as well as an experimental set-up that is immediately implementable with currently available technology.
According to usual definitions, entangled states cannot be given a separable decomposition in terms of products of local density operators. If we relax the requirement that the local density operators be positive, then an entangled quantum state may admit a separable decomposition in terms of more general sets of single-system operators. This form of separability can be used to construct classical models and simulation methods when only restricted set of measurements are available. With such motivations in mind, we ask what are the smallest such sets of local operators such that a pure biparti…
Ising interaction between qubits could produce distortion in entangled pairs generated for engineering purposes (as in quantum computation) in presence of parasite magnetic fields, destroying or altering the expected behavior of process in which is projected to be used. Quantum control could be used to correct that situation in several ways. Sometimes the user should be make some measurement upon the system to decide which is the best control scheme; other posibility is try to reconstruct the system using similar procedures without perturbate it. In the complete pictures both schemes are prese…
Self-testing refers to a method with which a classical user can certify the state and measurements of quantum systems in a device-independent way. Especially, the self-testing of entangled states is of great importance in quantum information process. A comprehensible example is that violating the CHSH inequality maximally necessarily implies the bipartite shares a singlet. One essential question in self-testing is that, when one observes a non-maximum violation, how close is the tested state to the target state (which maximally violates certain Bell inequality)? The answer to this question des…
Entangled quantum states can be given a separable decomposition if we relax the restriction that the local operators be quantum states. Motivated by the construction of classical simulations and local hidden variable models, we construct `smallest' local sets of operators that achieve this. In other words, given an arbitrary bipartite quantum state we construct convex sets of local operators that allow for a separable decomposition, but that cannot be made smaller while continuing to do so. We then consider two further variants of the problem where the local state spaces are required to contai…
The geometric phase (GP) for bipartite systems in transverse external magnetic fields is investigated in this paper. Two different situations have been studied. We first consider two non-interacting particles. The results show that because of entanglement, the geometric phase is very different from that of the non-entangled case. When the initial state is a Werner state, the geometric phase is, in general, zero and moreover the singularity of the geometric phase may appear with a proper evolution time. We next study the geometric phase when intra-couplings appear and choose Werner states as th…
Entanglement plays a crucial role in quantum information protocols, thus the dynamical behavior of entangled states is of a great importance. In this paper we suggest a useful scheme that permits a direct measure of entanglement in a two-qubit cavity system. It is realized in the cavity-QED technology utilizing atoms as fying qubits. To quantify entanglement we use the concurrence. We derive the conditions, which assure that the state remains entangled in spite of the interaction with the reservoir. The phenomenon of sudden death entanglement (ESD) in a bipartite system subjected to squeezed v…
Building upon the results of [R. Augusiak et al., Phys. Rev. Lett. 115, 030404 (2015)] we develop a general approach to the generation of genuinely entangled multipartite states of any number of parties from genuinely entangled states of a fixed number of parties, in particular, the bipartite entangled ones. In our approach, certain isometries whose output subspaces are either symmetric or genuinely entangled in some multipartite Hilbert spaces are applied to local subsystems of bipartite entangled or multipartite genuinely entangled quantum states. To prove that entanglement of the resulting …
The relations of antilinear maps, bipartite states and quantum channels is summarized. Antilinear maps are applied to describe bipartite states and entanglement. Teleportation is treated in this general formalism with an emphasis on conditional schemes applying partially entangled pure states. It is shown that in such schemes the entangled state shared by the parties, and those measured by the sender should ``match'' each other.
We consider the entanglement manipulation capabilities of the universal covariant quantum cloner or quantum processor circuit for quantum bits. We investigate its use for cloning a member of a bipartite or a genuine tripartite entangled state of quantum bits. We find that for bipartite pure entangled states a nontrivial behavior of concurrence appears, while for GHZ entangled states a possibility of the partial extraction of bipartite entanglement can be achieved.
We prove, using a new method based on map-state duality, lower bounds on entanglement resources needed to deterministically implement a bipartite unitary using separable (SEP) operations, which include LOCC (local operations and classical communication) as a particular case. It is known that the Schmidt rank of an entangled pure state resource cannot be less than the Schmidt rank of the unitary. We prove that if these ranks are equal the resource must be uniformly (maximally) entangled: equal nonzero Schmidt coefficients. Higher rank resources can have less entanglement: we have found numerica…
Witness Operator Provides Better Estimate of the Lower Bound of Concurrence of Bipartite Bound Entangled States in $d_{1}\otimes d_{2}$ Dimensional System
We illustrate a procedure to generate a bipartite, entangled compass state, which shows sub-Planck structure. The proposed method uses the interaction of a standing wave laser field, with two, two-level atoms and relies on the ability of this system to choose certain mesoscopic bipartite states to couple with the internal degrees of freedom. An appropriate measurement on the internal degrees of freedom then leads to the entangled state, which shows sub-Planck structures, desired for quantum metrology.
Subtle quantum properties offer exciting new prospects in optical communications. Quantum entanglement enables the secure exchange of cryptographic keys and the distribution of quantum information by teleportation. Entangled bright beams of light attract increasing interest for such tasks, since they enable the employment of well-established classical communications techniques. However, quantum resources are fragile and undergo decoherence by interaction with the environment. The unavoidable losses in the communication channel can lead to a complete destruction of useful quantum properties -- …
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…
Fully entangled fraction is a definition for bipartite states, which is tightly related to bipartite maximally entangled states, and has clear experimental and theoretical significance. In this work, we generalize it to multipartite case, we call the generalized version multipartite fully entangled fraction (MFEF). MFEF measures the closeness of a state to GHZ states. The analytical expressions of MFEF are very difficult to obtain except for very special states, however, we show that, the MFEF of any state is determined by a system of finite-order polynomial equations. Therefore, the MFEF can …
A large number of inequalities have been proposed for the detection of bipartite continuous variable entanglement and Einstein-Podolsky-Rosen steering. Many of these are based on either measured or inferred variances and are relatively easily measured via homodyne detection of optical outputs. In this work we examine a number of bipartite inequalities in the process of cascaded downconversion and sum-frequency generation inside an optical cavity. This has previously been predicted to be a potential source of continuous-variable tripartite entanglement, therefore we know that there are no separ…
We propose a scheme for distillation of free bipartite entanglement from bipartite bound-entangled states. The crucial element of our scheme is an ancillary system that is coupled to the initial bound-entangled state via appropriate weak measurements. We show that in this protocol free entanglement can be always generated with nonzero probability by using a single copy of the bound-entangled state. We also derive a lower bound on the entanglement cost of the protocol, and conclude that, on average, applying weaker measurements results in relatively higher values of free entanglement as well as…
We investigate the asymptotic rates of entanglement transformations for bipartite mixed states by local operations and classical communication (LOCC). We analyse the relations between the rates for different transitions and obtain simple lower and upper bound for these transitions. In a transition from one mixed state to another and back, the amount of irreversibility can be different for different target states. Thus in a natural way, we get the concept of "amount" of irreversibility in asymptotic manipulations of entanglement. We investigate the behaviour of these transformation rates for di…
It has been recently shown (Bartlett et al. 2003) that information encoded into relative degrees of freedom enables communication without a common reference frame using entangled bipartite states. In this case the relative information stored in the two-qubit system is shared between the polarization degrees of freedom and the degree of entanglement. In the present article a specific state discrimination problem is envisioned where the degree of entanglement carries the only relative parameter, so that certain maximally entangled states are perfectly distinguishable, while discrimination of pro…
Certain quantum information tasks require entanglement of assistance, namely a reduction of a tripartite entangled state to a bipartite entangled state via local measurements. We establish that 'concurrence of assistance' (CoA) identifies capabilities and limitations to producing pure bipartite entangled states from pure tripartite entangled states and prove that CoA is an entanglement monotone for $(2\times2\times n)$-dimensional pure states. Moreover, if the CoA for the pure tripartite state is at least as large as the concurrence of the desired pure bipartite state, then the former may be t…
