We study how well answer to the question ``Is the given quantum state equal to a certain maximally entangled state?'' using LOCC, in the context of hypothesis testing. Under several locality and invariance conditions, optimal tests will be derived for several special cases by using basic theory of group representations. Some optimal tests are realized by performing quantum teleportation and checking whether the state is teleported. We will also give a finite process for realizing some optimal tests. The performance of the tests will be numerically compared.
Group symmetric LOCC measurement for detecting maximally entangled state is considered. Usually, this type measurement has continuous-valued outcomes. However, any realizable measurement has finite-valued outcomes. This paper proposes discrete realizations of such a group symmetric LOCC measurement.
We show that there exist sets of three mutually orthogonal $d$-dimensional maximally entangled states which cannot be perfectly distinguished using one-way local operations and classical communication (LOCC) for arbitrarily large values of $d$. This contrasts with several well-known families of maximally entangled states, for which any three states can be perfectly distinguished. We then show that two-way LOCC is sufficient to distinguish these examples. We also show that any three mutually orthogonal $d$-dimensional maximally entangled states can be perfectly distinguished using measurements …
In the asymptotic setting, the optimal test for hypotheses testing of the maximally entangled state is derived under several locality conditions for measurements. The optimal test is obtained in several cases with the asymptotic framework as well as the finite-sample framework. In addition, the experimental scheme for the optimal test is presented.
In this paper, we study the one-way local operations and classical communication (LOCC) problem. In $\mathbb{C}^d\otimes\mathbb{C}^d$ with $d\geq4$, we construct a set of $3\lceil\sqrt{d}\rceil-1$ one-way LOCC indistinguishable maximally entangled states which are generalized Bell states. Moreover, we show that there are four maximally entangled states which cannot be perfectly distinguished by one-way LOCC measurements for any dimension $d\geq 4$.
Entanglement is a resource in quantum information theory when state manipulation is restricted to Local Operations assisted by Classical Communication (LOCC). It is therefore of paramount importance to decide which LOCC transformations are possible and, particularly, which states are maximally useful under this restriction. While the bipartite maximally entangled state is well known (it is the only state that cannot be obtained from any other and, at the same time, it can be transformed to any other by LOCC), no such state exists in the multipartite case. In order to cope with this fact, we in…
It is shown that while entanglement remains a significant factor in discriminating a set of mutually orthogonal entangled states perfectly by local operations and classical communication (LOCC), entanglement content is not. In particular, for a class of entangled multi-qubit states, the maximum number of perfectly LOCC distinguishable orthogonal states is shown to be independent of the average entanglement of the states, and the spatial configuration with respect to which LOCC operations may be carried out. It is also pointed out that the make-up of an ensemble, that is, whether it consists on…
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…
Entanglement is a resource to overcome the natural restriction of operations used for state manipulation to Local Operations assisted by Classical Communication (LOCC). Hence, a bipartite maximally entangled state is a state which can be transformed deterministically into any other state via LOCC. In the multipartite setting no such state exists. There, rather a whole set, the Maximally Entangled Set of states (MES), which we recently introduced, is required. This set has on the one hand the property that any state outside of this set can be obtained via LOCC from one of the states within the …
In classical information theory one can, in principle, produce a perfect copy of any input state. In quantum information theory, the no cloning theorem prohibits exact copying of nonorthogonal states. Moreover, if we wish to copy multiparticle entangled states and can perform only local operations and classical communication (LOCC), then further restrictions apply. We investigate the problem of copying orthogonal, entangled quantum states with an entangled blank state under the restriction to LOCC. Throughout, the subsystems have finite dimension D. We show that if all of the states to be copi…
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…
A way to characterize quantum nonlocality is to see difference in the figure of merit between LOCC optimal protocol and globally optimal protocol in doing certain task, e.g., state estimation, state discrimination, cloning and broadcasting. Especially, we focus on the case where $n$ tensor of unknown states. Our conclusion is that separable pure states are more non-local than entangled pure states. More specifically, the difference in the figure of the merit is exponentially small if the state is entangled, and the exponent is log of the largest Schmidt coefficient. On the other hand, in many …
Entanglement theory is formulated as a quantum resource theory in which the free operations are local operations and classical communication (LOCC). This defines a partial order among bipartite pure states that makes it possible to identify a maximally entangled state, which turns out to be the most relevant state in applications. However, the situation changes drastically in the multipartite regime. Not only do there exist inequivalent forms of entanglement forbidding the existence of a unique maximally entangled state, but recent results have shown that LOCC induces a trivial ordering: almos…
A novel criterion of extracting thermodynamical work from a bipartite pure qudit-entangled state by means of local operation and classical communication (LOCC) has been presented. We have shown that non-vanishing $G$-concurrence is a necessary condition to extract work from an higher dimensional entangled state in LOCC paradigm.
Local operations assisted by classical communication (LOCC) constitute the free operations in entanglement theory. Hence, the determination of LOCC transformations is crucial for the understanding of entanglement. We characterize here almost all LOCC transformations among pure multipartite multilevel states. Combined with the analogous results for qubit states shown by Gour \emph{et al.} [J. Math. Phys. 58, 092204 (2017)], this gives a characterization of almost all local transformations among multipartite pure states. We show that nontrivial LOCC transformations among generic, fully entangled…
Entanglement is the resource to overcome the restriction of operations to Local Operations assisted by Classical Communication (LOCC). The Maximally Entangled Set (MES) of states is the minimal set of n-partite pure states with the property that any truly n-partite entangled pure state can be obtained deterministically via LOCC from some state in this set. Hence, this set contains the most useful states for applications. In this work we characterize the MES for generic three qutrit states. Moreover, we analyze which generic three qutrit states are reachable (and convertible) under LOCC transfo…
We consider generic pure $n$-qubit states and a general class of pure states of arbitrary dimensions and arbitrarily many subsystems. We characterize those states which can be reached from some other state via Local Operations assisted by finitely many rounds of Classical Communication ($LOCC_{\mathbb{N}}$). For $n$ qubits with $n>3$ we show that this set of states is of measure zero, which implies that the maximally entangled set is generically of full measure if restricted to the practical scenario of $LOCC_{\mathbb{N}}$. Moreover, we identify a class of states for which any $LOCC_{\mathbb{N…
Mutual convertibility of bound entangled states under local quantum operations and classical communication (LOCC) is studied. We focus on states associated with unextendible product bases (UPB) in a system of three qubits. A complete classification of such UPBs is suggested. We prove that for any pair of UPBs $S$ and $T$ the associated bound entangled states $ρ_S$ and $ρ_T$ can not be converted to each other by LOCC, unless $S$ and $T$ coincide up to local unitaries. More specifically, there exists a finite precision $ε>0$ such that for any LOCC protocol mapping $ρ_S$ into a probabilistic ense…
Given two two-qubit pure states characterized by their Schmidt numbers we investigate an optimal strategy to convert the states between themselves with respect to their local unitary invariance. We discuss the efficiency of this transformation and its connection to LOCC convertibility properties between two single-copy quantum states. As an illustration of the investigated transformations we present a communication protocol where in spite of all expectations a shared maximally entangled pair between two participants is the worst quantum resource.
We show that a set of linearly independent quantum states $\{(U_{m,n}\otimes I)ρ^{AB}(U_{m,n}^{\dagger}\otimes I)\}_{m,n=0}^{d-1}$, where $U_{m,n}$ are generalized Pauli matrices, cannot be discriminated deterministically or probabilistically by local operations and classical communications (LOCC). On the other hand, any $l$ maximally entangled states from this set are locally distinguishable if $l(l-1)\le 2d$. The explicit projecting measurements are obtained to locally discriminate these states. As an example, we show that four Werner states are locally indistinguishable.
Entangled states can help in quantum state discrimination by local operations and classical communication (LOCC). For example, a Bell state is necessary (and sufficient) to perfectly discriminate a set of either three or four Bell states by LOCC. In this paper, we consider the task of LOCC discrimination of the states of noisy Bell ensembles, where a given ensemble consists of the states obtained by mixing the Bell states with an arbitrary two-qubit state with nonzero probabilities. It is proved that a Bell state is required for optimal discrimination by LOCC, even though the ensembles do not …
In order to cope with the fact that there exists no single maximally entangled state (up to local unitaries) in the multipartite setting, we introduced in [J. I. de Vicente, C. Spee and B. Kraus, Phys. Rev. Lett. 111, 110502 (2013)] the maximally entangled set of n-partite quantum states. This set consists of the states that are most useful under conversion of pure states via Local Operations assisted by Classical Communication (LOCC). We will review our results here on the maximally entangled set of three- and generic four-qubit states. Moreover, we will discuss the preparation of arbitrary (…
We study the problem of distinguishing quantum states using local operations and classical communication (LOCC). A question of fundamental interest is whether there exist sets of $k \leq d$ orthogonal maximally entangled states in $\mathbb{C}^{d}\otimes\mathbb{C}^{d}$ that are not perfectly distinguishable by LOCC. A recent result by Yu, Duan, and Ying [Phys. Rev. Lett. 109 020506 (2012) -- arXiv:1107.3224 [quant-ph]] gives an affirmative answer for the case $k = d$. We give, for the first time, a proof that such sets of states indeed exist even in the case $k < d$. Our result is constructive …
We consider the question of perfect local distinguishability of mutually orthogonal bipartite quantum states, with the property that every state can be specified by a unitary operator acting on the local Hilbert space of Bob. We show that if the states can be exactly discriminated by one-way LOCC where Alice goes first, then the unitary operators can also be perfectly distinguished by an orthogonal measurement on Bob's Hilbert space. We give examples of sets of N<=d maximally entangled states in $d \otimes d $ for d=4,5,6 that are not perfectly distinguishable by one-way LOCC. Interestingly fo…
We study the problem of distinguishing maximally entangled quantum states by using local operations and classical communication (LOCC). A question of fundamental interest is whether any three maximally entangled states in $\mathbb{C}^d\otimes\mathbb{C}^d (d\geq 4)$ are distinguishable by LOCC. In this paper, we restrict ourselves to consider the generalized Bell states. And we prove that any three generalized Bell states in $\mathbb{C}^d\otimes\mathbb{C}^d (d\geq 4)$ are locally distinguishable.
Recently, entanglement concentration was explicitly shown to be irreversible. However, it is still not clear what kind of states can be reversibly converted in the asymptotic setting by LOCC when neither the initial nor the target state is maximally entangled. We derive the necessary and sufficient condition for the reversibility of LOCC conversions between two bipartite pure entangled states in the asymptotic setting. In addition, we show that conversion can be achieved perfectly with only local unitary operation under such condition except for special cases. Interestingly, our result implies…
We consider the problem of establishing entangled states of optimal singlet fraction and negativity between two remote parties for every use of a noisy quantum channel and trace-preserving LOCC under the assumption that the parties do not share prior correlations. We show that for a family of quantum channels in every finite dimension $d\geq3$, one-shot optimal singlet fraction and entanglement negativity are attained only with appropriate nonmaximally entangled states. We further show that the generalization of the formula that exactly computes one-shot optimal singlet fraction for qubit chan…
We study maximally entangled states and fully entangled fraction in general d'\otimes d (d'\geq d) systems. Necessary and sufficient conditions for maximally entangled pure and mixed states are presented. As a natural generalization of the usual fully entangled fraction for d\otimes d systems, we define the maximal overlap between a given quantum state and the maximally entangled states as the fully entangled fraction in d'\otimes d systems. The properties of this fully entangled fraction and its relations to quantum teleportation have been analyzed. The witness for detecting maximally entangl…
We demonstrate that one maximally entangled state is sufficient and necessary to distinguish a complete basis of maximally entangled states by local operation and classical communication.
In a recent work [Phys. Rev. Lett. 120, 170502 (2018)], Pallister et al. proposed an optimal strategy to verify non-maximally entangled two-qubit pure states under the constraint that the accessible measurements being locally projective and non-adaptive. Their nice result leads naturally to the question: What is the optimal strategy among general LOCC measurements? In this Letter, we answer this problem completely for two-qubit pure states. To be specific, we give the optimal strategy for each of the following available classes of measurements: (i) local operations and one-way classical commun…
We study unextendible maximally entangled basis in arbitrary bipartite spaces. A systematic way of constructing a set of $d^{2}$ orthonormal maximally entangled states in $\mathbb{C}^{d}\bigotimes\mathbb{C}^{d'}(\frac{d'}{2}<d<d')$ is provided. The complementary space of the set of these $d^{2}$ orthonormal maximally entangled states contains no maximally entangled states that are orthogonal to all of them. Furthermore, we investigate mutually unbiased bases in which all the bases are unextendible maximally entangled ones. We present two unextendible maximally entangled bases in $\mathbb{C}^{2…
We explore the question of using an entangled state as a universal resource for implementing quantum measurements by local operations and classical communication (LOCC). We show that for most systems consisting of three or more subsystems, there is no entangled state from the same space that can enable all measurements by LOCC. This is in direct contrast to the bipartite case, where a maximally entangled state is an universal resource. Our results are obtained showing an equivalence between the problem of local state transformation and that of entanglement-assisted local unambiguous state disc…
In this paper, we study the superposition invariance of unitary operators and maximally entangled state respectively. Furthermore, we discuss the set of orthogonal maximally entangled states. We find that orthogonal basis of maximally entangled states can be divided into k subspaces. It is shown that some entanglement properties of superposed state in every subspace are invariant.
It is known that from entangled states that have positive partial transpose it is not possible to distill maximally entangled states by local operations and classical communication (LOCC). A long-standing open question is whether maximally entangled states can be distilled from every state with a non-positive partial transpose. In this paper we study a possible approach to the question consisting of enlarging the class of operations allowed. Namely, instead of LOCC operations we consider k-extendible operations, defined as maps whose Choi-Jamiolkowski state is k-extendible. We find that this c…
We generalize the notion of unextendible maximally entangled basis from bipartite systems to multipartite quantum systems. It is proved that there do not exist unextendible maximally entangled bases in three-qubit systems. Moreover,two types of unextendible maximally entangled bases are constructed in tripartite quantum systems and proved to be not mutually unbiased.
We introduce the notion of the unextendible maximally entangled basis (UMEB), a set of orthonormal maximally entangled states in d \times d consisting of fewer that d^2 vectors which have no additional maximally entangled vectors orthogonal to all of them. We prove that UMEBs don't not exist for d=2 and give an explicit constructions for a 6-member UMEB with d=3 and a 12-member UMEB with d=4.
Paolo Facchi et al.[Phys. Rev. A. 77, 060304(R) (2009)] presented a maximally multipartite entangled state(MMES). Here we give the criterion for four-qubit state.and some new characterizations of the maximally entangled four-qubit state are given.
A proof of Bell's theorem without inequalities for two maximally entangled particles is proposed using the technique of quantum teleportation. It follows Hardy's arguments for a non-maximally entangled state with the help of two auxiliary particles without correlation. The present proof can be tested by measurements with 100% probability.
We present a generalized criterion for maximally entangled nine-qubit states, whose minimum averaged subsystem purity should be equal to 1/14. In this note, we prove that absolutely maximally entangled state for nine qubits does not exist. Further, we construct a new genuine nine-qubit maximally entangled states of 110,15 and 1 balanced purities equal to 1/16, , 1/8 and and 1/4, respectively. We found that, the marginal density matrices for subsystems of 1,2,3- qubits all completely mixed in these states, therefore, maximally entangled nine-qubit state is a 3-uniform state.
Maximally entangled states should maximally violate the Bell inequality. In this paper, it is proved that all two-qubit states that maximally violate the Bell-Clauser-Horne-Shimony-Holt inequality are exactly Bell states and the states obtained from them by local unitary transformations. The proof is obtained by using the certain algebraic properties that Pauli's matrices satisfy. The argument is extended to the three-qubit system. Since all states obtained by local unitary transformations of a maximally entangled state are equally valid entangled states, we thus give the characterizations of …
We investigate the conditions under which a set $\SC$ of pure bipartite quantum states on a $D\times D$ system can be locally cloned deterministically by separable operations, when at least one of the states is full Schmidt rank. We allow for the possibility of cloning using a resource state that is less than maximally entangled. Our results include that: (i) all states in $\SC$ must be full Schmidt rank and equally entangled under the $G$-concurrence measure, and (ii) the set $\SC$ can be extended to a larger clonable set generated by a finite group $G$ of order $|G|=N$, the number of states …
We introduce the notion of a task-oriented maximally entangled state (TMES). This notion depends on the tasks for which a quantum state is used as the resource. This concept may be more fruitful than that of a general maximally entangled state in the case of a multipartite system. We illustrate this idea by giving an operational definition of maximally entangled states on the basis of communication tasks of teleportation and superdense coding. We also give examples and a procedure to obtain such TMESs for {\em n}-qubit systems.
Gram matrix approach to an entanglement analysis of pure states describing $(d,\infty)$ -- quantum systems is being introduced. In particular, maximally entangled states are described as those having a special forms of the corresponding Gram matrices.
We find that the mixed maximally entangled states exist and prove that the form of the mixed maximally entangled states is unique in terms of the entanglement of formation. Moreover, even if the entanglement is quantified by other entanglement measures, this conclusion is still proven right. This result is a supplementary to the generally accepted fact that all maximally entangled states are pure. These states possess important properties of the pure maximally entangled states, for example, these states can be used as a resource for faithful teleportation and they can be distinguished perfectl…
By going beyond Hubbard Hamiltonian we reflected correlation effects accurately in the wavefunctions of H2. Using ab initio e-e interaction parameters resulted maximally entangled ground and third excited states. We assigned this maximally entangled character to the nonmagnetic (S=0) property of these states and also the minimally entangled character of the first excited states to its magnetic property. By switching on a magnetic field an entangled state with Sz = 0 can be extracted from a minimally entangled degenerate magnetic state. We suggest that presence of a moderate correlated system a…
We introduce the concept of the locally unextendible non-maximally entangled basis (LUNMEB) in $H^d \bigotimes H^d$. It is shown that such a basis consists of $d$ orthogonal vectors for a non-maximally entangled state. However, there can be a maximum of $(d-1)^2$ orthogonal vectors for non-maximally entangled state if it is maximally entangled in $(d-1)$ dimensional subspace. Such a basis plays an important role in determining the number of classical bits that one can send in a superdense coding protocol using a non-maximally entangled state as a resource. By constructing appropriate POVM oper…
Motivated by the desire to better understand the class of quantum operations on bipartite systems that preserve positivity of partial transpose (PPT operations) and its relation to the class LOCC (local operations and classical communication), we present some results on deterministic bipartite pure state transformations by PPT operations. Restricting our attention to the case where we start with a rank K maximally entangled state, we give a necessary condition for transforming it into a given pure state, which we show is also sufficient when K is two and the final state has Schmidt rank three.…
A theoretical scheme for bidirectional swapping quantum controlled teleportation is presented using the entanglement property of maximally entangled five-qubit state, i.e., Alice wants to transmit a entangled state of particle a to Bob and Bob wants to transmit a entangled state of particle b to Alice via the control of the supervisor Charlie.
Relative and center of mass cordinates are used to generalize mutually unbiased bases (MUB) and define mutually unbiased bases (MUCB). Maximal entangled states are given as product staes in the collective varibles
