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arXiv:0707.4295 · Journal of Physics A Mathematical and Theoretical

Task-oriented maximally entangled states

Pankaj Agrawal, B Pradhan

Published May 17, 2010• 14 citations
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Abstract

We introduce the notion of a task-oriented maximally entangled state (TMES). This notion depends on the task for which a quantum state is used as the resource. TMESs are the states that can be used to carry out the task maximally . This concept may be more useful 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 n -qubit systems.

DOI
10.1088/1751-8113/43/23/235302

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We introduce the notion of a task-oriented maximally entangled state (TMES). This notion depends on the task for which a quantum state is used as the resource. TMESs are the states that can be used to carry out the task maximally . This concept may be more useful 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 n -qubit systems.
We explicate an example of the so called Task oriented Maximally Entangled states (TMES) in the context of teleportation and superdense coding. In physical situation, this state can emerge from decoherence of more entangled state or may be prepared for the purpose at hand, with less entanglement resource. We find that a variant of the Brown state can be utilized for perfect teleportation of single and two qubit states. In case of superdense coding, it is observed that five cbits can be transmitted by sending only three qubits.
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 introduce here the notion of the maximally entangled set (MES) of n-partite states. This is the set of states which are maximally useful under LOCC manipulation; i.e., any state outside of this set can be obtained via LOCC from one of the states within the set and no state in the set can be obtained from any other state via LOCC. We determine the MES for states of three and four qubits and provide a simple characterization for them. In both cases, infinitely many states are required. However, while the MES is of measure zero for 3-qubit states, almost all 4-qubit states are in the MES. This is because, in contrast to the 3-qubit case, deterministic LOCC transformations are almost never possible among fully entangled four-partite states. We determine the measure-zero subset of the MES of LOCC convertible states. This is the only relevant class of states for entanglement manipulation.
We introduce the notion of maximally multipartite entangled states of nnn qubits as a generalization of the bipartite case. These pure states have a bipartite entanglement that does not depend on the bipartition and is maximal for all possible bipartitions. They are solutions of a minimization problem. Examples for small nnn are investigated, both analytically and numerically.
Bell-state measurements are essential ingredients in many protocols for quantum information processing, ranging from quantum teleportation and dense coding to entanglement distribution in quantum networks. Their power relies on the fact that they are measurements in an entangled basis of a two-particle system and that the used Bell-state basis can be generated from a single Bell state by local unitary transformations. How can these measurements be generalized to more particles? We develop a general framework for this state-to-measurement problem: We introduce a hierarchy of classes of measurement bases, distinguished by the local transformations the parties may use for their generation from a single state. This leads to a generalization of the concept of maximally entangleable (or weighted hypergraph) states and the identification of a novel maximally entangled basis of four qubits, being a candidate for data-hiding tasks or distillation protocols. Finally, we prove that not all forms of entanglement can be encoded in an entire measurement basis.
For many applications the presence of a quantum advantage crucially depends on the availability of resourceful states. Although the resource typically depends on the particular task, in the context of multipartite systems entangled quantum states are often regarded as resourceful. We propose an algorithmic method to find maximally resourceful states of several particles for various applications and quantifiers. We discuss in detail the case of the geometric measure, identifying physically interesting states and delivering insights to the problem of absolutely maximally entangled states. Moreover, we demonstrate the universality of our approach by applying it to maximally entangled subspaces, the Schmidt-rank, the stabilizer rank as well as the preparability in triangle networks.