On two maximally entangled couples

In this work we study the entanglement of pure four-qubit quantum states. The analysis is realized, firstly, through the comparison between two different entanglement measures: the Groverian entanglement measure and the residual entanglement calculated with negativities. After, we use the last to measure the entanglement of several four-qubit states and the variation of the entanglement when the four-qubit state is processed by a two-qubit gate.

We derive absolutely maximally entangled (AME) states from phase states for a multi-qudit system whose dynamics is governed by a two-qudit interaction Hamiltonian of Heisenberg type. AME states are characterized by being maximally entangled for all bipartitions of the multi-qudit system and present absolute multipartite entanglement. The key ingredient of this approach is the theory of phase states for finite-dimensional systems (qudits). We define further the unitary phase operators of [Formula: see text]-qudit systems and we give next the corresponding separable phase states. Using a qudit–qudit Hamiltonian acting as entangling operator on separable phase states, we generate entangled phase states. Finally, from the labeled entangled phase states, we derive the absolutely maximally entangled states.

We construct a large family of Planar Maximally Entangled (PME) states which are a wider class of multi-partite entangled states than Absolutely Maximally Entangled (AME) states. These are states in which any half of the qudits are in a maximally mixed state, provided that they form a connected subset. We show that in contrast to AMEs, PMEs are easier to find and there are various PMEs for any even number of qudits. In particular, while it is known that no AME state of four qubits exists, we show that there are two distinct multi-parameter classes of four qubit PMEs. We also give explicit families of PMEs for any even number of particles and for any dimension.

Recently, Doroudiani and Karimipour [Phys. Rev. A \textbf{102} 012427(2020)] proposed the notation of planar maximally entangled (PME) states which are a wider class of multipartite entangled states than absolutely maximally entangled (AME) states. There they presented their constructions in the multipartite systems but the number of particles is restricted to be even. Here we first solve the remaining cases, i.e., constructions of planar maximally entangled states on systems with odd number of particles. In addition, we generalized the PME to the planar $k$-uniform states whose reductions to any adjacent $k$ parties along a circle of $N$ parties are maximally mixed. We presented a method to construct sets of planar $k$-uniform states which have minimal support.

Recently Liu et al.(Int. J. Theor. Phys. 53:4079, 2014) had shown that a special form of three-qubit entangled state can be teleported by using a five‐ qubit cluster state. However, we demonstrate that five-qubit cluster state and five-qubit von-Neumann projective measurements are not necessary to complete the protocol. The special three-qubit state can be teleported via a four-qubit entangled state, by introducing one ancillary qubit and one controlled-not operation. Physical realization of the proposed four-qubit entangled state and the generalization to teleport the multi-qubit state is also presented. Introduction Quantum teleportation is one of the most astonishing features of quantum mechanics. An unknown quantum state can be teleported from one site to another via previously shared entanglement assisted by classical communications and local operations. In 1993, Bennett et al. [1] proposed the first protocol of quantum teleportation of an arbitrary single-qubit state using a maximally entangled two-qubit state. Four years later, this protocol was experimentally demonstrated [2]. Thereafter, teleportation of an arbitrary single-qubit state was proposed using tripartite GHZ state [3], four-partite GHZ state [4], SLOCC equivalent W-class state [5], cluster state [6], etc. Teleportation of an arbitrary two-qubit state was proposed using tensor product of two Bell states [7], tensor product of two orthogonal states [8], genuinely entangled five-qubit state [9], five-qubit cluster state [10], six-qubit genuinely entangled state [11], etc. Recently, Liu et al. [12] had shown that a special form of three-qubit entangled state can be teleported by using a five-qubit cluster state based on five-qubit von-Neumann projective measurements and local unitary operations. However, we demonstrate that five-qubit cluster state and five-qubit von-Neumann projective measurements are not necessary to complete the protocol. The special three-qubit state can be teleported via a four-qubit entangled state, by introducing one ancillary qubit and one controlled-not (CNOT) operation. The proposed four-qubit entangled state can be physically realized by a pair of Bell states and two single-qubit states. The generalization of the protocol to teleport a multi-qubit state is also presented. Quantum Teleportation of a Three-qubit State In [12], a special form of three-qubit state is given by (1) where . Our teleportation scheme can be described as follows. Suppose Alice has an unknown three-qubit state . She wants to send this state to a distant receiver Bob. Alice and Bob share a four-qubit entangled state

Absolutely Maximally Entangled (AME) states are those multipartite quantum\nstates that carry absolute maximum entanglement in all possible partitions. AME\nstates are known to play a relevant role in multipartite teleportation, in\nquantum secret sharing and they provide the basis novel tensor networks related\nto holography. We present alternative constructions of AME states and show\ntheir link with combinatorial designs. We also analyze a key property of AME,\nnamely their relation to tensors that can be understood as unitary\ntransformations in every of its bi-partitions. We call this property\nmulti-unitarity.\n

It has been shown by Versraete et. al [F. Versraete, J. Dehaene, B. De Moor, and H. Verschelde, Phys. Rev. A65, 052112 (2002)] that by stochastic local operations and classical communication (SLOCC), a pure state of four qubits can be transformed to a state belonging to one of a set of nine families of states. By using selective partial transposition, we construct partial K-way negativities to measure the genuine 4-partite, tripartite, and bi-partite entanglement of single copy states belonging to the nine families of four qubit states. Partial K-way negativities are polynomial functions of local invariants characterizing each family of states as such entanglement monotones.

There is a connection between classical codes, highly entangled pure states (called k-uniform or absolutely maximally entangled (AME) states), and quantum error correcting codes (QECCs). This leads to a systematic method to construct stabilizer QECCs by starting from a k-uniform state or the corresponding classical code and tracing out one party at each step. We provide explicit constructions for codewords, encoding procedure and stabilizer formalism of the QECCs by describing the changes that partial traces cause on the corresponding generator matrix of the classical codes. We then modify the method to produce another set of stabilizer QECCs that encode a logical qudit into a subspace spanned by AME states. This construction produces quantum codes starting from an AME state without tracing out any party. Therefore, quantum stabilizer codes with larger codespace can be constructed.

We test a new four-qubit entangled state by the former Bell-like inequalities and find that it violates these inequalities, but not maximally. According to this entangled state, we build a new Bell-like inequality, which is violated by this new state maximally. We also determine the nonlocality of some other four-qubit states by the new inequality. It is found that the new inequality acts as a strong entanglement witness for the new state. This can be used to test new entangled states experimentally.

Highly entangled multipartite states such as $k$-uniform ($k$-UNI) and absolutely maximally entangled (AME) states serve as critical resources in quantum networking and other quantum information applications. However, there does not yet exist a complete classification of such states, and much remains unknown about their entanglement structure. Here we substantially broaden the class of known $k$-UNI and AME states by introducing a method for explicitly constructing such states that combines classical error correcting codes and qudit graph states. This method in fact constitutes a general recipe for obtaining multipartitite entangled states from classical codes. Furthermore, we show that at least for a large subset of the class of $k$-UNI states that we present, the states are inequivalent under stochastic local operations and classical communication. This subset is defined by an iterative procedure for constructing a hierarchy of $k$-UNI graph states.

The classification of multipartite entanglement is essential as it serves as a resource for various quantum information processing tasks. This study concerns a particular class of highly entangled multipartite states, the so-called absolutely maximally entangled (AME) states. 
These are characterized by maximal entanglement across all possible bipartitions. In particular we analyze the local unitary equivalence among AME states using invariants. One of our main findings is that the existence of special irredundant orthogonal arrays implies the existence of an infinite number of equivalence classes of AME states constructed from these. In particular, we show that there are infinitely many local unitary inequivalent three-party AME states for local dimension $d > 2$ and five-party AME states for $d \geq 2$.

Absolutely maximally entangled (AME) states are pure multi-partite generalizations of the bipartite maximally entangled states with the property that all reduced states of at most half the system size are in the maximally mixed state. AME states are of interest for multipartite teleportation and quantum secret sharing and have recently found new applications in the context of high-energy physics in toy models realizing the AdS/CFT-correspondence. We work out in detail the connection between AME states of minimal support and classical maximum distance separable (MDS) error correcting codes and, in particular, provide explicit closed form expressions for AME states of n parties with local dimension a power of a prime for all . Building on this, we construct a generalization of the Bell-basis consisting of AME states and develop a stabilizer formalism for AME states. For every prime, we show how to construct stabilizer QECCs that encode a logical qudit into a subspace spanned by AME states. Under a conjecture for which we provide numerical evidence, this construction produces a family of quantum error correcting codes for n even with the highest distance allowed by the quantum Singleton bound.

The absolutely maximally entangled (AME) states play key roles in quantum information processing. We provide an explicit expression of the generalized Bloch representation of AME states for general dimension $d$ of individual subsystems and arbitrary number of partite $n$. Based on this analytic formula, we prove that the trace of the squared support for any given weight is given by the so-called hyper-geometric function and is irrelevant with the choices of the subsystems. The optimal point for the existence of AME states is obtained.

We extend the relation between absolutely maximally entangled (AME) states and quantum maximum distance separable (QMDS) codes by constructing whole families of QMDS codes from their parent AME states. We introduce a reduction-friendly form for the generator set of the stabilizer representation of an AME state, from which the stabilizer form for children codes, all QMDS, can be obtained. We then relate this to optimal codes for one-way quantum repeaters, by minimizing the short-term infrastructure cost as well as the long-term running cost of such quantum repeaters. We establish that AME states provide a framework for a class of QMDS codes that can be used in quantum repeaters.

Absolutely maximally entangled (AME) states of k qudits (also known as perfect tensors) are quantum states that have maximal entanglement for all possible bipartitions of the sites/parties. We consider the problem of whether such states can be decomposed into a tensor network with a small number of tensors, such that all physical and all auxiliary spaces have the same dimension D. We find that certain AME states with k=6 can be decomposed into a network with only three 4-leg tensors; we provide concrete solutions for local dimension D=5 and higher. Our result implies that certain AME states with six parties can be created with only three two-site unitaries from a product state of three Bell pairs, or equivalently, with six two-site unitaries acting on a product state on six qudits. We also consider the problem for k=8, where we find similar tensor network decompositions with six 4-leg tensors.