We construct a large family of planar maximally entangled (PME) states, which are a wider class of multipartite 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 multiparameter classes of four-qubit PMEs. We also give explicit families of PMEs for any even number of particles and for any dimension. We also briefly mention the relevance of connectivity and underlying geometry to definition of a PME state. Here we have considered the simplest case of a one-dimensional graph.
We construct a large family of planar maximally entangled (PME) states, which are a wider class of multipartite 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 multiparameter classes of four-qubit PMEs. We also give explicit families of PMEs for any even number of particles and for any dimension. We also briefly mention the relevance of connectivity and underlying geometry to definition of a PME state. Here we have considered the simplest case of a one-dimensional graph.
Absolutely maximally entangled (AME) states are multipartite entangled states that are maximally entangled for any possible bipartition. In this paper, we study the description of AME states within the graph state formalism. The graphical representation provides an intuitive framework to visualize the entanglement in graph states, which makes them a natural candidate to describe AME states. We show two different methods of determining bipartite entanglement in graph states and use them to define various AME graph states. We further show that AME graph states exist for all number of parties, and that any AME graph states shared between an even number of parties can be used to describe quantum secret sharing schemes with a threshold or ramp access structure directly within the graph states formalism.
Ordering and classifying multipartite quantum states by their entanglement content remains an open problem. One class of highly entangled states, useful in quantum information protocols, the absolutely maximally entangled (AME) ones, are especially hard to compare as all their subsystems are maximally random. While it is well known that there is no AME state of four qubits, many analytical examples and numerically generated ensembles of four-qutrit AME states are known. However, we prove the surprising result that there is truly only one AME state of four qutrits up to local unitary equivalence. In contrast, for larger local dimensions, the number of local unitary classes of AME states is shown to be infinite. Of special interest is the case of local dimension 6, where it was established recently that a four-party AME state does exist, providing a quantum solution to the classically impossible Euler problem of 36 officers. Based on this, an infinity of quantum solutions are constructed, and we prove that these are not equivalent. The methods developed can be usefully generalized to multipartite states of any number of particles.
We demonstrate that absolutely maximally entangled (AME) states consisting of N=4n qudits with n∈{1,2,3,…}, each of even local dimension d, cannot be realized as canonical graph states over the ring Zd. This result imposes strong constraints on AME states in composite local dimensions and characterizes the limitations of graph-state constructions for highly entangled multipartite quantum systems. Furthermore, for d≡2(mod4) this obstruction, combined with the prime-power decomposition of stabilizer states, excludes all pure stabilizer AME states (including four quhexes), while clarifying the distinction between canonical Zd graph states and stabilizer constructions over prime-power factors. We also discuss a composite construction of mixed k-uniform states. For (N,d)=(4,6), the construction yields a rank-two mixed 2-uniform stabilizer state of purity 1/2, which is optimal among normalized stabilizer projectors.
We investigate absolutely maximally entangled (AME) states, which are multipartite quantum states that are maximally entangled with respect to any possible bipartition. These strong entanglement properties make them a powerful resource for a variety of quantum information protocols. In this paper, we show the existence of AME states for any number of parties, given that the dimension of the involved systems is chosen appropriately. We prove the equivalence of AME states shared between an even number of parties and pure state threshold quantum secret sharing (QSS) schemes, and prove necessary and sufficient entanglement properties for a wider class of ramp QSS schemes. We further show how AME states can be used as a valuable resource for open-destination teleportation protocols and to what extend entanglement swapping generalizes to AME states.
In entanglement theory, there are different methods to consider one state being more entangled than another. The “maximally” entangled states in a multipartite system can be defined from an axiomatic perspective. According to different criteria for selection, there are many specific types of quantum maximally entangled states, such as absolutely maximally entangled state, planar maximally entangled state and so on. In this paper we propose a new type of maximally entangled states, the planar two-region multipartite maximally entangled state. The requirement condition of this maximally entangled state is weak than that of the absolutely maximally entangled state and different from that of the planar maximally entangled state. We show that there are the two-region four-partite maximally entangled states in 4-qubit and 7-qubit planar systems, although there is no absolutely maximally entangled state in these systems. It is proved that there are the planar two-region four-partite maximally entangled states in both even particle quantum systems and odd particle quantum systems. Additionally, based on some planar two-region four-partite maximally entangled states, the new planar two-region four-partite maximally entangled states are generated. We also provide some important examples of the planar two-region multi-partite maximally entangled states.
Absolutely Maximally Entangled States of 2q Parties in Every Odd Prime-Power Dimension q
Absolutely maximally entangled (AME) states represent an extreme form of multipartite entanglement: every reduced system containing at most half of the parties is maximally mixed. These states provide perfect tensors and optimal quantum error-correcting codes, yet their existence is known only in restricted parameter regimes. For every odd prime power q=pe≥3, we construct a stabilizer AME(2q,q) state whose normalized one-party projection yields a stabilizer AME(2q−1,q) state. A closed-form q×q bordered-circulant matrix Aq over Fq2 generates a Hermitian self-dual maximum distance separable (MDS) code [2q,q,q+1]q2, which lies outside the extended Reed-Solomon classes. In suitable bases, the amplitude tensors define normalized q-unitary complex Hadamard matrices of order qq with pth-root phases. Additional constructions yield AME(q+3,q) states and families at intermediate particle numbers through explicit rescalings of selected submatrices. We also provide nine explicit parent matrices and the corresponding one-party projections.