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.
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.
Absolutely maximally entangled (AME) states are those multipartite quantum states that carry absolute maximum entanglement in all possible bipartitions. AME states are known to play a relevant role in multipartite teleportation, in quantum secret sharing, and they provide the basis novel tensor networks related to holography. We present alternative constructions of AME states and show their link with combinatorial designs. We also analyze a key property of AME states, namely, their relation to tensors, which can be understood as unitary transformations in all of their bipartitions. We call this property multiunitarity.
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.
Dual unitary gates are highly non-local two-qudit unitary gates that have been studied extensively in quantum many-body physics and quantum information in the recent past. A special class of dual unitary gates consists of rank-four perfect tensors that are equivalent to highly entangled multipartite pure states called absolutely maximally entangled (AME) states. In this work, numerical and analytical constructions of dual unitary gates and perfect tensors that are diagonal in a special maximally entangled basis are presented. The main ingredient in our construction is a phase-valued (unimodular) two-dimensional array whose discrete Fourier transform is also unimodular. We obtain perfect tensors for several local Hilbert space dimensions, particularly, in dimension six. A perfect tensor in local dimension six is equivalent to an AME state of four qudits, denoted as AME(4,6). Such a state cannot be constructed from existing constructions of AME states based on error-correcting codes and graph states. An explicit construction of AME(4,6) states is provided in this work using two-qudit controlled and single-qudit gates making it feasible to generate such states experimentally.
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.
Understanding and classifying multipartite entanglement is fundamental to quantum information processing. This work focuses on absolutely maximally entangled (AME) states, a class of highly entangled states characterized by their maximal entanglement across any bipartitions. To analyze and classify AME states, we employ the minimal decomposition entropy, defined as the minimum Rényi entropy Sq associated with the state's decomposition over all local product bases. This quantity identifies the product bases in which the state is maximally localized, thereby yielding optimal representations for analyzing properties of AME states.
We develop an efficient algorithm for computing the minimal decomposition entropy for finite q>1 and compare AME and Haar-random states for q=2 and q=∞ in qubit, qutrit, and ququad systems. For q=2, AME states of four qutrits and ququads show lower minimal entropy than generic states, indicating sparser optimal forms. For q=∞ -- related to the geometric measure of entanglement -- AME states exhibit higher entanglement. The algorithm also simplifies known AME states into sparser representations, aiding in distinguishing genuinely quantum AME states from those constructible from classical combinatorial designs. Our results advance the classification of AME states and demonstrate the utility of minimal decomposition entropy as both a local unitary invariant and a tool for state simplification.
We prove that an absolutely maximally entangled state of seven qudits exists if and only if the local dimension satisfies d≥3. Prior to this work, to the best of our knowledge, AME(7,d) states were known to exist only when d is a prime power other than 2, or when d can be expressed as a product of dimensions for which existence was already known. Since it has been proved that no AME(7,2) state exists, it remains to establish existence for all d≥3. We construct cyclic quadratic-phase states for every odd local dimension and develop a coupled binary--odd-dimensional construction for every dimension congruent to 2 modulo 4. Together with the known power-of-two cases and the product property of AME states, these constructions cover every local dimension d≥3.