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Laser Physics

Searching for an absolutely maximally five-level entangled state

Xin-Wei Zha

Published Nov 1, 2024• 0 citations
Publisher

Abstract

Abstract Entangled states play an important role in quantum computation and information. In this paper, for a five-level entangled state, we define some unitary operations and give an operator multiplication. With the help of the unitary operation, we obtain some local unitary equivalent bipartite and tripartite five-level absolutely maximally entangled states (AMES). Furthermore, by using recurrence relations, some four-partite and five-partite five-level AMES can be constructed.

DOI
10.1088/1555-6611/ad873c

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Abstract Entangled states play an important role in quantum computation and information. In this paper, for a five-level entangled state, we define some unitary operations and give an operator multiplication. With the help of the unitary operation, we obtain some local unitary equivalent bipartite and tripartite five-level absolutely maximally entangled states (AMES). Furthermore, by using recurrence relations, some four-partite and five-partite five-level AMES can be constructed.
We classify the local unitary equivalence classes of absolutely maximally entangled (AME) states of five qubits. We show that every 5-qubit AME state is equivalent to a state within the unique ((5,2,3)) quantum error-correcting code C\mathcal{C}C, and that two such states are equivalent if and only if they are related by the action of a transversal gate of C\mathcal{C}C. Furthermore, we exhibit a set of three invariant polynomials that separates these equivalence classes. As auxiliary results, we construct a 3-uniform nnn-qubit state for even n≥6n\geq 6n≥6, determine the local symmetries of the 6-qubit AME state, and explain how these symmetries are related to the transversal gates of both the ((5,2,3)) and ((4,4,2)) codes. Additionally, we demonstrate that every 4-qubit pure code of distance 2 is equivalent to a subspace of a ((4,4,2)) code. Our approach leverages an embedding of the 4-qubit state space into the Lie algebra of 8×88\times 88×8 skew-symmetric matrices, allowing us to apply results from Vinberg's theory of graded Lie algebras.
Abstract 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 (LU) 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 LU inequivalent three-party AME states for local dimension d > 2 and five-party AME states for d ⩾ 2 .
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 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.