Absolutely maximally entangled state equivalence and the construction of infinite quantum solutions to the problem of 36 officers of Euler

Abstract

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 specially 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 {\em 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.

arXiv commentRewritten as a regular article and few changes in the title from first version. Close to the published version

References in corpus (35)

• Mixed State Entanglement and Quantum Error Correction
• Separability Criterion for Density Matrices
• Three qubits can be entangled in two inequivalent ways
• Concentrating Partial Entanglement by Local Operations
• Experimental loophole-free violation of a Bell inequality using entangled electron spins separated by 1.3 km
• Significant-loophole-free test of Bell's theorem with entangled photons
• Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence
• Generalized Schmidt decomposition and classification of three-quantum-bit states
• Exact and Asymptotic Measures of Multipartite Pure State Entanglement
• Multipartite entanglement, quantum-error-correcting codes, and entangling power of quantum evolutions
• Entanglement of Quantum Evolutions
• Exact Correlation Functions for Dual-Unitary Lattice Models in 1+1 Dimensions
• Absolute Maximal Entanglement and Quantum Secret Sharing
• How entangled can two couples get?
• Unitary circuits of finite depth and infinite width from quantum channels
• Absolutely Maximally Entangled states, combinatorial designs and multi-unitary matrices
• Absolutely maximally entangled states of seven qubits do not exist
• Ergodic and non-ergodic dual-unitary quantum circuits with arbitrary local Hilbert space dimension
• Local unitary equivalence of multipartite pure states
• Entanglement and quantum combinatorial designs
• From dual-unitary to quantum Bernoulli circuits: Role of the entangling power in constructing a quantum ergodic hierarchy
• Creating ensembles of dual unitary and maximally entangling quantum evolutions
• Transformations among Pure Multipartite Entangled States via Local Operations Are Almost Never Possible
• Thirty-six entangled officers of Euler: Quantum solution to a classically impossible problem
• Entangling Power of Permutations
• Can local dynamics enhance entangling power?
• Matrix realignment and partial transpose approach to entangling power of quantum evolutions
• Construction and local equivalence of dual-unitary operators: from dynamical maps to quantum combinatorial designs
• Stochastic Local Operations with Classical Communication of Absolutely Maximally Entangled States
• Orthogonality for Quantum Latin Isometry Squares
• Existence of Universal Entangler
• Maximally Entangled States of Four Nonbinary Particles
• 9 $\times$ 4 = 6 $\times$ 6: Understanding the quantum solution to the Euler's problem of 36 officers
• Tensor rank and entanglement of pure quantum states
• Genuinely quantum SudoQ and its cardinality

Cited by in corpus (10)

• Exactly solvable many-body dynamics from space-time duality
• Tensor network decompositions for absolutely maximally entangled states
• Introduction to quantum entanglement in many-body systems
• Entanglement classification and \emph{non-k}-separability certification via Greenberger-Horne-Zeilinger-class fidelity
• Two-Unitary Complex Hadamard Matrices of Order $36$
• Construction of perfect tensors using biunimodular vectors
• Quantum convolutional channels and multiparameter families of 2-unitary matrices
• Quantum Circuits for High-Dimensional Absolutely Maximally Entangled States
• Absolutely maximally entangled pure states of multipartite quantum systems
• Identifying local unitary equivalence based on reduction of quantum states