Persistent Tensors and Multiqudit Entanglement Transformation

Abstract

We construct a lower bound of the tensor rank for a new class of tensors, which we call persistent tensors. We present three specific families of persistent tensors, of which the lower bound is tight. We show that there is a chain of degenerations between these three families of minimal-rank persistent tensors that can be used to study the entanglement transformation between them. In addition, we show that these three families of persistent tensors are indeed different generalizations of multiqubit $\mathrm{W}$ states within multiqudit systems and are geometrically in the orbit closure of multiqudit $\mathrm{GHZ}$ states. Consequently, we show that one can obtain every one of the generalizations of $\mathrm{W}$ state from a multiqudit $\mathrm{GHZ}$ state via asymptotic Stochastic Local Operations and Classical Communication (SLOCC) with rate one. Finally, we extend the obtained lower bound of the tensor rank to direct sums with persistent summands and to even more general combinations of tensors, which we call block pyramidal tensors. As a result, we show that the tensor rank is multiplicative under the Kronecker and tensor products of minimal-rank persistent tensors with the $\mathrm{GHZ}$ tensor.

arXiv comment19 pages. Your comments are more than welcome

References in corpus (25)

• Quantum entanglement
• Three qubits can be entangled in two inequivalent ways
• Security of quantum key distribution using d-level systems
• Classification of mixed three-qubit states
• Qudits and high-dimensional quantum computing
• Security Proof for Quantum Key Distribution Using Qudit Systems
• Tensor Rank and Stochastic Entanglement Catalysis for Multipartite Pure States
• Local Conversion of Greenberger-Horne-Zeilinger States to Approximate W States
• Tripartite entanglement transformations and tensor rank
• Quantum gates on hybrid qudits
• Universal qudit gate synthesis for transmons
• Tensor rank is not multiplicative under the tensor product
• Obtain $W$-state from three-qubit $GHZ$-state on rate 1
• The moduli space of three qutrit states
• Quantum Stabilizer Codes Embedding Qubits Into Qudits
• Asymptotic entanglement transformation between W and GHZ states
• The Tensor Rank of the Tripartite State ${|W⟩}^{\otimes n}$}
• The Simple Criteria of SLOCC Equivalence Classes
• Tensor Rank: Some Lower and Upper Bounds
• Entanglement distillation from Greenberger-Horne-Zeilinger shares
• Fine-Structure Classification of Multiqubit Entanglement by Algebraic Geometry
• Optimization at the boundary of the tensor network variety
• Tripartite to Bipartite Entanglement Transformations and Polynomial Identity Testing
• An Algebraic-Geometric Characterization of Tripartite Entanglement
• Three-qutrit entanglement and simple singularities

Cited by in corpus (1)

• Classifying Entanglement by Algebraic Geometry