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A Photonic System for Generating Unconditional Polarization-Entangled Photons Based on Multiple Quantum Interference
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Tripartite entanglement transformations and tensor rank.

Eric Chitambar1, Runyao Duan, Yaoyun Shi

  • 1Physics Department, University of Michigan, 450 Church Street, Ann Arbor, Michigan 48109-1040, USA. echitamb@umich.edu

Physical Review Letters
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PubMed
Summary
This summary is machine-generated.

Converting quantum entangled states probabilistically is complex for three systems. Determining feasibility is NP-hard, linking quantum entanglement to matrix multiplication algorithms and algebraic complexity.

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Area of Science:

  • Quantum Information Theory
  • Algebraic Complexity Theory
  • Quantum Computing

Background:

  • The conversion of quantum entangled states via local operations and classical communication (LOCC) is a fundamental question in quantum information.
  • While understood for bipartite systems, the multipartite case, particularly tripartite systems, remains largely unexplored.
  • This problem connects to deep questions in mathematics and computer science.

Purpose of the Study:

  • To investigate the feasibility of probabilistic conversion between quantum entangled states in tripartite systems.
  • To establish the computational complexity of determining such conversions.
  • To explore the relationship between multipartite entanglement and algebraic complexity theory.

Main Methods:

  • Analysis of multipartite entanglement properties, specifically focusing on tripartite systems.
  • Connections drawn between entanglement measures and tensor rank (Schmidt rank).
  • Investigation of the computational complexity class NP-hardness.

Main Results:

  • The problem of determining the feasibility of converting tripartite entangled states via LOCC is NP-hard.
  • No simple general criterion exists for assessing conversion feasibility.
  • Equivalence established between finding the most efficient matrix multiplication algorithm and determining the conversion rate of Greenberger-Horne-Zeilinger (GHZ) states to a specific tripartite entangled state distribution.

Conclusions:

  • The conversion of tripartite quantum entangled states is computationally challenging.
  • The study highlights profound links between quantum entanglement, algebraic complexity, and fundamental problems in computer science.
  • Results suggest new avenues for research at the intersection of quantum information and theoretical computer science.