Effective criteria for entanglement witnesses in small dimensions

2025/06/10 by Łukasz Grzelka, Grzelka, Łukasz, Łukasz Skowronek +3
Engineering · #FOS: Physical sciences #Image Processing Techniques and Applications #Integrated Circuits and Semiconductor Failure Analysis #Mathematical Physics (math-ph) #Non-Destructive Testing Techniques #Quantum Physics (quant-ph)

paper · doi:10.48550/arxiv.2506.09298

openalex publication_date 2025/06/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We present an effective set of necessary and sufficient criteria for block-positivity of matrices of order 4 over ℂ. The approach is based on Sturm sequences and quartic polynomial positivity conditions presented in recent literature. The procedure allows us to test whether a given 4× 4 complex matrix corresponds to an entanglement witness, and it is exact when the matrix coefficients belong to the rationals, extended by i. The method can be generalized to H₂\otimesH_d systems for d>2 to provide necessary but not sufficient criterion for block-positivity. We also outline an alternative approach to the problem relying on Gröbner bases.

Citations

• Symmetry reduction for testing k-block-positivity via extendibility
• On one-parameter families of hermiticity-preserving superoperators which are not positive
• A class of entanglement witnesses and a realignment-like criterion
• Positivity Conditions for Cubic, Quartic and Quintic Polynomials
• Extremal entanglement witnesses
• General theory of detection and optimality
• Entanglement detection
• Quantum entanglement
• Completely positive linear maps on complex matrices

Related

• A class of entanglement witnesses and a realignment-like criterion
• Entanglement witnesses and separability criteria based on generalized equiangular tight frames
• The detection power of real entanglement witnesses under local unitary equivalence
• Entanglement certification from moments of positive maps
• A hierarchy of entanglement criteria for four qubit symmetric Greenberger-Horne-Zeilinger diagonal states