Distillability Problem of 4×4 Monomial Matrices

Abstract

Abstract It is a fundamental problem in quantum information whether a particular quantum state of a composite system is entangled. It has enormous potential in quantum error correction, quantum cryptography, and quantum teleportation applications. This problem can be transferred in the form of a mathematical conjecture called the distillation conjecture. In the first section of this paper, relevant physical and mathematical information is presented, including basic linear algebra knowledge, the statement, and concrete applications of multiple mathematical knowledge like conjugate, eigenvalue, and singular value. Then, we introduce the distillation conjecture in a mathematical version for a more precise mathematical analysis. In an effort to make more significant headway in proving the conjecture, we selected some theories and findings relating to the Kronecker product, Kronecker sum, eigenvalue, and singular value, then evaluated and grouped them. In addition, we provided multiple proofs of the conjecture under varying conditions and made numerous attempts and hypotheses regarding how to establish the conjecture.