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Journal of Physics Conference Series

Distillability Problem of 4×4 Monomial Matrices

Yijin Wang, Ao Zhang, Hong Zhang

Published Dec 1, 2022• 1 citation
Publisher

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.

DOI
10.1088/1742-6596/2381/1/012014

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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.
The pure entangled state is of vital importance in the field of quantum information. The process of asymptotically extracting pure entangled states from many copies of mixed states via local operations and classical communication is called entanglement distillation. The entanglement distillability problem, which is a long-standing open problem, asks whether such process exists. The 2-copy undistillability of 4×44\times44×4 undistillable Werner states has been reduced to the validness of the a matrix inequality, that is, the sum of the squares of the largest two singular values of matrix A⊗I+I⊗BA\otimes I + I \otimes BA⊗I+I⊗B does not exceed (3d−4)/d2(3d-4)/d^2(3d−4)/d2 with A,BA,BA,B traceless d×dd\times dd×d matrices and ∣∣A∣∣F2+∣∣B∣∣F2=1/d||A||_F^2+||B||_F^2=1/d∣∣A∣∣F2​+∣∣B∣∣F2​=1/d when d=4d=4d=4. The latest progress, made by Ł.~Pankowski~ et al~[IEEE Trans. Inform. Theory, 56, 4085 (2010)], shows that this conjecture holds when both matrices AAA and BBB are normal. In this paper, we prove that the conjecture holds when one of matrices AAA and BBB is normal and the other one is arbitrary. Our work makes solid progress towards this conjecture and thus the distillability problem.
Quantum mechanics has a history more than 100 years, and there are still many challenging open-ended questions. The problems encountered in modern physics such as matrix mechanics and quantum information often involve both quantum theory and open-ended problems in modern mathematics such as linear algebra. Distillability problem is an important open-ended problem in the field of quantum information and linear algebra. This article recalls the development history and application of linear algebra, summarizes the background and progress of distillability conjecture, and attempts to explore this problem by summarizing and using the knowledge of linear algebra and matrix theory learned in the course.
Quantum mechanics is already 100 years old, but remains alive and full of challenging open problems. On one hand, the problems encountered at the frontiers of modern theoretical physics like quantum gravity, string theories, etc. concern quantum theory, and are at the same time related to open problems of modern mathematics. But even within non-relativistic quantum mechanics itself there are fundamental unresolved problems that can be formulated in elementary terms. These problems are also related to challenging open questions of modern mathematics; linear algebra and functional analysis in particular. Two of these problems will be discussed in this article: (a) the separability problem, i.e. the question when the state of a composite quantum system does not contain any quantum correlations or entanglement; and (b) the distillability problem, i.e. the question when the state of a composite quantum system can be transformed to an entangled pure state using local operations (local refers here to component subsystems of a given system). Although many results concerning the above mentioned problems have been obtained (in particular in the last few years in the framework of quantum information theory), both problems remain until now essentially open. We will present a primer on the current state of knowledge concerning these problems, and discuss the relation of these problems to one of the most challenging questions of linear algebra: the classification and characterization of positive operator maps.
The distillability conjecture of two-copy 4 by 4 Werner states is one of the main open problems in quantum information. We prove two special cases of the conjecture. The first case occurs when two 4 by 4 matrices A, B are both unitarily equivalent to block diagonal matrices with 2 by 2 blocks. The second case occurs when B is unitarily equivalent to either -A or the transpose of -A. Plus, we propose a simplified version of the distillability conjecture when both A and B are matrices with distinct eigenvalues.
An important open problem in quantum information theory is the question of the existence of NPT bound entanglement. In the past years, little progress has been made, mainly because of the lack of mathematical tools to address the problem. (i) In an attempt to overcome this, we show how the distillability problem can be reformulated as a special instance of the separability problem, for which a large number of tools and techniques are available. (ii) Building up to this we also show how the problem can be formulated as a Schmidt number problem. (iii) A numerical method for detecting distillability is presented and strong evidence is given that all 1-copy undistillable Werner states are also 4-copy undistillable. (iv) The same method is used to estimate the volume of distillable states, and the results suggest that bound entanglement is primarily a phenomenon found in low dimensional quantum systems. (v) Finally, a set of one parameter states is presented which we conjecture to exhibit all forms of distillability.
Abstract This paper not only explores in detail the basic properties of the Kronecker product, such as the application of the union law, the distributive law, the transpose and the conjugate transpose, as well as the trace and the rank of the Kronecker product, but also rigorously proves these properties by means of mathematical induction or direct proofs. In addition, this paper extends these properties to Kronecker n-matrices and explores the applicability of inverse matrices and determinants to Kronecker products. More importantly, we introduce a conjecture related to the long-standing problem of distillability, which is partially solved when the number of nonzero elements is restricted. By showing the framework of the proof of the conjecture and a special case, we provide new perspectives for further research in this area.