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 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.
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 Quantum information science is a new and rapidly developing discipline that studies the general laws governing the transmission, storage, and processing of information in systems that obey the laws of quantum mechanics. It makes extensive use of matrix theory and operator theory on Hilbert spaces to investigate the capabilities of such systems and to develop principles for their rational design. An important consequence of the development of quantum information science is a substantial clarification of the logical structure and foundations of quantum mechanics and of its relation to reality. These lectures are devoted to the foundations of the mathematical theory of quantum systems: states, observables, communication channels, their entropic and information characteristics, and quantum information-processing protocols. The lectures provide a detailed introduction to the statistical structure of quantum theory and the necessary background from classical information theory; hence, knowledge of the basic mathematical subjects—analysis, linear algebra, and probability theory—is sufficient for understanding the material. In the first part of the course, we use the “standard” statistical model of a quantum system, but consider it in comparison with the classical statistical description. A consistent development of this approach leads to a generalized statistical model of quantum mechanics, in which observables (in general, “unsharp”) are described by probability operator-valued measures, while evolutions (in general, irreversible) are described by completely positive maps. Introductory material on these concepts, which underlie mathematical models of quantum information systems in the presence of noise and errors and have numerous applications in quantum information science, is presented in the second part of the course.
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 4×4 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⊗B does not exceed (3d−4)/d2 with A,B traceless d×d matrices and ∣∣A∣∣F2+∣∣B∣∣F2=1/d when d=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 A and B are normal. In this paper, we prove that the conjecture holds when one of matrices A and B is normal and the other one is arbitrary. Our work makes solid progress towards this conjecture and thus the distillability problem.
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.