Physical systems are typically multipartite — that is, they are composite systems made up of two or more subsystems that may be labelled as A, B, etc. Numerous properties of a multipartite system arise as a consequence of the interactions between its subsystems.
If the system is quantum mechanical (QM) rather than classical, several new and interesting properties ensue. An important one is entanglement, a uniquely quantum mechanical phenomenon that is now recognised as a valuable resource in quantum information, communication and computation.
All the information about a QM system is contained in its state, which comes in two varieties. In a so-called pure state, the descriptor is a state vector. More generally, we have mixed state, in which case the descriptor is called a density matrix. A mixed state occurs when we have incomplete information about a system. A typical example is that of a system of interest, A, and its environment B (e.g., a heat bath), which may be regarded as a bipartite system comprising the subsystems A and B. In such a situation, information about B is of a statistical nature, which in turn implies that A itself is in a mixed state described by a so-called reduced density matrix.
A scenario that is very relevant for practical applications occurs when A and B are comparable subsystems, in the sense that their respective state vectors span spaces of a finite number of dimensions. Now suppose the full bipartite system AB is in some pure state, and we only have access to measurements on one of the subsystems (say, A) alone. A remarkable property ensues now. By ‘summing over’ the possible states of the other subsystem B, we obtain a reduced density matrix for A that shows that A is in fact in a mixed state in general! The latter is quantified by its so-called subsystem Von Neumann entropy, (SVNE). The SVNE is a quantity of central importance, as it is also a measure of the entanglement between the subsystems. It is expressible in terms of the eigenvalues of the reduced density matrix.
In most instances, one does not know the specific pure state in which the bipartite system AB happens to be. One must therefore treat this pure state itself as a random state, selected from an appropriate distribution (most often, a Gaussian distribution) of such states. In turn, this makes the eigenvalues of the reduced density matrix random variables whose statistical properties are to be deduced, and hence those of the SVNE.
The chief technical difficulty lies in the fact that the eigenvalues of the reduced density matrix are not independently varying random variables, but rather, strongly correlated ones. Several results are already known, including the joint probability distribution of the eigenvalues and the distributions of the smallest and largest eigenvalues. In this paper, the authors B. Sharmila, V. Balakrishnan, and S. Lakshmibala from the Department of Physics, Indian Institute of Technology (IIT) Madras, Chennai, India, the present work completes this task in a significant manner by deriving the full order statistics of the eigenvalues–namely, the distributions of each member of the full ordered sequence of the eigenvalues from the smallest to the largest one. In turn, this enables one to derive exact expressions for the statistical properties of a variety of entanglement indicators and related quantities. The details may be found in the link to the paper given below.
Prof. A. R. P. Rau from Louisiana State University, USA, gave his analysis and appreciated the work done by the others by giving the following comments: “Quantum systems are described by what is called a density matrix, an n X n array of numbers. In today’s active field of quantum information, which holds great promise of secure communication and quantum computers in the future that will be much more powerful than today’s classical computers (for certain applications in searches and simulations), the simplest and basic component is a “qubit,” which is described by a n=2 density matrix. Multiple qubits will have larger n, n = 2q2^q for q qubits. Other values of n are also of interest for other generalizations of qubits termed qutrits (n=3), etc. Binary systems of two such entities form the basic logic gates used in the field. The n “eigenvalues” of the matrix (positive numbers between 0 and 1, standardized so that they sum to 1) and the probability density function of them are of great interest. Both the number of eigenvalues and the probability density functions which are polynomials of order n2n^2 -2 grow rapidly with n. Even their enumeration is a major challenge. The authors tackle this problem, with both analytical and numerical results for n values up to 7, the algebraic expressions becoming very complicated already for n=5. There is impressive agreement between their numerical calculations and the analytical expressions they derive. These results will be of great interest in the field of quantum information.”
Article by Akshay Anantharaman
Here is the original link to the paper:
https://www.sciencedirect.com/science/article/pii/S000349162200210X?dgcid=rss_sd_all
