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IOSR Journal of Applied Physics

A Density Matrix Approach Towards Quantum Computing

Vandana Arora, Jagneet Kaur Anand

Published Jun 1, 2025• 1 citation
Publisher

Abstract

In this article, the authors start with a discussion of various quantum states and their associated density matrices. The evaluation of the density matrices for pure, mixed, and entangled states is done elaborately. The concept of the trace of the density matrices is then explored and its relation with the purity of any state is discussed. The mathematical process of diagonalization of a matrix is studied for pure, mixed, and entangled states.

DOI
10.9790/4861-1703030610

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In this article, the authors start with a discussion of various quantum states and their associated density matrices. The evaluation of the density matrices for pure, mixed, and entangled states is done elaborately. The concept of the trace of the density matrices is then explored and its relation with the purity of any state is discussed. The mathematical process of diagonalization of a matrix is studied for pure, mixed, and entangled states.
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The density matrix is a very convenient way to do calculations for a large number of spin systems and is developed from the fundamental ideas of quantum mechanics. The idea of pure and mixed states is introduced and used to develop an expression for the single-spin density matrix as it applies to NMR spectroscopy. Multiple-spin density matrices are produced by using the outer product of single-spin matrices and their commutation properties are examined. A very useful form of the scalar coupling density operator is developed and used, along with the chemical shift operator, to calculate the spectrum of a weakly coupled two-spin system. The density operator of spin-1 is explored briefly. Density matrix calculations in perturbed systems are investigated in preparation for relaxation calculations. Strong coupling is discussed and compared to the weak coupling approximation, using the full scalar coupling Hamiltonian and matrix diagonalization techniques.
Quantum computing is among the most far-reaching technologies of the 21st century, tackling challenges at the cutting edge of physics. This new paradigm in computer science harnesses quantum entanglement, one striking non-intuitive feature of quantum mechanics and a cornerstone of quantum information, to provide computation with a quantum speed-up over the best-known classical algorithms and to enable encrypted data communication against eavesdropping. The bulk of this article is focused on providing a deep and abiding understanding of non-relativistic matrix mechanics by demonstrating the fundamental mathematical identities of the contemporary postulatory approach of quantum mechanics within the state vector and density operator formalism in multipartite systems. In addition to that, we derive and analyze the respective 1-qubit, 1-qutrit, 2-qubit, and 2-qudit coherent and incoherent density operators using Bloch's parametrization for generalized ddd-dimensional NNN-qudit states embedded in the SU(d)SU(d)SU(d) Lie group with associate generalized Gell Mann's matrices spanning the su(d)\mathfrak{su}(d)su(d) Lie algebra. We also address the fundamental concepts of quantum nondemolition measurements, quantum decoherence and, particularly, quantum entanglement providing for the latter a systematic view on its historical development and mathematical description in multipartite systems. We conclude our review by introducing some of the ubiquitous quantum quantifiers required to measure degrees of quantum entanglement and quantum coherence, deriving the ppp-norm quantum coherence measure for a 1-qubit state.
Representing graphs as quantum states is becoming an increasingly important approach to study entanglement of mixed states, alternate to the standard linear algebraic density matrix-based approach of study. In this paper, we propose a general weighted directed graph framework for investigating properties of a large class of quantum states which are defined by three types of Laplacian matrices associated with such graphs. We generalize the standard framework of defining density matrices from simple connected graphs to density matrices using both combinatorial and signless Laplacian matrices associated with weighted directed graphs with complex edge weights and with/without self-loops. We also introduce a new notion of Laplacian matrix, which we call signed Laplacian matrix associated with such graphs. We produce necessary and/or sufficient conditions for such graphs to correspond to pure and mixed quantum states. Using these criteria, we finally determine the graphs whose corresponding density matrices represent entangled pure states which are well known and important for quantum computation applications. We observe that all these entangled pure states share a common combinatorial structure.