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Chinese Control Conference (CCC)

Quantum system Schmidt decomposition and its geometric analysis

Shuang Cong, Feng Xian-yong

Published Jul 1, 2008• 1 citation
Publisher

Abstract

Quantum entangled state and some common methods to measure entanglement are introduced, and Schmidt decomposition of 2-qubit quantum state and the method to identify the entangled state are induced. A method to calculate the degree of entanglement is also given, which is obtained very easily by using the Schmidt decomposition. An example about NMR quantum computation is discussed to illustrate the application of the Schmidt decomposition. The evolution of the state of sub-system is described by using the Bloch vector, which demonstrates the geometric prospect of quantum system.

DOI
10.1109/chicc.2008.4604906

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Quantum entangled state and some common methods to measure entanglement are introduced, and Schmidt decomposition of 2-qubit quantum state and the method to identify the entangled state are induced. A method to calculate the degree of entanglement is also given, which is obtained very easily by using the Schmidt decomposition. An example about NMR quantum computation is discussed to illustrate the application of the Schmidt decomposition. The evolution of the state of sub-system is described by using the Bloch vector, which demonstrates the geometric prospect of quantum system.
In the standard geometric approach, the entanglement of a pure state is sin⁡2θ\sin^2θsin2θ, where θθθ is the angle between the entangled state and the closest separable state of products of normalised qubit states. We consider here a generalisation of this notion by considering separable states that consist of products of unnormalised states of different dimension. The distance between the target entangled state and the closest unnormalised product state can be interpreted as a measure of the entanglement of the target state. The components of the closest product state and its norm have an interpretation in terms of, respectively, the eigenvectors and eigenvalues of the reduced density matrices arising in the Schmidt decomposition of the state vector. For several cases where the target state has a large degree of symmetry, we solve the system of equations analytically, and look specifically at the limit where the number of qubits is large.
When two or more subsystems of a quantum system interact with each other they can become entangled. In this case the individual subsystems can no longer be described as pure quantum states. For systems with only 2 subsystems this entanglement can be described using the Schmidt decomposition. This selects a preferred orthonormal basis for expressing the wavefunction and gives a measure of the degree of entanglement present in the system. The extension of this to the more general case of n subsystems is not yet known. We present a review of this process using the standard representation and apply this method to the geometric algebra representation. This latter form has the advantage of suggesting a generalisation to n subsystems.
When two or more subsystems of a quantum system interact with each other they can become entangled. In this case the individual subsystems can no longer be described as pure quantum states. For systems with only two subsystems this entanglement can be described using the Schmidt decomposition. This selects a preferred orthonormal basis for expressing the wavefunction and gives a measure of the degree of entanglement present in the system. The extension of this to the more general case of n subsystems is not yet known. We present a review of this process using the standard representation and apply this method in the geometric algebra setting, which has the advantage of suggesting a generalisation to n subsystems. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Quantum information theory is a rapidly growing area of math and physics that combines two independent theories, quantum mechanics and information theory. Quantum entanglement is a concept that was first proposed in the EPR paradox. In quantum mechanics, particles can be in superposition, meaning they are in multiple different states at once. It is not until the particle is measured that it is forced into a single state. However, it is possible that particles can be tied to other particles, meaning that the measurement of one particle will determine the measurement of the other particle. Entanglement is at the very core of quantum information theory. It is one of the core pieces that allows for the massive increase in computing power. For this paper, we decided to focus on demonstrating the mathematical method (the Schmidt decomposition) for determining if a system is entangled, and a demonstration of quantum entanglement's use (quantum teleportation) as well as a quick look at how to extend the uses of the Schmidt decomposition.
The thesis includes the original results of our articles [30, 37, 40, 42, 51, 53, 75]. A method is developed to compute analytically entanglement measures of three-qubit pure states. Owing to it closed-form expressions are presented for the geometric measure of entanglement for three-qubit states that are linear combinations of four orthogonal product states. It turns out that the geometric measure for these states has three different expressions depending on the range of definition in parameter space and the Hilbert space of three-qubits consists of three different entangled regions. The states that lie on joint surfaces separating different entangled regions, designated as shared states, have particularly interesting features and are dual quantum channels for the perfect teleportation and superdense coding [42]. A powerful method is developed to compute analytically multipartite entanglement measures. The method uses the duality concept and creates a bijection between highly entangled quantum states and their nearest separable states. The bijection gives explicitly the geometric entanglement measure of arbitrary generalized W states of n qubits [30, 75]. Additionaly, the behavior of the geometric entanglement measure of many-qubit W states is analyzed and an interpolating formula is derived [51]. Generalized Schmidt decomposition of pure three-qubit states has four positive and one complex coefficients. In contrast to the bipartite case, they are not arbitrary and the largest Schmidt coefficient restricts severely other coefficients. It is derived a non-strict inequality between three-qubit Schmidt coefficients, where the largest coefficient defines the least upper bound for the three nondiagonal coefficients or, equivalently, the three nondiagonal coefficients together define the greatest lower bound for the largest coefficient [53].