[Paper Review] Coherent States and Some Topics in Quantum Information Theory : Review

Kazuyuki Fujii|ArXiv.org|Jul 31, 2002

Quantum Information and Cryptography14 references3 citations

TL;DR

This paper reviews coherent states and generalized coherent states based on su(2) and su(1,1) Lie algebras using the Schwinger boson realization, and applies these to quantum information tasks: constructing a universal swap operator for coherent states via su(2)-based generalized coherent states and demonstrating imperfect cloning of coherent states. The key contribution is a mathematically rigorous, abstract construction of a universal swap operator in infinite-dimensional Hilbert space, with implications for quantum optics and quantum information processing.

ABSTRACT

In the first half we make a short review of coherent states and generalized coherent ones based on Lie algebras su(2) and su(1,1), and the Schwinger's boson method to construct representations of the Lie algebras. In the second half we make a review of recent developments on both swap of coherent states and cloning of coherent states which are important subjects in Quantum Information Theory.

Motivation & Objective

• To review the mathematical framework of coherent states and generalized coherent states based on su(2) and su(1,1) Lie algebras using the Schwinger boson method.
• To apply generalized coherent state formalism to quantum information tasks, specifically swap and cloning of coherent states.
• To construct a universal swap operator for coherent states in an infinite-dimensional Hilbert space using su(2)-based generalized coherent states.
• To investigate the feasibility of realizing the universal swap operator in quantum optics.

Proposed method

• Constructs coherent states via the displacement operator $ D(z) = e^{za^{ aisebox{1pt}{\dag}} - \bar{z}a} $ acting on the harmonic oscillator vacuum.
• Applies the Schwinger boson method to realize unitary representations of su(1,1) and su(2) algebras using bosonic operators.
• Defines generalized coherent states for su(1,1) as $ |z\rangle = e^{zK_{+} - \bar{z}K_{-}} |K,0\rangle $, with $ K_{+}, K_{-}, K_{3} $ generators of su(1,1).
• Defines generalized coherent states for su(2) as $ |z\rangle = e^{zJ_{+} - \bar{z}J_{-}} |J,0\rangle $, with $ J_{+}, J_{-}, J_{3} $ generators of su(2).
• Derives the universal swap operator in infinite dimensions via matrix representation $ U_{ij,kl} = \delta_{il}\delta_{jk} $, valid for all $ i,j,k,l \in \mathbb{N} $.
• Demonstrates imperfect cloning of coherent states using the generalized coherent state formalism and the su(2) representation.

Experimental results

Research questions

• RQ1 Can a universal swap operator for coherent states be constructed in an infinite-dimensional Hilbert space using generalized coherent states?
• RQ2 How can the su(2)-based generalized coherent state formalism be used to implement quantum operations like state swap and cloning?
• RQ3 Is the abstract universal swap operator, defined via $ U_{ij,kl} = \delta_{il}\delta_{jk} $, physically realizable in quantum optics?
• RQ4 What is the role of generalized coherent states in enabling imperfect cloning of coherent states?
• RQ5 Can the structure of the swap operator in finite dimensions be generalized to infinite dimensions using coherent state theory?

Key findings

• The universal swap operator is constructed as $ U_{ij,kl} = \delta_{il}\delta_{jk} $, which swaps the tensor product $ a \otimes b $ into $ b \otimes a $ for any vectors $ a, b $ in an infinite-dimensional Hilbert space.
• The swap operator is derived from a matrix decomposition involving Controlled-NOT gates in finite dimensions, and the structure generalizes to infinite dimensions via the Kronecker delta form.
• Imperfect cloning of coherent states is achieved using the su(2)-based generalized coherent state formalism, though the paper does not provide a quantitative fidelity value.
• The construction of the universal swap operator is purely mathematical and abstract, with no explicit physical implementation in quantum optics provided.
• The generalized coherent states for su(1,1) and su(2) are shown to be essential tools for formulating quantum operations in quantum information theory.
• The paper establishes a formal bridge between Lie algebraic coherent states and quantum information protocols like state swap and cloning.