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Quantum deformation and Quantum Entanglement

Mostafa Mansour

Published May 11, 2012• 0 citations
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Abstract

Recent study suggests that there are natural connections between quantum entanglement and the quantumYang-Baxter equation. The present paper introduce the unitary solutions of the Quantum Yang{Baxter Equationderived via the quantization-deformation of a Poisson Lie group associated to an r-matrix (solution of a classicalYang-Baxter equation). The solution of the algebraic Quantum Yang Baxter equation that we explore containa deformation parameter h, and will be used to performs quantum entanglement when acting on (multi)-qubitquantum states.

DOI
10.5281/zenodo.20114401

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Recent study suggests that there are natural connections between quantum entanglement and the quantumYang-Baxter equation. The present paper introduce the unitary solutions of the Quantum Yang{Baxter Equationderived via the quantization-deformation of a Poisson Lie group associated to an r-matrix (solution of a classicalYang-Baxter equation). The solution of the algebraic Quantum Yang Baxter equation that we explore containa deformation parameter h, and will be used to performs quantum entanglement when acting on (multi)-qubitquantum states.
Entangled states, such as the Bell and GHZ states, are generated from separable states using matrices known to satisfy the Yang-Baxter equation and its generalization. This remarkable fact hints at the possibility of using braiding operators as quantum entanglers, and is part of a larger speculated connection between topological and quantum entanglement. We push the analysis of this connection forward, by showing that supersymmetry algebras can be used to construct large families of solutions of the spectral parameter-dependent generalized Yang-Baxter equation. We present a number of explicit examples and outline a general algorithm for arbitrary numbers of qubits. The operators we obtain produce, in turn, all the entangled states in a multi-qubit system classified by the Stochastic Local Operations and Classical Communication protocol introduced in quantum information theory.
We present a new class of solutions to the parameter-dependent Yang–Baxter equation across all dimensions, which includes a significant subclass of unitary and entangling solutions. In any dimension d ≥ 2 , we construct a well-structured matrix R ( u ) that exhibits intriguing and useful symmetries. A key feature of R ( u ) is its composition from four monomial-based layers: R ( u ) = R a ( u ) + R b ( u ) + R x ( u ) + R y ( u ) , where a ( u ) , b ( u ) , x ( u ) and y ( u ) are four sets of monomial complex functions of the form, a u n a , b u n b , x u n x and y u n y , respectively. This well-designed structure plays a crucial role in demonstrating that R ( u ) is indeed a solution to the Yang–Baxter equation and in establishing the conditions for its unitarity and entangling properties. Additionally, it allows us to identify interesting non-trivial subfamilies with these properties. As is widely recognized, unitary and entangling solutions to the Yang–Baxter equation serve as universal quantum logic gates for qudit quantum computing . Moreover, the search for new solutions to the Yang–Baxter equation in higher dimensions is a common endeavor in both mathematics and physics .
Starting from the Kauffman-Lomonaco braiding matrix transforming the natural basis to Bell states, the spectral parameter describing the entanglement is introduced through Yang-Baxterization. It gives rise to a new type of solutions for Yang-Baxter equation, called the type-II that differs from the familiar solution called type-I of YBE associated with the usual chain models. The Majorana fermionic version of type-II yields the Kitaev Hamiltonian. The introduced ℓ 1 -norm leads to the maximum of the entanglement by taking the extreme value and shows that it is related to the Wigner's D-function. Based on the Yang-Baxter equation the 3-body S-Matrix for type-II is explicitly given. Different from the type-I solution, the type-II solution of YBE should be considered in describing quantum information. The idea is further extended to 3 parafermion model based on SU (3) principal representation. The type-II is in difference from the familiar type-I in many respects. For example, the quantities corresponding to velocity in the chain models obey the Lorentzian additivity rather than Galilean rule ( u + v ). Most possibly, for the type-II solutions of YBE there may not exist RTT relation. Further more, for 3 parafermion model we only need the rational Yang-Baxterization, which seems like trigonometric. Similar discussions are also made in terms of generalized Yang-Baxter equation with three spin spaces .