Non-standard quantum algebra $$\mathcal {U}_h (\mathfrak {sl}(2, \mathbb {R}))$$ and h-Dicke states

Abstract We discuss the application of the Jordanian quantum algebra $$\mathcal {U}_h (\mathfrak {sl}(2, \mathbb {R}))$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>h</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>sl</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi>R</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , a Hopf algebra deformation of the Lie algebra $$\mathfrak {sl}(2, \mathbb {R})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>sl</mml:mi> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi>R</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , in order to generate sets of N qubit quantum states. We construct the associated h -deformed Dicke states using the Clebsch–Gordan coefficients for $$\mathcal {U}_h (\mathfrak {sl}(2, \mathbb {R}))$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>h</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>sl</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi>R</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , showing that the former exhibit completely different features than the q -Dicke states obtained from the standard quantum deformation $${\mathcal {U}}_q (\mathfrak {sl}(2, \mathbb {R}))$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>q</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>sl</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi>R</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . Moreover, the density matrices of these h -deformed Dicke states are compared to the experimental realizations of those of Dicke states, and several similarities are observed, indicating that the h -deformation could be used to describe noise and decoherence effects in experimental settings, as well as to control the degree of entanglement of the state in quantum computing protocols. In particular, h -Dicke states for $$N=2,3,4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>3</mml:mn> <mml:mo>,</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> are presented, a method to construct the h -deformed analogs of W -states for arbitrary N is given, and some algebraic considerations for the explicit derivation of generic h -Dicke states are provided.