[Paper Review] Exact analytical solution of Entanglement of Formation and Quantum Discord for Werner state and Generalized Werner-Like states

S. Díaz-Solórzano, E. Castro|arXiv (Cornell University)|Feb 16, 2018

Quantum optics and atomic interactions3 citations

TL;DR

This paper presents exact analytical solutions for Entanglement of Formation (EoF) and Quantum Discord (QD) in Werner and Generalized Werner-Like (GWL) states, using f-deformed coherent states derived from Pöschl-Teller, Morse, and quantum dot potentials. The key contribution is a closed-form optimization of the conditional entropy via projective measurements, revealing that the structure of GWL states is preserved under measurement while the mixing parameter is renormalized, enabling exact computation of both EoF and QD without numerical approximation.

ABSTRACT

We obtained analytical expressions for Entangled of Formation (EoF) and Quantum Discord (QD) of Werner states and Generalized Werner-Like states. The optimization problem involved under the exact analytical form is obtained for both quantities. In order to illustrate the importance of our results we studied the EoF and the QD of these states. Using discrete formulation of continue states with the $f-$deformed coherent states obtained as deformed annihilation operator coherent states and as deformed displacement operator coherent states. The EoF and QD of bipartite Werner-Like states $f-$deformed coherent states are studied for the Pöschl-Teller, Morse and quantum dot deformed potentials. The result obtained are compared with the case of bipartite Werner-Like coherent states.

Motivation & Objective

• To derive exact analytical expressions for Entanglement of Formation (EoF) and Quantum Discord (QD) in Werner and Generalized Werner-Like (GWL) states.
• To resolve the optimization challenge in computing QD by analytically minimizing the conditional entropy under local projective measurements.
• To extend the analysis to f-deformed coherent states based on Pöschl-Teller, Morse, and quantum dot potentials, comparing results with standard GWL states.
• To demonstrate that the projective measurement preserves the structural form of GWL states while renormalizing the mixing parameter.

Proposed method

• Derives exact analytical expressions for EoF and QD using the von Neumann entropy and mutual information formalism for bipartite mixed states.
• Applies projective measurements to the A-partition of GWL states, deriving the post-measurement density matrices via the operator formalism.
• Introduces a mixing parameter $ x_m(p) = \frac{p\langle\widehat{\Pi}_m^A\rangle_\psi}{p_m^A} $ that renormalizes the state after measurement, preserving the GWL structure.
• Uses the conditional entropy $ S_{B|\{\Pi_m^A\}}(\rho_{AB}) = \min_{\{\Pi_m^A\}} \sum_m p_m^A S_{B|\Pi_m^A}(\rho_{AB}) $ and minimizes it by optimizing $ \langle\widehat{\Pi}_m^A\rangle_\psi $.
• Expresses the measurement probability $ \langle\widehat{\Pi}_m^A\rangle_\psi $ in terms of Pauli matrix expectations and trigonometric functions of angles $ \theta $, $ \phi $, and $ m\pi $.
• Derives the amplitude of oscillations in $ \langle\widehat{\Pi}_m^A\rangle_\psi $ as $ A = \frac{1}{2}\sqrt{\langle\sigma_z\rangle^2 + \langle\sigma_x\rangle^2 + \langle\sigma_y\rangle^2} $, enabling identification of the minimum probability and thus the minimal conditional entropy.

Experimental results

Research questions

• RQ1 What is the exact analytical form of Entanglement of Formation (EoF) for Werner and Generalized Werner-Like (GWL) states?
• RQ2 How can the Quantum Discord (QD) be exactly computed for GWL states by resolving the minimization of conditional entropy?
• RQ3 What is the effect of local projective measurements on the structure and mixing parameter of GWL states?
• RQ4 How do f-deformed coherent states based on Pöschl-Teller, Morse, and quantum dot potentials affect the EoF and QD compared to standard GWL states?
• RQ5 What is the role of the oscillatory behavior in $ \langle\widehat{\Pi}_m^A\rangle_\psi $ in minimizing the conditional entropy and thus determining the QD?

Key findings

• The paper derives an exact analytical expression for the Quantum Discord (QD) of Werner and GWL states by solving the minimization of the conditional entropy via projective measurements.
• The optimal measurement basis is determined by minimizing $ \langle\widehat{\Pi}_m^A\rangle_\psi $, which corresponds to the minimum of the function $ F(x_m(p)) $, and the minimum occurs at $ \langle\widehat{\Pi}_m^A\rangle_\psi^{\min} = \frac{1}{2} - A $, where $ A $ is the amplitude of oscillation in the measurement probability.
• The projective measurement does not alter the structural form of GWL states; instead, it renormalizes the mixing parameter $ p $ to $ x_m(p) $, preserving the state as a GWL state with modified mixing.
• The amplitude of oscillations in the measurement probability is given by $ A = \frac{1}{2}\sqrt{\langle\sigma_z\rangle^2 + \langle\sigma_x\rangle^2 + \langle\sigma_y\rangle^2} $, which depends on the reduced density matrix $ \rho_A(\psi) $ and the state's spin expectation values.
• For f-deformed coherent states based on Pöschl-Teller, Morse, and quantum dot potentials, the EoF and QD are computed exactly and compared to standard GWL states, showing deviations due to deformation.
• The exact solution for EoF and QD is obtained without numerical approximation, providing a benchmark for future studies on mixed-state quantum correlations.