[Paper Review] Quantum discord for two-qubit CS states: Analytical solution

M. A. Yurishchev|arXiv (Cornell University)|Feb 21, 2013

Quantum Information and Cryptography3 citations

TL;DR

This paper establishes a universal local orthogonal transformation using the Hadamard gate that maps any two-qubit centrosymmetric (CS) density matrix to an X-type density matrix, enabling analytical calculation of quantum discord for all CS states via known formulas for X states. The key contribution is a general analytical solution for quantum discord in CS states, applicable to physical systems like XXZ spin chains, spin-carrying particles in nanopores, and pseudopure states.

ABSTRACT

We found an universal local unitary (orthogonal) transformation which transforms any centrosymmetric (CS) density matrix ρ_{CS} into the X density matrix ρ_X: H\otimes Hρ_{CS}H\otimes H=ρ_X, where H is the Hadamard transformation. Because quantum discord is invariant under the local unitary transformations, this remarkable property allows to get the discord of general two-qubit CS states in the analytical form using the corresponding well-known formulas for the X states. Examples of systems with CS density matrices provided from the recent literature, including XXZ spin model with the Dzyaloshinsky-Moriya interaction, a gas of spin-carrying particles in closed nanopore, and a family of pseudopure states.

Motivation & Objective

• To develop a general analytical method for computing quantum discord in two-qubit centrosymmetric (CS) density matrices.
• To address the challenge that quantum discord is typically calculable only for X-states, not for general CS states.
• To establish a universal local orthogonal transformation that maps CS states to X-states, preserving discord values.
• To apply the method to physical systems such as XXZ spin models with Dzyaloshinsky-Moriya interaction, spin-carrying particles in nanopores, and pseudopure states.
• To provide exact analytical expressions for quantum discord in these physically relevant systems.

Proposed method

• Utilize the Hadamard transformation $ H \otimes H $ to transform any two-qubit CS density matrix $ \rho_{CS} $ into an X-state $ \rho_X $, with $ H\otimes H \rho_{CS} H\otimes H = \rho_X $.
• Leverage the invariance of quantum discord under local unitary operations, so $ Q(\rho_{CS}) = Q(\rho_X) $, enabling use of existing analytical formulas for X-states.
• Express the parameters of $ \rho_X $ in terms of those of $ \rho_{CS} $ via explicit algebraic relations (e.g., $ q_1 = \frac{1}{4} + p_2 + p_4 + \frac{1}{2}(p_6 + p_7) $).
• Apply the transformation to physical models: derive the X-form of the density matrix for XXZ spin chains with Dzyaloshinsky-Moriya interaction and for spin particles in nanopores.
• Use additional local unitary rotations (e.g., $ U = \exp(-i\varphi \sigma_z/2) $) to eliminate imaginary off-diagonal terms in X-states, reducing them to real X-states for which discord formulas are simplest.
• Apply known analytical formulas for discord in real X-states to compute $ Q = \min\{Q_1, Q_2\} $, with $ Q_1 $ and $ Q_2 $ defined via von Neumann entropy and eigenvalues of the density matrix.

Experimental results

Research questions

• RQ1 Can quantum discord be computed analytically for all two-qubit centrosymmetric (CS) density matrices, which are not in X-form?
• RQ2 Is there a universal local transformation that maps any CS state to an X-state while preserving quantum discord?
• RQ3 How can the analytical discord formula for X-states be extended to physical systems described by CS density matrices?
• RQ4 What is the behavior of quantum discord in a gas of spin-carrying particles confined in a closed nanopore?
• RQ5 Can the method be applied to pseudopure states, and what is the resulting discord expression?

Key findings

• The Hadamard transformation $ H \otimes H $ maps any two-qubit CS state to an X-state, establishing a one-to-one correspondence between the two classes.
• Quantum discord is invariant under this transformation, so $ Q(\rho_{CS}) = Q(\rho_X) $, enabling analytical computation of discord for all CS states.
• For the XXZ spin model with Dzyaloshinsky-Moriya interaction, the transformed density matrix takes the X-form $ \rho' $, allowing exact discord evaluation.
• In the nanopore model with $ N=20 $ particles and $ \beta=1 $, the quantum discord oscillates in time and saturates at $ Q \approx 0.15 $ when $ p=u=0 $, $ r=q=\frac{1}{8}\tanh^2(\beta/2) $.
• For pseudopure states $ \rho_{PP} = \alpha|\psi\rangle\langle\psi| + \frac{1-\alpha}{4}I $, if $ |\psi\rangle $ is of the form $ a(|00\rangle + |11\rangle) + b(|01\rangle + |10\rangle) $, the state is CS and its discord is analytically computable via the proposed method.
• The final expression for discord in the nanopore system is $ Q = \min\{Q_1, Q_2\} $, with $ Q_1 $ and $ Q_2 $ defined via eigenvalues and entropies of the real X-state after local rotation to eliminate $ xy $-terms.