Orthogonal measurements are {\it almost} sufficient for quantum discord of two qubits

TL;DR

This paper investigates the effectiveness of orthogonal measurements in calculating quantum discord for two-qubit states, showing they are nearly optimal for most cases and providing new bounds and methods for computation.

Contribution

It proves the optimality of orthogonal measurements for rank 2 states and offers strong numerical evidence of their near-optimality for higher ranks, along with a new approach based on Bloch vectors.

Findings

Orthogonal measurements are optimal for rank 2 states.

For higher rank states, orthogonal measurements are nearly optimal with minimal corrections.

Provides a tight upper bound for entanglement of formation in certain mixed states.

Abstract

The common use in literature of orthogonal measurements in obtaining quantum discord for two-qubit states is discussed and compared with more general measurements. We prove the optimality of orthogonal measurements for rank 2 states. While for rank 3 and 4 mixed states they are not optimal, we present strong numerical evidence showing that they give the correct quantum discord up to minimal corrections. Based on the connection, through purification with an ancilla, between discord and entanglement of formation (EoF), we give a tight upper bound for the EoF of a $2\otimes N$ mixed state of rank 2, given by an optimal decomposition of 2 elements. We also provide an alternative way to compute the quantum discord for two qubits based on the Bloch vectors of the state.