Rank dependent bounds on mixedness and entanglement for quantum teleportation

TL;DR

This paper derives rank-dependent bounds on mixedness and entanglement measures for bipartite states, enhancing understanding of their usefulness in quantum teleportation, with specific results for Werner and Werner-like states.

Contribution

It generalizes existing bounds on mixedness and entanglement measures by incorporating the rank of bipartite states, providing new theoretical limits for quantum teleportation resources.

Findings

Upper bounds on mixedness increase with rank.

Rank-dependent lower bounds on concurrence for two-qubit states.

Werner states reach the maximum mixedness among two-qubit states.

Abstract

Entanglement and mixedness of a bipartite mixed state resource are crucial for the success of quantum teleportation. Upper bounds on measures of mixedness, namely, von Neumann entropy and linear entropy beyond which the bipartite state ceases to be useful for quantum teleportation are known in the literature. In this work, we generalize these bounds and obtain rank dependent upper bounds on von Nuemann entropy and linear entropy for an arbitrary bipartite mixed state resource. We observe that the upper bounds on measures of mixedness increase with increase in the rank. For two qubit mixed states, we obtain rank dependent lower bound on the concurrence, a measure of entanglement, below which the state is useless for quantum teleportation. Werner state, which is a fourth rank state, exhibits the theoretical upper bound on mixedness among two qubit mixed states. We construct Werner like states of lower ranks and show that these states possess the theoretical rank dependent upper bounds obtained on the measures of mixedness and theoretical rank dependent lower bounds on the concurrence.