We construct for a Schur concave function
f on the set of quantum states a tight upper bound on the difference
f(ρ)−f(σ) for a quantum state
ρ with finite
f(ρ) and any quantum state
σ m-partially majorized by the state
ρ in the sense described in [1]. We also obtain a tight upper bound on this difference under the additional condition
21∥ρ−σ∥1≤ε and find simple sufficient conditions for vanishing this bound with
min{ε,1/m}→0.
The obtained results are applied to the von Neumann entropy. The concept of
ε-sufficient majorization rank of a quantum state with finite entropy is introduced and a tight upper bound on this quantity is derived and applied to the Gibbs states of a quantum oscillator.
We also show how the obtained results can be reformulated for Schur concave functions on the set of probability distributions with a finite or countable set of outcomes.
The general conditions for the orthogonal product states of the multistate systems to be used in quantum key distribution (QKD) are proposed, and a novel QKD scheme with orthogonal product states in the $3\ifmmode\times\else\texttimes\fi{}3$ Hilbert space is presented. We show that this protocol has many distinct features such as great capacity and high efficiency. The generalization to $n\ifmmode\times\else\texttimes\fi{}n$ systems is also discussed and an asymptotic limit of 1/2 for the eavesdropper's success probability is obtained.