Generalized Teleportation Fidelity and Singlet Fraction and their Relation for (In)-distinguishable Particles and Its Applications

TL;DR

This paper develops generalized formulas for teleportation fidelity and singlet fraction applicable to both distinguishable and indistinguishable particles with various degrees of freedom, providing new insights and practical applications in quantum information.

Contribution

It introduces generalized expressions and relations for teleportation fidelity and singlet fraction for diverse particle types and degrees of freedom, extending previous models.

Findings

Derived an upper bound for the generalized singlet fraction for distinguishable particles.

Demonstrated counter-intuitive values of generalized singlet fraction using optical circuits.

Showed advantages of using an additional degree of freedom as an ancilla in quantum cryptography.

Abstract

Quantum teleportation efficiently transfers quantum information between distant locations by utilizing a pre-established composite system. Assessing the effectiveness of teleportation hinges on its fidelity, representing the similarity between input and output states. This fidelity, in turn, relies on a singlet fraction, quantifying the resemblance of the composite system to maximally entangled states. The relation between teleportation fidelity and singlet fraction given by [Horodecki \textit{et al}., Phy. Rev. A \textbf{60}, 1888 (1999)] does not hold for distinguishable particles with multiple degrees of freedom or indistinguishable particles with single or multiple degrees of freedom. In this paper, we propose generalized expressions for teleportation fidelity and singlet fraction and derive their relations, applicable for both distinguishable and indistinguishable particles with single or multiple degrees of freedom. We derive an upper bound for the generalized singlet fraction for distinguishable particles using the monogamy of singlet fraction by [Kay \textit{et al.} Phys. Rev. Lett. \textbf{103}, 050501 (2009)]. We also show how our relation helps to characterize different types of composite states in terms of their distinguishability, separability, presence of maximally entangled structure, and the number of degrees of freedom. We complement our theory with two practical illustrations. First, we demonstrate two counter-intuitive values of generalized singlet fraction using our optical circuit and the circuit of [Li \textit{et al.}, Phys. Rev. Lett. \textbf{120}, 050404 (2018)]. Finally, we show that using an additional degree of freedom as an ancilla instead of a particle can be advantageous in quantum cryptographic protocols.