Entanglement of the Antisymmetric State

TL;DR

This paper investigates the antisymmetric state's entanglement properties, revealing it has low secrecy extractability but high entanglement cost, with implications for quantum information theory.

Contribution

It provides new bounds on the secrecy and entanglement cost of the antisymmetric state, using advanced representation theory techniques.

Findings

Secrecy extractable from the state is bounded by O(1/d)

Entanglement cost exceeds a constant independent of dimension d

Regularised relative entropy of entanglement is lower bounded by a constant

Abstract

We analyse the entanglement of the antisymmetric state in dimension d x d and present two main results. First, we show that the amount of secrecy that can be extracted from the state is low, more precisely, the distillable key is bounded by O(1/d). Second, we show that the state is highly entangled in the sense that a large number of ebits are needed in order to create the state: entanglement cost is larger than a constant, independent of d. The second result is shown to imply that the regularised relative entropy with respect to separable states is also lower bounded by a constant. Finally, we note that the regularised relative entropy of entanglement is asymptotically continuous in the state. Elementary and advanced facts from the representation theory of the unitary group, including the concept of plethysm, play a central role in the proofs of the main results.