Highly Entangled States With Almost No Secrecy

TL;DR

This paper demonstrates that a highly entangled quantum state can have almost no secrecy extractable from it, challenging assumptions about the relationship between entanglement and secrecy in quantum information.

Contribution

It provides the first example of a state that is highly entangled yet yields almost no secrecy, with bounds on secrecy and entanglement cost derived using advanced mathematical tools.

Findings

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

The state is highly entangled, requiring a large number of singlets to create.

The results clarify the relationship between squashed entanglement and relative entropy of entanglement.

Abstract

In this paper we illuminate the relation between entanglement and secrecy by providing the first example of a quantum state that is highly entangled, but from which, nevertheless, almost no secrecy can be extracted. More precisely, we provide two bounds on the bipartite entanglement of the totally antisymmetric state in dimension d x d. First, we show that the amount of secrecy that can be extracted from the state is low, to be precise it is bounded by O(1/d). Second, we show that the state is highly entangled in the sense that we need a large amount of singlets to create the state: entanglement cost is larger than a constant, independent of d. In order to obtain our results we use representation theory, linear programming and the entanglement measure known as squashed entanglement. Our findings also clarify the relation between the squashed entanglement and the relative entropy of entanglement.