Highly Entangled States with Almost No Secrecy

In this Letter 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 $d\times d$. 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.

Figure 1

Illustration of the properties of the state ${\Omega }_{A^n{B}^n{A}^{\prime n}{B}^{\prime n}}$. Systems connected by a curly line are in the state ${\alpha }_d$. Each group $A_i{B}_i{A}_i^{\prime }{B}_i^{\prime }$ is $g^{\otimes 4}$ invariant.