Entanglement cost of quantum state

Alice and Bob want to make n copies of a Quantum_state: ρ^⊗n.

They apply a LOCC Λ that consumes entanglement and produces a copy of ρ.

How many bottles of distilled entanglement do they require for each ρ?

import LOCC

/-- "The rate of entanglement needed 
to generate a bipartite quantum state ρ." -/
def entanglement_cost [quantum_state ρ] : ℝ :=
inf {r | lim n→∞ inf {∥ρ^⊗n - Λ(Φ^⊗(r*n))∥₁ | LOCC Λ} = 0}

• Lower bounds on entanglement cost of quantum state;

• Entanglement cost of quantum state is equal to its regularized entanglement of formation;

• Dual measure to Distillable entanglement of quantum state.

• “Entanglement cost of quantum channel is the quantum channel analog of the Entanglement cost of quantum state.”

References

• Lami2021;

• Lami2022;

• Wehner2015;

• Horodecki2007;