Squashed entanglement - addenda Nicholas LeCompte The other Fannes - - PowerPoint PPT Presentation

Squashed entanglement - addenda

Nicholas LeCompte

The other Fannes inequality

◮ In the proof sketch that the distillable entanglement

lower-bounds the squashed entanglement, we mentioned a Fannes inequality.

◮ Lemma: For any ε > 0 and d-dimensional density matrices ρ

and σ satisfying |ρ − σ|1 ≤ ε, we have |S(ρ) − S(σ)| ≤ η(ε) + ε log d, where η(ε) = −ε log ε ε ≤ 1/4 1/2

• therwise

◮ By invoking the bound I(A; B|E) ≥ I(A; B) − 2S(AB), we

can show that ED(ρ) ≤ Esq(ρ).

State redistribution

We briefly outline the procedure for state redistribution:

◮ Suppose Alice and Bob share some state ρABC, where Alice

has ρAC and Bob has ρB.

◮ Alice wants to send ρA to Bob coherently. ◮ Let “Calice” denote the party which holds rhoC and suppose

Alice and Calice have a quantum channel, with R as a reference system in the purification of ρABC.

State redistribution

◮ Alice merges ρA to Calice, extracting a rate of I(A : C)/2

ebits, and using a rate of I(A : RB)/2 qubits.

◮ As Calice and Bob share ebits, Calice can replace the ebits

generated with ebits shared between she and Bob.

◮ Calice then sends the remaining qubits to Bob with rate

I(A : CR)/2 − I(A : C)/2, leaving her with a rate of I(A : B)/2 ebits, and Bob now has ρA.

◮ Functionally, Calice acts as a relay between Alice and Bob.