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 � − ε log ε ε ≤ 1 / 4 η ( ε ) = 1 / 2 otherwise ◮ By invoking the bound I ( A ; B | E ) ≥ I ( A ; B ) − 2 S ( AB ), we can show that E D ( ρ ) ≤ E sq ( ρ ).
State redistribution We briefly outline the procedure for state redistribution: ◮ Suppose Alice and Bob share some state ρ ABC , where Alice has ρ AC and Bob has ρ B . ◮ Alice wants to send ρ A to Bob coherently. ◮ Let “Calice” denote the party which holds rho C and suppose Alice and Calice have a quantum channel, with R as a reference system in the purification of ρ ABC .
State redistribution ◮ Alice merges ρ A 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 ρ A . ◮ Functionally, Calice acts as a relay between Alice and Bob.
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