Distillation of secret key and entanglement from quantum states

TL;DR

The paper addresses distillation of secret key and entanglement from quantum correlations using one-way public communication. It proves a one-way secret-key coding theorem that achieves the wire-tapper bound $I(X;B)-I(X;E)$ for cqq correlations and shows a coherent protocol yielding the hashing inequality $D_\rightarrow(A\rangle B) \ge I_c(A\rangle B)$, linking secrecy with entanglement. This coherent implementation implies information-theoretic formulas for distillable entanglement and quantum capacities, unifying cryptographic secrecy with LOCC entanglement distillation and channel coding. It also discusses open questions on one-way versus two-way protocols, potential gaps between key and entanglement rates, and the forward communication cost of distillation.

Abstract

We study and solve the problem of distilling secret key from quantum states representing correlation between two parties (Alice and Bob) and an eavesdropper (Eve) via one-way public discussion: we prove a coding theorem to achieve the "wire-tapper" bound, the difference of the mutual information Alice-Bob and that of Alice-Eve, for so-called cqq-correlations, via one-way public communication. This result yields information--theoretic formulas for the distillable secret key, giving ``ultimate'' key rate bounds if Eve is assumed to possess a purification of Alice and Bob's joint state. Specialising our protocol somewhat and making it coherent leads us to a protocol of entanglement distillation via one-way LOCC (local operations and classical communication) which is asymptotically optimal: in fact we prove the so-called "hashing inequality" which says that the coherent information (i.e., the negative conditional von Neumann entropy) is an achievable EPR rate. This result is well--known to imply a whole set of distillation and capacity formulas which we briefly review.

Figures (1)

• Figure 1: A schematic view of the anatomy of the code: the typical sequences are covered by sets ${\cal C}_\ell$, which are good transmission codes for $B$. A magnified view of one ${\cal C}_\ell$ (to the lower right) reveals its inner structure: it is composed of ${\cal S}_{\ell m}$, which are good privacy amplification codes against $E$.

Theorems & Definitions (20)

• Theorem 1
• Remark 2
• Lemma 3: "Operator Chernoff bound" Ahlswede:Winter
• Proposition 4
• Proposition 5: HSW theorem
• Theorem 6
• Remark 7
• Theorem 8
• Remark 9
• Theorem 10: Hashing inequality