Relating Quantum Privacy and Quantum Coherence: An Operational Approach

Given many realizations of a state or a channel as a resource, two parties can generate a secret key as well as entanglement. We describe protocols to perform the secret key distillation (as it turns out, with optimal rate). Then we show how to achieve optimal entanglement generation rates by “coherent” implementation of a class of secret key agreement protocols, proving the long-conjectured “hashing inequality.”

Figure 1

HSW code. The arrows represent the map $x^n\mapsto {\varphi }_{x^n}^B$. Intuitively, the ${\psi }_{x^n}^B$, supported on the respective ${\mathcal{T}}_{B\mid X}\left(x^n\right)$ of dimension $2^{nH(B\mid X{)}^{+}}$, can be “packed” into ${\mathcal{T}}_B$ of dimension $2^{nH(B{)}^{-}}$ with negligible overlap if we take $\left|\mathcal{C}\right|=2^{nI(X;B{)}^{-}}$. The HSW theorem [15] confirms this geometric picture.

Figure 2

PA set. The arrows represent the map $x^n\mapsto {\varphi }_{x^n}^E$. Since ${\sigma }^{\otimes n}$ and ${\psi }_{x^n}^E$ are “almost” uniformly supported on ${\mathcal{T}}_E$ and ${\mathcal{T}}_{E\mid X}\left(x^n\right)$, respectively, satisfying (3) amounts to “covering” ${\mathcal{T}}_E$, of dimension $2^{nH(E{)}^{+}}$ with the ${\mathcal{T}}_{E\mid X}\left(x^n\right)$, $x^n\in \mathcal{S}$, each of which has dimension $2^{nH(E\mid X{)}^{-}}$.

Figure 3

A secret key generation code (left) viewed as a building block for a key distillation code (right). Note how the rate $R$ comes about as the difference between Bob's and Eve's information about $X$.