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Quantum Wiretap Channel Coding Assisted by Noisy Correlation

Minglai Cai, Andreas Winter

TL;DR

A lower bound on the assisted private capacity of a quantum wiretap channel, which asymptotically meets the multi-letter converse and which encompasses all sorts of previous results as special cases.

Abstract

We consider the private classical capacity of a quantum wiretap channel, where the users (sender Alice, receiver Bob, and eavesdropper Eve) have access to the resource of a shared quantum state, additionally to their channel inputs and outputs. An extreme case is maximal entanglement or a secret key between Alice and Bob, both of which would allow for onetime padding the message. But here both the wiretap channel and the shared state are general. In the other extreme case that the state is trivial, we recover the wiretap channel and its private capacity [N. Cai, A. Winter and R. W. Yeung, Probl. Inform. Transm. 40(4):318-336, 2004]. We show how to use the given resource state to build a code for secret classical communication. Our main result is a lower bound on the assisted private capacity, which asymptotically meets the multi-letter converse and which encompasses all sorts of previous results as special cases.

Quantum Wiretap Channel Coding Assisted by Noisy Correlation

TL;DR

A lower bound on the assisted private capacity of a quantum wiretap channel, which asymptotically meets the multi-letter converse and which encompasses all sorts of previous results as special cases.

Abstract

We consider the private classical capacity of a quantum wiretap channel, where the users (sender Alice, receiver Bob, and eavesdropper Eve) have access to the resource of a shared quantum state, additionally to their channel inputs and outputs. An extreme case is maximal entanglement or a secret key between Alice and Bob, both of which would allow for onetime padding the message. But here both the wiretap channel and the shared state are general. In the other extreme case that the state is trivial, we recover the wiretap channel and its private capacity [N. Cai, A. Winter and R. W. Yeung, Probl. Inform. Transm. 40(4):318-336, 2004]. We show how to use the given resource state to build a code for secret classical communication. Our main result is a lower bound on the assisted private capacity, which asymptotically meets the multi-letter converse and which encompasses all sorts of previous results as special cases.
Paper Structure (5 sections, 1 theorem, 21 equations, 2 figures)

This paper contains 5 sections, 1 theorem, 21 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Communication scenario
  3. Preliminaries
  4. Main result
  5. Discussion

Key Result

Theorem 5

Let $(\mathcal{N},\zeta)$ be an assisted quantum wiretap channel with the wiretap channel $\mathcal{Z}:A'\rightarrow B'E'$ defined as above. Assume furthermore a cq-channel $\mathcal{C}:\mathcal{U} \rightarrow A\otimes A'$ and probabilities $q(u)$ such that $\sum_u q(u) \operatorname{Tr}_A\mathcal{C where

Figures (2)

  • Figure 1: Communication diagramme of the wiretap coding problem over $\mathcal{N}$ with assistance by a tripartite resource state shared by sender (Alice), receiver (Bob) and eavesdropper (Eve). Unlike the plain wiretap channel, here each message is encoded not into a state, but rather a modulation $\mathcal{E}_m:A'\rightarrow A$.
  • Figure 2: Reformulation of the assisted wiretap code for $(\mathcal{N},\zeta)$ in terms of the tensor product wiretap channel $\mathcal{N}\otimes\mathcal{Z}$ with a restriction on the marginals of the signal states $\eta_m$ on ${A'}^n$.

Theorems & Definitions (6)

  • Definition 1
  • Definition 2
  • Definition 3
  • Remark 4
  • Theorem 5
  • proof