Private quantum codes: introduction and connection with higher rank numerical ranges

TL;DR

This paper introduces private quantum codes, explaining their role in quantum cryptography, and connects their characterization to higher rank numerical ranges and quantum error correction techniques.

Contribution

It establishes a novel connection between private quantum codes, higher rank numerical ranges, and conjugate channels, bridging quantum cryptography and error correction.

Findings

Private quantum codes are characterized by indistinguishability under quantum channels.

Higher rank numerical ranges describe private code states.

The connection between quantum error correction and cryptography is elucidated.

Abstract

We give a brief introduction to private quantum codes, a basic notion in quantum cryptography and key distribution. Private code states are characterized by indistinguishability of their output states under the action of a quantum channel, and we show that higher rank numerical ranges can be used to describe them. We also show how this description arises naturally via conjugate channels and the bridge between quantum error correction and cryptography.