Complementarity of Private and Correctable Subsystems in Quantum Cryptography and Error Correction

Dennis Kretschmann; David W. Kribs; Robert W. Spekkens

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Complementarity of Private and Correctable Subsystems in Quantum Cryptography and Error Correction

Dennis Kretschmann, David W. Kribs, Robert W. Spekkens

arXiv:0711.3438v1Nov 21, 2007quant-ph

TL;DR

Finding private subsystems matters for cryptographic applications, but the problem has received little attention compared with finding correctable subsystems. The paper relates privacy and correctability through complementary channels, showing an exact duality that also extends to approximate notions.

Problem
Finding private subsystems has significant cryptographic applications, yet little work has addressed the problem compared with finding correctable subsystems.
Source: p. 1
Method
The paper relates subsystem privacy for a channel to subsystem correctability for its complementary channel.
Source: p. 1
Results
A subsystem is correctable for a channel if and only if it is private for the complementary channel, and approximate versions have dimension-independent bounds.
Source: p. 3
Takeaways & Limitations
Techniques for finding correctable subsystems can be applied to finding private subsystems, while cryptographic results may also inform error correction.
Source: p. 1
The complementarity theorem is presented for finite-dimensional matrix algebras; extension to infinite-dimensional ones remains prospective.
Source: p. 3

Abstract

from arXiv · show

We make an explicit connection between fundamental notions in quantum cryptography and quantum error correction. Error-correcting subsystems (and subspaces) for quantum channels are the key vehicles for contending with noise in physical implementations of quantum information-processing. Private subsystems (and subspaces) for quantum channels play a central role in cryptographic schemes such as quantum secret sharing and private quantum communication. We show that a subsystem is private for a channel precisely when it is correctable for a complementary channel. This result is shown to hold even for approximate notions of private and correctable defined in terms of the diamond norm for superoperators.