Private Quantum Subsystems

TL;DR

This paper develops a comprehensive theory of private quantum subsystems, providing new conditions for their existence and revealing that private subsystems can exist without private subspaces, challenging previous assumptions.

Contribution

It introduces testable conditions for private quantum subsystems, establishes an analogue of Knill-Laflamme conditions, and shows private subsystems can exist independently of private subspaces.

Findings

Private subsystems can exist without private subspaces.

Established conditions for private quantum subsystems using Kraus operators.

Discovered private subsystems not complemented by quantum error correcting codes.

Abstract

We investigate the most general notion of a private quantum code, which involves the encoding of qubits into quantum subsystems and subspaces. We contribute to the structure theory for private quantum codes by deriving testable conditions for private quantum subsystems in terms of Kraus operators for channels; establishing an analogue of the Knill-Laflamme conditions in this setting. For a large class of naturally arising quantum channels, we show that private subsystems can exist even in the absence of private subspaces. In doing so, we also discover the first examples of private subsystems that are not complemented by operator quantum error correcting codes; implying that the complementarity of private codes and quantum error correcting codes fails for the general notion of private quantum subsystem.