Invertible Quantum Operations and Perfect Encryption of Quantum States

TL;DR

This paper characterizes invertible quantum operations as unitary transformations with ancillas, impacting quantum encryption by slightly broadening existing schemes while reaffirming key size lower bounds.

Contribution

It provides a precise characterization of invertible quantum operations and extends known encryption bounds to more general schemes.

Findings

Invertible quantum operations are unitary transformations with ancillas.

One-way quantum encryption schemes can be more general than previously thought.

The 2n-bit key lower bound for encrypting n qubits still applies.

Abstract

In this note, we characterize the form of an invertible quantum operation, i.e., a completely positive trace preserving linear transformation (a CPTP map) whose inverse is also a CPTP map. The precise form of such maps becomes important in contexts such as self-testing and encryption. We show that these maps correspond to applying a unitary transformation to the state along with an ancilla initialized to a fixed state, which may be mixed. The characterization of invertible quantum operations implies that one-way schemes for encrypting quantum states using a classical key may be slightly more general than the ``private quantum channels'' studied by Ambainis, Mosca, Tapp and de Wolf (FOCS 2000). Nonetheless, we show that their results, most notably a lower bound of 2n bits of key to encrypt n quantum bits, extend in a straightforward manner to the general case.