Dealing with unknown quantum operations

TL;DR

This paper investigates how unknown unitary transformations, or twirling, impact quantum communication, revealing the limitations on representing operations after twirling and identifying the conditions for their correct form.

Contribution

It establishes the minimal conditions for representing operations after twirling and characterizes the form of twirled quantum operations.

Findings

Not all operations are representable after twirling.

Minimal requirements for representability are identified.

The correct form of twirled operations is characterized.

Abstract

In the context of quantum communications between two parties (here Alice and Bob), Bob's lack of knowledge about the communications channel can affect the purity of the states that he receives. The operation of applying an unknown unitary transformation to a state, thus reducing its purity, is called "twirling". As twirling affects the states that Bob receives, it also affects his perception of the operations that Alice applies to her states. In this work we find that not every operation is representable after a twirling, we show the minimal requirement for this to be possible, and we identify the correct form of the "twirled" operations.