Perfect NOT transformation and conjugate transformation

TL;DR

This paper investigates perfect and probabilistic NOT transformations in quantum systems, providing criteria, constructing probabilistic gates, and extending the concept to conjugate transformations with efficiency bounds.

Contribution

It derives necessary and sufficient conditions for perfect NOT transformations, constructs probabilistic NOT gates, and generalizes to conjugate transformations with efficiency bounds.

Findings

Criteria for perfect NOT transformation on qubit sets.

Construction of probabilistic perfect NOT gates.

Lower bounds for efficiencies of conjugate transformations.

Abstract

The perfect NOT transformation, probabilistic perfect NOT transformation and conjugate transformation are studied. Perfect NOT transformation criteria on a quantum state set $S$ of a qubit are obtained. Two necessary and sufficient conditions for realizing a perfect NOT transformation on $S$ are derived. When these conditions are not satisfied we discuss a probabilistic perfect NOT transformation (gate). We construct a probabilistic perfect NOT machine (gate) by a general unitary-reduction operation. With a postselection of the measurement outcomes, the probabilistic NOT gate yields perfectly complements of the input states. We prove that one can realize probabilistically the NOT gate of the input states secretly chosen from a certain set $S=\{|\Psi_1>, |\Psi_2>,..., |\Psi_n>\}$ if and only if $|\Psi_1>, |\Psi_2>,...,$ and $|\Psi_n>$ are linearly independent. We also generalize the probabilistic NOT transformation to the conjugate transformation in the multi-level quantum system. The lower bound of the best possible efficiencies attained by a probabilistic perfect conjugate transformation are obtained.