Perfect NOT transformation and conjugate transformation

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.

6 pages

References in corpus (8)

• Measurement-based quantum computation with cluster states
• Quantum cloning
• Probabilistic cloning and identification of linearly independent quantum states
• Programmable quantum gate arrays
• Incoherent and Coherent Eavesdropping in the 6-state Protocol of Quantum Cryptography
• Long-distance entanglement purification for quantum communication
• Cloning and quantum computation
• Achievable efficiencies for probabilistically cloning the states