Universal complex conjugation of quantum states and unitaries: implementation algorithm and implications
Jisho Miyazaki, Akihito Soeda, Mio Murao · arXiv (Cornell University) · 2017
Not all mathematical funcitions used to define physical quantities are guaranteed to be implementable; complex conjugation is one such. We show that universal state conjugation, i.e., complex conjugation of unknown quantum states, is not implementable, even with nonzero failure probability admitted and finitely many state clones supplied. Complex conjugation can also be defined on unitaries, for which we present a deterministic, universal quantum algorithm with a blackbox quantum gate as the input unitary. Multiple uses of the oracle is shown to be necessary for unitary dimensions larger than 2. An operator used to define this algorithm is exploited to generalize the two-qubit concurrence for pure states. The generalized concurrence is based on complex conjugation of states, much like the original concurrence. It is shown to be equivalent to the $G$-concurrence, a previously known generalization of the original concurrence, derived from a separate mathematical observation and a member of a family of concurrence monotones. We show that our approach also reproduces all these concurrence monotones. Finally, the unitary conjugation algorithm is interpreted in terms of particles and holes and their mode transformation.