We describe a unified framework of phase-covariant multiuser quantum transformations for d-dimensional quantum systems. We derive the optimal phase-covariant cloning and transposition transformations for multiphase states. We show that for some particular relations between the input and output number of copies, they correspond to economical transformations, which can be achieved without the need of auxiliary systems. We prove a relation between the optimal phase-covariant cloning and transposition maps and optimal estimation of multiple phases for equatorial states.
We construct a unique optimal quantum device for turning a finite number of
d-level quantum systems in the same unknown pure state \ensuremath{\sigma} into
M systems of the same kind, in an approximation of the
M-fold tensor product of the state \ensuremath{\sigma}.
We present the universal cloning transformation of states in arbitrary-dimensional Hilbert spaces. This unitary transformation attains the optimal fidelity of cloning as specified by Werner [Phys. Rev. A 58, 1827 (1998)]. With this cloning transformation, pure as well as impure states can be optimally copied, and the quality of the copies does not depend on the state being copied. We discuss the properties of quantum clones. In particular, we show that in the limit of high dimension the fidelity of clones does not converge to zero but attains the limit
1/2. We also show that our cloning transformation is most suitable for cloning of entanglement.
State cloning and state transposition are fundamental transformations that, despite being desirable, cannot be perfectly realised as a consequence of two distinct requirements of quantum theory: cloning is forbidden by linearity, while transposition is ruled out by complete positivity. In this work, we show that, in spite of these disparate conceptual restrictions, the best physically allowed realisation of both transformations arises from a single physical process described by an isometry, which simultaneously implements their best possible approximations. This complementary relation allows us to determine the exact quantum and private classical capacities of the universal symmetric
N→N+K cloning channel in the fully general multicopy qudit setting, thereby closing the remaining gap in the quantum capacity problem for universal cloning. We provide an explicit circuit that simultaneously implements optimal pure-state
N→K transposition and optimal
N→N+K symmetric cloning, both attaining the same fidelity. Additionally, we prove that the optimal
N→K pure-state transposition channel is unique and realised by an estimation strategy. We also obtain the optimal mixed-state
N→1 transposition, and discuss its differences from its pure-state analogue.