Group representation approach to 1 - N universal quantum cloning machines

Abstract

In this work, we revisit the problem of finding an admissible region of fidelities obtained after an application of an arbitrary $1\to N$ universal quantum cloner which has been recently solved in [A. Kay et al., Quant. Inf. Comput 13, 880 (2013)] from the side of cloning machines. Using group-theory formalism, we show that the allowed region for fidelities can be alternatively expressed in terms of overlaps of pure states with recently found irreducible representations of the commutant $U\otimes U\otimes \dots \otimes U\otimes U^{\ast }$, which gives the characterization of the allowed region where states being cloned are figure of merit. Additionally, it is sufficient to take pure states with real coefficients only, which makes calculations simpler. To obtain the allowed region, we make a convex hull of possible ranges of fidelities related to a given irrep. Subsequently, two cases: $1\to 2$ and $1\to 3$ cloners, are studied for different dimensions of states as illustrative examples.

15 pages, 3 figures, comments welcome. To appear in PRA

References in corpus (4)

• Optimal Cloning and Singlet Monogamy
• A Ferromagnetic Lieb-Mattis Theorem
• Structure and properties of the algebra of partially transposed permutation operators
• Generalised Asymmetric Quantum Cloning

Cited by in corpus (10)

• Joint measurability of quantum effects and the matrix diamond
• Simplified formalism of the algebra of partially transposed permutation operators with applications
• RepLAB: a computational/numerical approach to representation theory
• Units of rotational information
• Compatibility of quantum measurements and inclusion constants for the matrix jewel
• Explicit constructions of unitary transformations between equivalent irreducible representations
• The compatibility dimension of quantum measurements
• A geometrical description of the universal $1\to 2$ asymmetric quantum cloning region
• The Asymmetric Quantum Cloning Region
• Universal Asymmetric Quantum Cloning Revisited