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Structure of the algebra generated by a noncommutative operator graph which demonstrates the superactivation phenomenon for zero-error capacity
Shirokov [1] recently suggested a construction of a noncommutative operator graph, depending on
a complex parameter θ, which enables one to construct channels with positive quantum capacity for
which the n-shot capacity is zero. We study the algebraic structure of this graph. Relations for the
algebra generated by the graph are derived. In the limit case θ = ??1, the graph becomes commutative
and degenerates into the direct sum of four one-dimensional irreducible representations of the Klein
group.
The superactivation of the capacity of quantum channels was discovered in [2]. It turned out that
the quantum capacity for the tensor product of two quantum channels can be positive, whereas the
quantum capacity of each of the channels in the product is zero. As was shown in [3] and [4], the value of
the quantum capacity is closely related to the so-called noncommutative operator graph of the quantum
channel. In [5], a similar property was discovered for the classical capacity with zero error. In [6] and [7],
a technique of studying superactivation, which uses noncommutative operator graphs, was developed.
This enables one to construct low-dimensional examples of superactivation for quantum capacity. In
the present paper, the algebra generated by the noncommutative operator graph constructed in [1] is
studied.1