Contraction and Expansion Values of Quantum Channels

The contraction coefficient of the trace distance is a central tool in quantum information, quantifying how strongly a quantum channel degrades the distinguishability of states.

However, being an extremal ratio, it captures only the most optimistic behaviour of the channel and is often trivial, even for very noisy channels.

Moreover, a single scalar is poorly suited to describe how contraction accumulates under channel composition.

In this work we introduce the \emph{contraction and expansion values}, two monotone sequences that refine the contraction and expansion coefficients in the same way singular values refine the operator norm.

They arise from a min--max variational principle over subspaces of traceless Hermitian operators, admit an operational interpretation in terms of two state-discrimination games, and are shown to coincide with the Gel'fand or Bernstein numbers of the channel restricted to traceless operators.

This identification places the sequences within Pietsch's theory of $s$-numbers and yields, in particular, bounds under channel composition that the contraction coefficient alone cannot provide.

We establish their main structural properties and compute or estimate them for single-qubit channels, $d$-dimensional amplitude damping channels, and direct-sum channels.