  Channel-state duality is a central result in quantum information science. It
refers to the correspondence between a dynamical process (quantum channel) and
a static quantum state in an enlarged Hilbert space. Since the corresponding
dual state is generally mixed, it is described by a Hermitian matrix. In this
article, we present a randomized channel-state duality. In other words, a
quantum channel is represented by a collection of $N$ pure quantum states that
are produced from a random source. The accuracy of this randomized duality
relation is given by $1/N$, with regard to an appropriate distance measure. For
large systems, $N$ is much smaller than the dimension of the exact dual matrix
of the quantum channel. This provides a highly accurate low-rank approximation
of any quantum channel, and, as a consequence of the duality relation, an
efficient data compression scheme for mixed quantum states. We demonstrate
these two immediate applications of the randomized channel-state duality with a
chaotic $1$-dimensional spin system.
