Finite-dimensional quantum compression converse ($F_e(\rho ,\mathcal{D}\mathcal{C})\le {\sum }_{j<k}{\lambda }_j$)

If encoding and decoding quantum channels factor through a $k$-dimensional space, their composite Kraus operators all have matrix rank at most $k$. The rank bound for a weighted operator trace and the Kraus formula for entanglement fidelity implyThus high operation fidelity requires the leading $k$ eigenvalues of the source density operator to carry almost all its probability. The bound applies to arbitrary completely positive trace-preserving encoders and decoders.