= Finite-dimensional quantum compression converse
{title2=$F_e(\rho,\mathcal D\mathcal C)\leq\sum_{j<k}\lambda_j$}
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> imply
$$
F_e(\rho,\mathcal D\mathcal C)\leq\sum_{j<k}\lambda_j(\rho).
$$
Thus 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.
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