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We show how the efficiency of a logical Bell measurement (BM) can becalculated for arbitrary CSS codes with the experimentally important constraintof using only transversal static linear-optical BMs on the physicalsingle-photon qubit level. For this purpose, we utilize the codes' descriptionin terms of stabilizers in order to calculate general efficiencies for theloss-free case, but also for specific cases including photon loss. Theseefficiencies can be, for instance, used for obtaining transmission rates ofall-optical quantum repeaters. In the loss-free case, we demonstrate that theimportant class of CSS codes with identical physical-qubit support for the twological Pauli ($Z$ and $X$) operators can only achieve a logical BM efficiencyof $\frac{1}{2}$ if one always employs the same ancilla-free static linearoptical BMs on the physical level. We apply our methods to various CSS codesincluding two-dimensional planar color and planar surface codes. We then findthat in many cases, the fixed use of the standard linear optical BM for allphysical BMs is suboptimal and performing linear optical transformations beforedoing the standard linear optical BM (still without any ancillary photons andwithout any feedforward) can increase the efficiency enormously. In fact, usingthis generalization in the no-loss (or sometimes also in the low-loss) caseallows us, on the one hand, to improve the logical BM efficiency of quantumparity codes compared to previously known results and, on the other hand, italso enables us to enhance the efficiency of two-dimensional planar colorcodes, whose efficiency is otherwise subject to the above $\frac{1}{2}$ bound,to arbitrarily close to unity.

Efficiencies of logical Bell measurements on CSS codes with static linear optics