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There are various physical scenarios in which one can only implementoperations with a certain symmetry. Under such restriction, a system in asymmetry-breaking state can be used as a catalyst, e.g. to prepare anothersystem in a desired symmetry-breaking state. This sort of (approximate)catalytic state transformations are relevant in the context of (i) statepreparation using a bounded-size quantum clock or reference frame, where theclock or reference frame acts as a catalyst, (ii) quantum thermodynamics, whereagain a clock can be used as a catalyst to prepare states which containcoherence with respect to the system Hamiltonian, and (iii) cloning of unknownquantum states, where the given copies of state can be interpreted as acatalyst for preparing the new copies. Using a recent result of Fawzi andRenner on approximate recoverability, we show that the achievable accuracy inthis kind of catalytic transformations can be determined by a single function,namely the relative entropy of asymmetry, which is equal to the differencebetween the entropy of state and its symmetrized version: if the desired statetransition does not require a large increase of this quantity, then it can beimplemented with high fidelity using only symmetric operations. Our lower boundon the achievable fidelity is tight in the case of cloners, and can be achievedusing the Petz recovery map, which interestingly turns out to be the optimalcloning map found by Werner.

From clocks to cloners: Catalytic transformations under covariant operations and recoverability