Abstract:
There are various physical scenarios in which one can only implement
operations with a certain symmetry. Under such restriction, a system in a
symmetry-breaking state can be used as a catalyst, e.g. to prepare another
system in a desired symmetry-breaking state. This sort of (approximate)
catalytic state transformations are relevant in the context of (i) state
preparation using a bounded-size quantum clock or reference frame, where the
clock or reference frame acts as a catalyst, (ii) quantum thermodynamics, where
again a clock can be used as a catalyst to prepare states which contain
coherence with respect to the system Hamiltonian, and (iii) cloning of unknown
quantum states, where the given copies of state can be interpreted as a
catalyst for preparing the new copies. Using a recent result of Fawzi and
Renner on approximate recoverability, we show that the achievable accuracy in
this kind of catalytic transformations can be determined by a single function,
namely the relative entropy of asymmetry, which is equal to the difference
between the entropy of state and its symmetrized version: if the desired state
transition does not require a large increase of this quantity, then it can be
implemented with high fidelity using only symmetric operations. Our lower bound
on the achievable fidelity is tight in the case of cloners, and can be achieved
using the Petz recovery map, which interestingly turns out to be the optimal
cloning map found by Werner.
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