Infinitesimal reference frames suffice to determine the asymmetry properties of a quantum system

Abstract

Symmetry principles are fundamental in physics, and while they are well understood within Lagrangian mechanics, their impact on quantum channels has a range of open questions. The theory of asymmetry grew out of information-theoretic work on entanglement and quantum reference frames, and allows us to quantify the degree to which a quantum system encodes coordinates of a symmetry group. Recently a complete set of entropic conditions was found for asymmetry in terms of correlations relative to infinitely many quantum reference frames. However these conditions are difficult to use in practice and the physical implications unclear. Here we show that this set of conditions has extensive redundancy, and one can restrict to reference frames forming any closed surface in the state space that has the maximally mixed state in its interior. This in turn implies that asymmetry can be reduced to just a single entropic condition evaluated at the maximally mixed state. Contrary to intuition, this shows that we do not need macroscopic, classical reference frames to determine the asymmetry properties of a quantum system, but instead infinitesimally small frames suffice. Building on this analysis, we provide simple, closed conditions to estimate the minimal depolarization needed so as to make a given quantum state accessible under G-covariant quantum channels.

Operational Interpretation of Quantum Fisher Information in Quantum Thermodynamics

“…Understanding this tradeoff remains an open question. Also, generalizing the present results to the case of non-Abelian groups, such as SO(3) will be interesting (See, e.g., [58,59] for progress in this direction). Our resource-theoretic approach enabled us to clarify a previously unnoticed relation between fundamental properties of QFI, which is arguably the most studied quantity in quantum metrology and estimation theory.…”

Constraints on magic state protocols from the statistical mechanics of Wigner negativity

“…Beyond this, another promising route is to formulate the majorization conditions for a subset of operations as a semi-definite program and then attempt to estimate realizable distillation rates by exploiting dualities and appropriate coarse-grainings. This approach has recently been applied by one of the authors to estimate the set of quantum states accessible by covariant channels in the resource theory of asymmetry 139 where one allows depolarizing noise that is then minimized. It would be of interest to see if a similar approach could be followed for magic state distillation.…”