Abstract
The imposition of symmetry upon the nature and structure of quantum observables has recently been extensively studied, with quantum reference frames playing a crucial role. In this paper, we extend this work to quantum transformations, giving quantitative results showing, in direct analogy to the case of observables, that a “large” reference frame is required for non-covariant channels to be well approximated by covariant ones. We apply our findings to the concrete setting of SU(2) symmetry.
Highlights
The role played by symmetry in the understanding and development of physics can hardly be overstated
As is common in gauge theories, e.g., [1], and postulated in, e.g., [2, 3] in the context of quantum reference frames, theoretical quantities which do not commute with a symmetry action are unobservable, even in principle
Concluding remarks We have seen that there is a positive lower bound on the difference between an arbitrary quantum channel and the restriction of a covariant channel, and that in order to reduce this discrepancy a large spread in the generator of the symmetry is needed in the reference system
Summary
The role played by symmetry in the understanding and development of physics can hardly be overstated. We see that in order to make possible good agreement between Λ and ΦρR, a highly “asymmetric” reference state ρR is necessary, since F (ρR, UR(l : s)ρRUR(l : s)∗) must decrease rapidly with respect to |s| as otherwise the left-hand side of the inequality can be large for non-covariant Λ. This asymmetry, or coherence factor, can be bounded by the “spread” of the (symmetry) generator LR: Corollary 2. A ≤ 1 − Γ(Φ (US (s0) ⊗ 1R ⊗ 1Z ))∗Γ(Φ (US (s0) ⊗ 1R ⊗ 1Z ))
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