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arXiv:1812.10019 · Journal of Mathematical Physics

Are almost-symmetries almost linear?

Javier Cuesta, Michael M. Wolf

Published Aug 1, 2019• 2 citations
arXivHTMLPDF

Abstract

It d-pends. Wigner’s symmetry theorem implies that transformations that preserve transition probabilities of pure quantum states are linear maps on the level of density operators. We investigate the stability of this implication. On the one hand, we show that any transformation that preserves transition probabilities up to an additive ε in a separable Hilbert space admits a weak linear approximation, i.e., one relative to any fixed observable. This implies the existence of a linear approximation that is 4εd–close in Hilbert-Schmidt norm, with d the Hilbert space dimension. On the other hand, we prove that a linear approximation that is close in norm and independent of d does not exist in general. To this end, we provide a lower bound that depends logarithmically on d.

DOI
10.1063/1.5087539

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It d-pends. Wigner’s symmetry theorem implies that transformations that preserve transition probabilities of pure quantum states are linear maps on the level of density operators. We investigate the stability of this implication. On the one hand, we show that any transformation that preserves transition probabilities up to an additive ε in a separable Hilbert space admits a weak linear approximation, i.e., one relative to any fixed observable. This implies the existence of a linear approximation that is 4εd–close in Hilbert-Schmidt norm, with d the Hilbert space dimension. On the other hand, we prove that a linear approximation that is close in norm and independent of d does not exist in general. To this end, we provide a lower bound that depends logarithmically on d.
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We prove dimension-independent Hyers--Ulam stability for surjective approximate isometries of the positive trace-class cone in both the trace and Bures metrics. Here ε≥0\varepsilon\geq0ε≥0 is a uniform upper bound on the additive error in preserving distances. Such an approximate isometry is uniformly approximated by a Wigner symmetry (a unitary or antiunitary conjugation), with error at most 9ε/29\varepsilon/29ε/2 in the trace metric and 22ε2\sqrt2\varepsilon22​ε in the Bures metric, even if it does not fix zero. For maps fixing zero, the trace bound improves to 3ε3\varepsilon3ε even under the hypothesis of δδδ-surjectivity, independently of δδδ. In infinite dimensions, we construct spectrum-preserving homeomorphisms of the full state space, including mixed states, whose distortions tend to zero in both metrics but which stay a fixed uniform distance from every Wigner symmetry. For state spaces, stability holds in each fixed finite dimension with a dimension-dependent modulus tending to zero as ε→0\varepsilon\to0ε→0, but no such modulus can be chosen independently of dimension.
A symmetry witness is a subset of the space of selfadjoint trace class operators that allows one to ascertain whether a linear map acting in that space is a symmetry transformation. This notion arises from a certain type of linear preserver problems. Precisely, a symmetry witness is a suitable set which is invariant with respect to an injective linear map in the Banach space of selfadjoint trace class operators where the quantum states live if and only if this map acts as a symmetry transformation. In particular, by a linear version of Wigner's classical theorem, the set of pure states — the rank-one projections — is a symmetry witness. Linearity entails that the usual assumption of preservation of the transition probability between pure states becomes superfluous. This result extends to every set of projections of a fixed (finite) rank, with some suitable constraint on this rank. One then obtains a classification of the sets of projections of a fixed rank that are symmetry witnesses. These symmetry witnesses are projectable. Namely, formulating the mentioned result in terms of quantum states, the sets of 'uniform' density operators of a suitable fixed rank are symmetry witnesses as well.
We explore the relationship between approximate symmetries of a gapped Hamiltonian and the structure of its ground space. We start by considering approximate symmetry operators, defined as unitary operators whose commutators with the Hamiltonian have norms that are sufficiently small. We show that when approximate symmetry operators can be restricted to the ground space while approximately preserving certain mutual commutation relations. We generalize the Stone-von Neumann theorem to matrices that approximately satisfy the canonical (Heisenberg-Weyl-type) commutation relations and use this to show that approximate symmetry operators can certify the degeneracy of the ground space even though they only approximately form a group. Importantly, the notions of “approximate” and “small” are all independent of the dimension of the ambient Hilbert space and depend only on the degeneracy in the ground space. Our analysis additionally holds for any gapped band of sufficiently small width in the excited spectrum of the Hamiltonian, and we discuss applications of these ideas to topological quantum phases of matter and topological quantum error correcting codes. Finally, in our analysis, we also provide an exponential improvement upon bounds concerning the existence of shared approximate eigenvectors of approximately commuting operators under an added normality constraint, which may be of independent interest.