A trivial observation on time reversal in random matrix theory

TL;DR

This paper reveals that in time reversal invariant random matrix ensembles, certain state-dependent quantities involving higher-order terms do not become independent of the state after averaging, contrary to common assumptions.

Contribution

It demonstrates that for time reversal invariant ensembles, higher-order state-dependent quantities are constant over symmetry group orbits, challenging previous beliefs.

Findings

Fidelity and decoherence are constant over symmetry orbits in specific models.

Higher-order state-dependent quantities do not always become ensemble-independent.

Time reversal symmetry imposes constraints on averaged state-dependent measures.

Abstract

It is commonly thought that a state-dependent quantity, after being averaged over a classical ensemble of random Hamiltonians, will always become independent of the state. We point out that this is in general incorrect: if the ensemble of Hamiltonians is time reversal invariant, and the quantity involves the state in higher than bilinear order, then we show that the quantity is only a constant over the orbits of the invariance group on the Hilbert space. Examples include fidelity and decoherence in appropriate models.