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Physical Review A

Adiabatic theorem independent of the quantum representations: Resolution of the Marzlin-Sanders inconsistency

Hsiang Shun Chou

Published Sep 12, 2022• 2 citations
Publisher

Abstract

We construct the adiabatic theorem independent of the quantum representations. The quantum-representation-independent geometrical phase, adiabatic condition, and fidelity of the adiabatic approximation are derived. As an example, we apply the quantum-representation-independent adiabatic theorem to provide a thorough resolution of the Marzlin-Sanders inconsistency. We show that the Marzlin-Sanders inconsistency results from the wrong identification of the time-dependent Hamiltonian as the instantaneous energy operator and the quantum representation dependence of the Marzlin-Sanders transformation.

DOI
10.1103/physreva.106.032210

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We construct the adiabatic theorem independent of the quantum representations. The quantum-representation-independent geometrical phase, adiabatic condition, and fidelity of the adiabatic approximation are derived. As an example, we apply the quantum-representation-independent adiabatic theorem to provide a thorough resolution of the Marzlin-Sanders inconsistency. We show that the Marzlin-Sanders inconsistency results from the wrong identification of the time-dependent Hamiltonian as the instantaneous energy operator and the quantum representation dependence of the Marzlin-Sanders transformation.
The adiabatic theorem states that if we prepare a quantum system in one of the instantaneous eigenstates then the quantum number is an adiabatic invariant and the state at a later time is equivalent to the instantaneous eigenstate at that time apart from phase factors. Recently, Marzlin and Sanders have pointed out that this could lead to apparent violation of unitarity. We resolve the Marzlin-Sanders inconsistency within the quantum adiabatic theorem. Yet, our resolution points to another inconsistency, namely, that the cyclic as well as non-cyclic adiabatic Berry phases may vanish under strict adiabatic condition. We resolve this inconsistency and develop an unitary operator decomposition method to argue for the validity of the adiabatic approximation.
A consensus that questions the perfunctory use of the quantum adiabatic theorem has emerged since Marzlin and Sanders [Phys. Rev. Lett. 93, 160408 (2004)] showed the existence of an inconsistency in the applicability of the theorem. Further analysis proved that the inconsistency may arise from the existence of resonant terms in the Hamiltonian, but recent work indicates that the debate about the full extent of the problem remains open. Here, we first show that key premises required in the standard demonstration of the theorem do not hold for a dual Hamiltonian involved in the Marzlin-Sanders inconsistency. Also, we show that two simple conditions can identify systems for which the adiabatic approximation fails, in spite of satisfying traditional quantitative conditions that were believed to guarantee its validity. Finally, we prove that the inconsistency only arises for Hamiltonians that contain resonant terms whose amplitudes go asymptotically to zero.
Marzlin and Sanders \cite{marzlin} have shown rigorously that the adiabatic approximation can be very inaccurate when applied to a Hamiltonian H(t)H(t)H(t) that generates the evolution U†(t)U^{\dagger} (t)U†(t) even if it gives an excellent approximation to the evolution U(t)U(t)U(t) generated by a dual Hamiltonian h(t)h(t)h(t). We show that this is not inconsistent with the adiabatic theorem and find that in general even if h(t)h(t)h(t) satisfies the conditions of the adiabatic theorem, H(t)H(t)H(t) will likely violate those conditions.
The quantum adiabatic theorem is a widely applied result in quantum mechanics. However, recently, counterexamples were produced, indicating that the transition probabilities in some systems do not converge as predicted by the adiabatic theorem [K. P. Marzlin and B. C. Sanders, Phys. Rev. Lett. 93, 160408 (2004)]. In this paper, we examine a proof of the adiabatic theorem, paying specific attention to the validity of the assumptions made in the proof. We show that the convergence of the transition probabilities is path dependent when the change in the Hamiltonian is large. Finally, we consider the application of this reformulated form of the adiabatic theorem to the adiabatic evolution paradigm in quantum computation.