© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:1111.5195 · Physical Review A

Quantum adiabatic theorem in light of the Marzlin-Sanders inconsistency

Juan Ortigoso

Published Sep 26, 2012• 24 citations

Abstract

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.

DOI: 10.1103/physreva.86.032121

Related papers

Sufficient Condition for Validity of Quantum Adiabatic Theorem

Yong Tao

In this paper, we attempt to give a sufficient condition of guaranteeing the validity of the proof of the quantum adiabatic theorem. The new sufficient condition can clearly remove the inconsistency and the counterexample of the quantum adiabatic theorem pointed out by Marzlin and Sanders [Phys. Rev. Lett. 93 (2004) 160408].

arXiv:1010.1326•Communications in Theoretical Physics•Mar 1, 2012•4 citations

Quantitative Conditions Do Not Guarantee the Validity of the Adiabatic Approximation

D. M. Tong, Kuldip Singh, L. C. Kwek +1 more

In this Letter, we point out that the widely used quantitative conditions in the adiabatic theorem are insufficient in that they do not guarantee the validity of the adiabatic approximation. We also reexamine the inconsistency issue raised by Marzlin and Sanders [Phys. Rev. Lett. 93, 160408 (2004)] and elucidate the underlying cause.

arXiv:quant-ph/0509073•Physical Review Letters•Sep 9, 2005•153 citations

Review of inconsistency of quantum adiabatic theorem

Yong Tao

In this paper, we note that there are two different types of inconsistencies of quantum adiabatic theorem in work of K. P. Marzlin and B. C. Sanders [Phys. Rev. Lett. 93, 160408, 2004], MS inconsistency and MS counterexample. Nevertheless, these two types are often confused as one by many authors. We shall point out that the inconsistencies of quantum adiabatic theorem raised by the relevant references just can be classified as these two types.

arXiv:1010.1321•Oct 7, 2010•0 citations

Is the Adiabatic Approximation Inconsistent?

Solomon Duki, Harsh Mathur, Onuttom Narayan

Marzlin and Sanders \cite{marzlin} have shown rigorously that the adiabatic approximation can be very inaccurate when applied to a Hamiltonian $H(t)$ that generates the evolution $U^{\dagger} (t)$ even if it gives an excellent approximation to the evolution $U(t)$ generated by a dual Hamiltonian $h(t)$. We show that this is not inconsistent with the adiabatic theorem and find that in general even if $h(t)$ satisfies the conditions of the adiabatic theorem, $H(t)$ will likely violate those conditions.

arXiv:quant-ph/0510131•Oct 17, 2005•2 citations

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

Hsiang Shun Chou

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.

Physical Review A•Sep 12, 2022•2 citations

Experimental Study of the Validity of Quantitative Conditions in the Quantum Adiabatic Theorem

Jiangfeng Du, Ling-Zhi Hu, Ya Wang +3 more

The quantum adiabatic theorem plays an important role in quantum mechanics. However, counter-examples were produced recently, indicating that their transition probabilities do not converge as predicted by the adiabatic theorem [K. P. Marzlin et al., Phys. Rev. Lett. 93, 160408 (2004); D. M. Tong et al., Phys. Rev. Lett. 95, 110407 (2005)]. For a special class of Hamiltonians, we examine the standard criterion for adiabatic evolution experimentally and theoretically, as well as three newly suggested adiabatic conditions. We show that the standard criterion is neither sufficient nor necessary.

Physical Review Letters•Aug 7, 2008•73 citations

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)

that generates the evolution

U^{\dagger} (t)

even if it gives an excellent approximation to the evolution

U(t)

generated by a dual Hamiltonian

h(t)

. We show that this is not inconsistent with the adiabatic theorem and find that in general even if

h(t)

satisfies the conditions of the adiabatic theorem,

H(t)

will likely violate those conditions.

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 quantum adiabatic theorem plays an important role in quantum mechanics. However, counter-examples were produced recently, indicating that their transition probabilities do not converge as predicted by the adiabatic theorem [K. P. Marzlin et al., Phys. Rev. Lett. 93, 160408 (2004); D. M. Tong et al., Phys. Rev. Lett. 95, 110407 (2005)]. For a special class of Hamiltonians, we examine the standard criterion for adiabatic evolution experimentally and theoretically, as well as three newly suggested adiabatic conditions. We show that the standard criterion is neither sufficient nor necessary.