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Journal of Physics A Mathematical and Theoretical

Energetic consequences of decoherence at small times for coupled systems

Bernard Gaveau, Lawrence S. Schulman

Published Jan 14, 2010• 10 citations
Publisher

Abstract

Disentangling two systems can cause an increase in energy, as discussed in Schulman and Gaveau (2006 Phys. Rev. Lett. 97 240405). We here prove a critical inequality used in that letter. Let A and B be two systems and suppose that the initial density matrix for the combined system is a product. Let them have coupling V and let them evolve for a time t under the full Hamiltonian (including V). For the now-entangled full density matrix, the energy is unchanged. Next, disentangle, obtaining the new density matrix for A by tracing over the B variables, and similarly for B. We show that for a large class of V's the expected energy obtained using the product of the disentangled density matrices exceeds the original energy.

DOI
10.1088/1751-8113/43/5/055308

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Disentangling two systems can cause an increase in energy, as discussed in Schulman and Gaveau (2006 Phys. Rev. Lett. 97 240405). We here prove a critical inequality used in that letter. Let A and B be two systems and suppose that the initial density matrix for the combined system is a product. Let them have coupling V and let them evolve for a time t under the full Hamiltonian (including V). For the now-entangled full density matrix, the energy is unchanged. Next, disentangle, obtaining the new density matrix for A by tracing over the B variables, and similarly for B. We show that for a large class of V's the expected energy obtained using the product of the disentangled density matrices exceeds the original energy.
Decoherence of quantum entangled particles is observed in most systems, and is usually caused by system-environment interactions. Disentangling two subsystems A and B of a quantum system AB is tantamount to erasure of quantum phase relations between A and B. It is widely believed that this erasure is an innocuous process, which e.g. does not affect the energies of A and B. Surprisingly, recent theoretical investigations by different groups showed that disentangling two systems, i.e. their decoherence, can cause an increase of their energies. Applying this result to the context of neutron Compton scattering from H2 molecules, we provide for the first time experimental evidence which supports this prediction. The results reveal that the neutron-proton collision leading to the cleavage of the H-H bond in the sub-femtosecond timescale is accompanied by larger energy transfer (by about 3%) than conventional theory predicts. It is proposed to interpreted the results by considering the neutron-proton collisional system as an entangled open quantum system being subject to decoherence owing to the interactions with the “environment” (i.e., two electrons plus second proton of H2).
Nuclei and electrons in condensed matter and/or molecules are usually entangled, due to the prevailing (mainly electromagnetic) interactions. However, the "environment" of a microscopic scattering system (e.g. a proton) causes ultrafast decoherence, thus making atomic and/or nuclear entanglement effects not directly accessible to experiments. However, our neutron Compton scattering experiments from protons (H-atoms) in condensed systems and molecules have a characteristic collisional time about 100–1000 attoseconds. The quantum dynamics of an atom in this ultrashort, but finite, time window is governed by non-unitary time evolution due to the aforementioned decoherence. Unexpectedly, recent theoretical investigations have shown that decoherence can also have the following energetic consequences. Disentangling two subsystems A and B of a quantum system AB is tantamount to erasure of quantum phase relations between A and B. This erasure is widely believed to be an innocuous process, which e.g. does not affect the energies of A and B. However, two independent groups proved recently that disentangling two systems, within a sufficiently short time interval, causes increase of their energies. This is also derivable by the simplest Lindblad-type master equation of one particle being subject to pure decoherence.
Decoherence is well understood, in contrast to disentanglement. According to common lore, irreversible coupling to a dissipative environment is the mechanism for loss of entanglement. Here, we show that, on the contrary, disentanglement can in fact occur at large enough temperatures TTT even for vanishingly small dissipation (as we have shown previously for decoherence). However, whereas the effect of TTT on decoherence increases exponentially with time, the effect of TTT on disentanglement is constant for all times, reflecting a fundamental difference between the two phenomena. Also, the possibility of disentanglement at a particular TTT increases with decreasing initial entanglement.
For a solvable pure-decoherence model, we confirm by an explicit model calculation that the decay of entanglement of two two-state systems (two qubits) is approximately governed by the product of the suppression factors describing decoherence of the subsystems, provided that they are subject to uncorrelated sources of quantum noise. This demonstrates an important physical property that separated open quantum systems can evolve quantum mechanically on time scales larger than the times for which they remain entangled.
We consider here a disentanglement process which transforms a state $\ensuremath{\rho}$ of two subsystems into an unentangled (i.e., separable) state, while not affecting the reduced density matrix of either subsystem. Recently, Terno [Phys. Rev. A 59, 3320 (1999)] showed that an arbitrary state cannot be disentangled, by a physically allowable process, into a tensor product of its reduced density matrices. In this Letter we show that there are sets of states which can be disentangled, but only into separable states other than the product of the reduced density matrices, and other sets of states which cannot be disentangled at all. Thus, we prove that a universal disentangling machine cannot exist.