Scientific RepoRts | 7: 176 | DOI:10.1038/s41598-017-00194-9
Fundamental Limitation on Cooling
under Classical Noise
, Ravindra W. Chhajlany
& Lian-Ao Wu
We prove a general theorem that the action of arbitrary classical noise or random unitary channels can
not increase the maximum population of any eigenstate of an open quantum system, assuming initial
system-environment factorization. Such factorization is the conventional starting point for descriptions
of open system dynamics. In particular, our theorem implies that a system can not be ideally cooled
down unless it is initially prepared as a pure state. The resultant inequality rigorously constrains the
possibility of cooling the system solely through temporal manipulation, i.e., dynamical control over
the system Hamiltonian without resorting to measurement based cooling methods. It is a substantial
generalization of the no-go theorem claiming that the exact ground state cooling is forbidden given
initial system-thermal bath factorization, while here we prove even cooling is impossible under classical
Cooling and, more generally, pure-state preparation
of a microscopic or mesoscopic open system (small
is of paramount importance to many intriguing quantum technologies and engineering of low
temperature quantum phases, in general. Examples of applications include quantum simulations of many-body
on a variety of platforms such as cold atoms and molecules, trapped ions and nanophotonic systems.
Similarly, quantum computers
, the promising quantum adiabatic computing (QAC) paradigm
, dynamically enhanced nuclear polarization
, small quantum devices
metrology are merely a few prominent examples of contemporary applications where signicant control over
quantum states needs to be exercised. More specically, for perfect realization of quantum logic operations, qubits
initially need to be cooled down to the ground state of motion prior to coherent manipulation
. Any cooling
scheme, e.g., bang-bang cooling
, single-shot state-swapping cooling
, and sideband cooling
, cannot be
performed when the system is isolated
e quantum adiabatic computation is an interesting paradigm for universal quantum computation. Here
the solution to a hard problem is encoded in the ground state of a many-body Hamiltonian, i.e., the computer.
To reach the solution, the computer is initialized to the ground state of some Hamiltonian that can be easily
prepared. e initial ground state is then transported adiabatically
to the target ground state encoding the
solution. In principle, adiabaticity suppresses errors in the preparation of the nal ground state by overcoming
the problem of energy relaxation
as the system at all times is kept in the ground state of the instantaneous
Hamiltonian during evolution. However, the changing-rate of the Hamiltonian control parameters, and so the
protocol’s running time, scale inversely with the square of the spectral gap to the lowest excitations. In practice,
the system is always excited during the protocol, most seriously because one must generically move through
regions in parameter space where the gap is very small or closed completely. Apart from this, the system is never
truly isolated from its environment, which also results in excitations. One solution to overcome this problem is
to combine the quasi-adiabatic evolution with active cooling to suppress such errors generated via excitations
during the running of the protocol.
Addressing feasibility of such schemes motivates the search for a better understanding of the description
of cooling eects in open system dynamics, in particular as described below, with classical noise. Cooling set-
ups consist of a small target object (e.g., a mechanical resonator) and an ancillary system (e.g., a qubit) as the
entire system which is embedded in a quantum environment. e three entities seem to be equally crucial in
a cooling process. e dynamics of the entire system is supposed to be governed by the conventional quantum
Institute of Atomic and Molecular Physics and Jilin Provincial Key Laboratory of Applied Atomic and Molecular
Spectroscopy, Jilin University, Changchun, 130012, Jilin, China.
Department of Theoretical Physics and History of
Science, The Basque Country University (EHU/UPV), PO Box 644, 48080, Bilbao, Spain.
Department of Physics,
Zhejiang University, Hangzhou, 310027, Zhejiang, China.
Faculty of Physics, Adam Mickiewicz University,
Umultowska 85, 61-614, Poznań, Poland.
Ikerbasque, Basque Foundation for Science, 48011, Bilbao, Spain.
Correspondence and requests for materials should be addressed to L.-A.W. (email: firstname.lastname@example.org)
Received: 28 October 2016
Accepted: 14 February 2017
Published: xx xx xxxx