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arXiv:1609.02224 · Scientific Reports

Fundamental Limitation on Cooling under Classical Noise

Jun Jing, Ravindra W. Chhajlany, Lian-Ao Wu

Published Mar 8, 2017• 3 citations
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Abstract

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 noise.

DOI
10.1038/s41598-017-00194-9

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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 noise.
Ground-state cooling and pure state preparation of a small object that is embedded in a thermal environment is an important challenge and a highly desirable quantum technology. This paper proves, with two different methods, that a fundamental constraint on the cooling dynamic implies that it is impossible to cool, via a unitary system-bath quantum evolution, a system that is embedded in a thermal environment down to its ground state, if the initial state is a factorized product of system and bath states. The latter is a crucial but artificial assumption included in numerous tools that treat system-bath dynamics, such as master equation approaches and Kraus operator based methods. Adopting these approaches to address ground state and even approximate ground state cooling dynamics should therefore be done with caution, considering the fundamental theorem exposed in this work.
Preparing a quantum system in a pure state is ultimately limited by the nature of the system's evolution in the presence of its environment and by the initial state of the environment itself. We show that, when the system and environment are initially uncorrelated and arbitrary joint unitary dynamics is allowed, the system may be purified up to a certain (possibly arbitrarily small) threshold if and only if its environment, either natural or engineered, contains a "virtual subsystem" which has the same dimension and is in a state with the desired purity. Beside providing a unified understanding of quantum purification dynamics in terms of a "generalized swap process," our results shed light on the significance of a no-go theorem for exact ground-state cooling, as well as on the quantum resources needed for achieving an intended purification task.
Algorithmic cooling techniques provide tools to increase the purity of quantum states. It is known that the cooling with these techniques is limited. However, the physical root of the limit is still unclear. Here we show that the unitarity of the compression operation imposes the cooling limit of heat-bath algorithmic cooling. Specifically, we prove that the unitarity of the compression operation does not allow increasing the purity beyond the maximum of the individual purities. We formalize the limitations imposed by the unitarity of the compression operation in two theorems. We then introduce an optimal cooling technique and show that without the limitations of the unitary operations the cooling technique would exceed the limit of heat-bath algorithmic cooling. We also show that the limitations of unitarity lead to the exact cooling limit of algorithmic cooling. This work opens up new avenues to understanding the limits of dynamic cooling.
The Shannon's bound for compression is one of the key restrictions for the compression of quantum information. Here we show that the unitarity of the compression operation imposes new bounds on the compression that are more limiting than Shannon's compression bound. This translates to a no-go theorem for the purification of quantum states. For a specific case of a two-qubit system, our results indicate that it is not possible to distill purity beyond the maximum of the individual purities. We show that this restriction results in the cooling limit of the heat-bath algorithmic cooling techniques. We formalize the limitations imposed by the unitarity of the compression operation in two theorems and use the theorems to show that the limitations of unitarity lead to the cooling limit of heat-bath algorithmic cooling. To this end, we introduce a new optimal cooling technique and show that without the limitations of the unitary operations, the new cooling technique would have exceeded the limit of Heat-bath algorithmic cooling. This work opens up new avenues to understanding the limits of dynamic cooling.
Cooling an arbitrary mixed state to a pure ground state requires transferring its entropy to an environment. Can a single coherent bath contact accomplish this without repeated bath resets? We develop a new cooling mechanism that transfers the input information into bath degrees of freedom that subsequently decouple, while the system and the remaining bath follow a ground-state path. The bath is discarded only at the end, and the system Hamiltonian remains on and unmodified throughout. We rigorously realize this mechanism for a class of weakly interacting, gapped fermionic systems with local interactions. Starting from any mixed state, our protocol prepares the interacting ground state in total physical time polylogarithmic in the system size and inverse global trace-norm error. It uses two initially empty bath modes per system mode and a single pulse shared across all onsite system--bath couplings. The same pulse prepares each admissible system's own ground state without knowledge of its microscopic parameters. The guarantees follow directly from a non-Markovian system--bath dynamics without an effective Lindbladian description.
I present an explicit classical simulation of arbitrary quantum noise for quantum models in which one qubit interacts with a quantum bath. The classical model simulates the interaction of the bath and the qubit by random unitary evolutions. I show that any arbitrary quantum dynamics, including quantum dissipation, recurrence, and dephasing, can be simulated classically when one allows the unitary operators in the classical model to depend on the initial state of the system and bath. For initial mixed states of the system and nonproduct states of the system and bath, I demonstrate that random unitary expansion is still possible, in terms of a set of pure states.