## Final Report Summary - QUANTUMLANDAUER (Beating Landauer’s Limit in the Quantum Regime)

Ultimately, the only task in information processing that must by necessity dissipate heat, given certain natural assumptions, is information erasure – the irreversible process of preparing a logical register in some definite state, irrespective of its prior configuration. Moreover, our best theory of matter is quantum mechanics. As such, the project “QuantumLandauer” set out to investigate the ultimate physical limits of heat dissipation due to information erasure carried out on a quantum register. Specifically, we aimed to determine how to minimise heat dissipation as far as permissible within the Landauer framework of information erasure. Then, we discuss logically consistent departures from the Landauer framework that permit more energy-efficient information erasure. The findings of this work are published in New. J. Phys, 18 (2016) 015011. In a related endeavor, we propose a protocol that uses a local probe, modeled as a Heisenberg spin chain, for the purpose of both cooling (erasing the information of), and measuring the temperature of, a collection of thermal qubits.

Landauer’s framework for information erasure is as follows: let the object whose information we wish to erase – the register – be brought in contact with a thermal reservoir which, initially, is in thermal equilibrium and has a well-defined temperature. Subsequently, let the two systems interact in a reversible manner, as a result of which the object is prepared in some pre-assigned state.This will by necessity increase the average energy of the reservoir; energy will be lost in the form of heat dissipation in the reservoir. If the systems are quantum mechanical, this means that, initially, they are uncorrelated, and evolve jointly by some global unitary operator. Moreover, (probabilistic) information erasure on the object is tantamount to fixing the probability of finding it in some pre-assigned pure state after the joint evolution.Landauer’s principle,then, will determine the minimum heat dissipation as a function of the reduction of entropy in the object, modulated by the initial temperature of the reservoir. This lower bound of heat dissipation is Landauer’s limit.The content of Landauer’s principle is that, for a certain entropy reduction obtained on an object, and some temperature, there exists an optimal reservoir such that the heat dissipated therein will equal Landauer’s limit. In practically all physical situations, however, nature fixes the structure of the reservoir and, consequently, the actual minimal heat dissipation permissible will exceed Landauer’s limit. Therefore, we approach the problem from a dynamical perspective – what is the optimal interaction between object and reservoir (characterized by a unitary operator) which, given a fixed reservoir, will minimise the heat dissipation due to probabilistic information erasure. We give a general prescription for characterising these optimal unitary operators. Our results will prove useful in engineering quantum computers in the future. Our work is of a sufficiently general nature that it will inform the optimal process of information erasure, irrespective of the specific architectures used.

The quantities dealt with by Landauer’s principle are entropy reduction in the object, temperature of the reservoir, and heat dissipated to the reservoir; Landauer’s principle establishes a relation between these. Any departure from Landauer’s framework, with the purpose of dissipating less heat than allowed by Landauer’s principle, must allow these quantities to be defined consistently with Landauer’s framework. This is not a problem for entropy reduction; this will be characterized by the von Neumann entropy, as before. For temperature and heat to be defined in a consistent manner, there must be within the framework a system that, initially, is in thermal equilibrium and is uncorrelated from all other systems considered. The heat dissipation will, again, be the increment in average energy of this system, i.e. the reservoir. We identify two possible departures from Landauer’s framework that satisfy these conditions. In the first, we introduce a third, auxiliary system which may or may not be initially correlated with the object. Provided that the state of the compound system of object-plus-auxiliary is less than or equal to the dimension of the auxiliary Hilbert space, we may perform information erasure without involving the reservoir and, thus, dissipate zero heat. The nature of correlations between object and auxiliary are shown to be irrelevant for this purpose. The second departure is to let the global system initially be in thermal equilibrium. The information of the object – a subsystem of this global thermal system – would be erased by a global unitary evolution, and the heat dissipation would be evaluated on the entire global system. Provided the eigenstates of the global Hamiltonian are product vectors across the object-remainder partition, it will be possible to dissipate heat that is smaller than Landauer’s limit.

In our second line of research, we focus on the task of erasing the information of a collection of two-level systems, or qubits, by use of local probe that can be modeled as a Heisenberg spin chain. Specifically, given that the qubits are initially thermal, we cool them by letting them interact with the spin chain. This protocol has an additional benefit, as it also allows us to gauge the temperature of the qubits. The protocol is very simple, and requires a minimal degree of control. Provided that the spin chain is initialised in a highly polarised pure state, letting the thermal qubits interact with the spin at the tip of the probe allows them to always be cooled. With a sufficient number of such interactions, the probe pseudo-thermalises, whereby all spins therein will become identical to the thermal qubits. Measurement of these spins, after the removal of the probe, will allow for the temperature of the qubits to be estimated. We show that this protocol is robust to dephasing and the strength of interaction between the qubits and the probe. This protocol is within reach of current experimental capabilities.

Landauer’s framework for information erasure is as follows: let the object whose information we wish to erase – the register – be brought in contact with a thermal reservoir which, initially, is in thermal equilibrium and has a well-defined temperature. Subsequently, let the two systems interact in a reversible manner, as a result of which the object is prepared in some pre-assigned state.This will by necessity increase the average energy of the reservoir; energy will be lost in the form of heat dissipation in the reservoir. If the systems are quantum mechanical, this means that, initially, they are uncorrelated, and evolve jointly by some global unitary operator. Moreover, (probabilistic) information erasure on the object is tantamount to fixing the probability of finding it in some pre-assigned pure state after the joint evolution.Landauer’s principle,then, will determine the minimum heat dissipation as a function of the reduction of entropy in the object, modulated by the initial temperature of the reservoir. This lower bound of heat dissipation is Landauer’s limit.The content of Landauer’s principle is that, for a certain entropy reduction obtained on an object, and some temperature, there exists an optimal reservoir such that the heat dissipated therein will equal Landauer’s limit. In practically all physical situations, however, nature fixes the structure of the reservoir and, consequently, the actual minimal heat dissipation permissible will exceed Landauer’s limit. Therefore, we approach the problem from a dynamical perspective – what is the optimal interaction between object and reservoir (characterized by a unitary operator) which, given a fixed reservoir, will minimise the heat dissipation due to probabilistic information erasure. We give a general prescription for characterising these optimal unitary operators. Our results will prove useful in engineering quantum computers in the future. Our work is of a sufficiently general nature that it will inform the optimal process of information erasure, irrespective of the specific architectures used.

The quantities dealt with by Landauer’s principle are entropy reduction in the object, temperature of the reservoir, and heat dissipated to the reservoir; Landauer’s principle establishes a relation between these. Any departure from Landauer’s framework, with the purpose of dissipating less heat than allowed by Landauer’s principle, must allow these quantities to be defined consistently with Landauer’s framework. This is not a problem for entropy reduction; this will be characterized by the von Neumann entropy, as before. For temperature and heat to be defined in a consistent manner, there must be within the framework a system that, initially, is in thermal equilibrium and is uncorrelated from all other systems considered. The heat dissipation will, again, be the increment in average energy of this system, i.e. the reservoir. We identify two possible departures from Landauer’s framework that satisfy these conditions. In the first, we introduce a third, auxiliary system which may or may not be initially correlated with the object. Provided that the state of the compound system of object-plus-auxiliary is less than or equal to the dimension of the auxiliary Hilbert space, we may perform information erasure without involving the reservoir and, thus, dissipate zero heat. The nature of correlations between object and auxiliary are shown to be irrelevant for this purpose. The second departure is to let the global system initially be in thermal equilibrium. The information of the object – a subsystem of this global thermal system – would be erased by a global unitary evolution, and the heat dissipation would be evaluated on the entire global system. Provided the eigenstates of the global Hamiltonian are product vectors across the object-remainder partition, it will be possible to dissipate heat that is smaller than Landauer’s limit.

In our second line of research, we focus on the task of erasing the information of a collection of two-level systems, or qubits, by use of local probe that can be modeled as a Heisenberg spin chain. Specifically, given that the qubits are initially thermal, we cool them by letting them interact with the spin chain. This protocol has an additional benefit, as it also allows us to gauge the temperature of the qubits. The protocol is very simple, and requires a minimal degree of control. Provided that the spin chain is initialised in a highly polarised pure state, letting the thermal qubits interact with the spin at the tip of the probe allows them to always be cooled. With a sufficient number of such interactions, the probe pseudo-thermalises, whereby all spins therein will become identical to the thermal qubits. Measurement of these spins, after the removal of the probe, will allow for the temperature of the qubits to be estimated. We show that this protocol is robust to dephasing and the strength of interaction between the qubits and the probe. This protocol is within reach of current experimental capabilities.