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arXiv:1906.04076 · Physical Review Research

Coherence cost for violating conservation laws

Hiroyasu Tajima, Naoto Shiraishi, Keiji Saito

Published Dec 15, 2020• 32 citations
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Abstract

In isolated quantum systems that have conserved quantities, such as energy and total magnetization, the conserved quantities cannot vary their values with their own dynamics.To violate conservation laws of the system keeping the unitary nature in the system's dynamics, quantum coherence must be supplied from the outer environment.This provides a general and abstract interpretation of the underlying mechanism of unitary operations on systems of interest.In this paper, we consider the amount of quantum coherence cost to realize the desired unitary operation.We derive upper and lower bounds for the coherence cost to implement arbitrary unitary operation within the desired error.These two bounds asymptotically match each other in a small error regime, and give an asymptotic equality.We present applications of our theories to several physical situations such as quantum heat engines and entanglement erasure.

DOI
10.1103/physrevresearch.2.043374

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In isolated quantum systems that have conserved quantities, such as energy and total magnetization, the conserved quantities cannot vary their values with their own dynamics.To violate conservation laws of the system keeping the unitary nature in the system's dynamics, quantum coherence must be supplied from the outer environment.This provides a general and abstract interpretation of the underlying mechanism of unitary operations on systems of interest.In this paper, we consider the amount of quantum coherence cost to realize the desired unitary operation.We derive upper and lower bounds for the coherence cost to implement arbitrary unitary operation within the desired error.These two bounds asymptotically match each other in a small error regime, and give an asymptotic equality.We present applications of our theories to several physical situations such as quantum heat engines and entanglement erasure.
We consider physical situations where the resource theories of coherence and thermodynamics play competing roles. In particular, we study the creation of quantum coherence using unitary operations with limited thermodynamic resources. We find the maximal coherence that can be created under unitary operations starting from a thermal state and find explicitly the unitary transformation that creates the maximal coherence. Since coherence is created by unitary operations starting from a thermal state, it requires some amount of energy. This motivates us to explore the trade-off between the amount of coherence that can be created and the energy cost of the unitary process. We also find the maximal achievable coherence under the constraint on the available energy. Additionally, we compare the maximal coherence and the maximal total correlation that can be created under unitary transformations with the same available energy at our disposal. We find that when maximal coherence is created with limited energy, the total correlation created in the process is upper bounded by the maximal coherence, and vice versa. For two-qubit systems we show that no unitary transformation exists that creates the maximal coherence and maximal total correlation simultaneously with a limited energy cost.
Despite being one of the hallmarks of quantum physics, there is a lack of operational interpretations of quantum coherence. Here we provide an operational interpretation of coherence of a quantum system, in terms of the amount of noise that is to be injected in order to fully decohere it. In particular, we show that in the asymptotic limit, the minimum amount of noise that is required to fully decohere a quantum system, is equal to the relative entropy of coherence. This quantifies the erasure cost of quantum coherence. We employ the entropy exchanged between system and environment during the decohering operation and the memory required to store the information about the decohering operation as the quantifiers of noise. We show that both the quantifiers yield the same cost of erasing coherence in the asymptotic limit. The relative entropy of coherence, hence, is endowed with a thermodynamical and operational interpretation.
Uncertainty relations are one of the fundamental principles in physics. It began as a fundamental limitation in quantum mechanics, and today the word {\it uncertainty relation} is a generic term for various trade-off relations in nature. In this letter, we improve the Kennard-Robertson uncertainty relation, and clarify how much coherence we need to implement quantum measurement under conservation laws. Our approach systematically improves and reproduces the previous various refinements of the Kennard-Robertson inequality. As a direct consequence of our inequalities, we improve a well-known limitation of quantum measurements, the Wigner-Araki-Yanase-Ozawa theorem. This improvement gives an asymptotic equality for the necessary and sufficient amount of coherence to implement a quantum measurement with the desired accuracy under conservation laws.
It is important to improve the accuracy of quantum measurements and operations both in engineering and fundamental physics. It is known, however, that the achievable accuracy of measurements and unitary operations are generally limited by conservation laws according to the Wigner-Araki-Yanase theorem (WAY theorem) and its generalizations. Although many researches have extended the WAY theorem quantitatively, most of them, as well as the original WAY theorem, concern only additive conservation laws like the angular momentum conservation law. In this paper, we explore the limitation incurred by the energy conservation law, which is universal but is one of the non-additive conservation laws. We present a lower bound for the error of a quantum measurement using a scattering process satisfying the energy conservation law. We obtain conditions that a control system Hamiltonian must fulfill in order to implement a controlled unitary gate with zero error when a scattering process is considered. We also show the quantitative relationship between the upper bound of the gate fidelity of a controlled unitary gate and the energy fluctuation of systems when a target system and a control system are both one qubit.
Gibbs-preserving operations have been studied as one of the standard free processes in quantum thermodynamics. Although they admit a simple mathematical structure, their operational significance has been unclear due to the potential hidden cost to implement them using an operationally motivated class of operations, such as thermal operations. Here, we show that this hidden cost can be infinite-we present a family of Gibbs-preserving operations that cannot be implemented by thermal operations aided by any finite amount of quantum coherence. Our result implies that there are uncountably many Gibbs-preserving operations that require unbounded thermodynamic resources to implement, raising a question about employing Gibbs-preserving operations as available thermodynamics processes. This finding is a consequence of the general lower bounds we provide for the coherence cost of approximately implementing a certain class of Gibbs-preserving operations with a desired accuracy. We find that our lower bound is almost tight, identifying a quantity-related to the energy change caused by the channel to implement-as a fundamental quantifier characterizing the coherence cost for the approximate implementation of Gibbs-preserving operations.
The Landauer principle states that any logically irreversible information processing must be accompanied by dissipation into the environment. In this Letter, we investigate the heat dissipation associated with finite-time information erasure and the effect of quantum coherence in such processes. By considering a scenario wherein information is encoded in an open quantum system whose dynamics are described by the Markovian Lindblad equation, we show that the dissipated heat is lower bounded by the conventional Landauer cost, as well as a correction term inversely proportional to the operational time. To clarify the relation between quantum coherence and dissipation, we derive a lower bound for heat dissipation in terms of quantum coherence. This bound quantitatively implies that the creation of quantum coherence in the energy eigenbasis during the erasure process inevitably leads to additional heat costs. The obtained bounds hold for arbitrary operational time and control protocol. By following an optimal control theory, we numerically present an optimal protocol and illustrate our findings by using a single-qubit system.