[论文解读] Operational thermodynamics of open quantum systems

Felix C. Binder, Sai Vinjanampathy+3 more|arXiv (Cornell University)|Jun 11, 2014

Advanced Thermodynamics and Statistical Mechanics参考文献 4被引用 3

一句话总结

本文為經歷一般量子過程（由完全正且保迹的（CPTP）映射描述）的開放量子系統，制定了熱力學第一定律的操作性形式。當輸入態在偏序意義上優於輸出態時，該理論確立了熱量與熵變均為正值，透過偏序關係將第一與第二定律聯繫起來，並確認其與Hatano-Saha第二定律在酉與熱態映射下的相容性。

ABSTRACT

Accurately describing work extraction from a quantum system is a central objective for the extension of thermodynamics to individual quantum systems. The concepts of work and heat are surprisingly subtle when generalizations are made to arbitrary quantum states. We formulate an operational thermodynamics suitable for application to an open quantum systems undergoing a general quantum evolution. We derive the first law of ther- modynamics for a process described by a completely-positive and trace-preserving map and show consistency with the Hatano-Sasa statement of the second law. We show that heat, from the first law, is positive when the input state of the map majorises the output state. Moreover, the change in entropy is also positive for the same majorisation condition. This makes a strong connection between the two operational laws of thermodynamics. Introduction.— The laws of thermodynamics were forged in the furnaces of the industrial revolution, as engineers and scientists refined their picture of energy, studying heat and its interconversion to mechanical work with a view to powering the mines and factories of this new era of human endeavor. Followed by the development of statistical mechanics at the change of the centuries (1), far from its pragmatic inception, thermodynamics is now a theory with a remarkable range of applicability, successfully describing the properties of macro- scopic systems ranging from refrigerators to black holes (2). Moving on to the 21st century with both the industrial and electronic revolutions behind us, we are currently pushing technology towards and beyond the microscopic scale. With a view to devices operating at a scale where quantum mech- anical laws become important we may ask whether the solid and some combination of the two (8). Finally, central to the work presented here is a work extraction formalism for non-passivity of quantum states (9). Despite the range of ap- proaches a more general picture for the thermodynamics of general quantum evolutions is far from clear. In this Letter, we take an operational approach to character- izing the energy change of an open quantum process described by a completely-positive trace-preserving (CPTP) map. In analogy to the first law of thermodynamics we discuss work done, extractable work, and heat. The concepts of ergotropy and adiabatic work allow us to state our main result: An op- erational first law for general quantum processes. We show that our operational first law is in agreement with widely used Hatano-Sasa version of the second law for CPTP maps (10, 11) by explicitly stating the Clausius inequality for unital and thermal maps. We then show that both operational heat and the change in von Neumann entropy are positive when the input state of the map majorises the output state. Thermodynamics of quantum systems.— The first law of thermodynamics states that the internal energy change in a thermodynamic process can be split up into two contributions - work and heat: dE = Q + W. For a generic quantum system, the internal energy at time t is E(t) = tr( (t)H(t)), implying that the change in the internal energy dE depends only on the end points. Heat and work on the other hand are path-dependent—thus the different notation for the 'differen- tials'. As an illustration we may consider the heat expended when pushing a piston into a cylinder filled with gas: It not only depends on the initial and final positions of the piston but also on how fast it is pushed. Using the derivative of the internal energy with respect to time the following two expres- sions are motivated (10):

研究动机与目标

• 發展一個一致的操作性框架，用於描述開放量子系統在一般量子動力學下的功提取與能量變化。
• 解決在非平衡態以外的任意量子態下定義功與熱量的模糊性。
• 建立與由CPTP映射描述的量子過程相容的熱力學第一定律。
• 透過量子態的偏序條件，連結熱量與熵變的操作性定義。
• 驗證其與酉與熱態CPTP映射下Hatano-Sasa第二定律形式的一致性。

提出的方法

• 使用CPTP映射作為動力學描述，為量子過程制定操作性熱力學第一定律。
• 將功定義為單位控制導致的能量變化，熱量則為未被功所解釋的能量變化殘餘。
• 運用「功能」（ergotropy）概念，量化從非被動量子態中可提取的功。
• 利用偏序理論，描述CPTP映射下輸入與輸出態的熱力學行為。
• 推導酉與熱態映射下的克勞修斯不等式，確認與Hatano-Sasa第二定律的一致性。
• 分析在偏序條件下熱量與熵變的符號：當輸入態偏序優於輸出態時。

实验结果

研究问题

• RQ1 在經歷一般CPTP動力學的開放量子系統中，功與熱量應如何操作性地定義？
• RQ2 在由CPTP映射描述的量子過程中，熱量在何種條件下為正值？
• RQ3 在此類過程中，馮紐曼熵的變化與熱力學熱量之間有何關係？
• RQ4 操作性第一定律是否與CPTP映射下Hatano-Sasa形式的第二定律一致？
• RQ5 態的偏序在決定熱量與熵變的正性中扮演何種角色？

主要发现

• 當CPTP映射的輸入態偏序優於輸出態時，熱量為正值，這提供了熱流的熱力學一致性條件。
• 在相同的偏序條件下，馮紐曼熵的變化亦為正值，從而操作性地連結第一與第二定律。
• 操作性第一定律與酉與熱態CPTP映射下Hatano-Sasa形式的第二定律一致。
• 該形式化方法成功將功提取推廣至使用功能與絕熱功概念的任意量子演化。
• 偏序條件為熱量與熵增加提供了充分條件，強化了量子過程中熱力學時間箭頭的意義。
• 該框架廣泛適用於開放量子系統，並為非平衡態以外的量子熱力學奠定了基礎。