Quantum Work Statistics with Initial Coherence
Abstract
:1. Introduction
2. Background
2.1. Quantum Work Statistics
2.2. Coherence Theory
3. Main Results
3.1. Distance between the Averages of Work
3.2. Distance between the Variances of Work
3.3. Study of the Entropy Production
4. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Appendix A. Proof of Theorem 1
- Equation (8): The parameterisation of qudit states and unitaries makes finding an exact expression for the absolute difference between average works a difficult task to tackle. However, one can still find an upper bound to such difference as follows:
Appendix B. Proof of Theorem 2
- Equation (10): By using the triangle inequality, the definition of given above and the fact that , we can also provide the following upper bound
- When focusing on qubits, we have
Appendix C. Derivation of Table 1
Appendix D. Proof of Theorem 3
- Equation (17): The Jarzynski equality [42] is only fulfilled when the initial state is at equilibrium. For an arbitrary initial state , the following fluctuation Theorem applies [21]We use this result in the definition of entropy production and use a cumulant expansion of to find [44]In the linear response regime, the first term yields [44], where is the variance of the MH distribution of work. Expanding the second term givesThus, the average entropy production in the linear response regime amounts to
- Equation (18): Let us now have a close look at qubits. For convenience, we consider a qubit prepared in the state , with , and . This ensures that, for small enough , will also be small:Let us suppose the qubit is subjected to a real unitary transformation such that , for . The average entropy production in the linear response regime is then given by Equation (17)
Appendix E. Proof of Corollary 2
Appendix F. Why Can 〈Στ〉MH Be Negative?
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Díaz, M.G.; Guarnieri, G.; Paternostro, M. Quantum Work Statistics with Initial Coherence. Entropy 2020, 22, 1223. https://doi.org/10.3390/e22111223
Díaz MG, Guarnieri G, Paternostro M. Quantum Work Statistics with Initial Coherence. Entropy. 2020; 22(11):1223. https://doi.org/10.3390/e22111223
Chicago/Turabian StyleDíaz, María García, Giacomo Guarnieri, and Mauro Paternostro. 2020. "Quantum Work Statistics with Initial Coherence" Entropy 22, no. 11: 1223. https://doi.org/10.3390/e22111223
APA StyleDíaz, M. G., Guarnieri, G., & Paternostro, M. (2020). Quantum Work Statistics with Initial Coherence. Entropy, 22(11), 1223. https://doi.org/10.3390/e22111223