APS March Meeting 2020 U38.00008 : Nonequilibrium work relations of open quantum systems in one-time measurement scheme* Presenter: Akira Sone (Los Alamos National Laboratory) Authors: Akira Sone (Los Alamos National Laboratory) Yi-Xiang Liu (Massachusetts Institute of Technology) Paola Cappellaro (Massachusetts Institute of Technology) Abstract: Because of the ambiguity in defining work and heat in open quantum systems, it is challenging to formulate non-equilibrium work relations for general quantum channels, beyond the unital case. To overcome this challenge, here we introduce well-defined notions of quantum heat and work in open quantum systems, based on the notion of guessed state defined by the one-time measurement scheme, as developed in [Phys. Rev. E 94, 010103(R)]. This allows us to derive non-equilibrium work relations for general quantum channels, and formulate a modified maximum work theorem with respect to the defined “guessed” quantum work. Acknowledgement: *This work is in part supported by ARO MURI W911NF-11-1-0400 and MIT MIST-FVG. AS acknowledges Thomas G. Stockham Jr. Fellowship from MIT, and he is currently supported by the U.S. Department of Energy and by the Center for Nonlinear Studies at Los Alamos National Laboratory.

1. Nonequilibrium work relations in open quantum systems
from the one-time measurement scheme
Akira Sone1,2, Yi-Xiang Liu2, and Paola Cappellaro2
March, 2020
1. Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory
2. Department of Nuclear Science and Engineering and Research Laboratory of Electronics,
Massachusetts Institute of Technology
(arXiv: 2002.06332)
LA-UR-19-32640

2. Thermodynamics in closed quantum system
Defining work in closed quantum system (Two-time measurement)
: Unitary
Work along the trajectory 𝒏 ⟶ 𝒎
H. Tasaki, arXiv:cond-mat/0009244 (2000), J. Kurchan, arXiv:cond-mat/0007360 (2001), S. Mukamel, Phys. Rev. Lett. 90, 170604 (2003)

3. Nonequilibrium work relation (Jarzynski equality)
Average over all
nonequilibrium processes
Equilibrium free energy difference Second law of Thermodynamics
Verification in closed quantum systems
An et al, Nat. Phys. 11, 193 (2015)
Trapped ion system NMR system
Batalhão et al, PRL 113, 140601 (2014)
Thermodynamics in closed quantum system
C. Jarzynski, Phys. Rev. Lett 78, 2690 (1997)

4. Bath
System
Correlation
2nd measurement
No correlation!
(no information about the bath)
wavefunction collapses
Measured only from system
Cannot distinguish heat and
work by only measuring system
Distinguishing work & heat in open quantum systems is hard
M. Perarnau-Llobet, et al, Phys. Rev. Lett. 118, 070601 (2017)

5. State changes
due to the backaction
In the formalism, the backaction due to the 2nd measurement
has not been taken into account…
Two-time measurement scheme is thermodynamically inconsistent
Bath
System
Correlation
2nd measurement
No correlation!
(no information about the bath)
wavefunction collapses
S. Deffner, J. P. Paz, and W. H. Zurek, Phys. Rev. E 94, 010103(R) (2016)

6. One-time measurement scheme for closed quantum systems
Internal energy difference / Work along the trajectory
S. Deffner, J. P. Paz, and W. H. Zurek, Phys. Rev. E 94, 010103(R) (2016)
: Unitary

7. One-time measurement scheme for our scenario (Open quantum systems)
Internal energy difference along the trajectory
AS, Y-X Liu and P. Cappellaro, arXiv: 2002.06332
Unital / Non-unital
Assumption
(1) Initial state is
decoupled
thermal states.
(2) 𝑯 𝑩 is time-
independent
(3) We can only
measure the
system alone

8. Distribution of internal energy difference (good definition)
AS, Y-X Liu and P. Cappellaro, arXiv: 2002.06332
Assumption
Exact energy difference
One-time measurement scheme for our scenario (Open quantum systems)
(1) Initial state is
decoupled
thermal states.
(2) 𝑯 𝑩 is time-
independent
(3) We can only
measure the
system alone
Unital / Non-unital

9. We derived
AS, Y-X Liu and P. Cappellaro, arXiv: 2002.06332
Standard Jarzynski equality with respect to the system only
Result 1: Modified quantum Jarzysnki equality and 2nd law

10. We derived
AS, Y-X Liu and P. Cappellaro, arXiv: 2002.06332
What is Θ 𝑆𝐵 𝑡 ? What is 𝐷?
Best guessed (thermal) state
(1) Fixed average final energy of the system alone
Maximizing 𝑺 𝑺𝑩 subject to
(2) Evolve from post-first-measurement state
Quantum relative entropy: distance measure
Result 1: Modified quantum Jarzysnki equality and 2nd law
(3) Temperature of the system does not change

11. We derived
AS, Y-X Liu and P. Cappellaro, arXiv: 2002.06332
What is 𝒩(𝜏 𝐵)?
Time evolution of the bath as if the true process is Energy dissipation of the bath as if the true process is
: Guessed heat
Result 1: Modified quantum Jarzysnki equality and 2nd law
(Guessed process)

12. We derived
AS, Y-X Liu and P. Cappellaro, arXiv: 2002.06332
: Guessed heat
1st law of thermodynamics
: Guessed work
(*For closed quantum system, , we can recover the result of Deffner et al.)
A new notion of work
and heat from one-time
measurement only on
the system alone
Result 1: Modified quantum Jarzysnki equality and 2nd law

13. Guessed quantum process:
Classical isothermal process:
Quantum Stein’s lemma: Given 𝑛 i.i.d. copies of these two final states…
Probability True state Indicated state by measurement
𝐴 𝑛 (Type-I) Θ 𝑆𝐵(𝑡) 𝜏 𝑆(𝑡)⨂𝜏 𝐵
𝐵𝑛 (Type-II) 𝜏 𝑆(𝑡)⨂𝜏 𝐵 Θ 𝑆𝐵(𝑡)
where
Guessed work determines the scaling of quantum hypothesis testing:
Result 2: Meaning of the guessed work
T. Ogawa and H. Nagaoka, IEEE Trans. Inf. Theo. 46, 2428 (2000)
F. G. Brandao and M. B. Plenio, Commun. Math. Phys. 295, 829 (2010)
AS, Y-X Liu and P. Cappellaro, arXiv: 2002.06332

14. Conclusion and acknowledgement
(1) Introduce a new notion of work and heat via one-time measurement only on system alone
for open quantum systems.
(2) Derived modified quantum Jarzynski equality and second law of thermodynamics with
respect to the introduced notion of work, and relate them to the quantum hypothesis
testing involving the classical isothermal process.
(3) Generalize the result of Deffner et al.
Conclusion
Acknowledgement
ARO MURI, MIT MIST-FVG, Thomas G. Stockham Jr. Fellowship (MIT), DOE, LDRD and CNLS (LANL)