Many-Body Coherence in Quantum Thermodynamics

Transcription

1 Many-Body Coherence in Quantum Thermodynamics [] H. Kwon, H. Jeong, D. Jennings, B. Yadin, and M. S. Kim

2 Contents I. Introduction II. Resource theory of quantum thermodynamics 1. Thermal operation and the quantum free energy 2. Free energy correlation in a many-body system III. Work extraction from many-body coherence 1. Many-body coherence in quantum thermodynamics 2. Work extraction without changing classical energy distribution IV. Many-body coherence as a clock resource 1. Quantum coherence as time reference 2. Coherence constraint for quantum state transformation under TO V. Conclusion

3 Introduction Thermodynamics in quantum system Operational description of transformation between quantum states interacting with the thermal bath Equilibrium state (Gibbs state) with the partition function and inverse temperature Q1. Q2. Quantum coherence (correlation) can do extra work? Thermodynamic resource beyond work-yielding resource? in Many-body system

4 Contents I. Introduction II. Resource theory of quantum thermodynamics 1. Thermal operation and the quantum free energy 2. Free energy correlation in a many-body system III. Work extraction from many-body coherence 1. Many-body coherence in quantum thermodynamics 2. Work extraction without changing classical energy distribution IV. Many-body coherence as a clock resource 1. Quantum coherence as time reference 2. Coherence constraint for quantum state transformation under TO V. Conclusion

5 Thermal operation in quantum system Thermal operation : Bath System Energy preserving condition : (Quantum) Free energy : Gibbs(equilibrium) state : where Minimum free energy : completely passive state

6 Generalized free energies A family of free energies : Renyi Divergence : for Renyi divergence becomes Resource theory of thermodynamics (the 2 nd law) : Single-shot (deterministic) work extraction : Work yield : where Brandao et.al., PNAS (2015) (for energy-diagonal states) Brandao et.al., PNAS (2015)

7 Contents I. Introduction II. Resource theory of quantum thermodynamics 1. Thermal operation and the quantum free energy 2. Free energy correlation in a many-body system III. Work extraction from many-body coherence 1. Many-body coherence in quantum thermodynamics 2. Work extraction without changing classical energy distribution IV. Many-body coherence as a clock resource 1. Quantum coherence as time reference 2. Coherence constraint for quantum state transformation under TO V. Conclusion

8 Free energy correlation Free energy in many-body system : (Generalized) Free energy correlation : (total correlation) For product state :

9 Free energy transfer from correlation Two-level system : Local free energy : Free energy correlation : : Interconversion between local free energy and correlation

10 Free energy change under thermal operations System interacting with bath Regardless of the state of system-bath (not necessarily in Gibbs state & can be entangled) Free energy is additionally leaked through the system-bath correlation If the bath is initially in Gibbs state and uncorrelated with the system : (The 2 nd law of quantum thermodynamics)

11 Contents I. Introduction II. Resource theory of quantum thermodynamics 1. Thermal operation and the quantum free energy 2. Free energy correlation in a many-body system III. Work extraction from many-body coherence 1. Many-body coherence in quantum thermodynamics 2. Work extraction without changing classical energy distribution IV. Many-body coherence as a clock resource 1. Quantum coherence as time reference 2. Coherence constraint for quantum state transformation under TO V. Conclusion

12 Coherence in many-body system N-body quantum system : where (Classical) Energy distribution : where Classical thermodynamics Coherence between Energy eigenstates : When When : Equi-level coherence : Inter-level coherence Total Energy :

13 Free energy and correlation for many-body system N-partite non-interacting system with non-degenerate local Hamiltonians Quantum contribution of free energy correlation : Classical (diagonal) state with the same energy statistics : Energy block-diagonal state All the free energy differences are contained in correlations

14 Contents I. Introduction II. Resource theory of quantum thermodynamics 1. Thermal operation and the quantum free energy 2. Free energy correlation in a many-body system III. Work extraction from many-body coherence 1. Many-body coherence in quantum thermodynamics 2. Work extraction without changing classical energy distribution IV. Many-body coherence as a clock resource 1. Quantum coherence as time reference 2. Coherence constraint for quantum state transformation under TO V. Conclusion

15 Extractable work from quantum coherence Work extraction without changing energy statistics : where Extracting work from quantum coherence : (for energy-diagonal states) Q1. Quantum coherence (correlation) can do extra work? Possible if and only if for all

16 Extractable work from quantum coherence Work extraction without changing energy statistics : where Extracting work from quantum coherence : For equi-level coherence Classical energy distribution unchanged Purely quantum contribution on work extraction process

17 Extractable work from quantum coherence Interconversion between work and correlation(coherence) : Example : Dicke states (in N-partite two level systems) T.O. When and

18 Work-locking in coherence Inter-level coherence does not contribute to work extraction process Logtaglio et.al., Nature Commun. (2015) : energy block-diagonal state Q2. Thermodynamic resource beyond work-yielding resource? (contained in inter-level coherence)

19 Contents I. Introduction II. Resource theory of quantum thermodynamics 1. Thermal operation and the quantum free energy 2. Free energy correlation in a many-body system III. Work extraction from many-body coherence 1. Many-body coherence in quantum thermodynamics 2. Work extraction without changing classical energy distribution IV. Many-body coherence as a clock resource 1. Quantum coherence as time reference 2. Coherence constraint for quantum state transformation under TO V. Conclusion

20 Coherence as a clock resource Quantum metrology: High-precision parameter estimation beyond classical shot-noise limit (Quantum Cramer-Rao bound) Estimator of the parameter : Variance of estimator : (Quantum Fisher Information) Time evolution of a quantum state with inter-level coherence : Clock frequency : Time reference

21 Coherence as a clock resource Coherent thermal state : (No work can be extracted) High clock resolution : Heisenberg Limit Clock resource does not increase under TO :

22 Contents I. Introduction II. Resource theory of quantum thermodynamics 1. Thermal operation and the quantum free energy 2. Free energy correlation in a many-body system III. Work extraction from many-body coherence 1. Many-body coherence in quantum thermodynamics 2. Work extraction without changing classical energy distribution IV. Many-body coherence as a clock resource 1. Quantum coherence as time reference 2. Coherence constraint for quantum state transformation under TO V. Conclusion

23 Coherence second law of thermodynamics Coherence second law in terms of QFI : Previously studied coherence restrictions : (Asymmetry) Decreases under TO Logtaglio et.al., Nature Commun. (2015) Logtaglio et.al., Phys. Rev. X (2015)

24 QFI gives additional restriction for TO Four level Hamiltonian : T.O.? Free energy: Asymmetry: Possible? QFI conditions : For every energy difference Transition is impossible!

25 State transformation in many-copy Product state vs. GHZ-state : GHZ-state cannot be generated from product state under TO Many-body entanglement and QFI : for k-producible state Asymmetry vanishes on many-copy limit : QFI does not :

26 Catalytic transform State transformation rate in asymptotic limit For pure gapless states : Allowing catalytic transformation : Catalysis : (sub-linear dimension) Catalysis itself requires super-linear amount of coherence resource

27 Contents I. Introduction II. Resource theory of quantum thermodynamics 1. Thermal operation and the quantum free energy 2. Free energy correlation in a many-body system III. Work extraction from many-body coherence 1. Many-body coherence in quantum thermodynamics 2. Work extraction without changing classical energy distribution IV. Many-body coherence as a clock resource 1. Quantum coherence as time reference 2. Coherence constraint for quantum state transformation under TO V. Conclusion

28 Many-body coherence in quantum thermodynamics Energy Distribution Coherence Classical Thermodynamics T.O. Equi-level coherence Activation Inter-level coherence Work Clock Resource