âQuantumâ Quantum Thermodynamics
Thermodynamics beyond free energy relations or “Quantum” Quantum Thermodynamics Matteo Lostaglio Imperial College London David Jennings Kamil Korzekwa Terry Rudolph This is wind, he is actually slim What makes quantum thermodynamics really quantum? Can we develop a general framework to understand coherence? Single-shot Try to maximize average work 𝑊 Single-shot 𝑊 Truly work-like work extraction Johan Aberg Nature Communications 4, 1925 (2013) Single-shot 𝑊 Single-shot thermodynamics Single-shot 𝑊 Single-shot thermodynamics Classical sector “Classical” Quantum Thermodynamics system 𝜌 bath 𝛾𝐵 ∝ 𝑒 −𝛽𝐻𝐵 catalyst 𝑐 “Classical” Quantum Thermodynamics system 𝜌 bath 𝛾𝐵 ∝ 𝑒 −𝛽𝐻𝐵 𝑈, 𝐻 = 0 overall energy conservation catalyst 𝑐 “Classical” Quantum Thermodynamics system 𝜌 bath 𝛾𝐵 ∝ 𝑒 −𝛽𝐻𝐵 catalyst 𝑐 𝑈, 𝐻 = 0 overall energy conservation 𝜎 𝑐 “Classical” Quantum Thermodynamics 𝜌, 𝐻𝑆 = 0 𝑒 −𝛽𝐻𝐵 “Classical” Quantum Thermodynamics 𝜌, 𝐻𝑆 = 0 Theory becomes ``classical’’ : 𝐵 𝑒 −𝛽𝐻 1. Diagonalise everything in energy eigenbasis. 2. Hence, only probabilities of occupying different energies matters. 3. Thermal operations are particular classical stochastic processes. “Classical” Quantum Thermodynamics 𝜌, 𝐻𝑆 = 0 Theory becomes ``classical’’ : 𝐵 𝑒 −𝛽𝐻 1. Diagonalise everything in energy eigenbasis. 2. Hence, only probabilities of occupying different energies matters. 3. Thermal operations are particular classical stochastic processes. c.f. two-measurement protocol in fluctuation theorems (?) “Classical” Quantum Thermodynamics If 𝜌, 𝐻𝑆 = 0, 𝜌→𝜎 𝑒 −𝛽𝐻𝐵 𝐹𝛼 𝜌 ≥ 𝐹𝛼 𝜎 The second laws of quantum thermodynamics Fernando G.S.L. Brandao, Michał Horodecki, Nelly Huei Ying Ng, Jonathan Oppenheim, Stephanie Wehner PNAS 112, 3275 (2015) ∀𝛼 “Classical” Quantum Thermodynamics If 𝜌, 𝐻𝑆 = 0 𝜌→𝜎 𝑒 −𝛽𝐻𝐵 𝐹𝛼 𝜌 ≥ 𝐹𝛼 𝜎 ∀𝛼 where 𝐹𝛼 𝜌 = −𝑘𝑇 log 𝑍 + 𝑘𝑇𝑆𝛼 (𝜌||𝛾) 𝛼/|𝛼| 𝑆𝛼 (𝜌| 𝛾 = log 𝑇𝑟𝜌𝛼 𝛾 1−𝛼 𝛼−1 The second laws of quantum thermodynamics Fernando G.S.L. Brandao, Michał Horodecki, Nelly Huei Ying Ng, Jonathan Oppenheim, Stephanie Wehner PNAS 112, 3275 (2015) 𝑒 −𝛽𝐻𝑆 𝛾= 𝑍𝑆 “Quantum” Quantum Thermodynamics 𝑒 −𝛽𝐻𝐵 𝜎 = 𝜀𝑇 𝜌 = 𝑇𝑟𝐵 [𝑈 −𝛽𝐻 𝜌⊗ 𝑈ϯ] 𝐵 𝑍𝐵 𝑒 𝑈, 𝐻 = 0 Now 𝜌, 𝐻𝑆 ≠ 0 (superpositions) “Quantum” Quantum Thermodynamics |0> ? ? |1> 𝑒 −𝛽𝐻𝑆 𝛾= 𝑍𝑆 “Quantum” Quantum Thermodynamics |0> 𝑒 −𝛽𝐻𝐵 𝑒 −𝛽𝐻𝑆 𝛾= 𝑍𝑆 |1> “Quantum” Quantum Thermodynamics |0> 𝑒 −𝛽𝐻𝐵 ``classical’’ thermalisation dephasing |1> 𝑒 −𝛽𝐻𝑆 𝛾= 𝑍𝑆 “Quantum” Quantum Thermodynamics |0> 𝑒 −𝛽𝐻𝐵 𝑒 −𝛽𝐻𝑆 𝛾= 𝑍𝑆 No processing of coherence |1> Free energy is not enough If first law 𝑈, 𝐻 = 0, this is impossible: 𝑒 −𝛽𝐻𝐵 𝜎 Despite any condition on F being satisfied. What new ideas can we use to incorporate coherence in the description? Description of quantum coherence in thermodynamic processes requires constraints beyond free energy Matteo Lostaglio, David Jennings, Terry Rudolph Nature Communications 6, 6383 (2015) Quantum coherence, time-translation symmetry and thermodynamics Matteo Lostaglio, Kamil Korzekwa, David Jennings, Terry Rudolph Phys. Rev. X 5, 021001 (2015) “Quantum” Quantum Thermodynamics Simple but powerful symmetry consideration: Non-equilibrium thermodynamics 𝜀𝑇𝑒 −𝛽𝐻 ⊂ 𝐵𝜀𝑆 Time-translation symmetry “Quantum” Quantum Thermodynamics Simple but powerful symmetry consideration: 𝜀𝑇𝑒 −𝛽𝐻 ⊂ 𝐵𝜀𝑆 𝜀𝑆 𝑒 −𝑖𝐻𝑆𝑡 𝜌𝑒 𝑖𝐻𝑆𝑡 = 𝑒 −𝑖𝐻𝑆𝑡 𝜀𝑆 𝜌 𝑒 𝑖𝐻𝑆𝑡 time-translation symmetric maps Extending Noether's theorem by quantifying the asymmetry of quantum states Nature Communications 5, 3821 (2014) Modes of asymmetry: the application of harmonic analysis to symmetric quantum dynamics and quantum reference frames Phys. Rev. A 90, 062110 (2014) Iman Marvian, Robert W. Spekkens “Quantum” Quantum Thermodynamics Simple but powerful symmetry consideration: 𝜀𝑇𝑒 −𝛽𝐻 ⊂ 𝐵𝜀𝑆 𝜀𝑆 𝑒 −𝑖𝐻𝑆𝑡 𝜌𝑒 𝑖𝐻𝑆𝑡 = 𝑒 −𝑖𝐻𝑆𝑡 𝜀𝑆 𝜌 𝑒 𝑖𝐻𝑆𝑡 time-translation symmetric maps Symmetry that describes how energy and coherence are NOT conserved in the system. Extending Noether's theorem by quantifying the asymmetry of quantum states Nature Communications 5, 3821 (2014) Modes of asymmetry: the application of harmonic analysis to symmetric quantum dynamics and quantum reference frames Phys. Rev. A 90, 062110 (2014) Iman Marvian, Robert W. Spekkens “Quantum” Quantum Thermodynamics Simple but powerful symmetry consideration: 𝜀𝑇𝑒 −𝛽𝐻 ⊂ 𝐵𝜀𝑆 𝜀𝑆 𝑒 −𝑖𝐻𝑆𝑡 𝜌𝑒 𝑖𝐻𝑆𝑡 = 𝑒 −𝑖𝐻𝑆𝑡 𝜀𝑆 𝜌 𝑒 𝑖𝐻𝑆𝑡 time-translation symmetric maps Symmetry that describes how energy and coherence are NOT conserved in the system. 1. Even if closed, 𝐻𝑆 , 𝐻𝑆 2 , … , 𝐻𝑆 𝑘 not enough for mixed states 2. Open dynamics “Quantum” Quantum Thermodynamics Simple but powerful symmetry consideration: 𝜀𝑇𝑒 −𝛽𝐻 ⊂ 𝐵𝜀𝑆 𝜀𝑆 𝑒 −𝑖𝐻𝑆𝑡 𝜌𝑒 𝑖𝐻𝑆𝑡 = 𝑒 −𝑖𝐻𝑆𝑡 𝜀𝑆 𝜌 𝑒 𝑖𝐻𝑆𝑡 time-translation symmetric maps Symmetry that describes how energy and coherence are NOT conserved in the system. 1. Even if closed, 𝐻𝑆 , 𝐻𝑆 2 , … , 𝐻𝑆 𝑘 not enough for mixed states 2. Open dynamics THERMODYNAMICS! “Quantum” Quantum Thermodynamics Some consequences 𝐴𝛼 𝜌 = 𝑆𝛼 (𝜌||𝐷 𝜌 ) 𝐵 𝑍 + 𝑘𝑇𝑆 (𝜌||𝛾) c.f. 𝐹𝛼 𝜌 = −𝑘𝑇 𝑒 −𝛽𝐻log 𝛼 Under any thermal operation: ∆𝐴𝛼 𝜌 ≤ 0 ``second laws’’ for quantum coherence “Quantum” Quantum Thermodynamics Some consequences 𝐴𝛼 𝜌 = 𝑆𝛼 (𝜌||𝐷 𝜌 ) 𝐵 𝑍 + 𝑘𝑇𝑆 (𝜌||𝛾) c.f. 𝐹𝛼 𝜌 = −𝑘𝑇 𝑒 −𝛽𝐻log 𝛼 Under any thermal operation: ∆𝐴𝛼 𝜌 ≤ 0 ``second laws’’ for quantum coherence Macroscopically irrelevant: lim 𝑁→∞ 𝐴𝛼 𝜌⊗𝑁 𝑁 =0 ∆𝑨𝜶 ≤ 𝟎 ∆𝑭𝜶 ≤ 𝟎 “Quantum” Quantum Thermodynamics Some consequences 𝑒 −𝛽𝐻𝐵 Standard quantum free energy: 𝑆 𝜌 𝐹 𝜌 = 𝐻 − =𝐹 𝐷 𝜌 𝛽 both decrease 1 + 𝐴(𝜌) 𝛽 Work locked “Quantum” Quantum Thermodynamics Some consequences 𝑒 −𝛽𝐻𝐵 Standard quantum free energy: 𝑆 𝜌 𝐹 𝜌 = 𝐻 − =𝐹 𝐷 𝜌 𝛽 both decrease 1 + 𝐴(𝜌) 𝛽 Work locked Activation: 𝐹 𝐷 𝜌⊗2 ≥ 2𝐹 𝐷 𝜌 c.f. passive states… (strict if 𝜌, 𝐻𝑆 ≠ 0 ) “Quantum” Quantum Thermodynamics Concrete bounds? If 𝜌 → 𝜎 𝜌𝑘𝑙 𝑒 −𝛽 |𝜎𝑛𝑚 | ≤ 𝑘,𝑙 𝐸𝑘 <𝐸𝑛 𝐸𝑘 −𝐸𝑙 =𝐸𝑛 −𝐸𝑚 𝐸𝑛 −𝐸𝑘 𝑒 −𝛽𝐻𝐵 + |𝜌𝑘𝑙 | 𝑘,𝑙 𝐸𝑘 ≥𝐸𝑛 𝐸𝑘 −𝐸𝑙 =𝐸𝑛 −𝐸𝑚 “Quantum” Quantum Thermodynamics Concrete bounds? If 𝜌 → 𝜎 𝜌𝑘𝑙 𝑒 −𝛽 |𝜎𝑛𝑚 | ≤ 𝑘,𝑙 𝐸𝑘 <𝐸𝑛 𝐸𝑘 −𝐸𝑙 =𝐸𝑛 −𝐸𝑚 𝜌 𝜌56 𝜌23 𝐸𝑛 −𝐸𝑘 𝑒 −𝛽𝐻𝐵 + |𝜌𝑘𝑙 | 𝑘,𝑙 𝐸𝑘 ≥𝐸𝑛 𝐸𝑘 −𝐸𝑙 =𝐸𝑛 −𝐸𝑚 𝜎 𝜎56 “Quantum” Quantum Thermodynamics Concrete bounds? If 𝜌 → 𝜎 𝜌𝑘𝑙 𝑒 −𝛽 |𝜎𝑛𝑚 | ≤ 𝑘,𝑙 𝐸𝑘 <𝐸𝑛 𝐸𝑘 −𝐸𝑙 =𝐸𝑛 −𝐸𝑚 𝜌 𝐸𝑛 −𝐸𝑘 𝑒 −𝛽𝐻𝐵 + |𝜌𝑘𝑙 | 𝑘,𝑙 𝐸𝑘 ≥𝐸𝑛 𝐸𝑘 −𝐸𝑙 =𝐸𝑛 −𝐸𝑚 𝜎 𝜌56 𝜎23 “Quantum” Quantum Thermodynamics Concrete bounds? If 𝜌 → 𝜎 𝜌𝑘𝑙 𝑒 −𝛽 |𝜎𝑛𝑚 | ≤ 𝑘,𝑙 𝐸𝑘 <𝐸𝑛 𝐸𝑘 −𝐸𝑙 =𝐸𝑛 −𝐸𝑚 𝑒 −𝛽𝐻𝐵 + |𝜌𝑘𝑙 | 𝑘,𝑙 𝐸𝑘 ≥𝐸𝑛 𝐸𝑘 −𝐸𝑙 =𝐸𝑛 −𝐸𝑚 𝜎 𝜌 𝑒 −𝛽∆𝐸 𝜌23 𝐸𝑛 −𝐸𝑘 𝜎56 “Quantum” Quantum Thermodynamics Concrete bounds? If 𝜌 → 𝜎 𝑒 −𝛽𝐻𝐵 Kamil: Uououo!! This is explained better in my poster! 𝑒 −𝛽∆𝐸 𝜌23 𝜎56 Conclusions 1. 2. 3. 4. Coherence in thermodynamics can be understood through symmetry principles Coherence + Thermo > Free energy consideration Bias towards energetic considerations? More to be done! Thank you for your attention! More details: Description of quantum coherence in thermodynamic processes requires constraints beyond free energy Matteo Lostaglio, David Jennings, Terry Rudolph Nature Communications 6, 6383 (2015) Quantum coherence, time-translation symmetry and thermodynamics Matteo Lostaglio, Kamil Korzekwa, David Jennings, Terry Rudolph Phys. Rev. X 5, 021001 (2015) Work extraction from coherence Kamil Korzekwa, Matteo Lostaglio , Jonathan Oppenheim, David Jennings Soon on ArXiv! “Classical” Quantum Thermodynamics 𝜌𝛼 𝛾 1−𝛼 𝛼/|𝛼| 𝑆𝛼 (𝜌| 𝛾 = log 𝑇𝑟 1−𝛼 −𝛽𝐻𝐵 1−𝛼 𝑒 𝛼−1 (𝜌 2𝛼 𝛾𝜌 2𝛼 )𝛼 0≤𝛼≤1 𝛼>1 Quantum hypothesis testing and the operational interpretation of the quantum Renyi relative entropies Milan Mosonyi, Tomohiro Ogawa Communications in Mathematical Physics: Volume 334, Issue 3 (2015), Page 1617-1648
© Copyright 2024