Fundamental work cost of quantum processes

Transcription

1 Fundamental work cost of quantum processes Philippe Faist 1,2, Renato Renner 1 1 Institute for Theoretical Physics, ETH Zurich 2 Institute for Quantum Information and Matter, Caltech QIP 2018, Delft, January 2018

2 electromagnetic radiation steam engines gases chemistry solid state physics black holes QIP 2018, Delft, January

3 electromagnetic radiation steam engines gases chemistry solid state physics black holes Maxwell s demon anomalous heat flows Jennings & Rudolph, PRE, 2010 side information del Rio et al., Nature, 2011 QIP 2018, Delft, January

4 electromagnetic radiation steam engines gases chemistry solid state physics black holes Maxwell s demon anomalous heat flows Jennings & Rudolph, PRE, 2010 actually OK, just need to be careful side information del Rio et al., Nature, 2011 QIP 2018, Delft, January

5 electromagnetic radiation steam engines gases chemistry solid state physics black holes Maxwell s demon anomalous heat flows Jennings & Rudolph, PRE, 2010 side information del Rio et al., Nature, 2011 actually OK, just need to be careful What is the most general formulation of thermodynamics? QIP 2018, Delft, January

6 electromagnetic radiation steam engines gases chemistry solid state physics black holes Maxwell s demon anomalous heat flows Jennings & Rudolph, PRE, 2010 side information del Rio et al., Nature, 2011 actually OK, just need to be careful What is the most general formulation of thermodynamics? Idea: role of information QIP 2018, Delft, January

7 Information and Thermodynamics 1 bit of information can be traded for work Szilárd engine Szilárd, 1929 Landauer: Irreversible information processing incurs thermodynamic cost Landauer, 1961 Bennett, 1982, 2003 QIP 2018, Delft, January

8 Maxwell s Demon Demon lets particles go from right to left only QIP 2018, Delft, January

9 Maxwell s Demon Demon lets particles go from right to left only Net decrease in entropy of gas violation of the second law of thermodynamics? QIP 2018, Delft, January

10 Maxwell s Demon Demon lets particles go from right to left only Net decrease in entropy of gas violation of the second law of thermodynamics? Idea: The demon has access to microscopic information QIP 2018, Delft, January

11 Maxwell s Demon memory The demon stores the measurement results Resetting this memory costs work! Landauer, 1961 Bennett, 1982, 2003 QIP 2018, Delft, January

12 Resource Theory of thermal operations Allowed any ancilla in a Gibbs state QIP 2018, Delft, January

13 Resource Theory of thermal operations Allowed any ancilla in a Gibbs state Allowed any energy-conserving unitaries: QIP 2018, Delft, January

14 Resource Theory of thermal operations Allowed any ancilla in a Gibbs state Allowed any energy-conserving unitaries: Allowed to discard any system QIP 2018, Delft, January

15 Known Results Necessary and sufficient conditions for (thermo-majorization, block-diagonal states) Horodecki & Oppenheim, Nat. Comm Conversion rates Brandão et al., PRL, 2013 Rényi- entropies monotones: second laws Brandão et al., PNAS, 2015 Generalized thermodynamic baths Yunger Halpern & Renes, PRE, 2016,... Catalytical transformations, correlations... Ng et al., NJP, 2015; Lostaglio et al., PRL, QIP 2018, Delft, January

16 Thermodynamic cost of any process? mapping of input states to output states AND, XOR, gate any classical or quantum computation any physical process (completely positive, tracepreserving map) QIP 2018, Delft, January

17 Thermodynamic cost of any process? mapping of input states to output states AND, XOR, gate any classical or quantum computation Fundamental thermodynamic limit any physical process (completely positive, preserving erving trace- map) to the cost of implementing? QIP 2018, Delft, January

18 Our approach A restriction on what we can do QIP 2018, Delft, January

19 Our approach A restriction on what we can do Free operations must preserve the thermal state (most generous set of maps) QIP 2018, Delft, January

20 Our approach A restriction on what we can do Free operations must preserve the thermal state (most generous set of maps) A resource which we can use to overcome the restriction QIP 2018, Delft, January

21 Our approach A restriction on what we can do Free operations must preserve the thermal state (most generous set of maps) A resource which we can use to overcome the restriction Battery system QIP 2018, Delft, January

22 Framework (2) To each system of interest associated an operator is QIP 2018, Delft, January

23 Framework (2) To each system of interest associated an operator is Choose QIP 2018, Delft, January

24 Framework (2) To each system of interest associated an operator is Allowed only trace-nonincreasing CPMs satisfying Choose QIP 2018, Delft, January

25 Framework (2) To each system of interest associated an operator is Choose Allowed only trace-nonincreasing CPMs satisfying Can always be dilated to trace-preserving, QIP 2018, Delft, January

26 Framework (2) To each system of interest associated an operator is Choose Allowed only trace-nonincreasing CPMs satisfying Can always be dilated to trace-preserving, Information battery QIP 2018, Delft, January

27 Framework (2) To each system of interest associated an operator is Choose Allowed only trace-nonincreasing CPMs satisfying Can always be dilated to trace-preserving, Information battery Large family of battery models are equivalent QIP 2018, Delft, January

28 Information Battery stored work Landauer/Szilárd: 1 pure qubit = work QIP 2018, Delft, January

29 Step #1: Limit for an exact process QIP 2018, Delft, January

30 Step #1: Limit for an exact process QIP 2018, Delft, January

31 Step #1: Limit for an exact process QIP 2018, Delft, January

32 Step #1: Limit for an exact process QIP 2018, Delft, January

33 Step #1: Limit for an exact process QIP 2018, Delft, January

34 Step #1: Limit for an exact process QIP 2018, Delft, January

35 Step #2: Optimize effective process QIP 2018, Delft, January

36 Step #2: Optimize effective process QIP 2018, Delft, January

37 Step #2: Optimize effective process process matrix of QIP 2018, Delft, January

38 Step #2: Optimize effective process process matrix of (purified/fidelity distance) QIP 2018, Delft, January

39 Step #2: Optimize effective process process matrix of (purified/fidelity distance) limit for specific = Step #1 QIP 2018, Delft, January

40 Step #2: Optimize effective process process matrix of (purified/fidelity distance) fundamental limit = Step #2 Step #1 QIP 2018, Delft, January

41 Results fundamental limit = Step #2 Step #1 QIP 2018, Delft, January

42 Results fundamental limit = Steps #1 & #2 QIP 2018, Delft, January

43 Results coherent relative entropy Ultimate maximum extractable work for implementing a map with process matrix close to = QIP 2018, Delft, January

44 Results coherent relative entropy Ultimate maximum extractable work for implementing a map with process matrix close to = QIP 2018, Delft, January

45 Coherent relatve entropy: Special cases relative Datta, IEEE TIT (2009) Åberg, Nat Comm (2013) Horodecki & Oppenheim, Nat Comm (2013) QIP 2018, Delft, January

46 Coherent relatve entropy: Special cases relative Datta, IEEE TIT (2009) Åberg, Nat Comm (2013) Horodecki & Oppenheim, Nat Comm (2013) coherent ( conditional ) for pure QIP 2018, Delft, January

47 Examples (with trivial Hamiltonian) Pure information processing no internal energy cost discarded information output PhF, Dupuis, Oppenheim, Renner, Nat Comm, 2015 QIP 2018, Delft, January

48 Examples (with trivial Hamiltonian) Pure information processing no internal energy cost cost: 1.6 bits cost: 1 bit extract: 2 bits cost: 1 bit PhF, Dupuis, Oppenheim, Renner, Nat Comm, 2015 QIP 2018, Delft, January

49 The Coherent Relative Entropy Data processing inequality Chain rule Asymptotic equipartition property: For many independent copies QIP 2018, Delft, January

50 The Coherent Relative Entropy Data processing inequality Chain rule Asymptotic equipartition property: For many independent copies The coherent relative entropy: measure of information has desirable properties reduces to known special cases QIP 2018, Delft, January

51 Battery models Define for For : Reversibly interconvertable Some common battery models equivalent (information battery, wit, weight) Battery states are robust to smoothing (no need to smooth battery states) Brandão et al., PNAS (2015) QIP 2018, Delft, January

52 Emergence of Macro Thermodynamics For a certain class of states (e.g. microcanonical): QIP 2018, Delft, January

53 Emergence of Macro Thermodynamics For a certain class of states (e.g. microcanonical): derives from a potential! QIP 2018, Delft, January

54 Emergence of Macro Thermodynamics For a certain class of states (e.g. microcanonical): derives from a potential! free isolated system: within class contact with heat bath: no i.i.d. assumption QIP 2018, Delft, January

55 Observers in Thermodynamics QIP 2018, Delft, January

56 Observers in Thermodynamics QIP 2018, Delft, January

57 Observers in Thermodynamics Petz, CMP (1986); Fawzi & Renner CMP (2015); Wilde PRSA (2015);... QIP 2018, Delft, January

58 Observers in Thermodynamics implies Petz, CMP (1986); Fawzi & Renner CMP (2015); Wilde PRSA (2015);... QIP 2018, Delft, January

59 Observers in Thermodynamics implies What if? possible apparent violation of second law Petz, CMP (1986); Fawzi & Renner CMP (2015); Wilde PRSA (2015);... QIP 2018, Delft, January

60 Observers in Thermodynamics implies What if Criterion? for when coarse-grained laws of thermodynamics hold: possible apparent violation of second law Petz, CMP (1986); Fawzi & Renner CMP (2015); Wilde PRSA (2015);... QIP 2018, Delft, January

61 A picture of thermodynamics restriction: preserve information battery coherent relative entropy macroscopic states emergent thermodynamics QIP 2018, Delft, January

62 Physics Information Theory Hamiltonian time evolution quantum state unitary operation QIP 2018, Delft, January

63 Physics Information Theory Hamiltonian time evolution quantum state unitary operation energy, number of particles QIP 2018, Delft, January

64 Physics Information Theory Hamiltonian time evolution quantum state unitary operation energy, number of particles thermodynamics (at least 2nd law) QIP 2018, Delft, January

65 A simple yet general, single-instance, observer-dependent view of thermodynamics Outlook better understanding of universality of thermodynamics New measure of information non-i.i.d. version of relative entropy difference Achievability with thermal operations (+...)? Applications to information theory, coding? Applications to physical systems? QIP 2018, Delft, January

66 Thank you for your attention!

67 Example: Maxwell s Demon microscopic picture macroscopic picture implicit demon explicit demon QIP 2018, Delft, January

68 Example: Maxwell s Demon microscopic picture macroscopic picture implicit demon thermodynamic entropy is observer-dependent! explicit demon QIP 2018, Delft, January

69 General case: non-trivial Hamiltonian Fundamental work cost for any process coherent relative entropy process matrix, characterizes and input state measures information relative to Gibbs weights QIP 2018, Delft, January

70 Approaches to information thermodynamics statistical mechanics Piechocinska, PRA, 2000 resource theory approach Brandão et al., PRL, 2013 axiomatic approach Lieb & Yngvason, PR, 1999 Weilenmann et al., PRL, 2016 QIP 2018, Delft, January

71 Approaches to information thermodynamics statistical mechanics Piechocinska, PRA, 2000 work probability distributions, time evolution, fluctuation relations axiomatic approach Lieb & Yngvason, PR, 1999 Weilenmann et al., PRL, 2016 resource theory approach Brandão et al., PRL, 2013 inherently one-shot, epsilonwork, general processes abstract, first-principles approach, structure of thermodynamics QIP 2018, Delft, January

72 Smoothing QIP 2018, Delft, January

73 Smoothing QIP 2018, Delft, January

74 Smoothing QIP 2018, Delft, January

75 Smoothing Ignore unlikely events up to total probability QIP 2018, Delft, January

76 What About Statistical Mechanics? Model system s time evolution Average energy, von Neumann entropy; oneshot statements more tricky Count work? p(w) not well defined quantum Closer to applications than resource-theory approaches QIP 2018, Delft, January

77 Work? Count work using a battery system Horodecki & Oppenheim, Nat. Comm QIP 2018, Delft, January

78 Work? Count work using a battery system Horodecki & Oppenheim, Nat. Comm QIP 2018, Delft, January

79 Work? Count work using a battery system Work extraction : Horodecki & Oppenheim, Nat. Comm QIP 2018, Delft, January

80 Work? Count work using a battery system Work extraction : Work cost of formation : Horodecki & Oppenheim, Nat. Comm QIP 2018, Delft, January

81 Work? Count work using a battery system Work extraction : Work cost of formation : valid for single instance of the process macroscopic limit Horodecki & Oppenheim, Nat. Comm QIP 2018, Delft, January