A tutorial on non-markovian quantum processes. Kavan Modi Monash University Melbourne, Australia

Transcription

1 A tutorial on non-markovian quantum processes Kavan Modi Monash University Melbourne, Australia

2 Quantum Information Science Postdoc Felix Pollock PhD Students Francesco Campaioli Simon Milz

3 Open quantum systems

4 dynamics quantum to classical non-markovian Non-Markovian coherence environmental effects memory kernel heat spectroscopy entanglement system-bath correlations memory effects Redfield density matrix strong coupling temperature master equation transport decoherence evolution energy exchange dephasing weak coupling dissipation Markovian Lindblad positivity spin-boson model Hierarchical Equation of Motion

5 Stochastic process

6 What is Markovian?

7 Master equation

8 What is non-markovian?

9 How Markovianise do we deal with this?

10 Quantum version of stochastic dynamics

11 Dynamical map S Map S We can use the system to characterise the Blackbox.

12 Contracted unitary dynamics E S Unitary SE S

13 Properties of map Dynamical maps = Reduced unitary dynamics Linear Completely positive Contractivity

14 Complete positivity S S Map A

15 Charactering stochastic quantum processes

16 Quantum process tomography Map Modi, Rodríguez-Rosario, Aspuru-Guzik, Phys. Rev. A 86, (2012)

17 Quantum process tomography of a qubit Map Map Map Map

18 Sequence of dynamical maps=markovian process Divisibility = Lindblad, Redfield

19 Witnesses

20 Witness NM through contractivity Divisible Indivisible Time Breuer, Laine, Piilo, PRL 103, (2009)

21 Check divisibility directly Divisible Indivisible Time Rivas, Huelga, Plenio, PRL 105, (2010)

22 Question Suppose you have a system undergoing open dynamics. The entropy (purity) of the system is first increasing (decreasing) and later decreasing (increasing). Can we conclude that the dynamics are non- Markovian?

23 NM but monotonic in distance Pollock, Rodríguez-Rosario, Frauenheim, Paternostro, Modi, arxiv: (2015)

24 Divisible but NM Pollock, Rodríguez-Rosario, Frauenheim, Paternostro, Modi, arxiv: (2015)

25 Divisible but NM Lindblad dynamics Pollock, Rodríguez-Rosario, Frauenheim, Paternostro, Modi, arxiv: (2015)

26 Full description?

27 Classical to quantum? Convex and informationally complete Convex or informationally complete

28 An illustration

29 Contracted unitary dynamics E S Unitary 1:0

30 Contracted unitary dynamics Unitary 2:1

31 Initial correlations witnesses Time Mazzola, Rodrıguez-Rosario, Modi, Paternostro, Phys. Rev. A 86, (R) (2012) Rodríguez-Rosario, Modi, Mazzola, Aspuru-Guzik, Europhys. Lett (2012)

32 NM without correlations arxiv: Pollock, Rodríguez-Rosario, Frauenheim, Paternostro, Modi, arxiv: (2015)

33 Dealing with initial correlations

34 Initially correlated SE E SE S Unitary SE S SE S Pechukas: We must give up complete positivity or linearity This is the simplest non-markovian case study.

35 Giving up CP? Holevo bound Masillo, Scolarici, Solombrino, J Math Phys 52, (2011) Data processing inequality Buscemi, PRL. 113, (2014) Entropy production Argentieri, Benatti, Floreanini, EPL. 107, (2014)

36 What can we say about initial state?

37 Initially correlated SE E SE S Unitary SE S Grad student presses buttons We can give up states as inputs

38 Superchannel Completely positive and linear Super channel S Modi Sci. Rep. 2, 581 (2012) Ringbauer, Wood, Modi, Gilchrist, White, Fedrizzi, PRL 114, (2015)

39 Using CP Holevo bound [Masillo, Scolarici, Solombrino, J Math Phys 52, (2011)] Data processing inequality [Buscemi, PRL. 113, (2014)] Entropy production [Argentieri, Benatti, Floreanini, EPL. 107, (2014)] Vinjanampathy & Modi PRA 92, (2015) Vinjanampathy & Modi, Int. J. Quantum Inf. 14, (2016)

40 Characterising a generic quantum processes

41 What is missing? Can we characterise non-markovian quantum processes? If so, how? Can we do this without making assumptions? Can have our cake and eat it? Keep positivity & linearity Can we do this practically?

42 Next time. Thanks!!

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58 Taking a middle page from the lab notebook Open quantum evolution Completely positive and linear

59 Open quantum evolution

60 Universality Process tensor is the most general descriptor of quantum processes We have not assumed any model It has all of the desired properties: linear, complete positivity, containment It s operationally motivated, meaning it is designed to work hand-in-hand with experiments

61 How do we find?

62 Linear expansion

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64 Encode into state X X =

65 Encode into state

66 Encode into state is a matrix product density operator.

67 What is Markovian?

68 Shift switch of grad student Causal break

69 Markovian Divisible X Semigroup

70 Encode into state

71 Measuring non-markovianity with Relative entropy S A1B1 A2B2 A3B3 A4B4 A5B5 A6B6 A7B7 S A1B1 A2B2 A3B3 A4B4 A5B5 A6B6 A7B7 You have a Markovian model to describe a non-markov process. How surprised are you when model gives wrong answer

72 What now? Existing experiments, e.g. ultrafast spectroscopy Bounds on energy transport Derive new master equations Typicality of non-markovian process Quantum information theory / error correction Causal structures

73 Conclusions arxiv: We have a universal descriptor for arbitrary quantum processes We can encode the process into a many-body state, leading to an efficient characterisation Operational definition of non-markovianity

74 1. Introduction 2. Group (co-authors) 3. Dynamical maps (Complete Positivity) 4. Stinespring 5. Initial correlations/non-markovianity 6. Pechucas theorem 7. Resolution (give up states) 8. Superchannel 9. Experiment 10. What is a process? 11. What is characterisation? 12. Process tensor 13. Properties of PT 14. Proof of Universality OQE to PT 15. Proof of Universality PT to OQE 16. Direct tomography (scaling) 17. CJI states 18. Substates of CJI state 19. Causal break 20. Non-Markov condition 21. Measure for non-markovianity 22. Applications 23. Holevo, transport 24. Quantum information theory

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76 Can we generalise this?

77 Quantum process These actions are the history of buttons grad student pressed