1. DECOHERENCE AND QUANTUM INFORMATION JUAN PABLO PAZ Departamento de Fisica, FCEyN Universidad de Buenos Aires, Argentina paz@df.uba.ar Paraty August 2007

2. Lecture 1 & 2: Decoherence and the quantum origin of the classical world Lecture 2 & 3: Decoherence and quantum information processing, noise characterization (process tomography) Colaborations with: W. Zurek (LANL), M. Saraceno (CNEA), D. Mazzitelli (UBA), D. Dalvit (LANL), J. Anglin (MIT), R. Laflamme (IQC), D. Cory (MIT), G. Morigi (UAB), S. Fernandez-Vidal (UAB), F. Cucchietti (LANL), 1, 2 & 3. Decoherence, an overview. –Hadamard-Phase-Hadamard and the origin of the classical world! Information transfer from the system to the environment. –Bosonic environment. Complex environments. Spin environments (some new results) 3. Decoherence in quantum information. How to characterize it? –Quantum process tomography (some new ideas) Current/former students: D. Monteoliva, C. Miquel (UBA), P. Bianucci (UBA, UT), L. Davila (UEA, UK), C. Lopez (UBA, MIT), A. Roncaglia (UBA), C. Cormick (UBA), A. Benderski (UBA), F. Pastawski (UNC), C. Schmiegelow (UNLP, UBA)

3. DECOHERENCE: AN OVERVIEW SUMMARY: SOME BASIC POINTS ON DECOHERENCE POINTER STATES: W.Zurek, S. Habib & J.P. Paz, PRL 70, 1187 (1993), J.P. Paz & W. Zurek, PRL 82, 5181 (1999) TIMESCALES: J.P. Paz, S. Habib & W. Zurek, PRD 47, 488 (1993), J. Anglin, J.P. Paz & W. Zurek, PRA 55, 4041 (1997) CONTROLLED DECOHERENCE EXPERIMENTS: Zeillinger et al (Vienna) PRL 90 160401 (2003), Haroche et al (ENS) PRL 77, 4887 (1997), Wineland et al (NIST), Nature 403, 269 (2000). DECOHERENCE AND THE QUANTUM-CLASSICAL TRANSITION: YES: HILBERT SPACE IS HUGE, BUT MOST STATES ARE UNSTABLE!! (DECAY VERY FAST INTO MIXTURES) CLASSICAL STATES: A (VERY!) SMALL SUBSET. THEY ARE THE POINTER STATES OF THE SYSTEM DYNAMICALLY CHOSEN BY THE ENVIRONMENT

4. DECOHERENCE: AN OVERVIEW LAST DECADE: MANY QUESTIONS ON DECOHERENCE WERE ANSWERED NATURE OF POINTER STATES: QUANTUM SUPERPOSITIONS DECAY INTO MIXTURES WHEN QUANTUM INTERFERENCE IS SUPRESSED. WHAT ARE THE STATES SELECTED BY THE INTERACTION? POINTER STATES: THE MOST STABLE STATES OF THE SYSTEM, DYNAMICALLY SELECTED BY THE ENVIRONMENT: W.Zurek, S. Habib & J.P. Paz, PRL 70, 1187 (1993), J.P. Paz & W. Zurek, PRL 82, 5181 (1999) TIMESCALES: HOW FAST DOES DECOHERENCE OCCURS? J.P. Paz, S. Habib & W. Zurek, PRD 47, 488 (1993), J. Anglin, J.P. Paz & W. Zurek, PRA 55, 4041 (1997) INITIAL CORRELATIONS: THEIR ROLE, THEIR IMPLICTIONS L. Davila Romero & J.P. Paz, Phys Rev A 54, 2868 (1997), MORE REALISTIC PREPARATION OF INITIAL STATE (FINITE TIME PREPARATION, ETC) J. Anglin, J.P. Paz & W. Zurek, PRA 55, 4041 (1997) DECOHERENCE FOR CLASSICALLY CHAOTIC SYSTEMS: W. Zurek & J.P. Paz, PRL 72, 2508 (1994), D. Monteoliva & J.P. Paz, PRL 85, 3373 (2000). CONTROLLED DECOHERENCE EXPERIMENTS: S. Haroche et al (ENS) PRL 77, 4887 (1997), D. Wineland et al (NIST), Nature 403, 269 (2000), A. Zeillinger et al (Vienna) PRL 90 160401 (2003), ENVIRONMENT ENGENEERING: Cirac & Zoller, etc; see Nature 412, 869 (2001) (review of PRL work published by UFRJ group on environment engeneering with ions) SPIN ENVIRONMENTS: ASK CECILIA CORMICK (POSTER NEXT WEEK).

5. SYSTEM ENVIRONMENT INTERACTION CREATES “A RECORD” OF THE STATE OF THE SYSTEM IN THE ENVIRONMENT CLASSICALITY, OBJECTIVITY AND REDUNDANCY INTERACTION WITH ENVIRONMENT INDUCES DECAY OF FRINGE VISIBILITY: DECOHERENCE PURE STATEMIXED STATE

6. WHAT SHOULD YOU DO IF YOU WANT TO LEARN ABOUT THE SYSTEM? (THE INITIAL STATE, FOR EXAMPLE) CLASSICALITY, OBJECTIVITY AND REDUNDANCY PURE STATEMIXED STATE LOOK AT THE ENVIRONMENT! HOW MUCH CAN YOU LEARN ABOUT THE SYSTEM BY LOOKING AT THE ENVIRONMENT? TOOL: MUTUAL INFORMATION (NOT ALWAYS USEFUL BUT…) I(A,B)=S(A)+S(B)-S(A,B) FLOW OF INFORMATION BETWEEN SYSTEM AND ENVIRONMENT AND ITS RELATION WITH CLASSICALITY (A. Roncaglia & J.P.P, 2007) A= ONE of the oscillators of the environment (a band centered around a certain frequency); B= SYSTEM

7. CLASSICALITY, OBJECTIVITY AND REDUNDANCY PURE STATEMIXED STATE I(A,B)=S(A)+S(B)-S(A,B) Environmental fraction THE SYSTEM BECOMES CLASSICAL WHEN THE INFORMATION ABOUT IT IS “STORED” REDUNDANTLY IN THE ENVIRONMENT. RECOVER THE STATE OF THE SYSTEM ONLY WHEN YOU MEASURE THE COMPLETE ENVIRONMENT. OTHERWISE YOU ONLY KNOW WHERE THE CAT IS (NOT THE PHASE!)

8. 1. Classical random walk 0 1 2 ¿  ó  ? 0 1 2 Typical probability distribution Probability 1/2 0 1 2 0 1 2 EXAMPLE: DECOHERENCE IN A QUANTUM ALGORITHM

9. 2. Quantum walk algorithm: A quantum coin (spin 1/2) and a quantum walker (moving in a ring with N sites) : It could be a useful “subroutine” 0 1 2 move to the right move to the left Quantum walk “algorithm”: Initial state: Evolution operator: -1 0 1 2 -1 0 1 2 + The coin (spin) and the walker become entangled. The state of the walker after t-iterations is: EXAMPLE: DECOHERENCE IN A QUANTUM WALK

10. Initial state: “Impartial spin”, localized walker Classical and quantum walks have rather different properties: Probability distribution: Classical vs Quantum Quantum walker spreads faster than classical! Reason?: Quantum interference (more later!) EXAMPLE: DECOHERENCE IN A QUANTUM WALK

11. Key to the potential advantadge of quantum walks?: Use the quantum nature of the walk, that allows for faster spreading over the graph (this enables, for example, exponentially faster hiting times) NOTE Quantum walks on graphs have been proposed as potentially useful quantum subroutines Review: J. Kempe, Contemp Phys 44, 307 (2003) Proposed in: D. Aharonov, A. Ambainis, J. Kempe, U. Vazirani, Proc 33. ACM STOC-2001, 50-59 There are very few algorithms that use quantum walks as a central piece: * N. Shenvi, J. Kempe & B. Whaley, PRA 67 052307 (2003) (DISCRETE); * A. M. Childs et al, Proc 35 ACM STOC-2003, 59-68 (CONTINUOUS) EXAMPLE: DECOHERENCE IN A QUANTUM WALK

12. What happens if the coin (or walker) interacts with an environment? 0 1 2 0 1 2 + Simple model to simulate coupling to a spin environment (NMR) (G.Teklemarian, et al PRA67, 062316 (2003)) (2002, 2003: V. Kendon, B. Tregenna, H. Carteret, T. Brun, A. Ambainis, etc) 12 34 k t=0 t=T k Feature: model can be exactly solved (analytic solutions available @ C.Lopez & J.P.P, PRA 68, 052305 (2003) = coupling strength EXAMPLE: DECOHERENCE IN A QUANTUM WALK

13. RESULTS: Decoherence and quantum-classical transition for quantum walks Fixe time, vary system-environment coupling strength EXAMPLE: DECOHERENCE IN A QUANTUM WALK

14. QUANTUM WALKS IN PHASE SPACE Simple interpretation Exact formula valid in the case of full decoherence Quantum walk with decoherence in the coin looks like random walk. Effect: Diffusion along the position direction in phase space. Lesson I from decoherence studies: a) Environment couples to spin. b) Spin couples with walker via U (displacement operator). d) U is diagonal in momentum. Then: momentum states are pointer states! EXAMPLE: DECOHERENCE IN A QUANTUM WALK

15. QUANTUM WALKS IN PHASE SPACE Lesson II from decoherence: If the phase space picture of decoherence (position diffusion) is correct, we can make a prediction: Superposition of two positions should be robust agains decoherence! No decoherenceFull decoherence EXAMPLE: DECOHERENCE IN A QUANTUM WALK

16. QUANTUM WALKS IN PHASE SPACE Quantum coherence in this Schrodinger cat state is robust against decoherence. No entropy is produced from the decay of the coherent superposition Mixture of two positionsSuperposition of two positions

17. QUANTUM WALKS IN PHASE SPACE A surprise: Entropy curves cross each other. Why? Some decoherence may be useful! Superposition of two positions Fix time, vary coupling strength

18. USE WIGNER FUNCTIONS TO REPRESENT THE STATE AND THE EVOLUTION OF QUANTUM COMPUTERS see C. Miquel, J.P.P & M. Saraceno, Phys Rev A 65 (2002), 062309 Phase space representation of Grover’s algorithm See paper for other interesting (useful?) analogies between quantum algorithms and quantum maps PHASE SPACE TECHNIQUES IN QUANTUM COMPUTATION Ather approaches to Discrte Wigner functions in NxN grids with interesting connection with stabilizer formalism: Wootters et al; C.Cormick, A. Roncaglia, M. Saraceno, E. Galvao, J.P.P, et al.

19. MOST GENERAL EVOLUTION OF TWO INTERACTING SUBSYSTEMS REDUCED DENSITY MATRIX OF SUB-SYSTEM A KRAUSS REPRESENTATION OF EVOLUTION OF REDUCED DENSITY MATRIX (general) Valid if initial state of B is pure EVOLUTION OF QUANTUM OPEN SYSTEMS

20. KRAUS REPRESENTATION (PHASE DAMPING CHANNEL) EXAMPLE: CONSIDER A DECOHERENCE MECHANISM THAT SUPRESSES OFF DIAGONAL TERMS IN {0,1} BASIS: EVOLUTION OF QUANTUM OPEN SYSTEMS

21. MORE GENERAL NOISY CHANNEL STUDY HOW THE QUANTUM STATE IS DEGRADED BY A NOISY CHANNEL FIDELITY DECAY: PURITY DECAY: DECAY DEPENDS ON THE STATE DECAY IS LINEAR IN p

22. HOW TO FIGHT AGAINST DECOHERENCE ERROR CORRECTION CAN BE USED TO PROTECT QUANTUM INFORMATION EncodingDecodingRepairRefresh FIDELITY GOES FROM LINEAR TO QUADRATIC IN p!

23. TO CORRECT ERRORS YOU MUST CHARACTERIZE THEM: KNOW YOUR ERRORS! MOST GENERAL EVOLUTION OF A QUANTUM OPEN SYSTEM CAN BE WRITTEN IN KRAUSS FORM (Exercise: Think about this!) THUS, WE NEED TO FIND OUT WHAT ARE THE KRAUSS OPERATORS: CHARACTERIZE A QUANTUM CHANNEL HOW TO FIGHT AGAINST DECOHERENCE WHAT TO DO IF YOU DON’T KNOW THE KRAUSS OPERATORS? Exercise: Show that Krauss representation is not unique Find another Krauss representation for the phase damping channel

24. ARE DOMINANT KRAUSS OPERATORS ONE QUBIT ERRORS? WHAT IS THE WEIGHT OF THE IDENTITY IN THE KRAUSS REPRESENTATION? HOW TO FIGHT AGAINST DECOHERENCE DO NOT KRAUSS OPERATORS? WE CAN STILL CONSIDER AN ARBITRARY BASIS OF OPERATORS E’s AND WRITE: COEFFICIENTS DEPEND UPON THE BASIS WE CHOOSE

25. HOW TO CHARACTERIZE A QUANTUM CHANNEL QUANTUM PROCESS TOMOGRAPHY: OBTAIN COMPLETE KNOWLEDGE OF CHANNEL PARAMETERS (KRAUSS OPERATORS) EXPONENTIALLY HARD TASK (MEASURE SURVIVAL PROBABILITY OF EXPONENTIALLY MANY INITIAL STATES) EACH EXPERIMENT (i,j) DOES NOT GIVE INFORMATION ABOUT THE CHANNEL BUT ON THE ACTION OF THE CHANNEL ON SOME STATES (BUT STATE DEPENDENT). SIMILARLY STATE FIDELITY:

26. HOW TO CHARACTERIZE A QUANTUM CHANNEL QUESTION: HOW TO DETERMINE PROPERTIES OF THE CHANNEL? EFFICIENT ESTIMATION OF AVERAGE FIDELITY IS POSSIBLE! (Klappenecker & Roetteler 2005; see C. Dankert’s Thesis quant-ph/0512217) WHAT DOES THE AVERAGE FIDELITY MEASURES? USE A VERY NICE BASIS E (PAULI’s):

27. HOW TO CHARACTERIZE A QUANTUM CHANNEL d(d+1) STATES FORMING d+1 MUTUALLY UNBIASED BASIS THIS SEEMS TO BE VERY HARD! HOW DO YOU INTEGRATE OVER ALL HILBERT SPACE IN REAL LIFE? HOWEVER… MOREOVER THIS CAN BE COMPUTED AS A SUM OVER d(d+1) STATES!

28. ESTIMATE AVERAGE FIDELITY: AVERAGE OVER d(d+1) STATES TO FIND AN ESTIMATE (WITH PRECISION INDEPENDENT OF d) WE NEED TO SAMPLE A RANDOM SUBSET OF THE STATES OF THE MUB’S! (it is efficient if we are OK with a d-independent accuracy) HOW TO CHARACTERIZE A QUANTUM CHANNEL

29. SEE POSTER BY F. PASTAWSKI FOR A GENERALIZATION TO DETERMINE (EFFICIENTLY) ANY COEFFICIENT IN THE EXPANSION OF THE CHANNEL (SELECTIVE EFFICIENT PROCESS TOMOGRAPHY) HOW TO CHARACTERIZE A QUANTUM CHANNEL NEED A ‘CLEAN’ SOURCE OF STATES FROM MUBs (can be efficiently produced) EXAMPLE: TWO QUBITS, d=4, d+1=5 (5 Mutually Unbiased Bases) EACH BASIS FORMED BY EIGENSTATES OF OPERATORS IN EACH ROW

30. PHOTONIC QUANTUM PROCESS TOMOGRAPHY SEE POSTER BY CH. SCHMIEGELOW FOR AN EXPLANATION OF PHOTONIC IMPLEMENTATION (ALL GATES NEEDED TO PREPARE THE d(d+1) STATES ARE REALIZED WITH VARIATIONS OF PHASE SHIFTERS, QWP & HWP). RESULT: OBTAIN ALL DIAGONAL COEFFICIENTS OF THE CHANNEL

31. HOW TO CHARACTERIZE A QUANTUM CHANNEL METHOD TO DETERMINE ANY COEFFICIENT IN THE EXPANSION OF THE CHANNEL (SELECTIVE EFFICIENT PROCESS TOMOGRAPHY). AVERAGE SURVIVAL PROBABILITY OVER A SAMPLE OF STATES FROM MUBs IN AN INTERACTION OF THE FORM: C NEED ONE EXTRA QUBIT (BETTER THAN OTHER METHODS..). SEE F. PASTAWSKI’s POSTER