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. 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) 2 &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. HOW DOES THE WIGNER FUNCTION OF A QUANTUM STATE LOOK LIKE?: SUPERPOSITION OF TWO GAUSSIAN STATES OSCILLATIONS IN WIGNER FUNCTION: THE SIGNAL OF QUANTUM INTERFERENCE. HOW DOES DECOHERENCE AFFECTS THIS STATE? DECOHERENCE IN QUANTUM BROWNIAN MOTION (VI)

4. DECOHERENCE RATE: MUCH LARGER THAN RELAXATION RATE DECOHERENCE IN QUANTUM BROWNIAN MOTION (VII) MASTER EQUATION CAN BE REWRITTEN FOR THE WIGNER FUNCTION: IT HAS THE FORM OF A FOKER-PLANCK EQUATION

5. A DIGRESSION ON WIGNER FUNCTIONS Know: Use circuit to learn about (“spectrometer”) Know: Use circuit to learn about (“tomographer”) APLICATION: DAVIDOVICH-LUTTERBACH SCHEME FOR MEASURING WIGNER FUNCTION REMEMBER THE SIMPLE SCHEME TO USE ONE QUBIT TO LEARN ABOUT EITHER THE QUANTUM STATE OF ANOTHER SYSTEM OR ABOUT THE QUBIT-SYSTEM INTERACTION

6. REMEMBER THE DEFINITION OF WIGNER FUNCION : phase space point operators PROPERTIES OF WIGNERS ARE CONSEQUENCES OF Integral along lines give a projection operator: form a complete orthonormal set phase space displacement operator R: reflection operator A DIGRESSION ON WIGNER FUNCTIONS

7. TOMOGRAPHY OF (CONTINUOUS) WIGNER FUNCTION ALLA DAVIDOVICH-LUTTERBACH This quantum circuit describes the ENS-experiment to measure Wigner function Qbit: 2-level atom; System: em-field inside the cavity Displacement operator: inject classical field in cavity; Reflection: dispersive interaction between field & atom APLICATION II: CAN THIS TYPE OF TOMOGRAPHY BE GENERALIZED TO OTHER SYSTEMS? DISCRETE WIGNER FUNCTIONS A DIGRESSION ON WIGNER FUNCTIONS Wigner function is nothing but the matrix element of the density matrix in a special operator basis

8. A SYSTEM WITH A FINITE DIMENSIONAL (N) HILBERT SPACE Position basis (arbitrary) Momentum basis PHASE SPACE DISPLACEMENT OPERATORS ARE EASY TO CONSTRUCT Position shift Momentum shift PHASE SPACE POINT OPERATORS ARE NOT SO EASY TO CONSTRUCT Naive attempt Does not have all the right properties (for all values of N) A DIGRESSION ON WIGNER FUNCTIONS

9. 0 1 2 3 4 5 6 7 0 1 3 4 2 5 6 7 q+p=0 N=8 0 1 2 3 4 5 6 7 0 1 3 4 2 5 6 7 2q+2p=0 more Lines may have more than N pointsq & p are integers, arithmetic mod N WHY??: {1,2,…,N} is NOT a FINITE FIELD: 2 x 4 = 0 (mod 8) (it is a field if N is a prime number GF(N)) A DIGRESSION ON WIGNER FUNCTIONS

10. SOLUTION (M. Berry, 1980; U. Leonhardt,…) DISCRETE WIGNER FUNCTION Phase space operators: DFT of displacements In a grid of 2N by 2N SAME PROPERTIES THAN IN THE CONTINUOUS CASE: W(q,p) is real Sum over lines is always possitive! A DIGRESSION ON WIGNER FUNCTIONS

11. Discrete Wigner function: How does it look like? Position eigenstate Gaussian wavepacket positiveoscillatory OSCILLATIONS DUE TO PERIODIC BOUNDARY CONDITIONS A DIGRESSION ON WIGNER FUNCTIONS

12. DISCRETE WIGNER FUNCTIONS IN QUANTUM COMPUTATION; see “Quantum computers in phase space”, C. Miquel, J.P.P & M. Saraceno, PRA 65 (2002), 062309 TOMOGRAPHY OF (DISCRETE) WIGNER FUNCTION EFFICIENT QUANTUM CIRCUIT TO IMPLEMENT ‘CONTROLLED-A(x,p) EXISTS A DIGRESSION ON WIGNER FUNCTIONS

13. MEASUREMENT OF DISCRETE WIGNER FUNCTION OF FOUR COMPUTATIONAL STATES IN A TCE- SAMPLE IN A 500Mhz NMR SPECTROMETER (C. Miquel, J.P.P, M. Saraceno, E. Knill, R. Laflamme, C. Negrevergne, Nature 418, 59 (2002) A DIGRESSION ON WIGNER FUNCTIONS

14. DECOHERENCE RATE: MUCH LARGER THAN RELAXATION RATE DECOHERENCE IN QUANTUM BROWNIAN MOTION (VII) MASTER EQUATION CAN BE REWRITTEN FOR THE WIGNER FUNCTION: IT HAS THE FORM OF A FOKER-PLANCK EQUATION

15. EVOLUTION OF WIGNER FUNCTION NOTICE: NOT ALL STATES ARE AFFECTED BY THE ENVIRONMENT IN THE SAME WAY (SOME SUPERPOSITIONS LAST LONGER THAN OTHERS) POINTER STATES, DECOHERENCE TIMESCALE (I)

16. WARNING: DECOHERENCE TIMESCALE OBTAINED IN THE HIGH TEMPERATURE LIMIT FOR INITIAL “CAT STATE” IS ONLY AN APPROXIMATION! POINTER STATES, DECOHERENCE TIMESCALE (II) EXAMPLE 1: EVOLUTION OF FRINGE VISIBILITY FACTOR IN AN ENVIRONMENT AT ZERO TEMPERATURE FOR QUANTUM BROWNIAN MOTION (NON-EXPONENTIAL DECAY: CAN BE UNDERSTOOD FROM TIME-DEPENDENCE OF COEFFICIENTS OF MASTER EQUATION)

17. POINTER STATES, DECOHERENCE TIMESCALE (II’) Measure degradation of system’s state with entropy (von Neuman) or purity decay These quantities depend on time AND on the initial state USE MASTER EQUATION TO ESTIMATE PURITY DECAY OR ENTROPY GROWTH System & Environment become entangled (large environment: irreversible process) PURE STATEMIXED STATE

18. WARNING: DECOHERENCE TIMESCALE OBTAINED IN THE HIGH TEMPERATURE LIMIT FOR INITIAL “CAT STATE” IS ONLY AN APPROXIMATION! POINTER STATES, DECOHERENCE TIMESCALE (II) EXAMPLE 1: EVOLUTION OF FRINGE VISIBILITY FACTOR IN AN ENVIRONMENT AT ZERO TEMPERATURE FOR QUANTUM BROWNIAN MOTION (NON-EXPONENTIAL DECAY: CAN BE UNDERSTOOD FROM TIME-DEPENDENCE OF COEFFICIENTS OF MASTER EQUATION)

19. CLASSICALQUANTUM WIGNER FUNCTION DEVELOPS OSCILLATORY STRUCTURE AT SUB-PLANCK SCALES W.H. Zurek, Nature 412, 712 (2001); D. Monteoliva & J.P. Paz, PRE 64, 056238 (2002) NONTRIVIAL TIMESCALE: HARMONICALLY DRIVEN DOUBLE WELL (CHAOS) POINTER STATES, DECOHERENCE TIMESCALE (III)

20. DECOHERENCE DESTROYS THE FRINGES THAT ARE DYNAMICALLY PRODUCED CLASSICAL + NOISEQUANTUM + DECOHERENCE DECOHERENCE IMPLIES INFORMATION TRANSFER INTO CORRELATIONS BETWEEN SYSTEM AND ENVIRONMENT. WHAT IS THE RATE?: Lyapunov regime: Above a certain threshold, rate of entropy production fixed by the Lyapunov exponent of the system (W. Zurek & J.P.P., 1994) POINTER STATES, DECOHERENCE TIMESCALE (IV)

21. INTUITIVE EXPLANATION: WHY IS THERE A REGIME OF ENTROPY GROWTH FIXED BY THE LYAPUNOV EXPONENT? STRETCHING + FOLDING DECOHERENCE DIGRESSION: DECOHERENCE FOR CLASSICALLY CHAOTIC SYSTEMS STRETCHING DECOHERENCE

22. NUMERICAL EVIDENCE IS RATHER STRONG (D. Monteoliva and J.P.P., PRL (2000), PRE (2002), (2006)) DIGRESSION: DECOHERENCE FOR CLASSICALLY CHAOTIC SYSTEMS

23. time Log(D) Log(1/a) D. Monteoliva and J.P. P. (2006) DIGRESSION: DECOHERENCE FOR CLASSICALLY CHAOTIC SYSTEMS

24. POINTER STATES, DECOHERENCE TIMESCALE (V) NOT ALL STATES ARE AFFECTED BY DECOHERENCE IN THE SAME WAY QUESTION: WHAT ARE THE STATES WHICH ARE MOST ROBUST UNDER DECOHERENCE? (STATES WHICH ARE LESS SUSCEPTIBLE TO BECOME ENTANGLED WITH THE ENVIRONMENT). THE STATES THAT EXIST “OBJECTIVELY” POINTER STATES: STATES WHICH ARE MINIMALLY AFFECTED BY THE INTERACTION WITH THE ENVIRONMENT (MOST ROBUST STATES OF THE SYSTEM) Information initially ‘stored’ in the system flows into correlations with the environment Measure information loss by entropy growth (or purity decay) AN OPERATIONAL DEFINITION OF POINTER STATES: “PREDICTABILITY SIEVE” Initial state of the system (pure)State of system at time t (mixed) t

25. POINTER STATES, DECOHERENCE TIMESCALE (VI) Measure degradation of system’s state with entropy (von Neuman) or purity decay These quantities depend on time AND on the initial state PREDICTABILITY SIEVE: FIND THE INITIAL STATES SUCH THAT THESE QUANTITIES ARE MINIMIZED (FOR A DYNAMICAL RANGE OF TIMES) PREDICTABILITY SIEVE IN A PHYSICALLY INTERESTING CASE? ANALIZE QUANTUM BROWNIAN MOTION USE MASTER EQUATION TO ESTIMATE PURITY DECAY OR ENTROPY GROWTH

26. Minimize over initial state: Pointer states for QBM are minimally uncertainty coherent states! W.Zurek, J.P.P & S. Habib, PRL 70, 1187 (1993) A SIMPLE SOLUTION FROM THE PREDICTABILITY SIEVE CRITERIONApproximations I: Neglect friction, use asymptotic form of coefficients, assume initial state is pure and apply perturbation theory: POINTER STATES, DECOHERENCE TIMESCALE (VI) Approximations II: State remains approximately pure, average over oscillation period

27. Minimize over initial state: Pointer states for QBM are minimally uncertainty coherent states! W.Zurek, J.P.P & S. Habib, PRL 70, 1187 (1993) A SIMPLE SOLUTION FROM THE PREDICTABILITY SIEVE CRITERION Approximations I: Neglect friction, use asymptotic form of coefficients, assume initial state is pure and apply perturbation theory: POINTER STATES, DECOHERENCE TIMESCALE (VII) Approximations II: State remains approximately pure, average over oscillation period “momentum eigenstate” “position eigenstate”

28. 1) Dynamical regime (QBM): Pointer basis results from interplay between system and environment 3) “Slow environment” regime: The evolution of the environment is very “slow” (adiabatic environment): Pointer states are eigenstates of the Hamiltonian of the system! The environment only “learns” about properties of system which are non-vanishing when averaged in time. J.P. Paz & W.Zurek, PRL 82, 5181 (1999) 2) “Slow system” regime: (Quantum Measurement): System’s evolution is negligible, Pointer basis is determined by the interaction Hamiltonian (position in QBM) POINTER STATES, DECOHERENCE TIMESCALE (VIII) WARNING: DIFFERENT POINTER STATES IN DIFFERENT REGIMES! TAILOR MADE POINTER STATES? Environmental engeneering…

29. 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

30. 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).