DECOHERENCE AND QUANTUM INFORMATION

JUAN PABLO PAZDepartamento de Fisica, FCEyN

Universidad de Buenos Aires, Argentina

paz@df.uba.ar

Paraty

August 2007

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)

HOW DOES THE WIGNER FUNCTION OF A QUANTUM STATE LOOK LIKE?: SUPERPOSITION OF TWO GAUSSIAN STATES

€

˙ W = H0,W{ }MB

+ D∂ 2pp W +L

OSCILLATIONS IN WIGNER FUNCTION: THE SIGNAL OF

QUANTUM INTERFERENCE. HOW DOES DECOHERENCE AFFECTS THIS

STATE?

€

Distance L

Wavelength λ p =h

L

DECOHERENCE IN QUANTUM BROWNIAN MOTION (VI)

€

˙ W = H0,W{ }MB

+ D∂ 2pp W +L

€

Distance L

Wavelength λ p =h

L

€

˙ W = H0,W{ }MB

+ D∂ 2pp W +L

Wosc ≈ A(t)cos(kp p)⇒ A(t) ≈ exp(−Γt)

Γ = Dkp2

DECOHERENCE RATE: MUCH LARGER THAN

RELAXATION RATE

€

Γ=DL2 /h2, D = 2 mγ kBT, λ DB =h / 2m kBT

⇒ Γ =γ L /λ DB( )2

≈1040 γ, m =1gr, T = 300K, L =1cm

DECOHERENCE IN QUANTUM BROWNIAN MOTION (VII)

MASTER EQUATION CAN BE REWRITTEN FOR THE WIGNER FUNCTION: IT HAS THE FORM OF A FOKER-PLANCK EQUATION

€

˙ W = H0,W{ }MB

+ D∂ 2pp W +L

A DIGRESSION ON WIGNER FUNCTIONS

€

0

€

ρB

€

U€

H

€

X = Re TrB ρ BU( )( )

Y = −Im TrB ρ BU( )( )

Know : Use circuit to learn about (“spectrometer”)

Know : Use circuit to learn about (“tomographer”)

€

ρ

€

U

€

U

€

ρAPLICATION:

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

• REMEMBER THE DEFINITION OF WIGNER FUNCION

€

A(x, p) =1

πhD(x, p) R D+(x, p) =

dλ dζ

(2πh)2exp(−i(xζ − pλ ) /h) D(λ ,ζ )∫

€

W (x, p) =dy

2πhe ipy / h∫ < x −

y

2| ρ | x +

y

2> = Tr(ρ A(x, p) )

€

A(x, p) : phase space point operators

• PROPERTIES OF WIGNERS ARE CONSEQUENCES OF

Integral along lines give a projection operator:

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Tr(A(x, p) A( ′ x , ′ p ) =1

2πhδ(x − ′ x ) δ(p − ′ p )

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A(x, p) = A+(x, p)

€

A(x, p) form a complete orthonormal set

€

dx dp∫ A(x, p) = Ψ Ψ

€

ax + bp = c

€

(a ˆ X + b ˆ P ) Ψ = c Ψ

phase space displacement operator

R: reflection operator

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D(λ ,ζ ) = exp(−i(λ ˆ P −ζ ˆ X ) /h)

R x = −x

A DIGRESSION ON WIGNER FUNCTIONS

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

€

0

€

ρ

€

R€

H

€

Z = Prob(0) − Prob(1)

Z = Tr D+(x, p)ρD(x, p)R( ) = Tr ρD(x, p)RD+(x, p)( )

Z = πh W (x, p)

€

D(x, p)€

H

€

W (x, p) = Tr(ρ A(x, p) )

ρ = W (x, p)x,p

∑ A(x, p)

Wigner function is nothing but the matrix element of the density matrix in a special operator basis

€

ρ x, x '( ) = Tr(ρ x x ' )

• A SYSTEM WITH A FINITE DIMENSIONAL (N) HILBERT SPACE

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{ n , n =1,K ,N}

{ k , k =1,K ,N}

Position basis (arbitrary)

Momentum basis

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k =1

Nexp(i2π nk /N)

n=1

N

∑ n

• PHASE SPACE DISPLACEMENT OPERATORS ARE EASY TO CONSTRUCT

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U n = n +1, mod(N)

V k = k +1, mod(N)

Position shift

Momentum shift

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D(q, p) = U qV − p exp(iπ qp /N)

• PHASE SPACE POINT OPERATORS ARE NOT SO EASY TO CONSTRUCT

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A(q, p) =1

N 2n=1

N

∑ exp(i2π (np + mp) /N) D(n,m)n=1

N

∑Naive attempt

Does not have all the right properties (for all values of N)

A DIGRESSION ON WIGNER FUNCTIONS

0 1 2 3 4 5 6 7

0

1

3

4

2

5

6

7

q+p=0N=8

0 1 2 3 4 5 6 7

0

1

3

4

2

5

6

7

2q+2p=0

Lines may have moremore 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

• SOLUTION (M. Berry, 1980; U. Leonhardt,…)

• DISCRETE WIGNER FUNCTION

€

A(q, p) =1

2N n=1

2N

∑ exp(i2π (np + mp) /2N) D(n,m)n=1

2N

∑

= U qRV − p exp(iπ qp /N)

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!

€

Nq,p=1

2N

∑ W1(q, p) W2(q, p) = Tr(ρ1 ρ 2)

€

W (q, p)q,p=1

2N

∑ = ϕ ρ ϕ , D(b,a) ϕ = exp(i2πc /N) ϕ

€

ax − bp = c

€

W (q, p) =1

2NTr(ρ A(q, p)), 0 ≤ q, p ≤ 2N −1

A DIGRESSION ON WIGNER FUNCTIONS

• Discrete Wigner function: How does it look like?

Position eigenstate Gaussian wavepacket

positive oscillatory oscillatory

osci

llato

ry

OSCILLATIONS DUE TO PERIODIC BOUNDARY CONDITIONS

A DIGRESSION ON WIGNER FUNCTIONS

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

€

0

€

ρ

€

A(x, p)€

H

€

X = Tr ρA(x, p)( )

X = N W (x, p)

EFFICIENT QUANTUM CIRCUIT TO IMPLEMENT ‘CONTROLLED-A(x,p) EXISTS

A DIGRESSION ON WIGNER FUNCTIONS

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

€

˙ W = H0,W{ }MB

+ D∂ 2pp W +L

€

Distance L

Wavelength λ p =h

L

€

˙ W = H0,W{ }MB

+ D∂ 2pp W +L

Wosc ≈ A(t)cos(kp p)⇒ A(t) ≈ exp(−Γt)

Γ = Dkp2

DECOHERENCE RATE: MUCH LARGER THAN

RELAXATION RATE

€

Γ=DL2 /h2, D = 2 mγ kBT, λ DB =h / 2m kBT

⇒ Γ =γ L /λ DB( )2

≈1040 γ, m =1gr, T = 300K, L =1cm

DECOHERENCE IN QUANTUM BROWNIAN MOTION (VII)

MASTER EQUATION CAN BE REWRITTEN FOR THE WIGNER FUNCTION: IT HAS THE FORM OF A FOKER-PLANCK EQUATION

€

˙ W = H0,W{ }MB

+ D∂ 2pp W +L

EVOLUTION OF WIGNER FUNCTION

€

ν

NOTICE: NOT ALL STATES ARE AFFECTED BY THE ENVIRONMENT IN THE SAME WAY (SOME

SUPERPOSITIONS LAST LONGER THAN OTHERS)

POINTER STATES, DECOHERENCE TIMESCALE (I)

WARNING: DECOHERENCE TIMESCALE OBTAINED IN THE HIGH TEMPERATURE LIMIT FOR INITIAL “CAT STATE” IS ONLY AN APPROXIMATION!

POINTER STATES, DECOHERENCE TIMESCALE (II)

€

˙ W = H0,W{ }MB

+ D∂ 2pp W +L

Wosc ≈ A(t)cos(kp p)⇒ A(t) ≈ exp(−Γt)

Γ = Dkp2

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)

POINTER STATES, DECOHERENCE TIMESCALE (II’)

Measure degradation of system’s state with entropy (von Neuman) or purity decay

€

SVN (t) = −Tr ρ(t)ln ρ(t)( )( ), ζ (t) = Tr ρ 2(t)( )

These quantities depend on time AND on the initial state

USE MASTER EQUATION TO ESTIMATE PURITY DECAY OR ENTROPY GROWTH

€

˙ ρ =− i HR +m

2δω2(t)x 2,ρ

⎡ ⎣ ⎢

⎤ ⎦ ⎥− iγ(t) x, p,ρ{ }[ ]− D(t) x, x,ρ[ ][ ]− f (t) x, p,ρ[ ][ ]

€

˙ ζ = 2Tr ˙ ρ ρ( ) = 2γζ − 2DTr x,ρ[ ]2

( ) + 2 fTr x,ρ[ ] p,ρ[ ]( )

System & Environment become entangled

(large environment: irreversible process)

€

Ψ(0) Ψ(0)

€

ρ(t)

PURE STATE MIXED STATE

WARNING: DECOHERENCE TIMESCALE OBTAINED IN THE HIGH TEMPERATURE LIMIT FOR INITIAL “CAT STATE” IS ONLY AN APPROXIMATION!

POINTER STATES, DECOHERENCE TIMESCALE (II)

€

˙ W = H0,W{ }MB

+ D∂ 2pp W +L

Wosc ≈ A(t)cos(kp p)⇒ A(t) ≈ exp(−Γt)

Γ = Dkp2

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)

CLASSICAL QUANTUM

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)

€

λX ≈h /P, λ P ≈h /L ⇒ A ≈λ X λ P ≈hh

LP<< h

NONTRIVIAL TIMESCALE: HARMONICALLY DRIVEN DOUBLE WELL (CHAOS)

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V (x) = −a x 2 + b x 4 + cx cos(ω t)

POINTER STATES, DECOHERENCE TIMESCALE (III)

DECOHERENCE DESTROYS THE FRINGES THAT ARE DYNAMICALLY PRODUCED

CLASSICAL + NOISE QUANTUM + 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)

INTUITIVE EXPLANATION: WHY IS THERE A REGIME OF ENTROPY GROWTH FIXED BY THE LYAPUNOV EXPONENT?

€

SVN = −Tr ρ log ρ( )( ), SLIN = −log Tr ρ 2( )( ), Tr ρ 2

( ) = 2πh dx dpW 2(x, p)∫

STRETCHING

+ FOLDING

€

t ≈1

λ

DECOHERENCE

€

t ≈ tDECO ≈1

Γ<<

1

λ

€

S = 0 (pure)

€

S = 0 (pure)

€

S ≈1 (mixed)

DIGRESSION: DECOHERENCE FOR CLASSICALLY CHAOTIC SYSTEMS

€

eλt

€

e−λ t€

eλt

€

σ c = 2D /λ

€

Area ≈ σ c eλt

S ≈ log(Area) ≈ λ t

STRETCHING DECOHERENCE

NUMERICAL EVIDENCE IS RATHER STRONG

(D. Monteoliva and J.P.P., PRL (2000), PRE (2002), (2006))

DIGRESSION: DECOHERENCE FOR CLASSICALLY CHAOTIC SYSTEMS

time time

Log(D)

Log(1/a)

€

MC ≈aexp(−λ t)

€

a ≈ 1/D

€

MC ≈bexp(−Γt)

€

Γ ≈Dkp2(t)

D. Monteoliva and J.P. P. (2006)

DIGRESSION: DECOHERENCE FOR CLASSICALLY CHAOTIC SYSTEMS

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

€

SVN (t) = −Tr ρ(t)ln ρ(t)( )( ), ζ (t) = Tr ρ 2(t)( )

Measure information loss by entropy growth (or purity decay)

AN OPERATIONAL DEFINITION OF POINTER STATES:

“PREDICTABILITY SIEVE”

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Ψ(0) Ψ(0)

€

ρ(t)

Initial state of the system (pure) State of system at time t (mixed)

t

POINTER STATES, DECOHERENCE TIMESCALE (VI)

Measure degradation of system’s state with entropy (von Neuman) or purity decay

€

SVN (t) = −Tr ρ(t)ln ρ(t)( )( ), ζ (t) = Tr ρ 2(t)( )€

Ψ(0) Ψ(0)

€

ρ(t)

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

€

˙ ρ =− i HR +m

2δω2(t)x 2,ρ

⎡ ⎣ ⎢

⎤ ⎦ ⎥− iγ(t) x, p,ρ{ }[ ]− D(t) x, x,ρ[ ][ ]− f (t) x, p,ρ[ ][ ]

€

˙ ζ = 2Tr ˙ ρ ρ( ) = 2γζ − 2DTr x,ρ[ ]2

( ) + 2 fTr x,ρ[ ] p,ρ[ ]( )

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

€

˙ ζ = 2Tr ˙ ρ ρ( ) = 2γζ − 2DTr x,ρ[ ]2

( ) + 2 fTr x,ρ[ ] p,ρ[ ]( )

Approximations I: Neglect friction, use asymptotic form of coefficients, assume initial state is pure and apply perturbation theory:

€

ζ (T) −ζ (0) = 2D dt0

T

∫ Tr x(t),ρ[ ]2

( )

POINTER STATES, DECOHERENCE TIMESCALE (VI)

€

ζ (T) = ζ (0) − 2D Δx 2 +1

m2ω2Δp2 ⎛

⎝ ⎜

⎞

⎠ ⎟

Approximations II: State remains approximately pure, average over oscillation period

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

€

˙ ζ = 2Tr ˙ ρ ρ( ) = 2γζ − 2DTr x,ρ[ ]2

( ) + 2 fTr x,ρ[ ] p,ρ[ ]( )

Approximations I: Neglect friction, use asymptotic form of coefficients, assume initial state is pure and apply perturbation theory:

€

ζ (T) −ζ (0) = 2D dt0

T

∫ Tr x(t),ρ[ ]2

( )

POINTER STATES, DECOHERENCE TIMESCALE (VII)

€

ζ (T) = ζ (0) − 2D Δx 2 +1

m2ω2Δp2 ⎛

⎝ ⎜

⎞

⎠ ⎟

Approximations II: State remains approximately pure, average over oscillation period

€

ζ (t)

“momentum eigenstate”

“position eigenstate”

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)

€

ζ (t)

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…

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

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).