ENTROPY AND DECOHERENCE IN QUANTUM THEORIES

Tomislav Prokopec, ITP & Spinoza Institute, Utrecht University

Nikhef, Mar 30 2012

Based onBased on: : Jurjen F. Koksma, Tomislav Prokopec and Michael G. Schmidt,Jurjen F. Koksma, Tomislav Prokopec and Michael G. Schmidt, Phys. Rev. D (2011Phys. Rev. D (2011)) [ [arXiv:1102.4713 [hep-th]];arXiv:1101.5323 [quant-ph]; Annals Phys. (2011), arXiv:1012.3701 [quant-ph];

Phys. Rev. D Phys. Rev. D 8181 (2010) 065030 (2010) 065030 [[arXiv:0910.5733 [hep-th]]]

Annals Phys. Annals Phys. 325325 (2010) 1277 (2010) 1277 [[arXiv:1002.0749 [hep-th]]]

Tomislav Prokopec and Jan Weenink, [Tomislav Prokopec and Jan Weenink, [arXiv:1108.3994[gr-qc]]+ in preparation]+ in preparation

˚ 1˚

PLAN˚ 2˚

ENTROPY as a physical quantity and decoherence

ENTROPY of (harmonic) oscillators

ENTROPY and DECOHERENCE in relativistic QFT’s

APPLICATIONS

DISCUSSION

● bosonic oscillator● fermionic oscillator

● CB● neutrino oscillations and decoherence

VON NEUMANN ENTROPY ˚ 3˚

von Neumann entropy (of a closed system):

operatordensity)(ˆ)],ˆln(ˆTr[)(vN ttS

]ˆ,ˆ[)(ˆ Htt

..OBEYS A HEISENBERG EQUATION:

as a result, von Neumann entropy is conserved:

Consequently, von Neumann entropy is conserved, hence USELESS.

constant.)(0)( vNvN tStSdt

d

However: vN entropy is constant if applied to closed systems, whereall dof’s and their correlations are known. In practice: never the case!

CLOSED SYSTEM

const.)(vN tS

OPEN SYSTEMS ˚ 4˚

◙ OPEN SYSTEMS (S) interact with an environment (E).If observer (O) does not perceive SE correlations (entanglement),(s)he will detect a changing (increasing?) vN entropy.

von Neumann entropy is not any more conserved

timein(?)increases)(

0)(

vN

vN

tS

tSdt

d

NB: entropy/decoherence is an observer dependent concept. Hence, arguably there is no unique way of defining it. Some argue: useless. In practice: has shown to be very useful.

OPEN SYSTEM

Proposal: vN entropy (of S) is a quantitative measure for decoherence.

SE

ENTROPY, DECOHERENCE, ENTANGLEMENT

˚ 5˚

system (S) + environment (E) + observer (O)

E interacts very weakly with O: unobservable

O sees a reduced density matrix:

Tracing over E is not unitary: destroys entanglement;responsible for decoherence & entropy generation

1]ˆTr[]ˆTr[ red2red

]ˆ[Trˆ ESEred )]ˆln(ˆTr[)( redredvN tS

DIVISION S-E can be in physical space: traditional entropy; black holes; CFTsSrednicki, 1992Srednicki, 1992

CORRELATOR APPROACH TO DECOHERENCE

˚ 6˚

BASED ON (UNITARY, PERTURBATIVE) EVOLUTION OF 2-pt FUNCTIONS (in field theory or quantum mechanics)

Koksma, Prokopec, Schmidt (‘09, ‘10), Giraud, Serreau (‘09)Koksma, Prokopec, Schmidt (‘09, ‘10), Giraud, Serreau (‘09)

ADVANTAGES:

evolution is in principle unitary: reduction of does not affect the evolution, i.e. it happens in the channel: O-S, and not S-E

NEW INSIGHT: decoherence/entropy increase is due to unobservable higher order correlations (non-gaussianities) in the S-E sector:realisation of COARSE GRAINING.

(almost) classical systems tend to behave stochastically, i.e. there is a stochastic force, kicking particles in unpredictable ways.Examples: Solar Planetary System; Large scalar structure of the Universe

DECOHERENCE AND CLASSICIZATION˚ 7˚

A theory that explains how quantum systems become (more) classicalZeh (1970), Joost, Zurek (1981) & othersZeh (1970), Joost, Zurek (1981) & others

Decoherence has gained in relevance: EPR paradox; quantum computational systems

Phase space picture:

p(t)

x(t)

EARLY TIME t LATE TIME t’>t

x(t’)

p(t’)

EVOLUTION: IRREVERSIBLE! – in discord with quantum mechanics!

HARMONIC OSCILLATORS

˚ 8˚

BOSONIC OSCILLATORS (bHO)˚ 9˚

● HAMILTONIAN & HAMILTON EQUATIONS

i

N

ii qqtHtHqm

m

ptH ˆ)(ˆ,)(ˆˆ

2

1

2

ˆ)(ˆ

1intint

222

Htqdt

dHtp

dt

dpqˆ)(ˆ,ˆ)(ˆ

● GAUSSIAN DENSITY OPERATOR

)1(,]ˆ,ˆ[,}ˆ,ˆ){(ˆ)(ˆ)(exp1

)(ˆ 2221

g pqpqtqtptZ

t

NB: knowing (t), (t), (t) is equivalent to solving the problem exactly!

● THE FOLLOWING TRANSFORMATION DIAGONALISES :

1]ˆ,ˆ[,ˆˆ12

)(ˆ,ˆˆ12

)(ˆ

bbpqtbpqtb

BOSONIC OSCILLATOR: GAUSSIAN ENTROPY

˚10˚

● DIAGONAL DENSITY OPERATOR

σeZbbNNtZ

t 1,ˆˆˆ,,)2/1ˆ)((exp1

)(ˆ 12g

● Can relate parameters in (, , ) to correlators:

2

2

1ˆ,ˆ,

2

1)ln(2ˆ,

2

1)ln(2 22

npqnZqnZp

● INTRODUCE A FOCK BASIS: nnInnn

0

,,..1,0,

● IN THIS BASIS:1

1ˆ)(,)1(

)(,)(ˆ1g

e

Ntnn

ntnnt

n

n

nn

n

2

coth21ˆ,ˆˆˆ4 222

21222

npqpq

● AN INVARIANT OF A GAUSSIAN DENSITY OPERATOR

GAUSSIAN ENTROPY˚11˚

● in terms of and

2

1ln

2

1

2

1ln

2

1))(ˆln()(ˆTr gg ttS

● is an invariant measure (statistical particle number) of the phase space volume of the state in units of ħ/2.

Ntn ˆ)(

nnnnS ln1ln)1(

2

1ˆ)(

Ntn

p

q

ENTROPY GROWTH IS THUS PARAMETRIZED BY THE GROWTH OF THE PHASE SPACE AREA (in units of ħ) (t)

1)1/()( nnn nnnnt● is the

probability that there are n particles in the state.

ENTROPY FOR 1+1 bHO˚12˚

(ENTROPY)

NB: grey: UNPHYSICAL SECULAR GROWTH AT LATE TIMES

►UNITARY EVOLUTION (black); REDUCED EVOLUTION (gray)

► LEFT: nonresonant regime; RIGHT: resonant regime

TIME

NB: relatively small Poincaré recurrence time.

TIME

NB: If initial conditions are Gaussian, the evolution is linear and will preserve Gaussianity. Scorr will be generated by <xq>0 correlators.

ES

SS

E-SS

SS

SS

(PERT. MASTER EQ)

ENTROPY FOR 50+1 bHO˚13˚

(ENTROPY)

NB: gray: UNPHYSICAL SECULAR GROWTH AT LATE TIMES (PERT. MASTER EQ)

►UNITARY EVOLUTION (black); REDUCED EVOLUTION (gray)

► LEFT: nonresonant regime; RIGHT: resonant regime

TIME

NB: exponentially large Poincaré recurrence time.

TIME

SS(PERT. MASTER EQ)

SS

SS

FERMIONIC OSCILLATORS (fHO)˚14˚

● LAGRANGIAN & EQUATIONS OF MOTION FOR fHOs

i

N

iitt ttjjjttL ˆ)()(ˆ,ˆˆˆˆˆ)(

2

1

2

1ˆ)(ˆ

1

E

● DENSITY OPERATOR FOR fHO

aeZNNNNtaZ

t 1,ˆˆ,ˆˆˆ,1ˆ,ˆ,ˆ)(exp1

)(ˆ 2

..or:

jtjt ttˆˆ)(,ˆˆ)(

)(1

1,

1

1,ˆ)12()1()(ˆ th

a

en

enNnnt

a

● DENSITY OPERATOR IN THE FOCK SPACE REPRESENTATION

1,0,110)1(0)(ˆ :spaceFocknnt

Tomislav Prokopec and Jan Weenink, in preparationTomislav Prokopec and Jan Weenink, in preparation

ENTROPY OF FERMIONIC OSCILLATOR˚15˚

● ENTROPY OF fHO

)'(ˆ),(ˆ2

1)';(,

2tanh)(21);(2)( ttttF

atnttFt

● INVARIANT PHASE SPACE AREA:

: (statistical) number of particles)(tn

]1,1[,2

1ln

2

1

2

1ln

2

1)(

tS

]1,0[2

1),ln()1ln()1()(

nnnnntS

ALSO FOR FERMIONS: ENTROPY IS PARAMETRIZED BY THE PHASE SPACE INVARIANT (in units of ħ)(can be >0 or <0)

)(t

][

][

ENTROPY FOR 1+1 fHO˚16˚

ENTROPY

NB2: For 2 oscillators, small Poincare recurrence time: quick return to initial state.

► LEFT PANEL: WEAK COUPLING RIGHT: STRONG COUPLING

TIME

NB1: MAX ENTROPY ln(2) approached, but never reached.

TIME

ENTROPY

ENTROPY FOR 50+1 fHO˚17˚

ENTROPY

► LEFT: LOW TEMPERATURE RIGHT: HIGH TEMPERATURE

TIME

NB: exponentially large Poincaré recurrence time. When i<<1, Smax=ln(2) reached

TIME

random frequencies i[0,5]0

evenly distributed frequencies i[0,5]0

ENTROPY AND DECOHERENCE IN FIELD THEORIES

˚18˚

TWO INTERACTING SCALARS˚19˚

ACTION:

Can solve pertubatively for the evolution of (S) and (E)

O only sensitive to (near) coincident Gaussian (2pt) correlators. Cubic interaction generates non-Gaussian S-E correlations: Sng,corr, e.g. 3pt fn:

intSSSS

2222

2

1

2

1,

2

1

2

1

mxdSmxdS DD

,!32

1 32int

hxdS D

tot S corr S g,S ng,S corr g,corr ng,corr corrES S S S 0; S S S 0, S S S I 0

)()'()()'()()(~)''()'()( yxyxyxydhxxx D

NB: Expressible in terms of (non-coincident!) Gaussian S-E (2pt) correlators

EVOLUTION EQUATIONS˚20˚

baxxixyciyxyMdxxim Dab

c

cbacDab ,),'()';();()';( 322

In the in-in formalism: the keldysh propagator i is a 2x2 matrix:

ii

iii

► are the time ordered (Feynman) and anti-time ordered propagators ii ,

► are the Wightman functions ii ,

► is the self-energy (self-mass). At one loop:abM

2

2

);(2

);( yxiih

yxiM abab

Solve the above KB Eq.: spatially homogeneous limit; m=0

PROBLEM: scattering in presence of thermal bath

Kadanoff, Baym (1961); Hu (1987)

);( yxi ab► are the thermal correlators.

QUANTUM FIELD THEORY: 2 SCALARS˚20˚

1 LOOP SCHWINGER-DYSON EQUATION FOR & :

= +

= + +

NB: INITIALLY we put in a pure state at T=0 (vacuum) & in a thermal state at temp. T

0)',,(),,()',,(),,()',,( ccth,

'

cc

'

22222 tktkMdtktkZdkttkmkt

t

t

t

tc

t

1 LOOP KADANOFF-BAYM EQUATIONS (in Schwinger-Keldysh formalism):

)',,(),,()',,(),,()',,( cFc22222 tktkZtkFtkZdkttkFmkt

tt

0)',,(),,()',,(),,( cFth,

cth,

tktkMtkFtkMdt

► are the renormalised `wave function’ and self-massesFc,Fc, , MZ

iiiiiF2

1,

2

1 c STATISTICAL & CAUSAL CORRELATORS:

RESULTS FOR SCALARS

˚21˚

STATISTICAL CORRELATOR AT T>0˚23˚

► t-t’: DECOHERENCE DIRECTION

HIGH TEMPERATURE LOW TEMPERATURE

)';,( ttkF

˚24˚PHASE SPACE AREA AND

ENTROPY AT T>0

TIMETIME

HIGH TEMPERATURELOW TEMPERATURE

► Entropy reaches a value Sms we can (analytically) calculate.

DECOHERENCE RATE @ T>0˚25˚

2/)(exp1

2/)(exp1log

1632

1 22

χχφdec k

k

k

hh

► decoherence rate can be well approximated by perturbative one-particle decay rate:

0)( msdec

MIXING FERMIONS˚26˚

EQUATION OF MOTION (homogeneous space):

N

iioit tkmtkjtkjtkmk

1

0 ),(ˆ),(ˆ),,(ˆ),(ˆ)(

Helicity h is conserved: work with 2 spinors . Diagonalise:

223130 ||),,(ˆ),(ˆ)( mktkjmhk

tk hht

),(ˆ tkh

22

*12

1211

''''',

,),'(ˆ),,(ˆ,),(ˆ),(),(ˆexp1

)(ˆ

hkkhhhhhhk

hh tktktktktkZ

t

ENTROPY

..can be diagonalised

0

0,),(ˆ),(),(ˆexp

1)(ˆ

,

dh

dh

hk

dhh

d tktktkZ

t

a (diagonal) Fock representation:

),(ˆ),(ˆˆ,ˆ)12()1()(ˆ,)(ˆ)(ˆ tktkNNnnttt hhhkhkhkhkhkhk

hk

ENTROPY OF FERMIONIC FIELDS˚27˚

● FERMIONIC ENTROPY:

]1,0[2

1),ln()1ln()1()(,)()(

hkhkhkhkhkhkhk

hkhk

nnnnntStStS

FOR FERMIONIC FIELDS: ENTROPY PER DOF ALSO PARAMETRIZED BY THE PHASE SPACE INVARIANT

)(thk

][

hk

][

hk

)';(ˆ),;(ˆ2

1)';;(,

2tanh)(21);;(2)( tktkttkFtnttkFt hhh

hkhkhhk

]1,1[,2

1ln

2

1

2

1ln

2

1)(

hkhkhkhkhk

hktS

hk

RESULTS FOR FERMIONS

˚28˚

ENTROPY OF TWO MIXING FERMIONS˚29˚

● TOTAL ENTROPY OF THE SYSTEM FIELD

► LEFT PANEL: LOW TEMP. 0=1 RIGHT: HI TEMP: 0=1/2

HI TEMP: 0=1/10 ● TERMALISATION RATE

T

APPLICATIONS TO NEUTRINOS

˚30˚

NEUTRINOS ˚31˚

There are 3 active (Majorana) left-handed neutrino species, that mix and possibly violate CP symmetry.

Majorana condition implies that each neutrino has 2 dofs (helicities):

IN GAUSSIAN APPROXIMATION, ONE CAN DEFINE GENERAL INITIAL CONDITIONS FOR NEUTRINOS IN TERMS OF EQUAL TIME STATISTICAL CORRELATORS:

*2)(

Tc C

tt'(flavour),2,1,,)';(ˆ),;(ˆ2

1);;( '

jitktkttkF tthjhihij

Mark Pinckers, Tomislav Prokopec, in preparation

NEUTRINO OSCILLATIONS ˚32˚

IF INITIALLY PRODUCED IN A DEFINITE FLAVOUR, NEUTRINOS DO OSCILLATE:

eV1032.2,eV106.7 3213

223

5212

mmm

861.0)2(sin,97.0)2(sin,10.0)2(sin 122

232

132

BLUE = MUON ; RED = TAU ; BLACK=ELECTRON

INITIAL ELECTRON

INITIAL MUON

OSCILLATIONOS ARE A MANIFESTATION OF QUANTUM COHERENCE, BUT ARE NOT GENERIC!

ExmP 4/sin)2(sin 2212

212

˚34˚

),(2

1ˆˆ),(,0ˆˆ&)2,1(,1 tkFtkninn ihihihihiiii

EXAMPLE A:

other (mixed) correllators vanish.

Q: can one construct such a state in laboratory?

NB: albeit neutrinos coming e.g. from the Sun are coherent and do oscillate, when averaged over the source localtion, oscillations tend to cancel,and one observes neutrino deficit, but no oscillations.

NEUTRINOS NEED NOT OSCILLATE WE FOUND GENERAL CONDITIONS ON F’s UNDER WHICH NEUTRINOS DO NOT OSCILLATE.

EXAMPLES (WHEN MAJORANA NEUTRINOS DO NOT OSCILLATE):

COSMIC NEUTRINO BACKGROUND˚34˚

)2,1(,1ˆ,ˆ2

1ˆ,ˆ

2

1

inn

khFF

kh iiiiiiii

EXAMPLE B: thermal cosmic neutrino background (CB):

NB1: CB neutrinos do not oscillates (by assumption)

NB2: CB violates both lepton number and helicity and CB contains a calculable lepton neutrino condensate.

NB3: A similar story holds for supernova neutrinos (they are believed to be approximately thermalised).

NB4: Can construct a diagonal thermal density matrix for CB (that is neither diagonal in helicity nor in lepton number)

In flavour diagonal basis:

K73.2,K95.1)11/4( 3/1 TTTCurrent temperature:

APPLICATIONS: Need to understand better how neutrinos affect CB

CONCLUSIONS˚35˚

Correlator approach to decoherence is based on perturbative evolution of 2 point functions & neglecting observationally inaccessible (non-Gaussian) correlators.

DECOHERENCE: the physical process by which quantum systems become (more) classical, i.e. they become classical stochastic systems.

Our methods permit us to study decoherence/classicization in realistic (quantum field theoretic) settings.There is no classical domain in the usual sense: phase space area – and therefore the `size’ of the system – never decreases in time.Particular realisations of a stochastic system (recall: large scale structure of our Universe) behave (very) classically.

Von Neumann entropy (of a suitable reduced sub-system) is a good quantitative measure of decoherence, and can be applied to both bosonic and fermionic systems.

APPLICATIONS˚35b

˚

Quantum information

Classicality of scalar & tensor cosmological perturbations (observable in CMB?)

Thermal cosmic neutrino background: - relation to lepton number and baryogenesis via leptogenesis

Lab experiments on neutrinos; neutrinos from supernovae

Baryogenesis: CP violation (requires coherence)

INTUITIVE PICTURE: WIGNER FUNCTION

˚36˚

GAUSSIAN STATE (momentum space: per mode):

WIGNER FUNCTION:

22t tt'2 t' t t' t

4F(t,t) F(t,t ' ) F(t,t ' )

g

1 1 1 1S Log Log

2 2 2 2

ENTROPY ~ effective phase space area of the state

2/)'(],,',[e)'(],,W[ )'( xxxtxxxxtpx xxip D

p

x

WIGNER FUNCTION: SQUEEZED STATES

˚37˚

PURE STATE (=1,Sg=0) MIXED STATE (>1,Sg>0)

EVOLUTI ON

NB: ORIGIN OF ENTROPY GROWTH: neglected S-E (nongaussian) correlators

STATISTICAL ENTROPY: g

1S (n 1)Log n 1 nLog n , n : uncorr.regions

2

GENERALISED UNCERTAINTY RELATION:

2

2 2 22

4 1x p x, p 1 n 0, S 0

2

WIGNER FUNCTION AS PROBABILITY˚38˚

WIGNER ENTROPY (Wigner function = quasi-probability)

GAUSSIAN ENTROPY:

2

1),(

2

11)ln()ln()1ln()1( 2

g

nnO

nnnnnnS

1)ln(W nS

THE AMOUNT OF QUANTUMNESS IN THE STATE: the difference of the two entropies:

1),(6

1

2

1 32g nnOnn

SSS W

WIGNER FUNCTION OF NONGAUSSIAN STATE

˚39˚

POSITIVE KURTOSIS : NEGATIVE KURTOSIS :

2

2 2 22

4 1x p x, p 1 n 0, S 0

2

Q: can non-Gaussianity – e.g. a negative curtosis – break the Heiselberg uncertainty relation?

Naïve Answer: YES(!?); but it is probably wrong.

CLASSICAL STOCHASTIC SYSTEMS

˚40˚

DISTRIBUTION OF GALAXIES IN OUR UNIVERSE (2dF): ● amplified vacuum fluctuations ● we observe one realisation (breaks homogeneity of the vacuum)

BROWNIAN PARTICLE (3 dim)● exhibits walk of a drunken man/woman● distance traversed: d ~ t

NB: first order phase transitions also spont.break spatial homogeneity of a state.

NB2: planetary systems are stochastic, and essentially unstable.

RESULTS: CHANGING MASS

˚41˚

CHANGING MASS CASE˚42˚

► RELEVANCE: ELECTROWEAK SCALE BARYOGENESIS: axial vector current is generated by CP violating scatterings of fermions off bubble walls in presence of a plasma. ► Since the effect vanishes when ħ0, quantum coherence is important.

► ANALOGOUS EFFECT: double slit with electrons in presence of air

BUBBLE WALL: m²(t)

TIME: t

PROBLEMS:

► non-equilibrium dynamics in a plasma at T>0;

► non-adiabatically changing mass term;

► apply to Yukawa coupled fermions.

CHANGING MASS: STATISTICAL PROPAGATOR

˚43˚

► NOTE: ADDITIONAL OSCILLATORY STRUCTURE

DELTA: FREE CASE, CHANGING MASS

˚44˚

► CONSTANT GAUSSIAN ENTROPY

TIME

► the state gets squeezed, but the phase space area is conserved

EXACT SOLUTION: in terms of hypergeometric functions

Pure + frequency mode at t- becomes a mixture of + & - frequencysolutions at t+ Mixing amplitude: (t)

Particle production:

inout

outin

22

2

1,

/sinh/sinh

/sinh

kn

outin

2

in

in

,,2

12

n

MASS CHANGE AT T>0 ˚45˚

LOW T MASS INCREASE: T=/2, k=, h=4, m= 2

LOW T MASS DECREASE: T=/2, k=, h=4, m=2

timetime

NB: ENTROPY CHANGES AT THE ONE PARTICLE DECAY RATEdec

NB2: MASS CHANGES MUCH FASTER THAN ENTROPY: dec/ mm

MASS CHANGE AT T>0 ˚46˚

HIGH T MASS INCREASE: T=2, k=, h=3, m= 2

HIGH T MASS DECREASE: T=2, k=, h=3, m=2

time

EVOLUTION OF SQUEEZED STATES ˚47˚

HIGH T: 2r=ln(5), =/2 T=2m, h=3m, k=m

LOW T: 2r=ln(5), =0T=2m, h=3m, k=m

timetime

NB: ADDITIONAL OSCILLATIONS DECAY AT THE RATE = dec.

► of relevance for baryogenesis: changing mass induces squeezing (coherent effect)

QUANTUM COHERENCE IS NOT DESTROYED BY THERMAL EFFECTS.

CONJECTURE: THIN WALL BG UNAFFECTED BY THERMAL EFFECTS.

► ► related work: Herranen, Kainulainen, Rahkila (2007-10) related work: Herranen, Kainulainen, Rahkila (2007-10)

KADANOFF-BAYM EQUATIONS˚48˚

0)',,(),,()',,(),,()',,( ccth,

'

cc

'

22222 tktkMdtktkZdkttkmkt

t

t

t

tc

t

IMPORTANT STEPS: calculate 1 loop self-masses renormalise using dim reg solve for the causal and statistical correlators (must be done numerically, since it involves memory effects)

calculate the (gaussian) entropy of (S)

)',,(2

1)',,(),',,()',,( ttkiittkFttkiittki c

► here: m² is the renormalised mass term (the only renormalisation needed at 1loop)

Fc,Fc, , MZ

KB equations can be written in a manifestly causal and real form:Berges, Cox (1998); Koksma, TP, Schmidt (2009)

22t tt'2 t' t t' t

4F(t,t) F(t,t ' ) F(t,t ' )

g

1 1 1 1S Log Log

2 2 2 2

)',,(),,()',,(),,()',,( cFc22222 tktkZtkFtkZdkttkFmkt

tt

0)',,(),,()',,(),,( cFth,

cth,

tktkMtkFtkMdt

► are the renormalised `wave function’ and self-masses

SELF-MASSES˚49˚

)'()4)(3(16

12

)';(2/

42

cct, xxi

DD

Dih

xxiM DD

D

► there are also thermal contributions to the self-masses (which are complicated)

LOCAL VACUUM MASS COUNTERTERM

RENORMALISED VACUUM SELF-MASSES

)',,()()',,( 22 ttkiZkttkiM abt

ab

|)|2si(|)|2ci(2||2

log64

||2

||2

2

tkitkeit

ke

hiZ tik

Etik

|)|2si()sign(|)|2ci()sign(2||2

log64 22

2

tktitketit

ke

k

hiZ tik

Etik

► CURIOUSLY: we could not find these expressions in literature or textbooks

► there is also the subtlety with KB eqs: in practice t0=- should be made finite. But then there is a boundary divergence at t=t0, which can be cured by (a) adiabatically turning on coupling h, or (b) by modifying the initial state.

˚50˚

h=4m, k= m

PHASE SPACE AREA AND ENTROPY AT T=0

ENTROPY

TIMETIME

TIME

► evolution towards the new (interacting) vacuum with stationary ms (calculated)

ms

ms

► initial conditions `forgotten’

► ms reached at perturbative rate=decoherence/entropy growth rate:

2

treepert,dec 32

1 h

► wiggles (in part) due to imperfect memory kernel

˚51˚ENTROPY AT T>0

NB: COUPLING h IS PERTURBATIVE UP TO h~3 (k²+m² )

● ms as a function of coupling h, T=2m, k=m

LOW TEMPERATURE vs VACUUM CASE:T= m /10 (black) & T=0 (gray), h=4m, k=m

● ENTROPY

TWO POINT FUNCTION˚52˚

QUANTUM COMPUTATION˚53˚

CLASSICAL LOGICAL GATES

Feynman; Shore (factoring into primes)Feynman; Shore (factoring into primes)

E.g. NAND GATE 2 STATE SYSTEM WAVE FUNCTION:

0111

1010

1101

1000

QUANTUM LOGICAL GATES

NOT GATE

10

01

1,1022

quantum NOT GATE

01

10NOT 1

* general q-gate: any `rotation’ on the Bloch sphere; e.g. Pauli matrices: rotation around x, y and z axes)

Bloch sphere: {{,} | ||²+||²=1}

{1,0}

{0,1}

MAIN PROBLEM of quantum computation: how to reduce decoherence of q-gates

A MEASURE OF DECOHERNECE: GAUSSIAN VON NEUMANN ENTROPY

˚54˚

STATISTICAL (HADAMARD) 2-pt GREEN FUNCTION:

CAUSAL (SPECTRAL) FUNCTION (PAULI-JORDAN, SCHWINGER) 2-pt GREEN FUNCTION:

one solves the perturbative dynamical equations for of S+E c &F

PROGRAM:

one calculates the Gaussian von Neumann entropy Sg of S:

g

1 1 1 1S Log Log

2 2 2 2

Gaussian density matrix:

)])x(t'x(t),[Tr();'( tti c

)}x(t'x(t),{

2

1Tr);'( ttF

2

'''22 )';F()';F();F(

2)( ttttttt ttttttt

')(2')()(exp);',( 22

gauss xxtcxtbxtaNtxx

INTERMEDIATE SUMMARY˚55˚

CONVENTIONAL APPROACH:

S+E E weakly coupledEvolve Ered Tr S red redS Tr log 0

NEW FRAMEWORK:

E weakly coupledS+E Evolve 2pt correlators for S & E: perturbatively c, F

g,S S SS Tr log 0

tot S corr S g,S ng,S corr g,corr ng,corr corrES S S S 0; S S S 0, S S S I 0

BROWNIAN PARTICLE ˚56˚

])(/[,1/log)2/1( 2200 TkmtttS B LATE TIME ENTROPY: grows without limit

)'(2)'()(),()(' ttTktFtFtFxVvvm B

DYNAMICS: LANGEVIN EQUATION

► Describes motion of a Brownian particle (Einstein); of a drunken man/woman; also: inflaton fluctuations during inflation (Starobinsky; Woodard; Tsamis; TP)

► v=dx/dt; F(t)=Markovian (noise), V(x)= potential, = friction coefficient

Q: How can we understand this unlimited growth of phase space area?

WHEN V(x)=0: t

BROWNIAN PARTICLE 2˚57˚

Consider a free moving quantum particle (described by a wave packet)

But x keeps growing!: explains the (unlimited) growth of phase space area.

Quantum evolution: preserves the minimum phase space area xp=ħ/2

p(t)

x(t)

EARLY TIME t LATE TIME t’>t

x(t’)

p(t’)

BROWNIAN PARTICLE gets thermal kicks: keeps p constant! 22

2 Tk

m

p B