Entropy and decoherence in quantum theories
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˚ 1˚. ENTROPY AND DECOHERENCE IN QUANTUM THEORIES. Based on : Jurjen F. Koksma, Tomislav Prokopec and Michael G. Schmidt, Phys. Rev. D (2011 ) [ arXiv:1102.4713 [hep-th]]; arXiv:1101.5323 [quant-ph]; Annals Phys. (2011), arXiv:1012.3701 [quant-ph];

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ENTROPY AND DECOHERENCE IN QUANTUM THEORIES

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˚ 1˚

ENTROPY AND DECOHERENCE IN QUANTUM THEORIES

Based on:

Jurjen F. Koksma, Tomislav Prokopec and Michael G. Schmidt,

Phys. Rev. D (2011) [arXiv:1102.4713 [hep-th]];

arXiv:1101.5323 [quant-ph];

Annals Phys. (2011), arXiv:1012.3701 [quant-ph];

Phys. Rev. D 81 (2010) 065030 [arXiv:0910.5733 [hep-th]]

Annals Phys. 325 (2010) 1277 [arXiv:1002.0749 [hep-th]]

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

Tomislav Prokopec, ITP & Spinoza Institute, Utrecht University

Nikhef, Mar 30 2012


˚ 2˚

PLAN

ENTROPY as a physical quantity and decoherence

ENTROPY of (harmonic) oscillators

● bosonic oscillator

● fermionic oscillator

ENTROPY and DECOHERENCE in relativistic QFT’s

APPLICATIONS

● CB

● neutrino oscillations and decoherence

DISCUSSION


˚ 3˚

VON NEUMANN ENTROPY

 von Neumann entropy (of a closed system):

..OBEYS A HEISENBERG EQUATION:

CLOSED SYSTEM

 as a result, von Neumann entropy is conserved:

Consequently, von Neumann entropy is conserved, hence USELESS.

However: vN entropy is constant if applied to closed systems, where

all dof’s and their correlations are known. In practice: never the case!


˚ 4˚

OPEN SYSTEMS

◙ 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.

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

OPEN SYSTEM

  • von Neumann entropy is

  • not any more conserved

S

E

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.


˚ 5˚

ENTROPY, DECOHERENCE, ENTANGLEMENT

  • 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

  • DIVISION S-E can be in physical space: traditional entropy; black holes; CFTs

Srednicki, 1992


˚ 6˚

CORRELATOR APPROACH TO DECOHERENCE

BASED ON (UNITARY, PERTURBATIVE) EVOLUTION

OF 2-pt FUNCTIONS (in field theory or quantum mechanics)

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

ADVANTAGES:

  • NEW INSIGHT: decoherence/entropy increase is due to unobservable

  • higher order correlations (non-gaussianities) in the S-E sector:

  • realisation of COARSE GRAINING.

  • 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

  • (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


˚ 7˚

DECOHERENCE AND CLASSICIZATION

A theory that explains how quantum systems become (more) classical

Zeh (1970), Joost, Zurek (1981) & others

Phase space picture:

EARLY TIME t

LATE TIME t’>t

p(t)

p(t’)

x(t)

x(t’)

EVOLUTION: IRREVERSIBLE! – in discord with quantum mechanics!

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


˚ 8˚

HARMONIC OSCILLATORS


˚ 9˚

BOSONIC OSCILLATORS (bHO)

● HAMILTONIAN & HAMILTON EQUATIONS

● GAUSSIAN DENSITY OPERATOR

  • NB: knowing (t), (t), (t) is equivalent to solving the problem exactly!

● THE FOLLOWING TRANSFORMATION DIAGONALISES :


˚10˚

BOSONIC OSCILLATOR: GAUSSIAN ENTROPY

● DIAGONAL DENSITY OPERATOR

● INTRODUCE A FOCK BASIS:

● IN THIS BASIS:

● Can relate parameters in  (, , ) to correlators:

●  AN INVARIANT OF A GAUSSIAN DENSITY OPERATOR


˚11˚

GAUSSIAN ENTROPY

● in terms of  and

● is an invariant measure (statistical particle

number) of the phase space volume of the state in units of ħ/2.

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

p

q

● is the probability

that there are n particles in the state.


˚12˚

ENTROPY FOR 1+1 bHO

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

► LEFT: nonresonant regime; RIGHT: resonant regime

(PERT. MASTER EQ)

(ENTROPY)

TIME

TIME

 NB: relatively small Poincaré recurrence time.

  • NB: grey: UNPHYSICAL SECULAR

  • GROWTH AT LATE TIMES

  • NB: If initial conditions are Gaussian, the evolution is linear and

    will preserve Gaussianity. Scorr will be generated by <xq>0 correlators.


˚13˚

ENTROPY FOR 50+1 bHO

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

► LEFT: nonresonant regime; RIGHT: resonant regime

(PERT. MASTER EQ)

(ENTROPY)

TIME

TIME

  • NB: gray: UNPHYSICAL SECULAR

  • GROWTH AT LATE TIMES

  • (PERT. MASTER EQ)

 NB: exponentially large Poincaré recurrence time.


˚14˚

FERMIONIC OSCILLATORS (fHO)

Tomislav Prokopec and Jan Weenink, in preparation

● LAGRANGIAN & EQUATIONS OF MOTION FOR fHOs

● DENSITY OPERATOR FOR fHO

..or:

● DENSITY OPERATOR IN THE FOCK SPACE REPRESENTATION


˚15˚

ENTROPY OF FERMIONIC OSCILLATOR

● INVARIANT PHASE SPACE AREA:

: (statistical) number of particles

● ENTROPY OF fHO

ALSO FOR FERMIONS: ENTROPY IS PARAMETRIZED BY THE PHASE SPACE INVARIANT  (in units of ħ)

(can be >0 or <0)


˚16˚

ENTROPY FOR 1+1 fHO

► LEFT PANEL: WEAK COUPLING RIGHT: STRONG COUPLING

ENTROPY

ENTROPY

TIME

TIME

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

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


˚17˚

ENTROPY FOR 50+1 fHO

► LEFT: LOW TEMPERATURE RIGHT: HIGH TEMPERATURE

random frequencies i[0,5]0

ENTROPY

TIME

TIME

evenly distributed frequencies i[0,5]0

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


˚18˚

ENTROPY AND DECOHERENCE IN FIELD THEORIES


˚19˚

TWO INTERACTING SCALARS

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:

NB: Expressible in terms of (non-coincident!)

Gaussian S-E (2pt) correlators


˚20˚

EVOLUTION EQUATIONS

Kadanoff, Baym (1961); Hu (1987)

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

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

► are the Wightman functions

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

► are the thermal correlators.

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

PROBLEM: scattering in presence of thermal bath


˚20˚

QUANTUM FIELD THEORY: 2 SCALARS

 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

 STATISTICAL & CAUSAL CORRELATORS:

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

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


˚21˚

RESULTS FOR SCALARS


˚23˚

STATISTICAL CORRELATOR AT T>0

LOW TEMPERATURE

HIGH TEMPERATURE

► t-t’: DECOHERENCE DIRECTION


˚24˚

PHASE SPACE AREA AND ENTROPY AT T>0

TIME

TIME

HIGH TEMPERATURE

LOW TEMPERATURE

► Entropy reaches a value Smswe can (analytically) calculate.


˚25˚

DECOHERENCE RATE @ T>0

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


˚26˚

MIXING FERMIONS

EQUATION OF MOTION (homogeneous space):

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

 ENTROPY

..can be diagonalised

 a (diagonal) Fock representation:


˚27˚

ENTROPY OF FERMIONIC FIELDS

● FERMIONIC ENTROPY:

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


˚28˚

RESULTS FOR FERMIONS


˚29˚

ENTROPY OF TWO MIXING FERMIONS

● TOTAL ENTROPY OF THE SYSTEM FIELD

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

HI TEMP:0=1/10

● TERMALISATION RATE


˚30˚

APPLICATIONS TO NEUTRINOS


˚31˚

NEUTRINOS

Mark Pinckers, Tomislav Prokopec, in preparation

 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:


˚32˚

NEUTRINO OSCILLATIONS

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

INITIAL MUON 

  • BLUE = MUON ;

  • RED = TAU ;

  • BLACK=ELECTRON 

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

INITIAL ELECTRON 


˚34˚

NEUTRINOS NEED NOT OSCILLATE

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

 EXAMPLES (WHEN MAJORANA NEUTRINOS DO NOT OSCILLATE):

 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.


˚34˚

COSMIC NEUTRINO BACKGROUND

 EXAMPLE B: thermal cosmic neutrino background (CB):

Current temperature:

In flavour diagonal basis:

NB1: CB neutrinos do not oscillates (by assumption)

NB2: CB violates both lepton number and helicity

and CB 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 CB

(that is neither diagonal in helicity nor in lepton number)

APPLICATIONS: Need to understand better how neutrinos affect CB


˚35˚

CONCLUSIONS

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

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.

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

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.


APPLICATIONS

˚35b˚

Classicality of scalar & tensor cosmological perturbations

(observable in CMB?)

Baryogenesis: CP violation (requires coherence)

Quantum information

Thermal cosmic neutrino background:

- relation to lepton number and baryogenesis via leptogenesis

Lab experiments on neutrinos; neutrinos from supernovae


˚36˚

INTUITIVE PICTURE: WIGNER FUNCTION

WIGNER FUNCTION:

GAUSSIAN STATE (momentum space: per mode):

p

ENTROPY ~ effective phase space area of the state


˚37˚

WIGNER FUNCTION: SQUEEZED STATES

 PURE STATE (=1,Sg=0)

 MIXED STATE (>1,Sg>0)

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

 STATISTICAL ENTROPY:

 GENERALISED UNCERTAINTY RELATION:


˚38˚

WIGNER FUNCTION AS PROBABILITY

 GAUSSIAN ENTROPY:

 WIGNER ENTROPY (Wigner function = quasi-probability)

  • THE AMOUNT OF QUANTUMNESS IN THE STATE:

    the difference of the two entropies:


˚39˚

WIGNER FUNCTION OF NONGAUSSIAN STATE

POSITIVE KURTOSIS :

NEGATIVE KURTOSIS :

Q: can non-Gaussianity – e.g. a negative curtosis –

break the Heiselberg uncertainty relation?

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


˚40˚

CLASSICAL STOCHASTIC SYSTEMS

BROWNIAN PARTICLE (3 dim)

● exhibits walk of a drunken man/woman

● distance traversed: d ~ t

DISTRIBUTION OF GALAXIES IN OUR UNIVERSE (2dF):

● amplified vacuum fluctuations

● we observe one realisation (breaks homogeneity of the vacuum)

NB: first order phase transitions also spont.

break spatial homogeneity of a state.

NB2: planetary systems are stochastic,

and essentially unstable.


˚41˚

RESULTS: CHANGING MASS


˚42˚

CHANGING MASS CASE

► 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 ħ0, quantum coherence is important.

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

PROBLEMS:

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

►non-adiabatically changing mass term;

BUBBLE WALL:

m²(t)

►apply to Yukawa coupled fermions.

TIME: t


˚43˚

CHANGING MASS: STATISTICAL PROPAGATOR

► NOTE: ADDITIONAL OSCILLATORY STRUCTURE


˚44˚

DELTA: FREE CASE, CHANGING MASS

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

 EXACT SOLUTION:

in terms of hypergeometric functions

► CONSTANT GAUSSIAN

ENTROPY

 Pure + frequency mode at t-

becomes a mixture of + & - frequency

solutions at t+  Mixing amplitude: (t)

TIME

 Particle production:


˚45˚

MASS CHANGE AT T>0

LOW T MASS INCREASE:

T=/2, k=,h=4, m=2

LOW T MASS DECREASE:

T=/2, k=,h=4,m=2

time

time

NB: ENTROPY CHANGES AT THE ONE PARTICLE DECAY RATEdec

NB2: MASS CHANGES MUCH FASTER THAN ENTROPY:


˚46˚

MASS CHANGE AT T>0

HIGH T MASS INCREASE:

T=2, k=, h=3, m=2

HIGH T MASS DECREASE:

T=2, k=, h=3,m=2

time


˚47˚

EVOLUTION OF SQUEEZED STATES

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

HIGH T: 2r=ln(5), =/2

T=2m, h=3m, k=m

LOW T: 2r=ln(5), =0

T=2m, h=3m, k=m

time

time

NB: ADDITIONAL OSCILLATIONS DECAY AT THE RATE = dec.

 QUANTUM COHERENCE IS NOT DESTROYED BY THERMAL EFFECTS.

CONJECTURE: THIN WALL BG UNAFFECTED BY THERMAL EFFECTS.

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


˚48˚

KADANOFF-BAYM EQUATIONS

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)

 KB equations can be written in a manifestly causal and real form:

Berges, Cox (1998); Koksma, TP, Schmidt (2009)

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

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


˚49˚

SELF-MASSES

LOCAL VACUUM MASS COUNTERTERM

RENORMALISED VACUUM SELF-MASSES

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

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

► 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˚

PHASE SPACE AREA AND ENTROPY AT T=0

 h=4m, k= m

ms

ENTROPY

TIME

TIME

► evolution towards the new (interacting)

vacuum with stationary ms (calculated)

ms

► initial conditions `forgotten’

► msreached at perturbative rate

=decoherence/entropy growth rate:

TIME

► wiggles (in part) due to imperfect memory kernel


ENTROPY AT T>0

˚51˚

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

● ENTROPY

LOW TEMPERATURE vs VACUUM CASE:

T= m /10 (black) & T=0 (gray), h=4m, k=m

NB: COUPLING h IS PERTURBATIVE UP TO h~3 (k²+m²)


˚52˚

TWO POINT FUNCTION


˚53˚

QUANTUM COMPUTATION

Feynman; Shore (factoring into primes)

QUANTUM LOGICAL GATES

CLASSICAL LOGICAL GATES

E.g. NAND GATE

2 STATE SYSTEM WAVE FUNCTION:

{1,0}

Bloch sphere: {{,} | ||²+||²=1}

{0,1}

NOT GATE

quantum NOT GATE

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

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


˚54˚

A MEASURE OF DECOHERNECE: GAUSSIAN VON NEUMANN ENTROPY

CAUSAL (SPECTRAL) FUNCTION (PAULI-JORDAN, SCHWINGER)

2-pt GREEN FUNCTION:

STATISTICAL (HADAMARD) 2-pt GREEN FUNCTION:

PROGRAM:

 one solves the perturbative dynamical equations for of S+E

 one calculates the Gaussian von Neumann entropy Sgof S:

 Gaussian density matrix:


˚55˚

INTERMEDIATE SUMMARY

CONVENTIONAL APPROACH:

E weakly coupled

S+E

Evolve

NEW FRAMEWORK:

Evolve 2pt correlators for

S & E: perturbatively

E weakly coupled

S+E


˚56˚

BROWNIAN PARTICLE

  • 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

 WHEN V(x)=0:

 LATE TIME ENTROPY: grows without limit

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


˚57˚

BROWNIAN PARTICLE 2

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

Quantum evolution: preserves the minimum phase space area xp=ħ/2

EARLY TIME t

LATE TIME t’>t

p(t’)

p(t)

x(t)

x(t’)

BROWNIAN PARTICLE gets thermal kicks: keeps p constant!

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


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