AISAMP Nov’08 1/25 The cost of information erasure in atomic and spin systems Joan Vaccaro Griffith University Brisbane, Australia Steve Barnett University.

11/25/25AISAMP Nov’08AISAMP Nov’08

The cost of information erasure in atomic and spin systems

Joan VaccaroGriffith University Brisbane, Australia

Steve BarnettUniversity of Strathclyde Glasgow, UK


Introduction Energy Cost Angular MtIntroduction Energy Cost Angular Mtmm Cost Impact Summary Cost Impact Summary

IntroductioIntroductionn▀ Landauer erasureLandauer, IBM J. Res. Develop. 5, 183 (1961)

00

1

forward process:

0 0

1 0

time reversed:

?

Erasure is irreversible

Minimum cost

00/1

Process: maximise entropy subject to conservation of energy

BEFORE erasure AFTER erasure

env2 smicrostate # total N

)2ln( )ln( env kTNkT

)2ln(kTQ

# microstates

environment

)ln( envNkTQ

heat

)2ln( envNkTQ



▀ Exorcism of Maxwell’s demon

▀ Information is Physical information must be carried by physical system (not new)

its erasure requires energy expenditure

1871 Maxwell’s demon extracts work of Q from thermal reservoir by collecting only hot gas particles. (Violates 2nd Law: reduces entropy of whole gas)

Q

▀ Thermodynamic Entropy

1982 Bennet showed full cycle requires erasure of demon’s memory which costs at least Q :

Bennett, Int. J. Theor. Phys. 21, 905 (1982)

Cost of erasure is commonly expressed as entropic cost:

This is regarded as the fundamental cost of erasing 1 bit. BUT this result is implicitly associated with an energy cost:

)2ln(kS

STQ

Qwork



ImpactImpact

This talk

Energy CostEnergy Cost▀ from conservation of energy▀ simple 2-state atomic model ▀ re-derive Landauer’s minimum cost of kT ln2 per bit

▀ energy degenerate states of different spin ▀ conservation of angular momentum ▀ cost in terms of angular momentum only

Angular Momentum CostAngular Momentum Cost

▀ New mechanism▀ 2nd Law Thermodynamics

zJ

2

0

1dEE



Z

eP

kTE

E

/

▀ System:

0 1 0/1

Memory bit: 2 degenerate atomic states

Thermal reservoir: multi-level atomic gas at temperature T

E

Energy CostEnergy Cost

heat engine:

cold

work hot

heat pump:

work hot

cold 0/1

▀ recall heat pump

erasure



T

Z

eP

kTE

E

/

0 1

▀ Thermalise memory bit while increasing energy gap

0/1

2

11 P

2

10 P



T

Z

eP

kTE

E

/


raise energy of state(e.g. Stark or Zeeman shift) 0

1dE

0/1

1

dEPdW 1

kTE

kTE

e

eP

/

/

11

kTEeP

/01

1

Work to raise state from E to E+dE



T

Z

eP

kTE

E

/


0/1

dEPdW 1

01 P


2log10

/

/

01 kTdE

e

edEPW

EkTE

kTE

E

Total work

1

0

10 P

raise energy of state(e.g. Stark or Zeeman shift)

1



T

Z

eP

kTE

E

/


0/1

dEPdW 1

01 P


2log10

/

/

01 kTdE

e

edEPW

EkTE

kTE

E

Total work

1

0

10 P

raise energy of state(e.g. Stark or Zeeman shift)

Thermalisation of memory bit:

Bring the system to thermal equilibrium at each step in energy:i.e. maximise the entropy of the system subject to conservation of energy.

THUS erasure costs energy because the conservation law for energy is used to perform the erasure



• an irreversible process

• based on random interactions to bring the system to maximum entropy subject to a conservation law

• the conservation law restricts the entropy

• the entropy “flows” from the memory bit to the reservoir

▀ Principle of Erasure:

01

0/1

E

T

0

1dE

0/1

E

T

work



▀ System:● spin ½ particles● no B or E fields so spins states are energy degenerate● collisions between particles cause spin exchanges

0/1Memory bit: single spin ½ particle

Reservoir: collection of N spin ½ particles.

Possible states

,,

,,

Simple representation: ,n

# of spin up

multiplicity (copy): 1,2,…

n particles are spin up

Angular Momentum Angular Momentum CostCost

nN

21

21



0/1

zJ

▀ Angular momentum diagram

states

Memory bit:

Reservoir:,n

1,0,3,1,2,1,1,1

# of spin up

multiplicity (copy)

zJ

nN1,2,…

21

21state

number of states with

NnJ z 21



▀ Reservoir as “canonical” ensemble (exchanging not energy)

Maximise entropy of reservoir

subject to

,

,, lnn

nn PP

NN

nPJn

nz 21

,,reservoir 2

1

,,

nnP&

Total is conserved

zJ

zJ

1,0,1

zJ

1,0 ,1

,n

Reservoir:Bigger spin bath:

,nP



zJ

1,0 ,1

,n

1,0,1

zJ

1,0,1

▀ Reservoir as “canonical” ensemble (exchanging not energy)

Reservoir:Bigger spin bath:

Maximise entropy of reservoir

subject to

,

,, lnn

nn PP

1,

,

n

nP& NN

nPJn

nz 21

,,reservoir 2

21 zJ

10 1

Average spinZ

Z

J

Je2

2

Nn

ne

eP

1,

1

ln1



0/1

▀ Erasure protocolReservoir:

1

ln1

zJ 2

1P

Nn

ne

eP

1,

2

1P

Memory spin:

zJ

1,0,1



zJ

1,0,1

0/1


Nn

ne

eP

1,

1

ln1

zJ

Coupling

1,11,0

e

eP

1

eP

1

1

Memory spin:




Nn

ne

eP

1,

1

ln1

0/1

e

eP

1

e

P1

1

Increase Jz using ancilla in

memory(control)

ancilla (target)

zJ

2

this operation costs

Memory spin:

and CNOT operation

,2

zJ

1,02



,2

zJ

1,02

0/1


Nn

ne

eP

1,

1

ln1

zJ

2

2

2

1

e

eP

21

1

e

P

2

Coupling

1,21,0

Memory spin:



zJ

1,0

m

1,m

0/1


Nn

ne

eP

1,

1

ln1

zJ

m

0 P

1 P

m

Repeat

Final state of memory spin & ancilla

memory erased ancilla in initial state

Memory spin:



zJ

1,0

m

1,m


Nn

ne

eP

1,

1

ln1

m

Repeat

Final state of memory spin & ancilla

memory erased ancilla in initial state

0/1

zJ

Memory spin:

m

1P

2/

0 P

Total cost:The CNOT operation on state of memory spin consumes angular momentum. For step m:

m

m

e

eP

1

00 1mm

m

mz e

ePJ

memory (m-1) mth ancilla

mth ancilla

m=0 term includes cost of initial state

)1ln(2ln eJ z

Z

Z

J

Je2

2



Single thermal reservoir: - used for both extraction and erasure

ImpactImpact

Q

erased memorywork

work

Q

heat engine

cycle

entropy

No net gain

Recall: Bennett’s exorcism of Maxwell’s demon



cycle

Two Thermal reservoirs:

- one for extraction, - one for erasure

Q1heat engine

work

entropy

increased entropy

Net gain if T1 > T2

T1

T2

Q2

work

erased memory &Q energy decrease

Recall: heat engine



spin reservoir

zJ

,11,0

cycleentropy

Here:Thermal and Spin reservoirs:

- extract from thermal reservoir- erase with spin reservoir

spin

Qheat engine

workerased

memory &Q energy decrease

zJ

increased entropy

Gain?



zJ

,11,0

Shannon

cost work

entropy

E

thermal reservoir

spin reservoir New

mechanism:

2nd Law Thermodynamics

Clausius It is impossible to construct a device which will produce in a cycle no effect other than the transfer of heat from a colder to a hotter body.

Kelvin-Planck

It is impossible for a heat engine to produce net work in a cycle if it exchanges heat only with bodies at a single fixed temperature.

S 0

applies to thermal reservoirs only

obeyed for Shannon entropy



zJ

2zJ

2zJ

2zJ

2zJ

2zJ

2

▀ the cost of erasure depends on the nature of the reservoir and the conservation law

▀ energy cost

▀ angular momentum cost

2ln1

2lnkT where

kT

1

2ln

zJ

1

ln1

where

SummarySummary

0

1dEE

0

1dEE

0

11dEE

0

1dEE

0

1dEE

0

11dEE

▀ 2nd Law is obeyed: total entropy is not decreased

▀ New mechanism

zJ

,11,0

Shannon

cost work

entropy

E

thermal reservoir

spin reservoir

2626/25/25

Entropy CostEntropy Cost

AISAMP Nov’08AISAMP Nov’08

▀ physical system has states that are degenerate in energy, momentum, … e.g. encode in position of a particle:

logical 0 =logical 1 =

Memory bit: 1 “logical bit” with states Reservoir: many “logical bits”

Entropy CostEntropy Cost

010BA

1,0101

BA

1110, n

)ln(

)ln(

)ln(

2ln11

W

)ln(2ln H(increase in reservoir entropy)

NW

(microcanonical ensemble)

(canonical ensemble)

▀ define Hamming Weight ▀ define maximisation subject to fixed Hamming Weight▀ repeat the angular momentum protocol with W in place of Jz

▀ Shannon entropy cost:

)s1' logical of (# W

AISAMP Nov’08 1/25 The cost of information erasure in atomic and spin systems Joan Vaccaro Griffith University Brisbane, Australia Steve Barnett University.

Documents

cost of erasure

entropic cost

fundamental cost

cost of information

energy of state

energy gap

energy expenditure

irreversible minimum