Date post: | 21-Dec-2015 |
Category: |
Documents |
View: | 218 times |
Download: | 2 times |
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
22/25/25AISAMP Nov’08AISAMP Nov’08
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
33/25/25AISAMP Nov’08AISAMP Nov’08
Introduction Energy Cost Angular MtIntroduction Energy Cost Angular Mtmm Cost Impact Summary Cost Impact Summary
▀ 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
44/25/25AISAMP Nov’08AISAMP Nov’08
Introduction Energy Cost Angular MtIntroduction Energy Cost Angular Mtmm Cost Impact Summary Cost Impact Summary
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
55/25/25AISAMP Nov’08AISAMP Nov’08
Introduction Energy Cost Angular MtIntroduction Energy Cost Angular Mtmm Cost Impact Summary Cost Impact Summary
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
66/25/25AISAMP Nov’08AISAMP Nov’08
Introduction Energy Cost Angular MtIntroduction Energy Cost Angular Mtmm Cost Impact Summary Cost Impact Summary
T
Z
eP
kTE
E
/
0 1
▀ Thermalise memory bit while increasing energy gap
0/1
2
11 P
2
10 P
77/25/25AISAMP Nov’08AISAMP Nov’08
Introduction Energy Cost Angular MtIntroduction Energy Cost Angular Mtmm Cost Impact Summary Cost Impact Summary
T
Z
eP
kTE
E
/
▀ Thermalise memory bit while increasing energy gap
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
88/25/25AISAMP Nov’08AISAMP Nov’08
Introduction Energy Cost Angular MtIntroduction Energy Cost Angular Mtmm Cost Impact Summary Cost Impact Summary
T
Z
eP
kTE
E
/
▀ Thermalise memory bit while increasing energy gap
0/1
dEPdW 1
01 P
Work to raise state from E to E+dE
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
99/25/25AISAMP Nov’08AISAMP Nov’08
Introduction Energy Cost Angular MtIntroduction Energy Cost Angular Mtmm Cost Impact Summary Cost Impact Summary
T
Z
eP
kTE
E
/
▀ Thermalise memory bit while increasing energy gap
0/1
dEPdW 1
01 P
Work to raise state from E to E+dE
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
1010/25/25AISAMP Nov’08AISAMP Nov’08
Introduction Energy Cost Angular MtIntroduction Energy Cost Angular Mtmm Cost Impact Summary Cost Impact Summary
• 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
1111/25/25AISAMP Nov’08AISAMP Nov’08
Introduction Energy Cost Angular MtIntroduction Energy Cost Angular Mtmm Cost Impact Summary Cost Impact Summary
▀ 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
1212/25/25AISAMP Nov’08AISAMP Nov’08
Introduction Energy Cost Angular MtIntroduction Energy Cost Angular Mtmm Cost Impact Summary Cost Impact Summary
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
1313/25/25AISAMP Nov’08AISAMP Nov’08
Introduction Energy Cost Angular MtIntroduction Energy Cost Angular Mtmm Cost Impact Summary Cost Impact Summary
▀ 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
1414/25/25AISAMP Nov’08AISAMP Nov’08
Introduction Energy Cost Angular MtIntroduction Energy Cost Angular Mtmm Cost Impact Summary Cost Impact Summary
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
1515/25/25AISAMP Nov’08AISAMP Nov’08
Introduction Energy Cost Angular MtIntroduction Energy Cost Angular Mtmm Cost Impact Summary Cost Impact Summary
0/1
▀ Erasure protocolReservoir:
1
ln1
zJ 2
1P
Nn
ne
eP
1,
2
1P
Memory spin:
zJ
1,0,1
1616/25/25AISAMP Nov’08AISAMP Nov’08
Introduction Energy Cost Angular MtIntroduction Energy Cost Angular Mtmm Cost Impact Summary Cost Impact Summary
zJ
1,0,1
0/1
▀ Erasure protocolReservoir:
Nn
ne
eP
1,
1
ln1
zJ
Coupling
1,11,0
e
eP
1
eP
1
1
Memory spin:
1717/25/25AISAMP Nov’08AISAMP Nov’08
Introduction Energy Cost Angular MtIntroduction Energy Cost Angular Mtmm Cost Impact Summary Cost Impact Summary
▀ Erasure protocolReservoir:
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
1818/25/25AISAMP Nov’08AISAMP Nov’08
Introduction Energy Cost Angular MtIntroduction Energy Cost Angular Mtmm Cost Impact Summary Cost Impact Summary
,2
zJ
1,02
0/1
▀ Erasure protocolReservoir:
Nn
ne
eP
1,
1
ln1
zJ
2
2
2
1
e
eP
21
1
e
P
2
Coupling
1,21,0
Memory spin:
1919/25/25AISAMP Nov’08AISAMP Nov’08
Introduction Energy Cost Angular MtIntroduction Energy Cost Angular Mtmm Cost Impact Summary Cost Impact Summary
zJ
1,0
m
1,m
0/1
▀ Erasure protocolReservoir:
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:
2020/25/25AISAMP Nov’08AISAMP Nov’08
Introduction Energy Cost Angular MtIntroduction Energy Cost Angular Mtmm Cost Impact Summary Cost Impact Summary
zJ
1,0
m
1,m
▀ Erasure protocolReservoir:
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
2121/25/25AISAMP Nov’08AISAMP Nov’08
Introduction Energy Cost Angular MtIntroduction Energy Cost Angular Mtmm Cost Impact Summary Cost Impact Summary
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
2222/25/25AISAMP Nov’08AISAMP Nov’08
Introduction Energy Cost Angular MtIntroduction Energy Cost Angular Mtmm Cost Impact Summary Cost Impact Summary
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
2323/25/25AISAMP Nov’08AISAMP Nov’08
Introduction Energy Cost Angular MtIntroduction Energy Cost Angular Mtmm Cost Impact Summary Cost Impact Summary
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?
2424/25/25AISAMP Nov’08AISAMP Nov’08
Introduction Energy Cost Angular MtIntroduction Energy Cost Angular Mtmm Cost Impact Summary Cost Impact Summary
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
2525/25/25AISAMP Nov’08AISAMP Nov’08
Introduction Energy Cost Angular MtIntroduction Energy Cost Angular Mtmm Cost Impact Summary Cost Impact Summary
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