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Context Energy cost Angular Mtm cost Impact Summary
Erasure of information under conservation laws
Joan VaccaroCentre for Quantum DynamicsGriffith University Brisbane, Australia
Steve BarnettSUPAUniversity of StrathclydeGlasgow, UK
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Landauer erasure
Landauer, IBM J. Res. Develop. 5, 183 (1961)
00
1
forward process:
0 0
1 0
time reversed:
?
Erasure is irreversible
Minimum cost
00/1
BEFORE erasure AFTER erasure
env2 smicrostate # total N
)2ln( )ln( env kTNkT
)2ln(kTQ
# microstates
environment
)ln( envNkTQ
heat
)2ln( envNkTQ
Context
Hide the past of the memory in a reservoir (who’s past is unknown)
?
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Exorcism of Maxwell’s demon
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
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Different Paradigm all states are degenerate in energy maximisation of entropy subject to
conservation of angular momentum cost of erasure is angular momentum
Conventional Paradigm maximisation of entropy subject to
conservation of energy cost of erasure is work
S
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Example to set the stage… single-electron atoms with ground state spin angular momentum memory: spin-1/2 atoms in equal mixture reservoir: spin-1 atoms all in mj = 1 state (spin polarised)
independent optical trapping potentials (dipole traps) atoms exchange spin angular momentum via collisions when traps brought together erasure of memory by loss of spin polarisation of reservoir – the cost of erasure is spin angular momentum
1/2 1/2
1 1 1 1 1 1
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1/2 1/2
1 1 1 1 1 1
Example to set the stage… single-electron atoms with ground state spin angular momentum memory: spin-1/2 atoms in equal mixture reservoir: spin-1 atoms all in mj = 1 state (spin polarised)
independent optical trapping potentials (dipole traps) atoms exchange spin angular momentum via collisions when traps brought together erasure of memory by loss of spin polarisation of reservoir – the cost of erasure is spin angular momentum
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Impact
This talk
Energy CostConventional paradigm:▀ conservation of energy▀ simple 2-state atomic model
New paradigm:▀ conservation of angular momentum ▀ energy degenerate states of different spin
Angular Momentum Cost
▀ New mechanism▀ statements of the 2nd Law
zJ
2
0
1dEE
zJ
,11,0
Shannon
cost work
entropy
E
thermal reservoir spin reservoir
Proc. R. Soc. A 467 1770 (2011)
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System:
0 1 0/1
Memory bit: 2 degenerate atomic states
Thermal reservoir: multi-level atomic gas at temperature T
Z
eP
kTE
E
/
E
Energy cost
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T
Z
eP
kTE
E
/
0 1
Thermalise memory bit while increasing energy gap
0/1
2
11 P
2
10 P
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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
Thermalise memory bit while increasing energy gap
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T
Z
eP
kTE
E
/
0/1
dEPdW 1
01 P
Work to raise state from E to E+dE
/
/0 0
1log 2
log 21
E kT
E kTE E
eW P dE dE kT
e
Total work
1
0
10 P
raise energy of state(e.g. Stark or Zeeman shift)
1
Thermalise memory bit while increasing energy gap
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T
Z
eP
kTE
E
/
0/1
dEPdW 1
01 P
Work to raise state from E to E+dE
/
/0 0
1log 2
log 21
E kT
E kTE E
eW P dE dE kT
e
Total work
1
0
10 P
raise energy of state(e.g. Stark or Zeeman shift)
1
Thermalise memory bit while increasing energy gap
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.
This is erasure in the paradigm of thermal reservoirs
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• 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
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System:● spin ½ 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
nN
21
21
Angular Momentum Cost
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0/1
zJ
Angular momentum diagram
states
Memory bit:
Reservoir:,n
0,11,1 , 1,2 , 1,3 ,
# of spin up
multiplicity (copy)
zJ
N
n1,2,…
21
21state
number of states with
12z J n N
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Reservoir as “canonical” ensemble (exchanging not energy)
Maximise entropy of reservoir
subject to
,
,, lnn
nn PP
1, 2reservoir
, 2
z nn
NJ P n N 1
,,
nnP&
Total is conserved
zJ
zJ
0,11,
zJ
0,1 1,
,n
Reservoir:Bigger spin bath:
,nP
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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
Jz
zJ
0,11,
zJ
1,0 1,
,n
Reservoir:Bigger spin bath:
121zJ
10 1 1
Average spin
1
1
2
2Z
Z
J
Je
,1
n
n N
eP
e
1 1ln
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0/1
Erasure protocolReservoir:
zJ 2
1P
2
1P
Memory spin:
zJ
0,11,
,1
n
n N
eP
e
1 1ln
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zJ
0,11,
0/1
Reservoir:
zJ
Coupling
0,1 1,1
1
eP
e
1
1P
e
Memory spin:Erasure protocol
,1
n
n N
eP
e
1 1ln
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Reservoir:
0/1
Increase Jz using ancilla in
memory(control)
ancilla (target)
zJ
2
this operation costs
Memory spin:
and CNOT operation
2,
zJ
0,12
Erasure protocol
,1
n
n N
eP
e
1 1ln
1
eP
e
1
1P
e
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2,
zJ
0,12
0/1
Reservoir:
zJ
2
2
21
eP
e
2
1
1P
e
2
Coupling
0,1 2,1
Memory spin:Erasure protocol
,1
n
n N
eP
e
1 1ln
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zJ
0,1
m
,1m
0/1
Reservoir:
,1
n
n N
eP
e
1 1ln
zJ
m
0 P
1 P
m
Repeat
Final state of memory spin & ancilla
memory erased ancilla in initial state
Memory spin:Erasure protocol
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zJ
1,0
m
,1m
Reservoir:
Nn
ne
eP
1,
1 1ln
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:
1
m
m
eP
e
0 0 1
m
z mm m
eJ P
e
memory (m-1) mth ancilla
mth ancilla
m=0 term includes cost of initial state
ln 2z
J
1
1
2
2Z
Z
J
Je
Erasure protocol
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Single thermal reservoir: - used for both extraction and erasure
Impact
Q
erased memorywork
work
Q
heat engine
cycle
entropy
No net gain
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cycle
Two Thermal reservoirs:
- one for extraction, - one for erasure
Q1
work
entropy
increased entropy
Net gain if T1 > T2
T1
T2
Q2
work
erased memory &Q energy decreaseheat engine
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spin reservoir
zJ
,11,0
cycleentropy
Here:Thermal and Spin reservoirs:
- extract from thermal reservoir- erase with spin reservoir
spin
Q
workerased
memory &Q energy decrease
zJ
increased entropy
Gain if T1 > 0 heat engine
T1
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zJ
,11,0
Shannon
cost work
entropy
E
thermal reservoir
spin reservoir New
mechanism:
2nd Law Thermodynamics
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 Schumacher (yesterday) “There can be no physical process whose sole effect is the erasure of information”
applies to thermal reservoirs only
Shannon entropy
general
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▀ cost of erasure depends on the conservation law
▀ thermal reservoir is a resource for erasure:
cost is
▀ spin reservoir is a resource for erasure:
cost is
ln 2ln 2E kT
where
kT
1
ln 2J
z
1 1ln
where
▀ 2nd Law is obeyed: total entropy is not decreased
▀ New mechanism
Summary
zJ
,11,0
Shannon
cost work
entropy
E
thermal reservoir
spin reservoir
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Spinning as a resource…
xkcd.com