/ 29 ContextEnergy costAngular Mtm costImpactSummary 1 Erasure of information under conservation laws Joan Vaccaro Centre for Quantum Dynamics Griffith University Brisbane, Australia Steve Barnett SUPA University of Strathclyde Glasgow, UK

ContextEnergy costAngular Mtm costImpactSummary / 292 Landauer erasure Landauer, IBM J. Res. Develop. 5, 183 (1961) forward process: time reversed: ? Erasure is irreversible Minimum cost 0 0/1 BEFORE erasureAFTER erasure # microstates environment heat Context Hide the past of the memory in a reservoir (who’s past is unknown) ?

ContextEnergy costAngular Mtm costImpactSummary / 293 Exorcism of Maxwell’s demon 1871 Maxwell’s demon extracts work of  Q from thermal reservoir by collecting only hot gas particles. (Violates 2 nd Law: reduces entropy of whole gas) Q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: QQ work

ContextEnergy costAngular Mtm costImpactSummary / 294 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

ContextEnergy costAngular Mtm costImpactSummary / 295 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 m j =  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 11  1

ContextEnergy costAngular Mtm costImpactSummary / 296  1/2 1/2 11  1 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 m j =  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

ContextEnergy costAngular Mtm costImpactSummary / 297 Impact This talk Energy Cost Conventional 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 2 nd Law E Shannon cost work entropy E thermal reservoirspin reservoir Proc. R. Soc. A (2011)

ContextEnergy costAngular Mtm costImpactSummary / 298 System: 0/1 Memory bit: 2 degenerate atomic states Thermal reservoir: multi-level atomic gas at temperature T Energy cost

ContextEnergy costAngular Mtm costImpactSummary / 299 Thermalise memory bit while increasing energy gap 0/1

ContextEnergy costAngular Mtm costImpactSummary / 2910 raise energy of state (e.g. Stark or Zeeman shift) 0/1 Work to raise state from E to E+dE Thermalise memory bit while increasing energy gap

ContextEnergy costAngular Mtm costImpactSummary / /1 Work to raise state from E to E+dE Total work raise energy of state (e.g. Stark or Zeeman shift) Thermalise memory bit while increasing energy gap

ContextEnergy costAngular Mtm costImpactSummary / /1 Work to raise state from E to E+dE Total work raise energy of state (e.g. Stark or Zeeman shift) 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

ContextEnergy costAngular Mtm costImpactSummary / 2913 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: 0/1 E E work

ContextEnergy costAngular Mtm costImpactSummary / 2914 System: ●spin ½ ½ particles ●no B or E fields so spins states are energy degenerate ● collisions between particles cause spin exchanges 0/1 Memory bit: single spin ½ particle Reservoir: collection of N spin ½ particles. Possible states Simple representation: # of spin up multiplicity (copy): 1,2,… n particles are spin up Angular Momentum Cost

ContextEnergy costAngular Mtm costImpactSummary / /1 Angular momentum diagram states Memory bit: Reservoir: # of spin up multiplicity (copy) 1,2,… state number of states with

ContextEnergy costAngular Mtm costImpactSummary / 2916 Reservoir as “canonical” ensemble (exchanging not energy) Maximise entropy of reservoir subject to Total is conserved Reservoir : Bigger spin bath :

ContextEnergy costAngular Mtm costImpactSummary / 2917 Reservoir as “canonical” ensemble (exchanging not energy) Maximise entropy of reservoir subject to Total is conserved Reservoir : Bigger spin bath : Average spin

ContextEnergy costAngular Mtm costImpactSummary / /1 Erasure protocol Reservoir : Memory spin :

ContextEnergy costAngular Mtm costImpactSummary / /1 Reservoir : Coupling Memory spin : Erasure protocol

ContextEnergy costAngular Mtm costImpactSummary / 2920 Reservoir : 0/1 Increase J z using ancilla in memory (control) ancilla (target) this operation costs Memory spin : and CNOT operation Erasure protocol

ContextEnergy costAngular Mtm costImpactSummary / /1 Reservoir : Coupling Memory spin : Erasure protocol

ContextEnergy costAngular Mtm costImpactSummary / /1 Reservoir : Repeat Final state of memory spin & ancilla memory erased ancilla in initial state Memory spin : Erasure protocol

ContextEnergy costAngular Mtm costImpactSummary / 2923 Reservoir : Repeat Final state of memory spin & ancilla memory erased ancilla in initial state 0/1 Memory spin : Total cost: The CNOT operation on state of memory spin consumes angular momentum. For step m : memory ( m -1) m th ancilla m=0 term includes cost of initial state Erasure protocol

ContextEnergy costAngular Mtm costImpactSummary / 2924 Single thermal reservoir: - used for both extraction and erasure Impact QQ erased memory work QQ heat engine cycle entropy No net gain

ContextEnergy costAngular Mtm costImpactSummary / 2925 cycle Two Thermal reservoirs: - one for extraction, - one for erasure Q1Q1 work entropy increased entropy Net gain if T 1 > T 2 T1T1 T2T2 Q2Q2 work erased memory &  Q energy decrease heat engine

ContextEnergy costAngular Mtm costImpactSummary / 2926 spin reservoir cycle entropy Here: Thermal and Spin reservoirs: - extract from thermal reservoir - erase with spin reservoir spin QQ work erased memory &  Q energy decrease increased entropy Gain if T 1 > 0 heat engine T1T1

ContextEnergy costAngular Mtm costImpactSummary / 2927 Shannon cost work entropy E thermal reservoirspin reservoir New mechanism: 2 nd 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

ContextEnergy costAngular Mtm costImpactSummary / 2928 ▀ 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 where ▀ 2 nd Law is obeyed: total entropy is not decreased ▀ New mechanism Summary Shannon cost work entropy E thermal reservoirspin reservoir

ContextEnergy costAngular Mtm costImpactSummary / 2929 Spinning as a resource… xkcd.com