Erasure of information under conservation laws

1. Erasure of information under conservation laws Joan VaccaroCentre for Quantum DynamicsGriffith University Brisbane, Australia Steve Barnett SUPAUniversity of StrathclydeGlasgow, UK

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

3. 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) 1982 Bennet showed full cycle requires erasure of demon’s memory which costs at least Q: Q Q work Bennett, Int. J. Theor. Phys. 21, 905 (1982) Thermodynamic Entropy 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:

4. S Conventional Paradigm maximisation of entropysubject to conservation of energy cost of erasure is work Different Paradigm all states are degenerate in energy maximisation of entropysubject to conservation of angular momentum cost of erasure is angular momentum

5. 1 1 1 -1 0 -1 -1 0 0 1/2 -1/2 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

6. 1 1 1 -1 0 -1 0 -1 0 1/2 -1/2 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

7. E thermal reservoir spin reservoir Shannon entropy cost work E This talk Proc. R. Soc. A 467 1770 (2011) Energy Cost Conventional paradigm: • conservation of energy • simple 2-state atomic model Angular Momentum Cost New paradigm: • conservation of angular momentum • energy degenerate states of different spin Impact • New mechanism • statements of the 2nd Law

8. 0/1 Energy cost System: Memory bit: 2 degenerate atomic states Thermal reservoir: multi-level atomic gas at temperature T

9. 0/1 Thermalise memory bit while increasing energy gap

10. 0/1 Thermalise memory bit while increasing energy gap raise energy of state(e.g. Stark or Zeeman shift) Work to raise state from E to E+dE

11. 0/1 Thermalise memory bit while increasing energy gap raise energy of state(e.g. Stark or Zeeman shift) Work to raise state from E to E+dE Total work

12. 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 raise energy of state(e.g. Stark or Zeeman shift) Work to raise state from E to E+dE Total work 0/1

13. E E work 0/1 0/1 Principle of 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

14. 0/1 Angular Momentum Cost System: ●spin ½ ½ particles●no B or E fields so spins states are energy degenerate ●collisions between particles cause spin exchanges Memory bit: single spin ½ particle Reservoir: collection of N spin ½ particles. Possible states Simple representation: # of spin up n particles are spin up multiplicity (copy): 1,2,…

15. 0/1 Angular momentum diagram Memory bit: state Reservoir: states multiplicity (copy) 1,2,… # of spin up number of states with

16. Reservoir as “canonical” ensemble (exchanging not energy) Bigger spin bath: Reservoir: Total is conserved Maximise entropy of reservoir subject to

17. Average spin Reservoir as “canonical” ensemble (exchanging not energy) Bigger spin bath: Reservoir: Total is conserved Maximise entropy of reservoir subject to

18. 0/1 Erasure protocol Reservoir: Memory spin:

19. 0/1 Erasure protocol Reservoir: Memory spin: Coupling

20. 0/1 Erasure protocol Reservoir: Memory spin: Increase Jzusing ancilla in and CNOT operation this operation costs ancilla (target) memory(control)

21. 0/1 Erasure protocol Reservoir: Memory spin: Coupling

22. 0/1 Erasure protocol Reservoir: Memory spin: Repeat Final state of memory spin & ancilla memory erased ancilla in initial state

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

24. Impact Single thermal reservoir:- used for both extraction and erasure cycle entropy work Q erased memory work No net gain Q heat engine

25. work Q2 Two Thermal reservoirs: - one for extraction, - one for erasure increased entropy T2 cycle entropy T1 erased memory &Q energy decrease work Net gain if T1 > T2 Q1 heat engine

26. spin reservoir Here:Thermal and Spin reservoirs: increased entropy - extract from thermalreservoir- erase with spin reservoir spin cycle entropy T1 erased memory &Q energy decrease work Gain if T1 > 0 Q heat engine

27. thermal reservoir spin reservoir Shannon entropy cost work E Newmechanism: 2nd Law Thermodynamics applies to thermal reservoirs only  Kelvin-Planck It is impossible for a heat engine to produce net work in a cycle if it exchanges heat only with bodies at asingle fixed temperature. S  0  Schumacher (yesterday) “There can be no physical process whose sole effect is the erasure of information” Shannon entropy general

28. thermal reservoir spin reservoir Shannon entropy cost work E Summary • cost of erasuredepends on the conservation law • thermal reservoir is a resource for erasure: cost is • spin reservoir is a resource for erasure:cost is where where • 2nd Law is obeyed: total entropy is not decreased • New mechanism

29. Spinning as a resource… xkcd.com