The cost of information erasure in atomic and spin systems

1. The cost of information erasure in atomic and spin systems Joan VaccaroGriffith University Brisbane, Australia Steve Barnett University of Strathclyde Glasgow, UK

2. Introduction • Landauer erasure Landauer, IBM J. Res. Develop. 5, 183 (1961) Erasure is irreversible forward process: time reversed: 0 ? 0 0 0 0 1 1 Minimum cost environment BEFORE erasure AFTER erasure 0 0/1 heat # microstates Process:maximise entropy subject to conservation of energy

3. Information is Physical  information must be carried by physical system (not new)  its erasure requires energy expenditure • 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. E This talk Energy Cost • from conservation of energy • simple 2-state atomic model • re-derive Landauer’s minimum cost of kTln2 per bit Angular Momentum Cost • energy degenerate states of different spin • conservation of angular momentum • cost in terms of angular momentum only Impact • New mechanism • 2nd Law Thermodynamics

5. recall heat pump heat engine: work hot cold heat pump: work hot 0/1 0/1 cold erasure Energy Cost • System: Memory bit: 2 degenerate atomic states Thermal reservoir: multi-level atomic gas at temperature T

6. 0/1 • Thermalise memory bit while increasing energy gap

7. 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

8. 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

9. 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. THUSerasure costs energy because the conservation law for energy is used to perform the erasure raise energy of state(e.g. Stark or Zeeman shift) Work to raise state from E to E+dE Total work 0/1

10. 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

11. 0/1 Angular Momentum Cost • System: ● spin 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,…

12. 0/1 • Angular momentum diagram Memory bit: state Reservoir: states multiplicity (copy) 1,2,… # of spin up number of states with

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

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

15. 0/1 • Erasure protocol Reservoir: Memory spin:

16. 0/1 • Erasure protocol Reservoir: Memory spin: Coupling

17. 0/1 • Erasure protocol Reservoir: Memory spin: Increase Jzusing ancilla in and CNOT operation this operation costs ancilla (target) memory(control)

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

19. 0/1 • Erasure protocol Reservoir: Memory spin: Repeat Final state of memory spin & ancilla memory erased ancilla in initial state

20. 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

21. Impact Recall: Bennett’s exorcism of Maxwell’s demon Single thermal reservoir:- used for both extraction and erasure cycle entropy work Q erased memory work No net gain Q heat engine

22. work Q2 Recall: heat engine 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

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

24. Newmechanism: thermal reservoir spin reservoir Shannon entropy cost work E 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 asingle fixed temperature. S  0 applies to thermal reservoirs only obeyed for Shannon entropy

25. thermal reservoir spin reservoir Shannon entropy cost work E Summary • thecost of erasuredepends on the nature of the reservoirand the conservation law • energy cost • angular momentum cost where where • 2nd Law is obeyed: total entropy is not decreased • New mechanism

26. Entropy Cost • 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” • define Hamming Weight • define maximisation subject to fixed Hamming Weight • repeat the angular momentum protocol with W in place of Jz (canonical ensemble) • Shannon entropy cost: (microcanonical ensemble) (increase in reservoir entropy)