1. AISAMP Nov’08 1/25 The cost of information erasure in atomic and spin systems Joan Vaccaro Griffith University Brisbane, Australia Steve Barnett University of Strathclyde Glasgow, UK

2. AISAMP Nov’08 Introduction Energy Cost Angular Mt m Cost Impact Summary Introduction Energy Cost Angular Mt m Cost Impact Summary 2/25 Introductio n ▀ Landauer erasure Landauer, IBM J. Res. Develop. 5, 183 (1961) 0 0 1 forward process: 0 0 1 0 time reversed: ? Erasure is irreversible Minimum cost 0 0/1 Process: maximise entropy subject to conservation of energy BEFORE erasureAFTER erasure # microstates environment heat

3. AISAMP Nov’08 Introduction Energy Cost Angular Mt m Cost Impact Summary Introduction Energy Cost Angular Mt m Cost Impact Summary 3/25 ▀ 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 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

4. AISAMP Nov’08 Introduction Energy Cost Angular Mt m Cost Impact Summary Introduction Energy Cost Angular Mt m Cost Impact Summary 4/25 Impact This talk Energy 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 Cost ▀ New mechanism ▀ 2 nd Law Thermodynamics E

5. AISAMP Nov’08 Introduction Energy Cost Angular Mt m Cost Impact Summary Introduction Energy Cost Angular Mt m Cost Impact Summary 5/25 ▀ System: 0/1 Memory bit: 2 degenerate atomic states Thermal reservoir: multi-level atomic gas at temperature T Energy Cost heat engine: cold work hot heat pump: work hot cold 0/1 ▀ recall heat pump  erasure

6. AISAMP Nov’08 Introduction Energy Cost Angular Mt m Cost Impact Summary Introduction Energy Cost Angular Mt m Cost Impact Summary 6/25 ▀ Thermalise memory bit while increasing energy gap 0/1

7. AISAMP Nov’08 Introduction Energy Cost Angular Mt m Cost Impact Summary Introduction Energy Cost Angular Mt m Cost Impact Summary 7/25 ▀ Thermalise memory bit while increasing energy gap raise energy of state (e.g. Stark or Zeeman shift) 0/1 Work to raise state from E to E+dE

8. AISAMP Nov’08 Introduction Energy Cost Angular Mt m Cost Impact Summary Introduction Energy Cost Angular Mt m Cost Impact Summary 8/25 ▀ Thermalise memory bit while increasing energy gap 0/1 Work to raise state from E to E+dE Total work raise energy of state (e.g. Stark or Zeeman shift)

9. AISAMP Nov’08 Introduction Energy Cost Angular Mt m Cost Impact Summary Introduction Energy Cost Angular Mt m Cost Impact Summary 9/25 ▀ Thermalise memory bit while increasing energy gap 0/1 Work to raise state from E to E+dE Total work 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. T HUS  erasure costs energy because the conservation law for energy is used to perform the erasure

10. AISAMP Nov’08 Introduction Energy Cost Angular Mt m Cost Impact Summary Introduction Energy Cost Angular Mt m Cost Impact Summary 10/25 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

11. AISAMP Nov’08 Introduction Energy Cost Angular Mt m Cost Impact Summary Introduction Energy Cost Angular Mt m Cost Impact Summary 11/25 ▀ System: ●spin 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

12. AISAMP Nov’08 Introduction Energy Cost Angular Mt m Cost Impact Summary Introduction Energy Cost Angular Mt m Cost Impact Summary 12/25 0/1 ▀ Angular momentum diagram states Memory bit: Reservoir: # of spin up multiplicity (copy) 1,2,… state number of states with

13. AISAMP Nov’08 Introduction Energy Cost Angular Mt m Cost Impact Summary Introduction Energy Cost Angular Mt m Cost Impact Summary 13/25 ▀ Reservoir as “canonical” ensemble (exchanging not energy) Maximise entropy of reservoir subject to Total is conserved Reservoir : Bigger spin bath :

14. AISAMP Nov’08 Introduction Energy Cost Angular Mt m Cost Impact Summary Introduction Energy Cost Angular Mt m Cost Impact Summary 14/25 ▀ Reservoir as “canonical” ensemble (exchanging not energy) Reservoir : Bigger spin bath : Maximise entropy of reservoir subject to Average spin

15. AISAMP Nov’08 Introduction Energy Cost Angular Mt m Cost Impact Summary Introduction Energy Cost Angular Mt m Cost Impact Summary 15/25 0/1 ▀ Erasure protocol Reservoir : Memory spin :

16. AISAMP Nov’08 Introduction Energy Cost Angular Mt m Cost Impact Summary Introduction Energy Cost Angular Mt m Cost Impact Summary 16/25 0/1 ▀ Erasure protocol Reservoir : Coupling Memory spin :

17. AISAMP Nov’08 Introduction Energy Cost Angular Mt m Cost Impact Summary Introduction Energy Cost Angular Mt m Cost Impact Summary 17/25 ▀ Erasure protocol Reservoir : 0/1 Increase J z using ancilla in memory (control) ancilla (target) this operation costs Memory spin : and CNOT operation

18. AISAMP Nov’08 Introduction Energy Cost Angular Mt m Cost Impact Summary Introduction Energy Cost Angular Mt m Cost Impact Summary 18/25 0/1 ▀ Erasure protocol Reservoir : Coupling Memory spin :

19. AISAMP Nov’08 Introduction Energy Cost Angular Mt m Cost Impact Summary Introduction Energy Cost Angular Mt m Cost Impact Summary 19/25 0/1 ▀ Erasure protocol Reservoir : Repeat Final state of memory spin & ancilla memory erased ancilla in initial state Memory spin :

20. AISAMP Nov’08 Introduction Energy Cost Angular Mt m Cost Impact Summary Introduction Energy Cost Angular Mt m Cost Impact Summary 20/25 ▀ Erasure protocol 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

21. AISAMP Nov’08 Introduction Energy Cost Angular Mt m Cost Impact Summary Introduction Energy Cost Angular Mt m Cost Impact Summary 21/25 Single thermal reservoir: - used for both extraction and erasure Impact QQ erased memory work QQ heat engine cycle entropy No net gain Recall: Bennett’s exorcism of Maxwell’s demon

22. AISAMP Nov’08 Introduction Energy Cost Angular Mt m Cost Impact Summary Introduction Energy Cost Angular Mt m Cost Impact Summary 22/25 cycle Two Thermal reservoirs: - one for extraction, - one for erasure Q1Q1 heat engine work entropy increased entropy Net gain if T 1 > T 2 T1T1 T2T2 Q2Q2 work erased memory &  Q energy decrease Recall: heat engine

23. AISAMP Nov’08 Introduction Energy Cost Angular Mt m Cost Impact Summary Introduction Energy Cost Angular Mt m Cost Impact Summary 23/25 spin reservoir cycle entropy Here: Thermal and Spin reservoirs: - extract from thermal reservoir - erase with spin reservoir spin QQ heat engine work erased memory &  Q energy decrease increased entropy Gain?

24. AISAMP Nov’08 Introduction Energy Cost Angular Mt m Cost Impact Summary Introduction Energy Cost Angular Mt m Cost Impact Summary 24/25 Shannon cost work entropy E thermal reservoirspin reservoir New mechanism: 2 nd 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

25. AISAMP Nov’08 Introduction Energy Cost Angular Mt m Cost Impact Summary Introduction Energy Cost Angular Mt m Cost Impact Summary 25/25 ▀ the cost of erasure depends on the nature of the reservoir and the conservation law ▀ energy cost ▀ angular momentum cost where Summary ▀ 2 nd Law is obeyed: total entropy is not decreased ▀ New mechanism Shannon cost work entropy E thermal reservoirspin reservoir

26. Entropy Cost AISAMP Nov’08 26/25 ▀ 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 Cost (increase in reservoir entropy) (microcanonical ensemble) (canonical ensemble) ▀ define Hamming Weight ▀ define maximisation subject to fixed Hamming Weight ▀ repeat the angular momentum protocol with W in place of J z ▀ Shannon entropy cost: