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

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) forward process: time reversed: ? Erasure is irreversible Minimum cost 0 0/1 Process: maximise entropy subject to conservation of energy BEFORE erasureAFTER erasure # microstates environment heat

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

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

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

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

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

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)

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

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

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

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

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 :

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

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 :

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 :

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

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 :

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 :

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

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

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

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?

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

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

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: