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11.3 Two Qubit Machine

11.3.2 Repeated & Asymptotic Cycles

We here would like to go beyond the single cycle regime of the 2 qubit machine 1 qubit target joint system. Studying the repeated applications for each scenario is in this case still very much tractable due to the simplicity of the systems at hand.

The main analytical challenges have indeed already been dealt within the previous sections. The choice of the machine that cools in the incoherent scenario has been made. In the coherent scenario, we will see that after the first cycle of cooling is completed, the optimization problem is easily solved due to the restricted subspace that is still available for cooling.

Repeated Incoherent Scenario

After a cycle is completed, applying another Λincto the target amounts to rether-malizing M1 toTR and M2 to TH before repeating the energy conserving unitary operation

Uinc=|010i h101|SM+|101i h010|SM+1spanc{|010iSM,|101iSM}. (11.133) As before, the effect ofUincto the ground state population of the target is to remove the population of|010iSMand add that of|101iSM. If the ground state population of the target before the application ofUincisr, then after applyingUincit becomes

r0 =r−r(1−rM1)rMH2+ (1−r)rM1(1−rHM2). (11.134)

Chapter 11. Qubit System 61 One can massage the right-hand side of Eq.11.134and rewrite it as

r0 =rV+ (1−NV)(r−rV), (11.135) whereNVis the sum of the populations of|01iMand|10iM, i.e.,

NV =rM1(1−rHM2) + (1−rM1)rHM2, (11.136) andrVis the population of|01iM renormalized byNV, i.e.,

rV = rM1(1−rHM2)

NV = 1

1+eβREM1eβHEM2. (11.137) The rewriting may at first seem arbitrary but it proves to be very convenient to calculate the ground state population of the target after n applications of the incoherent scenario. Indeed,

rinc,n−rV = (1−NV)(rinc,n1−rV) (11.138) rinc,n1−rV = (1−NV)(rinc,n2−rV) (11.139)

... (11.140)

rinc−rV = (1−NV)(rS−rV), (11.141) such that

rinc,n=rV+ (1−NV)n(rS−rV). (11.142) And as 0< NV1,

rinc,∞= lim

nrinc,n=rV. (11.143)

From the expression ofrVandrinc, = 1

1+eβinc,∞ES we get

βinc,∞ES= −βREM1+βHEM2, (11.144) which yields

Tinc,∞ = E ES

M1

TRETM2

H

. (11.145)

Besides being convenient to calculate,rV and NV can also be given a physical interpretation. If one pretends that the two level system of the machine|01iM,|10iM is a qubit of energy gap EV = EM1 − EM2, then its norm is given by NV and its normalized ground state population byrV. As a reminder that this two level system is not exactly a real qubit but only a two level system, the name “virtual qubit” was coined to denote this two level system [69]. See also Appendix G of [91]. With this we have calculated the attainable ground state populations afternapplications of the incoherent scenario for anyn ∈ N. We next turn our attention to the work cost. Calculating the work cost amounts to calculatingqHi , the heat drawn from the hot bath to rethermalizeM2toTH at the beginning of stepi= 1, . . . ,n. Remember that by energy conservationqHi equals the population change effected inM2by the rethermalization timesEM2, i.e.,

qHi =EM2(τMH2σiM2), (11.146)

Chapter 11. Qubit System 62 whereσMi

2 is the state of M2 at the end of stepi−1. The total heat drawn after n applications ofΛincis therefore given by

QHn =EM2(rM2 −rMH2) +

n i=2

qiH (11.147)

=EM2(rM2 −rMH2) +

n i=2

EM2(rinc,i1−rinc,i2) (11.148)

=EM2(rM2 −rMH2) +EM2(rinc,n1−rS), (11.149) where in the second step we used that from the form ofUinc, the difference of population contributing toqHi is

rinc,i1−rinc,i2, (11.150)

the difference in ground state population that Uinc effects on S at step i−1. In summary we therefore have

∆Finc,n(TH) =EM2rM2 −rHM2+rinc,n1−rS 1− TR

TH

, (11.151)

rinc,n(TH) =rinc,∞+ (1−NV)n(rS−rinc,∞), (11.152) where

rinc,∞ = 1

1+eTinc,∞ES

, Tinc,∞ = E ES

M1

TRETM2

H

, (11.153)

NV =rM1(1−rHM2) + (1−rM1)rHM2. (11.154) Note in particular that

TlimHrinc,∞ =rM1, (11.155)

which is the temperature achieved by a one qubit machine of energy gapEM1 for the coherent scenario. This is an exemplification of a more general connection between the incoherent and coherent scenario that we will explore in Section12.

Repeated Coherent Scenario

Having studied a single application ofΛcohfor the special instance of the 2 qubit machine 1 qubit target such that EM1 = EM2 +ES, we now turn our attention to the repeated application ofΛncoh. As done for the single application case, out of all the possibleΛcohmaps, we would like to focus here on the maps that minimize the work cost. For temperatures of the targetrcoh ∈ [rS,rcoh ]we already know which mapsΛncoh= Λcoh(n)Λcoh(n1)◦ · · · ◦Λ(coh1) minimize the work cost. Indeed, they are the maps such that eachΛ(i)is as prescribed by Theorem6and Theorem7. This is not straightforward to see from the statement of both Theorems but upon looking at their proofs more closely, one realizes that they extend to the desired case. Indeed, given EM1 =EM2 +ES, the practical rewriting used in the proof of Theorem6, Eq.11.111, remains practical if one has an initial system state ˜ρSat hand such that its ground state population ˜rSfulfillsrM2 ≤r˜S≤ rM1. This is because in this case the ordering of

Chapter 11. Qubit System 63 D(ρ˜SρM)is the same as that ofD(ρSM), namely

[D(ρ˜SρM)]0 ≥[D(ρ˜SρM)]1 ≥[D(ρ˜SρM)]4≥[D(ρ˜SρM)]5 (11.156)

≥ [D(ρ˜SρM)]2 ≥[D(ρ˜SρM)]3 ≥[D(ρ˜SρM)]6≥[D(ρ˜SρM)]7. (11.157) We therefore have the following result.

Theorem 8. LetEM1 = EM2+ES. Letρ˜Sbe a system state such thatr˜S ∈[rM2,rM1]. Let rcoh ∈[r˜S,rcoh]. Letµ= rrcohr˜S

M1r˜S. Let t=arcsin(√

µ). Then

v =D(Uρ˜SρMU), with U=eitLSM1 (11.158) minimizes the optimization problem

vDmin(ρ˜SρM)v·D(HSM), s.t.

3 i=0

vi =rcoh. (11.159) and has an associated work cost of

∆Fcoh = (rcoh−r˜S)(EM1−ES). (11.160) Note that Theorem8holds forEM2 ≤ESas well as forEM2 > ES. This is because the rewriting ofv·D(HSM)of Eq.11.111does not depend on howEM2 andESrelate to one another. For the caseEM2 ≤ ES, as rM2 ≤ rS, the result of Theorem8 can straightforwardly be applied. It tells us that in this case it is of no advantage, from a work cost perspective, to rethermalize our machine before the target has been cooled torM1. Indeed, if we start withρS, anyΛcohthat cools the target to less thanrM1, does so at a work cost of at leastEM1 −ESper population change of the target. This is the content of Theorem6. Furthermore, the next application of the coherent scenario will just continue to cool at a work cost of at leastEM1 −ESper population change of the target and so on as long asrcoh ∈[rS,rM1]. One may therefore as well not rethermalize the machine and directly cool the target torM1 at a work cost of(rM1−rS)(EM2−ES) within a single application of the scenario, as prescribed by Theorem6.

In the caseEM2 > ES, the ordering ofD(ρSM)is different. And so, one cannot directly make use of Theorem8. But a similar reasoning applies here as well. If our initial state ˜ρSis such that ˜rS∈ [rS,rM2], then the ordering ofD(ρ˜SρM)is the same as that ofD(ρSM)and we can use the rewriting ofv·D(HSM)of Eq.11.120to get the following result.

Theorem 9. Let EM1 = EM2 +ES. Let ρ˜S be a system state such thatr˜S ≤ rM2. Let rcoh ∈[r˜S,rM2]. Letµ= rrcohr˜S

M2r˜S. Let t=arcsin(√

µ). Then

v =D(Uρ˜SρMU), with U=eitLSM2 (11.161) minimizes the optimization problem

vDmin(ρ˜SρM)v·D(HSM), s.t.

3 i=0

vi =rcoh. (11.162) and has an associated work cost of

∆Fcoh = (rcoh−r˜S)(EM2−ES). (11.163)

Chapter 11. Qubit System 64 Note that Theorem9also holds for bothEM2 ≤ ES andEM2 > ES. In the case EM2 ≤ESone may be worried that 2EM2− EM1 <0 might happen for some machines and that in that case one cannot make use of Eq.11.120to derive the desired result anymore. But as

(v5+v6+v5+v7)(2EM2− EM1) = (1−v0−v1−v2−v4)(EM1−2EM2), (11.164) the statement of Theorem9also holds for these machines. In any case, Theorem9 tells us that also in the case ofEM2 > ES, it is of no advantage to rethermalize the machine before one has cooled the target torM1.

Once one has cooled the target torM1, the only way to further cool the target is to rethermalize the machine. Once this is done, further applications of the coherent scenario are only able to cool by acting on the|011iSMand|100iSMsubspace of the joint systemSM. Performing a partial swap, gradually moving population form

|100iSMto|011iSM, is then the optimal operation to further cool. At the end of this partial swap one reaches a ground sate population of

r0 =r+ (1−r)rM1rM2−r(1−rM1)(1−rM2), (11.165) if the ground state population of the target before the swap wasr. Massaging the right-hand side of Eq.11.165as done for Eq.11.134we get

r0 =rV˜ + (1−NV˜)(r−rV˜), (11.166) whereNV˜ is the sum of population of|00iMand|11iM, i.e.,

NV˜ =rM1rM2+ (1−rM1)(1−rM2), (11.167) andrV˜ is the population of|00iM renormalized byNV˜, i.e.,

rV˜ = rM1rM2 NV˜

= 1

1+eβR(EM2+EM1). (11.168) One therefore gets

rcoh,n=rV˜ + (1−NV˜)n1(rB−rV˜). (11.169) And as again 0< NV˜ ≤1,

rcoh,∞= lim

nrcoh,n=rV˜. (11.170)

From the expression ofrV˜ andrcoh,∞ = 1

1+eβcoh,∞ES we get βcoh,∞ =βREM2+EM1

ES , (11.171)

which yields

Tcoh,∞ = ES

EM1+EM2TR. (11.172) The work cost associated to it is, forn∈ N∪ {+},

∆Fcoh,n =∆Fcoh +2EM2

n i=2

(rcoh,i−rcoh,ii) (11.173)

=∆Fcoh +2EM2(rcoh,n−rM1). (11.174)

Chapter 11. Qubit System 65

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