**11.3 Two Qubit Machine**

**11.3.1 Single Cycle**

Given a machine dimensiondMand a systemS, the main challenge of the
incoher-ent scenario is to idincoher-entify which specific machines allow cooling systemS. From
Section10.2we know that the machine has to have a tensor product structure to be
able to coolS. Indeed, if not, the only way it can use the hot thermal bath is to either
stay thermal atT_{R}or to entirely thermalize toT_{H}. But we know from Lemma9and
Lemma8that this doesn’t lead to any cooling ofS, whatever the degeneracies of

Chapter 11. Qubit System 52
H_{SM}. For machines of dimension 4, i.e.,d_{M} =4, this means that the useful machines
have to consist of 2 qubits. Furthermore, the only way of making good use of the hot
thermal bath is to leave one qubit atT_{R}and thermalize the other toT_{H}. W.l.o.g. let
M_{1}remain atT_{R}andM_{2}be the one thermalizing toT_{H}. We therefore have

*ρ*^{R}_{SM}^{,}^{H} =*τ*_{S}⊗*τ*_{M}_{1}⊗*τ*_{M}^{H}_{2}. (11.62)
The next step is to screen through the potential degeneracies ofH_{SM}to see which
ones enable to coolS. From Lemma10we know that degenerate subspaces consisting
of degeneracies of the machine only are of no use. This gets rid of the degeneracies
E_{M}_{1} =0,E_{M}_{2} =0, andE_{M}_{1}+E_{M}_{1} = 0 as potential candidates. Similarly, Lemma11
tells us thatE_{S} = 0 is also not a useful degeneracy to have inH_{SM}. This leaves us
with 5 potential degeneracies

1. E_{S}=E_{M}_{1}
2. E_{S}=E_{M}_{2}
3. E_{S}=E_{M}_{1} +E_{M}_{2}
4. E_{M}_{1} =E_{S}+E_{M}_{2}
5. E_{M}_{2} =E_{S}+E_{M}_{1}.

One can show explicitly, see Appendix A of [91], that none of these degeneracies lead to cooling except number4

E_{M}_{1} =E_{S}+E_{M}_{2}. (11.63)

This seems to fully take care of the degeneracy question ford_{M} =4. There is one
small subtlety, however. Degenerate subspaces can be defined by more than just one
degeneracy condition. To fully conclude the proof that one cannot cool incoherently
with machines of dimension 4 unless

E_{M}_{1} =E_{S}+E_{M}_{2}, (11.64)

we therefore have to make sure that combined subspaces cannot get activated, i.e., cool, although they individually were useless. For the 2 qubit machine this can be done explicitly. For the details we refer to Appendix A of [91]. We therefore have the following result

**Theorem 5.** Given a qubit target system and a machine of dimension 4, the incoherent
scenario can cool the target if and only if the following holds

1. The machine comprises two qubits M_{1}and M_{2}
2. E_{M}_{1} =E_{S}+E_{M}_{2}

3. Only M_{2}is brought into contact with the hot thermal bath.

Once the machine, as well as which part to heat up, has been identified, the
cooling part of the incoherent scenario is fairly straightforward. Indeed, given the
hot temperatureT_{H}, one heats upM_{2}toT_{H}and then performs the unitary cooling
maximally within the degenerate subspace. Note that the unitary operation is in
particular performed at no cost. In the case of the 2 qubit machine this unitary is
given by

Chapter 11. Qubit System 53

U= |010i h101|_{SM}+|101i h010|_{SM}+_{1}_{span}c{|010i,|101i}, (11.65)
where span^{c}{|010i,|101i}denotes the complement of the set span{|010i,|101i}.
Doing so we get

r_{inc}(T_{H}) =r_{S}r_{M}_{1}+ [(1−r_{S})r_{M}_{1} +r_{S}(1−r_{M}_{1})] (1−r^{H}_{M}_{2}). (11.66)
The associated temperatureT_{inc}(T_{H})is given by the usual formula,

T_{inc}(T_{H}) = ^{E}^{S}
ln

r_{inc}(T_{H})
1−rinc(TH)

. (11.67)

The work cost is calculated from the hot bath. According to Eq.8.4this amounts
to calculating the heat drawn from the bath, Q^{H}, which is equal to the change of
energy ofM_{2}upon heating it. We therefore get

Q^{H} =E_{M}_{2}(r_{M}_{2}−r^{H}_{M}_{2}) (11.68)
and

∆F_{inc}(T_{H}) =E_{M}_{2}(r_{M}_{2}−r^{H}_{M}_{2})

1− ^{T}^{R}
T_{H}

. (11.69)

From this we get the * quantities by taking the limitT_{H} →_{∞, i.e.,}

r^{∗}_{inc}= ^{1}

2(r+rM_{1}) (11.70)

T_{inc}^{∗} = ^{E}^{S}
ln _{r}

S+rM1

2−(rS+rM2)

(11.71)

∆F_{inc}^{∗} = E_{M}_{2}(r_{M}_{2} −^{1}

2). (11.72)

**Coherent Scenario**

We now turn our attention to the coherent scenario. Since we are mostly interested in seeing how it compares to the incoherent scenario, we will impose here the restriction

E_{M}_{1} =E_{S}+E_{M}_{2}, (11.73)

although this is not needed here for the machine to cool. This will also simplify the analysis, as will become clearer in the following. As in Section11.2.2, we are interested in solving a special instance of the general problem of Eq.10.28. More precisely, we are interested in solving

v≺minD(*ρ*_{SM})v·D(H_{SM}), s.t.

### ∑

3 i=0v_{i} =c, (11.74)

with

Chapter 11. Qubit System 54

We first would like to determiner^{∗}_{coh}to know what rangecis allowed to evolve
in, and also to know what the maximal cooling on the target is. For this we need to
determine what the 4 biggest entries ofD(*ρ*_{SM})are. Looking at this more closely, and
taking our restriction of Eq.11.73into account, we find the following ordering

[D(*ρ*_{SM})]_{0}>{[D(*ρ*_{SM})]_{1}_{,}[D(*ρ*_{SM})]_{4}}> [D(*ρ*_{SM})]_{2}= [D(*ρ*_{SM})]_{5}> _{(11.81)}

which gives an associated temperature of
T_{coh}^{∗} = ^{E}^{S}

E_{M}_{1}^{T}^{R}^{.} ^{(11.86)}

A transformation achieving this cooling is the unitaryUswapping the energy
eigenstates|011i_{SM}

Chapter 11. Qubit System 55
This is however not the unitary operating at the minimal work cost since the
state that one gets after performingUis not passive in the respective subspaces of
Lemma4. We will now work out the energetically most efficient transformation,
which depends on the ordering ofD(*ρ*_{SM}). As the ordering depends on whether
E_{M}_{2} ≤E_{S}orE_{M}_{2} > E_{S}, we treat both cases separately.

**a.**E_{M}_{2} ≤E_{S} In this case we have

[D(*ρ*_{SM})]_{0}>[D(*ρ*_{SM})]_{1},[D(*ρ*_{SM})]_{4} >[D(*ρ*_{SM})]_{5}= [D(*ρ*_{SM})]_{2}>

>[D(*ρ*_{SM})]_{3}_{,}[D(*ρ*_{SM})]_{6} >[D(*ρ*_{SM})]_{7}_{,} ^{(11.88)}
and so the vectorv≤ ∈_{R}^{8}minimizing Eq.11.74forc=r^{∗}_{coh}is given by

v= ([D(*ρ*_{SM})]_{0},[D(*ρ*_{SM})]_{1},[D(*ρ*_{SM})]_{4},[D(*ρ*_{SM})]_{5},[D(*ρ*_{SM})]_{2}, (11.89)
,[D(*ρ*_{SM})]_{3},[D(*ρ*_{SM})]_{6},[D(*ρ*_{SM})]_{7}). (11.90)
One checks that, indeed, the first half ofv≤,

a_{v}_{≤} = ([D(*ρ*_{SM})]_{0},[D(*ρ*_{SM})]_{1},[D(*ρ*_{SM})]_{4},[D(*ρ*_{SM})]_{5}), (11.91)
is inversely ordered with respect the first half ofD(H_{SM}),

a_{D}_{(}_{H}_{SM}_{)}= (_{0,}E_{M}_{2}_{,}E_{M}_{1}_{,}E_{M}_{1}+E_{M}_{2}), (11.92)
and that the second half ofv≤,

b_{v}_{≤} = ([D(*ρ*_{SM})]_{2},[D(*ρ*_{SM})]_{3},[D(*ρ*_{SM})]_{6},[D(*ρ*_{SM})]_{7}), (11.93)
is also inversely ordered with respect the second half ofD(H_{SM}),

b_{D}_{(}_{H}_{SM}_{)}= (E_{S},E_{S}+E_{M}_{2},E_{S}+E_{M}_{1},E_{S}+E_{M}_{1}+E_{M}_{2}). (11.94)
A unitaryU_{≤}^{∗} that achievesv≤is given by

U_{≤}^{∗} =U_{24}U35, (11.95)

with

U_{ij} =|i_{2}i hj_{2}|+|j_{2}i hi_{2}|+_{1}_{span}c{|i2i,|j2i}, (11.96)
wherei_{2}is the 3 digit display ofi=0, . . . , 7 in base 2, e.g., 0_{2} =000. This corresponds
to swapping the energy eigenstates|010i_{SM}with |100i_{SM} as well as|011i_{SM}with

|101i_{SM}. Physically it corresponds to swapping the population ofSwithM_{1}andU_{≤}^{∗}
can be written compactly as

U_{≤}^{∗} =e^{−}^{i}^{π}^{2}^{L}^{SM}^{1}, (11.97)
where

L_{SM}_{k} =i|01i h10|_{SM}

k−i|10i h01|_{SM}

k. (11.98)

The work cost associated toU_{≤}^{∗} is

∆F_{coh}^{∗} = (E_{M}_{1}−E_{S})(r_{M}_{1} −r_{S}) (11.99)

= E_{M}_{2}(r_{M}_{1} −r_{S}). (11.100)

Chapter 11. Qubit System 56
**b.**E_{M}_{2} >E_{S} In this case we have

[D(*ρ*_{SM})]_{0} >[D(*ρ*_{SM})]_{4},[D(*ρ*_{SM})]_{1}> [D(*ρ*_{SM})]_{5}= [D(*ρ*_{SM})]_{2}> (11.101)

>[D(*ρ*_{SM})]_{6},[D(*ρ*_{SM})]_{3} >[D(*ρ*_{SM})]_{7}, (11.102)
and so the vectorv> ∈_{R}^{8}minimizing Eq.11.74forc=r^{∗}_{coh}is given by

v>= ([D(*ρ*_{SM})]_{0},[D(*ρ*_{SM})]_{4},[D(*ρ*_{SM})]_{1},[D(*ρ*_{SM})]_{5},[D(*ρ*_{SM})]_{2}, (11.103)
,[D(*ρ*_{SM})]_{6},[D(*ρ*_{SM})]_{3},[D(*ρ*_{SM})]_{7}). (11.104)
As before, av> andbv> are indeed inversely ordered w.r.t. a_{D}(H_{SM}) andb_{D}(H_{SM})

respectively. A unitaryU^{∗}_{≥}that achievesv>is here given by

U_{>}^{∗} =U_{24}U35U_{14}U36. (11.105)
Physically,U_{14}andU36 together correspond to swapping the populations ofS
andM_{2}. Once this is done,U_{24}U_{35}is applied, which corresponds to swapping the
population ofSwithM_{1}. One can writeU_{>}^{∗} compactly as

U_{>}^{∗} =e^{−}^{i}^{π}^{2}^{L}^{SM}^{1}e^{−}^{i}^{π}^{2}^{L}^{SM}^{2}. (11.106)
The work cost associated toU_{>}^{∗} is

∆F_{coh,}^{∗} > = (E_{M}_{2}−E_{S})(r_{M}_{2} −r_{S}) +E_{M}_{2}(r_{M}_{1}−r_{M}_{2}). (11.107)
All the analysis of this section until now is a straight exemplification of Lemma4.

It allowed us to nevertheless gather some important intuition about the general case
ofr_{coh} ∈ [r_{S},r^{∗}_{coh}]. We also saw that in the caseE_{M}_{2} > E_{S}, one could adopt a better
strategy than swapping the populations ofSand M_{1}directly by swapping the
popu-lations ofSwith that ofM_{2}first. The achieved temperature on the target is the same
but the work cost differs between both strategies.

Next we turn our attention to the case, where r_{coh} ∈ [r_{S},r^{∗}_{coh}]. Here we again
distinguish between the caseE_{M}_{2} ≤ E_{S}andE_{M}_{2} >E_{S}. ForE_{M}_{2} ≤E_{S}we find a result
that is a complete analog of the 1 qubit machine result of Section11.2.2.

**Theorem 6.** LetE_{M}_{2} ≤ E_{S}. Let r_{coh} ∈ [r_{S},r^{∗}_{coh}]. Let*µ* = _{r}^{r}^{coh}^{−}^{r}^{S}

M1−rS. Let t = arcsin(√
*µ*).
Then

v=D(Uρ_{SM}U^{†}), with U =e^{−}^{it}^{L}^{SM}^{1} (11.108)
minimizes the optimization problem of Eq.11.74for c=r_{coh}and has an associated work cost
of

∆F_{coh} = (r_{coh}−r_{S})(E_{M}_{1}−E_{S})_{.} _{(11.109)}
Proof idea. The idea of the proof is exactly the same as that of Theorem4, namely to
rewrite

v·D(H_{SM}) (11.110)

such that the majorization conditions as well as the constraint can naturally be
expressed. The practical rewriting depends on the ordering ofD(*ρ*_{SM})as well as how
E_{S},E_{M}_{1}, andE_{M}_{2} relate to one another. In the caseE_{M}_{1} = E_{S}+E_{M}_{2} andE_{M}_{2} ≤ E_{S}the
ordering ofD(*ρ*_{SM})is fixed, and given by Eq.11.88, and the useful rewriting is the

Chapter 11. Qubit System 57 Using again that the minimum of the sum is greater than the sum of the minima we get

Q_{ij} : permutation matrix exchanging coordinatesiandjonly, (11.115)
*µ*(c) = ^{c}−r_{S}

r_{M}_{1}−r_{S}. (11.116)

As∑^{3}_{i}=0[v(c)]_{i} =c,v(r_{coh})is the solution of our problem. For further details, the
rewriting ofv(r_{coh})as in the statement, as well as the expression of∆F_{coh}we refer to
Appendix C of [91].

The solution of Theorem6corresponds to partially performing the swapping ofS
andM_{1}instead of performing the full swap. In the case ofE_{M}_{2} > E_{S}we also find a
minimizes the optimization problem of Eq.11.74for c=r_{coh}and has an associated work cost
of

∆Fcoh=

((r_{coh}−r_{S})(E_{M}_{2} −E_{S}) ,if r≤r_{coh} <r_{M}_{2}
(rM_{2}−r_{S})(E_{M}_{2} −E_{S}) + (r_{coh}−rM_{2})(E_{M}_{1}−E_{S}) ,if rM_{2} ≤r_{coh} <rM_{1}.

(11.119)

Chapter 11. Qubit System 58
Proof idea. In this case we have two practical rewritings ofv·D(H_{SM})depending on
the value ofr_{coh}. The first one is practical forr_{coh}∈ [r_{S},rM_{2}]and is the following

v·D(H_{SM}) =−v_{0}E_{M}_{2} + (−v_{0}−v_{1}−v_{4})(E_{M}_{1} − E_{M}_{2}) +v_{7}E_{M}_{2}

+ (v_{6}+v_{3}+v_{7})(E_{M}_{1}− E_{M}_{2}) + (v_{5}+v_{6}+v_{3}+v_{7})(2E_{M}_{2} − E_{M}_{1})
+E_{M}_{1}+E_{S}− E_{M}_{2} +

### ∑

3 i=_{0}

v_{i}(E_{M}_{2}−E_{S}).

(11.120) This leads to

min

v≺D(*ρ*SM)

∑^{3}i=0vi=_{c}

v·D(H_{SM})≥ v_{1}(c)·D(*ρ*_{SM}), (11.121)

where

v_{1}(c) =T_{14}(*µ*_{1}(c))T_{36}(*µ*_{1}(c))D(*ρ*_{SM}), (11.122)
with

*µ*_{1}(c) = ^{c}−r_{S}

r_{M}_{2}−r_{S}. (11.123)

As∑^{3}i=0[v_{1}(c)]_{i} = cforc ∈ [r_{S},rM_{2}], v_{1}(r_{coh})is the solution of our problem for
r_{coh}∈ [r_{S},r_{M}_{2}]. Forr_{coh}∈[r_{M}_{2},r_{M}_{1}]we use the following rewriting

v·D(H_{SM}) =−v_{0}E_{M}_{2} + (−v_{0}−v_{1})(E_{M}_{1}− E_{M}_{2}) + (v_{6}+v_{7})(E_{M}_{1} − E_{M}_{2}) (11.124)
+v_{7}E_{M}_{2} + (v_{3}+v_{5}+v_{6}+v_{7})E_{M}_{2}+E_{S}+

### ∑

3 i=0v_{i}(E_{M}_{1}−E_{S}). (11.125)
This leads to

min

v≺D(_{ρ}_{SM})

∑^{3}i=0vi=c

v·D(H_{SM})≥ v_{2}(c)·D(*ρ*_{SM}), (11.126)

where

v_{2}(c) =T_{24}(*µ*_{2}(c))T_{35}(*µ*_{2}(c))T_{14}(1)T_{36}(1)D(*ρ*_{SM}), (11.127)
with

*µ*_{2}(c) = ^{c}−r_{M}_{2}
rM_{1}−rM_{2}

. (11.128)

As∑^{3}i=0[v_{2}(c)]_{i} = cforc∈ [r_{M}_{2},r_{M}_{1}],v_{2}(r_{coh})is the solution of our problem for
r_{coh}∈ [rM_{2},rM_{1}]. For further details we refer to Appendix C of [91].

What Theorem7says is that forr_{coh} ∈ [r_{S},r_{M}_{2}], the best strategy is to partially
swap the populations ofSand M_{2}. Indeed, in that case*µ* ∈ [0,^{1}_{2}]and sog(*µ*) =0
and f(*µ*) =arcsin(^{p}2µ), so that

U=e^{−}^{i}^{arcsin}^{(}

√2µ)L_{SM}

2. (11.129)

Ifr_{coh} ∈(r_{M}_{2},r_{M}_{1}], Theorem7tells us that the best strategy is to fully swap the
populations ofSandM_{2}and then subsequently partially swap the populations ofS
andM_{1}. Indeed, in that case*µ*∈(^{1}_{2}, 1]so that f(*µ*) = ^{π}_{2} andg(*µ*) =arcsin(^{p}2µ−1).

Chapter 11. Qubit System 59 We therefore have

U =e^{−}^{i}^{arcsin}^{(}

√

2µ−1)L_{SM}_{1}

e^{−}^{i}^{π}^{2}^{L}^{SM}^{2}. (11.130)
Note that it is a priori not evident at all, at least to us, that out of all the 8×8
unitaries that one could potentially choose, that the ones performing the partial swaps
of the end result are the most efficient in terms of energy expenditure. The proof has
to deal with all these possibilities at first and makes sure that no other transformation
performs better. To do so it uses a lot of the fine-tuned information about the problem,
i.e., the specific ordering ofD(*ρ*_{SM})as well as the relationship of the different energy
levels of the joint systemSM. Given the end results and the fact that they are so
intuitive suggests, however, that this fine-tuned information might not, after all, be
that crucial. It would be interesting to find a proof that is based on a more general
principle. This might allow generalizing the result to more complicated machines
more easily as well.

**Coherent vs. Incoherent**

Now that we have analyzed both scenarios in detail for the two qubit machine
cool-ing a qubit target, we would like to see how they compare both in terms of coolcool-ing
performance and in terms of cooling for a given amount of injected work cost. This
comparison is summarized in Figure11.1, where the amount of cooling vs. the
as-sociated work cost is mapped out. The incoherent curve is generated by plotting
T_{inc}(TH), Eq.11.67, and∆Finc(TH), Eq.11.69, parametrically in the hot bath
tempera-tureT_{H} ∈ [T_{R},+_{∞}]. The coherent curve is generated by plottingT_{coh}(r_{coh}), Eq.11.4,
and ∆F_{coh}(r_{coh}), Eq. 11.163, parametrically in the target ground state population
r_{coh}∈ [r_{S},r_{coh}^{∗} ].

For the Figure we selectedE_{M}_{2} < E_{S}. But apart from the coherent curve having
a discontinuity in the first derivative atr_{coh} = r_{M}_{2}, the behavior of the curves for
E_{M}_{2} ≥E_{S}are qualitatively the same.

There are a few interesting observations that can be made from this comparison.

First of all, looking at the end points of the curve, we see that one can coherently cool more than incoherently and that one does so at a lower work cost. This is true in general as one can easily analytically prove that

T_{coh}^{∗} < T_{inc}^{∗} , (11.131)

∆F_{coh}^{∗} < _{∆}F_{inc}^{∗} , (11.132)
always holds, see [91]. The coherent scenario therefore always performs better than
the incoherent scenario for maximal cooling in the single cycle regime. However,
the coherent scenario is not universally superior to the incoherent one. Indeed, for
sufficiently low work cost, the incoherent scenario always outperforms the coherent
one. This is suggested by Figure11.1and can be easily proven in general by
com-puting the initial slope of each curve. The incoherent scenario starts with an infinite
slope while the coherent one with a finite one. The incoherent curve therefore always
lies below the coherent one at the beginning. The incoherent and coherent curves
must eventually cross since the endpoint of the coherent curve lies below that of the
incoherent one. There therefore exists a critical work cost∆Fcritsuch that if one only
has at its disposal some work cost∆F to invest that is smaller than∆F_{crit}, one can
cool more incoherently than coherently.

Chapter 11. Qubit System 60

FIGURE11.1: Parametric plot of the relative temperature of the target
qubit _{T}^{T}_{R} as a function of its work cost∆FforE_{M}_{2} =0.4 andTR =1.

The red solid curve corresponds to the incoherent scenario, the blue
dashed, to the coherent scenario. When the cooling is maximal (i.e., the
work cost is unrestricted), the coherent scenario always outperforms
the incoherent one,T_{coh}^{∗} <T_{inc}^{∗} and∆F_{coh}^{∗} <_{∆F}_{inc}^{∗} . However, below a
critical work cost∆Fcrit, the incoherent scenario always outperforms

the coherent one..