extends HBAC to a larger set that will be called Ex-
tended Heat-Bath Algorithmic Cooling (xHBAC). For
example, the recent protocol introduced in Ref. [10],
which proposes the use of the heat bath to implement
a non-local thermalization ‘state-reset’ (SR) process re-
lated to the Nuclear Overhauser Effect [11], can already
be seen as a (non-optimal) protocol within our extended
family of xHBAC.
For every finite dimensional target state in an arbi-
trary initial state, we optimize the cooling performance
over every xHBAC for any given number of rounds. We
give an analytical form for the optimal cooling opera-
tions, uncovering their elegant structure, and show that
the ground state population goes to 1 exponentially fast
in the number of rounds. This opens up a new avenue
in the experimental realization of algorithmic cooling
schemes, one requiring control over fewer ancillas, but
better control of the interaction with the environment.
Our results suggest that xHBAC schemes provide new
cooling protocols in practically relevant settings. We
show that, for a single qubit target and no ancillas, the
optimal protocol in xHBAC can be approximated by
coupling the qubit by a Jaynes-Cummings (JC) interac-
tion to a single bosonic thermal mode, itself weakly cou-
pled to the external bath. The memory effects present
in the JC interaction are crucial in approximating the
theoretical optimal cooling. The performance of the
ideal cooling protocol is robust to certain kinds of noise
and imperfections, and it outperforms HBAC schemes
with a small number of auxiliary qubits. This makes it,
in our opinion, the most promising proposal for an ex-
perimental demonstration of cooling through xHBAC,
as well as an experimental demonstration that non-
Markovianity can be harnessed to improve cooling.
1 Results and discussion
1.1 A general cooling theorem
A general xHBAC will consist of a number of rounds
and manipulate two types of systems, the target system
S to be purified and the auxiliary systems A. Further-
more, we will assume we can access a thermal environ-
ment at inverse temperature β = (kT )
−1
, with k Boltz-
mann’s constant. We denote by H
S
=
P
d−1
i=0
E
i
|iihi|,
E
0
≤ ··· ≤ E
d−1
, the Hamiltonian of S and by ρ
(k)
S
the state after round k. Also, we denote by H
A
the
Hamiltonian of the auxiliary systems, initially in state
ρ
A
, and by τ
X
= e
−βH
X
/Tr[e
−βH
X
] the thermal state
on X = S, A. A protocol is made of k rounds, each
allowing for (Fig. 1):
1. Unitary. Any unitary U
(k)
applied to SA.
Figure 1: A generic xHBAC protocol.
2. Dephasing thermalization. Any Λ
(k)
applied
to SA, where Λ
(k)
is any quantum map such that
i) Λ
(k)
(τ
S
⊗ τ
A
) = τ
S
⊗ τ
A
(thermal fixed point);
ii) If |Ei, |E
0
i are states of distinct energy on SA,
hE|Λ
(k)
(ρ
SA
)|E
0
i = 0 (dephasing).
At the end of the round, a refreshing stage returns the
auxiliary systems A to their original state ρ
A
. We de-
note the set of protocols whose rounds have this general
form by P
ρ
A
. Typically ρ
A
= τ
A
, and this operation
is simply a specific kind of dephasing thermalization.
However, more generally, ρ
A
may also be the output of
some previous cooling algorithm, subsequently used as
an auxiliary system. Note that A is distinguished by
the rest of the environment in that it is under complete
unitary control.
Some comments about xHBAC protocols are in or-
der. First, if we skipped the dephasing thermaliza-
tions and ρ
A
= τ
A
, we would get back to a standard
HBAC scheme. xHBAC protocols include thermaliza-
tion of subsets of energy levels, as in the mentioned SR
protocol of Ref. [10]. But they are by no means lim-
ited to these strategies. Second, for every k we wish
to find a protocol maximizing the ground state popu-
lation p
(k)
0
= h0|ρ
(k)
S
|0i over all sequences of k-round
operations. Note that often the literature on cooling
is restricted to finding asymptotically optimal proto-
cols (e.g., the PPA [8]), but here we find optimal k
rounds protocols for every k. Finally, we note in passing
that within this work we will not make use of auxiliary
‘scratch qubits’ [9], i.e. S itself is the target system to
be cooled.
To give the analytical form of the optimal protocols
we need to introduce two notions. First, that of maxi-
mally active states. Given a state ρ with Hamiltonian
H, the maximally active state ˆρ is formed by diagonal-
izing ρ in the energy eigenbasis and ordering the eigen-
values in increasing order with respect to energy. That
is, ˆρ is the most energetic state in the unitary orbit of
ρ.
Second, we define the thermal polytope in the follow-
ing way. Given an initial state ρ with populations in
the energy eigenbasis p, it is the set of populations
p
0
of Λ(ρ), with Λ an arbitrary dephasing thermaliza-
tion. Without loss of generality, we assume that ev-
ery energy eigenspace has been diagonalized by energy
preserving unitaries. The thermal polytope is convex
Accepted in Quantum 2019-09-16, click title to verify 2