
can be pumped into the system leading to trivial
“resource” theories. We are able to circumvent
this problem thanks to the notion of bound energy,
which intrinsically distinguishes accessible and non-
accessible energy. Our formalism results in a theory
in which systems and environments are treated on an
equal footing, or in other words, in a “temperature”-
independent formulation of thermodynamics.
Our work is complementary to other general ap-
proaches, where thermodynamics is obtained after in-
serting some form of thermal state(s) in general math-
ematical expressions, e.g. [41]. While in these works
the mathematical expressions for arbitrary states have
no a priori thermodynamic meaning, our construc-
tion is built on a physically motivated quantity, the
bound energy.
The present temperature-independent thermody-
namics is essential in contexts in which the state of
the bath can be affected by the system after exchange
of heat (see [42] for a review on the notion of temper-
ature). This can be due to the fact of either having a
relatively small environment compared to the system,
or an environment simply not being thermal. In fact,
in current experiments, environments do not have to
be necessarily thermal, but can possess quantum co-
herence or correlations.
The entropy preserving operations make all states
with equal energies and entropies thermodynamically
equivalent. This allows for representing all the states
and thermodynamic processes in a simple energy-
entropy diagram. We exploit this geometric ap-
proach and give a diagrammatic representation for
heat, work and other thermodynamic quantities. In
this way we are able to reproduce several results of
the literature, e.g. the resource theory of thermody-
namics applicable for arbitrary quantum systems and
environments [43]. Our formalism extends naturally
to scenarios with multiple conserved quantities.
The structure of the rest of the paper is as follows:
In the following Section 2, we discuss the class of
entropy preserving operations, on which our theory
is built. Then, in Section 3, we introduce the fun-
damental notions of bound and free energy, and in
Section 4 define the energy-entropy diagram of a sys-
tem. After that, we develop thermodynamics, from
the zeroth law (Section 5) and a discussion of the
max-entropy vs. min-energy principle (Section 6), to
the first law (Section 7) and the second law of ther-
modynamics (Section 8). In Section 9 we discuss if,
and why not, a third law can be formulated in our
setting. Then, in Section 10, we move to the devel-
opment of a resource theory based on the previous
formalism, which allows us to consider transforma-
tion rates between states and visualize the results in
the energy-entropy diagram. In Section 11, we out-
line how the theory so far developed generalizes to
the case of multiple conserved quantities. Finally, we
summarize our findings in Section 12, and discuss the
main obstructions towards an extension of thermo-
dynamics that would be valid both in the single-shot
scenario and for non-thermal environments.
2 Entropy preserving operations, en-
tropic equivalence class and intrinsic
temperature
The set of operations that we consider in this frame-
work is the set of, so called, entropy preserving oper-
ations. Given a system initially in a state ρ, the set of
entropy preserving operations are all the operations
that arbitrarily change the state, but keep its entropy
constant
ρ → σ s.t. S(ρ) = S(σ), (1)
where S(ρ) B −Tr ρ log ρ is the von Neumann en-
tropy. Importantly, an operation that acts on ρ and
produces a state with the same entropy, not necessar-
ily preserves entropy when acting on other states. In
fact, such entropy preserving operations are in gen-
eral not linear, since they have to be constrained to
some input state. It was shown in Ref. [36] that a
quantum channel Λ(·) that preserves entropy and, at
the same time, respects linearity, i. e. Λ(pρ
1
+ (1 −
p)ρ
2
) = pΛ(ρ
1
) + (1 −p)Λ(ρ
2
), has to be a unitary.
However, the entropy preserving operations can be
microscopically described by global unitaries in the
limit of many copies [43]. Given any two states ρ
and σ with equal entropies S(ρ) = S(σ), there exists
an additional system of O(
√
n log n) ancillary qubits
and a global unitary U such that, as n → ∞,
Tr
anc
Uρ
⊗n
⊗ ηU
†
− σ
⊗n
1
→ 0, (2)
where the partial trace is performed on the ancillary
qubits. Here, kXk
1
B Tr
√
X
†
X is the one-norm.
As shown in [43, Thm. 4], the reverse statement is
also true. In other words, if two states are related as
in Eq. (2), then they also have equal entropies.
It is important to restrict entropy preserving op-
erations that are also energy preserving, as we shall
see later. The energy and entropy preserving opera-
tions can also be implemented using a global energy
Accepted in Quantum 2018-11-28, click title to verify 3