Gibbs-preserving operations achieve strictly higher reliability than thermal operations in asymptotic work extraction.

Trace distance 𝑇(𝜌, 𝜎) between two states 𝜌, 𝜎∈ D (H ) is defined as 𝑇(𝜌, 𝜎):= 1 2 ∥𝜌−𝜎∥ 1

Distance measures and divergences We first review distance measures and information-theoretic quantities used in this work. Trace distance 𝑇(𝜌, 𝜎) between two states 𝜌, 𝜎∈ D (H ) is defined as 𝑇(𝜌, 𝜎):= 1 2 ∥𝜌−𝜎∥ 1. The (square) fidelity 𝐹(𝜌, 𝜎) between two states is defined as 𝐹(𝜌, 𝜎):=∥ √𝜌√𝜎∥ 2

The purified distance 𝑃(𝜌, 𝜎) and the infidelity 𝐼(𝜌, 𝜎) are defined using the fidelity as 𝑃(𝜌, 𝜎)=: √︁ 1−𝐹(𝜌, 𝜎) , 𝐼(𝜌, 𝜎):=𝑃 2 (𝜌, 𝜎) . The Petz 𝛼-R´enyi relative entropy 𝐷 𝛼 (𝜌∥𝜎) and the sandwiched 𝛼-R´enyi relative entropy e𝐷 𝛼 (𝜌∥𝜎) are defined for 𝜌∈ D ≤ (H ) and𝜎∈ P (H )as 𝐷 𝛼 (𝜌∥𝜎)= ( 1 𝛼−1 log Tr 𝜌 𝛼𝜎1−𝛼 if(𝛼 <1∧𝜌̸⊥𝜎) ∨supp(𝜌) ⊂supp(𝜎), +∞otherw...

Hypothesis testing is the task of discriminating between two hypotheses, namelynull hypothesis 𝜌∈ D (H ) andalternative hypothesis 𝜎∈ D (H )

Quantum hypothesis testing We now review standard notions in the information-theoretic task of quantum hypothesis testing. Hypothesis testing is the task of discriminating between two hypotheses, namelynull hypothesis 𝜌∈ D (H ) andalternative hypothesis 𝜎∈ D (H ) . Suppose that we are given either of the hypotheses, 𝜌 and 𝜎, and our goal is to guess which...

For any𝑛∈N,A 𝑛,B 𝑛 are convex and compact

The sequence A= {A𝑛}𝑛∈N ,B= {B𝑛}𝑛∈N are closed under taking tensor-product, that is, A and B satisfyA 𝑛 ⊗ A 𝑚 ⊂ A 𝑚+𝑛 andB 𝑛 ⊗ B𝑚 ⊂ B 𝑚+𝑛. In Ref. [54], it is shown that, for the composite hypotheses satisfying the conditions above, it holds that 𝐵𝐻 (A ∥B;𝑟)=lim 𝑛→∞ 1 𝑛 inf 𝜌𝑛 ∈ A𝑛 𝜎𝑛 ∈ B𝑛 sup 𝛼∈ (0,1) 𝛼−1 𝛼 𝑛𝑟− 𝐷 𝛼 (𝜌 𝑛 ∥𝜎𝑛) , 𝐵∗ 𝐻 (A ∥B;𝑟) ≥lim 𝑛→∞ 1 𝑛 ...

To define this quantity, we need to specify which set of matrices and which distance measure we use for the smoothing

Smoothed max relative entropy We review another important quantity employed throughout this paper, namely the smoothed max relative entropy. To define this quantity, we need to specify which set of matrices and which distance measure we use for the smoothing. For this purpose, we define 𝜀-balls around a quantum state 𝜌 with respect to a distance measure 𝑑...

The subset F(H ) is called the set offree states, the states which can be prepared without any cost in the given scenario

Quantum resource theories A quantum resource theory is identified by a subset F(H ) ⊂ D (H ) of states and a subset O(H 𝐴 → H 𝐵) ⊂ CPTP(A→B) of quantum channels such that for any Λ∈O(H 𝐴 → H 𝐵), Λ(F(H 𝐴)) ⊂F(H 𝐵). The subset F(H ) is called the set offree states, the states which can be prepared without any cost in the given scenario. Similarly, O(H 𝐴 → H...

We study the smallest error with which one can extract the resource state from a fixed reference resource state 𝜙𝑑 from the initial state 𝜌

One-shot optimal error of general resource distillation We are now in a position to discuss the main results. We study the smallest error with which one can extract the resource state from a fixed reference resource state 𝜙𝑑 from the initial state 𝜌. The figure of merit in this 15 scenario is defined as follows. Definition S.3.Let (F,O) be the resource th...

We would like to investigate how the error in resource distillation behaves when one attempts to distill a resource at a fixed rate𝑟

Error exponent and strong converse exponent of general resource distillation Now, we will discuss the asymptotic behavior of the optimal error for resource distillation in general resource theories. We would like to investigate how the error in resource distillation behaves when one attempts to distill a resource at a fixed rate𝑟. If the target rate 𝑟 is ...

First, we consider the one-shot optimal error, where one aims to obtain a target resource state 𝜌from a fixed reference resource state𝜙 𝑑

One-shot optimal error of general resource dilution We consider the fundamental limitation on the precision of the resource dilution in the framework of general resource theories. First, we consider the one-shot optimal error, where one aims to obtain a target resource state 𝜌from a fixed reference resource state𝜙 𝑑. Definition S.7.Let (F,O) be the resour...

Strong converse exponent of the smoothed max relative entropy As we will see later, the performance of the resource dilution is tightly connected to quantities such as the standard robustness and the generalized robustness, which is known to be equivalent to the max relative entropy. Before discussing the fundamental limitation on the precision of resourc...

Such a situation occurs when one attempts to obtain the target state from the initial reference state that is smaller than the optimal dimension required for noiseless dilution

Strong converse exponent of general resource dilution We consider the strong converse exponent of resource dilution, the exponent with which the error of the resource dilution converges to 1. Such a situation occurs when one attempts to obtain the target state from the initial reference state that is smaller than the optimal dimension required for noisele...

Preliminaries for quantum thermodynamics Free operations and free state We consider a quantum system associated with a Hilbert space H and Hamiltonian 𝐻, in contact with a thermal bath of the inverse temperature𝛽. To analyze the amount and precision of work extraction, we employ a resource-theoretic approach, which has proven successful in characterizing ...

First, we define the one-shot optimal error of work extraction by focusing on the fidelity error between the final state and the target excited state as follows

Reliability functions for work extraction (Proofs of Proposition 1 and Theorem 2) Here, we will characterize the optimal precision of work extraction. First, we define the one-shot optimal error of work extraction by focusing on the fidelity error between the final state and the target excited state as follows. Definition S.16.The one-shot optimal error o...

The zero-rate error exponent of work extraction is defined as follows

Zero-rate error exponent of work extraction (Proof of Theorem 3) In the subsequent discussion, we consider the zero-rate error decay of the work extraction, that is, the exponent of the error when we aim at extracting a constant amount of work from an increasing number of quantum states reliably. The zero-rate error exponent of work extraction is defined ...

Πsupp(𝜌) 𝑚∑︁ 𝑘=1 𝜆𝑘 𝐸𝑘 # =Tr

Separation in the optimal precision of work extraction So far, we have seen that the performance of work extraction can depend on the class of allowed operations. However, the expressions for the error exponents in Theorem S.18 and Theorem S.19 involve optimizations, and hence their behavior is not immediately transparent. In this subsection, we analyze t...

(E17), the optimization of the sandwiched R´enyi divergence in the interval𝛼∈ (0,1) appears

Data-processing inequality with covariance (Proof of Proposition 4) In Eq. (E17), the optimization of the sandwiched R´enyi divergence in the interval𝛼∈ (0,1) appears. However, the sandwiched R ´enyi divergence of order 𝛼∈ 0, 1 2 does not satisfy the data-processing inequality [ 38]. 34 Interestingly, despite this fact, we can show the data-processing ine...