## What is/are Incoherent Operations?

Incoherent Operations - In the unassisted setting, we introduce a variant of randomness extraction framework where free incoherent operations are allowed before the incoherent measurement and the randomness extractors.^{[1]}In the unassisted setting, we introduce a variant of randomness extraction framework where free incoherent operations are allowed before the incoherent measurement and the randomness extractors.

^{[2]}Moreover, using this notion and the majorization relation, we obtain a necessary condition for the conversion of general quantum states by means of incoherent operations.

^{[3]}Strictly incoherent operations (SIO) proposed in Winter and Yang (2016) [13] are promising to be a good candidate of free operations in the resource theory of quantum coherence, setting against the central role of local operations and classical communication in the resource theory of quantum entanglement.

^{[4]}Experimental demonstration of one-shot coherence distillation: realizing N-dimensional strictly incoherent operations Shao-Jie Xiong,1,2 Zhe Sun,1,* Qi-Ping Su,1 Zheng-Jun Xi,3 Li Yu,1 Jin-Shuang Jin,1 Jin-Ming Liu,2 Franco Nori,4,5,7 AND Chui-Ping Yang1,6,8 Department of Physics, HangzhouNormal University, Hangzhou, Zhejiang 310036, China State Key Laboratory of Precision Spectroscopy, Department of Physics, East China Normal University, Shanghai 200062, China College of Computer Science, Shaanxi Normal University, Xian 710062, China Theoretical QuantumPhysics Laboratory, RIKENCluster for Pioneering Research,Wako-shi, Saitama 351-0198, Japan Department of Physics, University ofMichigan, Ann Arbor, Michigan 48109-1040, USA Quantum Information Research Center, ShangraoNormal University, Shangrao 334001, China e-mail: fnori@riken.

^{[5]}We study the conversion between pure states and X states under incoherent operations.

^{[6]}, $\overline{C}_{\mathcal{R}}$), where the role of genuinely incoherent operations (GIO) is highlighted.

^{[7]}Here, we solve this question completely for qubit states by determining the optimal probabilities for mixed-state conversions via stochastic incoherent operations.

^{[8]}In this work, we have provided a classification of states under local incoherent operations.

^{[9]}First, we give a necessary and sufficient condition for when two pure multipartite states are equivalent under local incoherent operations assisted by classical communications (LICC), i.

^{[10]}We establish POVM-based coherence measures and POVM-incoherent operations that coincide for the case of von Neumann measurements with their counterparts in standard coherence theory.

^{[11]}Finally, we find that the creation of entanglement with bipartite incoherent operations is bounded by the logarithmic coherence number of the initial system during the process.

^{[12]}We compare two extremal levels of control: first, coherent operations, where the entropy of the resource is left unchanged, and, second, incoherent operations, where only energy at maximum entropy (i.

^{[13]}We consider asymptotic channel distillation and dilution with maximally incoherent operations and find the theory asymptotically irreversible, in contrast to the asymptotic reversibility of the coherence of states.

^{[14]}We compute analytically the maximal rates of distillation of quantum coherence under strictly incoherent operations (SIO) and physically incoherent operations (PIO), showing that they coincide for all states, and providing a complete description of the phenomenon of bound coherence.

^{[15]}In this Letter, we complete the deterministic distillation of quantum coherence for a finite number of coherent states under strictly incoherent operations.

^{[16]}We find that the $\epsilon$-smooth measure of any coherence monotone is still a coherence monotone, but it does not satisfy monotonicity on average under incoherent operations.

^{[17]}Specifically, we determine the distillable coherence with a given fidelity under incoherent operations (IO) through a generalization of the Winter-Yang protocol.

^{[18]}The amount of discord created by strictly incoherent operations is upper-bounded by the initial coherence.

^{[19]}Considering certain well-known quantum cloning machines (input state independent and dependent), we provide examples of coherent and incoherent operations performed by them.

^{[20]}We show that the coherence of a qubit state can be converted to the nonlocality of two-qubit states via incoherent operations.

^{[21]}We find a one to one mapping between genuinely incoherent operations and special one-way local operations and classical communication(LOCC) for density matrices with full rank.

^{[22]}Finally, we provide an operational interpretation of such correlations as those allowing two distant parties to increase their respective local quantum computational resources only using locally incoherent operations and classical communication.

^{[23]}

## Strictly Incoherent Operations

Strictly incoherent operations (SIO) proposed in Winter and Yang (2016) [13] are promising to be a good candidate of free operations in the resource theory of quantum coherence, setting against the central role of local operations and classical communication in the resource theory of quantum entanglement.^{[1]}Experimental demonstration of one-shot coherence distillation: realizing N-dimensional strictly incoherent operations Shao-Jie Xiong,1,2 Zhe Sun,1,* Qi-Ping Su,1 Zheng-Jun Xi,3 Li Yu,1 Jin-Shuang Jin,1 Jin-Ming Liu,2 Franco Nori,4,5,7 AND Chui-Ping Yang1,6,8 Department of Physics, HangzhouNormal University, Hangzhou, Zhejiang 310036, China State Key Laboratory of Precision Spectroscopy, Department of Physics, East China Normal University, Shanghai 200062, China College of Computer Science, Shaanxi Normal University, Xian 710062, China Theoretical QuantumPhysics Laboratory, RIKENCluster for Pioneering Research,Wako-shi, Saitama 351-0198, Japan Department of Physics, University ofMichigan, Ann Arbor, Michigan 48109-1040, USA Quantum Information Research Center, ShangraoNormal University, Shangrao 334001, China e-mail: fnori@riken.

^{[2]}We compute analytically the maximal rates of distillation of quantum coherence under strictly incoherent operations (SIO) and physically incoherent operations (PIO), showing that they coincide for all states, and providing a complete description of the phenomenon of bound coherence.

^{[3]}In this Letter, we complete the deterministic distillation of quantum coherence for a finite number of coherent states under strictly incoherent operations.

^{[4]}The amount of discord created by strictly incoherent operations is upper-bounded by the initial coherence.

^{[5]}

## Genuinely Incoherent Operations

, $\overline{C}_{\mathcal{R}}$), where the role of genuinely incoherent operations (GIO) is highlighted.^{[1]}We find a one to one mapping between genuinely incoherent operations and special one-way local operations and classical communication(LOCC) for density matrices with full rank.

^{[2]}

## Local Incoherent Operations

In this work, we have provided a classification of states under local incoherent operations.^{[1]}First, we give a necessary and sufficient condition for when two pure multipartite states are equivalent under local incoherent operations assisted by classical communications (LICC), i.

^{[2]}

## Free Incoherent Operations

In the unassisted setting, we introduce a variant of randomness extraction framework where free incoherent operations are allowed before the incoherent measurement and the randomness extractors.^{[1]}In the unassisted setting, we introduce a variant of randomness extraction framework where free incoherent operations are allowed before the incoherent measurement and the randomness extractors.

^{[2]}