Hierarchy of quantum operations in manipulating coherence and entanglement Hayata Yamasaki1,2,3, Madhav Krishnan Vijayan4, and Min-Hsiu Hsieh4,5 1Photon Science Center, Graduate School of Engineering, The University of Tokyo, 7–3–1 Hongo, Bunkyo-ku, Tokyo 113–8656, Japan 2Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria 3Atominstitut, Technische Universit¨at Wien, Stadionallee 2, 1020 Vienna, Austria 4Centre for Quantum Software & Information (UTS:QSI), University of Technology Sydney, Sydney NSW, Australia 5Hon Hai Quantum Computing Research Center, Taipei City, Taiwan Quantum resource theory under differ- whether in computation [28, 64], communication [66], ent classes of quantum operations advances or cryptography [45], arise from various inherent prop- multiperspective understandings of inherent erties of quantum mechanics, such as quantum co- quantum-mechanical properties, such as quan- herence and quantum entanglement. Quantum re- tum coherence and quantum entanglement. source theories [14, 34, 35] have grown to be an im- We establish hierarchies of different opera- portant theoretical framework for quantitative anal- tions for manipulating coherence and entan- yses of such properties from operational perspectives glement in distributed settings, where at least using information processing tasks. A resource the- one of the two spatially separated parties are ory is conventionally defined by specifying a class of restricted from generating coherence. In these allowed operations as free operations. One way to settings, we introduce new classes of opera- choose free operations may be to use practical or ex- tions and also characterize those maximal, i.e., perimental restrictions. For example, the resource the resource-non-generating operations, pro- theory of entanglement can be defined by consider- gressing beyond existing studies on incoherent ing a distributed setting for multiple parties with ac- versions of local operations and classical com- cess only to local operations on each party's quantum munication and those of separable operations. system [25, 32, 46]; then, local operations and clas- The maximal operations admit a semidefinite- sical communication (LOCC) [19, 23, 73] may arise programming formulation useful for numeri- as a natural candidate for free operations. Entan- cal algorithms, whereas the existing operations glement serves as a resource for distributed quan- not. To establish the hierarchies, we prove a tum information processing where spatially separated sequence of inclusion relations among the op- parties are restricted to LOCC, by enabling quan- erations by clarifying tasks where separation tum teleportation [5] and allowing for the imple- of the operations appears. We also demon- mentation of nonlocal operations to the shared sys- strate an asymptotically non-surviving sepa- tem [73, 75, 76]. Yet importantly, to deepen our ration of the operations in the hierarchy in understandings of entanglement, it is also crucial to terms of performance of the task of assisted introduce and exploit larger classes of free opera- coherence distillation, where a separation in a tions than LOCC, such as separable (SEP) opera- one-shot scenario vanishes in the asymptotic tions [49, 58] and positive-partial-transpose (PPT) limit. Our results serve as fundamental ana- operations [50], in analytical and numerical studies lytical and numerical tools to investigate in- of entanglement [25, 32, 46]. Along with the studies terplay between coherence and entanglement of entanglement, distributed settings also commonly arXiv:1912.11049v3 [quant-ph] 22 Jun 2021 under different operations in the resource the- arise in other resource theories, where each party has ory. a restricted power of manipulating given quantum re- sources rather than performing arbitrary local opera- tions, and needs assistance of another party in using 1 Introduction the given resources [10, 37, 43, 52, 59]. Advantages of quantum information processing In this paper, we investigate the distributed set- over conventional classical information processing, tings that involve two prominent resource theories, entanglement and coherence [16, 24, 41, 57, 71]. In Hayata Yamasaki: [email protected] particular, in the spirit of studying LOCC, SEP, and Madhav Krishnan Vijayan: [email protected] PPT operations in entanglement theory, we intro- Min-Hsiu Hsieh: [email protected] duce and study different natural classes of operations Accepted in Quantum 2021-06-15, click title to verify. Published under CC-BY 4.0. 1 in the distributed settings of manipulating coherence A B and entanglement, and compare their relative power in performing information theoretic tasks. Coher- CC ence, i.e., superposition of a certain set of quantum states, has been shown to play important roles in quantum biology [33], quantum thermodynamics [27] AB and photonic experiments [7], where certain states are <latexit sha1_base64="c5sCUn7HqKKbgcan9K1YemZl/z0=">AAACLHicbVDLSgNBEJyNrxiNSfToZTGInsKuig/wEPXiSSKYhyQxzE4myZCZnWWmVwhLvsKr/oIXf8WLiEc95hvczQYxxoKGoqqb6m7H40yDZb0Zibn5hcWl5HJqZTW9lsnm1ita+orQMpFcqpqDNeXMpWVgwGnNUxQLh9Oq07+I/Oo9VZpJ9wYGHm0K3HVZhxEMoXTb8DS7C87Oh61s3ipYY5izxJ6QfHFnNMqcvnyWWjkj3WhL4gvqAuFY67ptedAMsAJGOB2mGr6mHiZ93KX1kLpYUN0MxhsPze1QaZsdqcJywRyrvycCLLQeCCfsFBh6+q8Xif95dR86x82AuZ4P1CVxUMfnJkgzOt9sM0UJ8EFIMFEs3NUkPawwgfBJUymOwn0KU3cEUR5IyfW07DgifuBJhMOfd82Syl7B3i8cXNv54hWKkUSbaAvtIhsdoSK6RCVURgQJ9IAe0ZPxbLwa78ZH3JowJjMbaArG1zdX8K0F</latexit> easier to create than their superposition. The re- Quantum source theory of coherence [56] is a well-established or Incoherent resource theory that is useful for introducing classi- Incoherent fications, partial orders, and quantifications of quan- tum coherence. The resource theory of coherence con- Figure 1: Distributed manipulation of coherence and entan- glement of a quantum state ψAB shared between two spe- siders situations where coherence cannot be created cially separated parties A and B. The parties manipulate on a quantum system due to a restriction of opera- their local quantum systems, while they can use classical tions for manipulating the system. The free states communication (CC). Local operations and classical commu- in the resource theory of coherence are states rep- nication with one party B restricted to incoherent operations resented as diagonal density operators in some fixed are called LQICC, which can be regarded as a client-server basis. As is the case of entanglement, several different setting where the ability of the client B to generate coher- free operations that preserve diagonal density opera- ence is restricted while the server A can perform any local tors have been well investigated, such as incoherent quantum operation to assist B. Local operations and classi- operations (IO) [4, 67], maximally incoherent oper- cal communication with both parties A and B restricted to ations (MIO) [47, 48], strictly incoherent operations incoherent operations are called LICC, where the abilities of (SIO) [67, 71], and physically incoherent operations A and B are the same. (PIO) [12, 13], to name a few. The resource theory of coherence in the distributed settings has also attracted attentions of broad interests [16, 24, 41, 57, 71], as lem, more general classes of operations than LOCC, with the distributed settings in other resource theo- such as separable operations and PPT operations, ries. These resource theories for the distributed ma- are vital to investigating performances of information nipulation of coherence provide a framework for inves- processing tasks, which also yields bounds of the per- tigating an interplay between coherence and entangle- formance under LOCC. Especially, PPT operations ment in various information processing tasks such as provide numerical algorithms for calculating perfor- distillation and dilution of these resources [16], as- mance of the entanglement-assisted tasks by means sisted distillation of coherence [10, 52, 59], quantum of semidefinite programming (SDP) [65], even if the state merging [55], quantum state redistribution [2], corresponding tasks under LOCC are hard to analyze and multipartite state transformation [9, 39]. From due to its mathematical structure [26, 50, 53, 60{63]. a practical perspective, the distributed manipulation Similarly, in the resource theory of coherence, MIO of coherence naturally arises in photonic systems as serves as a class of operations beyond IO, and MIO demonstrated in recent experiments [68{70]. provides numerical algorithms based on SDP simi- larly to PPT operations [51]. Importantly, even if the In the distributed settings of manipulating coher- operations such as SEP, PPT, and MIO are defined ence, especially for two parties A and B, LOCC with mathematically, these different classes of operations one party restricted to IO are called LQICC, and provide efficiently calculable bounds in analyzing the those with both parties restricted to IO are LICC [57], information processing tasks and crucially help us to as depicted in Fig.1. LQICC can be regarded as a understand the properties of resources in the study client-server setting where the client's ability to gen- of the resource theories. In the same way, in our dis- erate coherence is restricted, while the abilities of two tributed settings of manipulating coherence and en- parties in LICC are the same. The set of free states tanglement, we may suffer from the difficulty if the for LQICC, that is,