One-shot operational quantum resource theory (With applications to quantum computation) Zi-Wen Liu Perimeter Institute QIST 2019, YITP, Kyoto 1904.05840, joint with Kaifeng Bu (Zhejiang, Harvard) and Ryuji Takagi (MIT) And several works in progress
Outline • Background and overview • Preliminaries: Theory of resource destroying maps, one-shot divergences and resource monotones • Framework: Resource currencies, golden states, modification coefficients • Main results: Collapse of modification coefficients, optimal rates of one-shot formation and distillation tasks, some general implications • Applications to quantum computation via e.g. magic states • Outlook
Resource theory • Useful • Hard to gain, easy to lose • The more, the better
<latexit sha1_base64="k/8s03dXTZdTCYdePwStsrGOeM=">ACIXicbZDLSgMxFIbP1Fut6pLN8EiuLHMVMEui25cVrAX6JSTNtaCYzJhmhTPsqbnwVNy4U6U58GdN2KNp6IPDx/+ckOb8Xca0bX9ZmbX1jc2t7HZuZ3dv/yB/eFRXYSwJrZGQh7LpYU5E7Smea0GUmKA4/Thje4nfqNJyoVC8WDHka0HeCeYD4jWBupky+7vsQkGdmuxKLH6chJAV2gBS/ceKqR6mT0njcyRfsoj0rtApOCgVIq9rJT9xuSOKACk04Vqrl2JFuJ1hqRszNOTdWNMJkgHu0ZVDgKp2MtwjM6M0kV+KM0RGs3U3xMJDpQaBp7pDLDuq2VvKv7ntWLtl9sJE1GsqSDzh/yYIx2iaVyoyQlmg8NYCKZ+SsifWwi0ybUnAnBWV5FeqlonNZLN1fFSo3aRxZOIFTOAcHrqECd1CFGhB4hld4hw/rxXqzPq3JvDVjpTPH8Kes7x/JA6SK</latexit> Resource theory | 0 i | 1 i � | 1 i | 0 i p 2 • Useful (communication, teleportation, wormholes … ) • Hard to gain, easy to lose (LOCC ⟶ separable states) • The more, the better (telep.: n ebits + 2n cbits ≥ n qubits)
Resource theory A mathematical framework aiming at rigorously, quantitatively characterizing the above resource features. • Building blocks, abstract formulations [Coecke/Fritz/Spekkens, IC ’ 16]: • Building blocks, abstract formulations [Coecke/Fritz/Spekkens, IC ’ 16]: • Free objects (quantum states/density operators): objects that • Free objects (quantum states/density operators): objects that carry no resource carry no resource • Free morphisms (quantum operations/cptp maps): manipulations • Free morphisms (quantum operations/cptp maps): manipulations that are considered easy that are considered easy • Central problem: quantification of resource • Central problem: quantification of resource • Axiomatic: basic criteria, e.g. vanish on free objects, monotonicity • Axiomatic: basic criteria, e.g. vanish on free objects, monotonicity under free morphisms under free morphisms • Operational: physical meanings of the resource measure • Operational: physical meanings of the resource measure • Performance/usefulness in specific tasks/scenarios • Performance/usefulness in specific tasks/scenarios • Value in direct trading between resource entities (more universal • Value in direct trading between resource entities (more universal and fundamental) and fundamental) In this talk, we focus on the state theory. Recently: quantum channels, GPTs [ZWL/Winter, 1904.04201 … ]
Resource theory This scheme has been used to understand and characterize many important quantum features and their power in many scenarios... Theory Free states Free operations Applications Q. communication, Entanglement Separable states LOCC, non-entangling ops … information scrambling … Thermal non- Thermal ops, Gibbs state Work extraction … equilibrium Gibbs-preserving ops … Incoherent (diagonal) Coherence IO, DIO, MIO … Q. transport, metrology … states Stabilizer states Stabilizer ops, Q. computation, Magic state (stabilizer polytope) stabilizer-preserving ops … classical simulation costs … Symmetric states Q. reference frames, Asymmetry Symmetry-preserving ops … (wrt some symm. group) metrology … Discord-type π -commuting ops, Classical-quantum states DQC1, heat transfer … correlation commutativity-preserving ops … Non- Gaussian states Gaussian ops... Q. (optical) computation … Gaussianity
This talk A general, unified quantitative theory of one-shot resource trading. Not specific to any particular resource or any particular task Only one or finite instances of resource are in play Conversion from/to some “ currency ” states ...And also, some explicit applications to the magic state theory, which plays key roles in many key developments on quantum computation.
General resource theory Unified machineries/ understandings Coherence Entanglement Corresponding results Magic states Different resource theories could share lots of common structures... → Let ’ s invent all-purpose resource theory juicers!
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One-shot You only get one shot Do not miss your chance to blow This opportunity comes Once in a lifetime yo — Eminem “Lose Yourself” *Credit to a talk by Nicole Yunger Halpern • Realistic scenario: i) Only finite instances of resource are available; ii) Certain extent of error/inaccuracy is allowed. • Contrast: “ asymptotic ” , i.e. infinite i.i.d. instances (a conventional setting of information theory — think about e.g. entropies, channel capacities; in resource theory: asymptotic reversibility [Brandao/Gour, PRL ’ 15]).
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