Catalytic Quantum Randomness

Randomness is a defining element of mixing processes in nature and an essential ingredient to many protocols in quantum information. In this work, we investigate how much randomness is required to transform a given quantum state into another one. Specifically, we ask whether there is a gap between t...

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Bibliographic Details
Published in:Physical review. X 2018-10, Vol.8 (4), p.041016, Article 041016
Main Authors: Boes, P., Wilming, H., Gallego, R., Eisert, J.
Format: Article
Language:English
Subjects:
Algorithms
Cryptography
Expanders
Information theory
Quantum computing
Quantum mechanics
Quantum phenomena
Quantum physics
Quantum theory
Random processes
Randomness
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Summary:Randomness is a defining element of mixing processes in nature and an essential ingredient to many protocols in quantum information. In this work, we investigate how much randomness is required to transform a given quantum state into another one. Specifically, we ask whether there is a gap between the power of a classical source of randomness compared to that of a quantum one. We provide a complete answer to these questions, by identifying provably optimal protocols for both classical and quantum sources of randomness, based on a dephasing construction. We find that in order to implement any noisy transition on ad-dimensional quantum system it is necessary and sufficient to have a quantum source of randomness of dimensiondor a classical one of dimensiond. Interestingly, coherences provided by quantum states in a source of randomness offer a quadratic advantage. The process we construct has the additional features to be robust and catalytic; i.e., the source of randomness can be reused. Building upon this formal framework, we illustrate that this dephasing construction can serve as a useful primitive in both equilibration and quantum information theory: We discuss applications describing the smallest measurement device, capturing the smallest equilibrating environment allowed by quantum mechanics, or forming the basis for a cryptographic private quantum channel. We complement the exact analysis with a discussion of approximate protocols based on quantum expanders deriving from discrete Weyl systems. This gives rise to equilibrating environments of remarkably small dimension. Our results highlight the curious feature of randomness that residual correlations and dimension can be traded against each other.
ISSN:2160-3308
2160-3308
DOI:10.1103/PhysRevX.8.041016