TY - JOUR

T1 - Catalytic Quantum Randomness

AU - Boes, P.

AU - Wilming, H.

AU - Gallego, R.

AU - Eisert, J.

N1 - Funding Information: P. B. thanks Lluis Masanes, Markus Müller, Jon Richens, and Ingo Roth for interesting conversations, and especially Jonathan Oppenheim for suggesting cryptographic applications of the results. We acknowledge funding from the ERC (TAQ), the DFG (EI 519/14-1, CRC183, FOR 2724), the Templeton Foundation, and the Studienstiftung des Deutschen Volkes. This work has also received funding from the European Union’s Horizon 2020 research and innovation program under grant agreement No. 817482 (PASQUANS).

PY - 2018/10/29

Y1 - 2018/10/29

N2 - 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 a d-dimensional quantum system it is necessary and sufficient to have a quantum source of randomness of dimension d or a classical one of dimension d. 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.

AB - 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 a d-dimensional quantum system it is necessary and sufficient to have a quantum source of randomness of dimension d or a classical one of dimension d. 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.

UR - http://www.scopus.com/inward/record.url?scp=85057332945&partnerID=8YFLogxK

U2 - 10.1103/PhysRevX.8.041016

DO - 10.1103/PhysRevX.8.041016

M3 - Article

AN - SCOPUS:85057332945

VL - 8

JO - Physical Review X

JF - Physical Review X

SN - 2160-3308

IS - 4

M1 - 041016

ER -