Abstract
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.Received 13 June 2018Revised 21 August 2018DOI:https://doi.org/10.1103/PhysRevX.8.041016Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasQuantum channelsQuantum correlations in quantum informationQuantum cryptographyQuantum quenchThermodynamicsQuantum InformationInterdisciplinary Physics
Highlights
Randomness is a central concept and resource in various fields of research in computer science, information theory, and physics, in both the classical and the quantum realm
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-dimensionpalffiffiffiquantum system it is necessary and sufficient to have a quantum source of randomness of dimension d or a classical one of dimension d
Summary
Randomness is a central concept and resource in various fields of research in computer science, information theory, and physics, in both the classical and the quantum realm. We give a complete answer to both of the above questions We provide, for both the implicit and explicit model, optimal and tight bounds on the amount of randomness required to implement physical processes on quantum systems. These processes include dephasing and equilibration [2,3], decoherence [4,5], the implementation of measurements [5,6,7], any transition between two quantum states that requires randomness [1], as well as the novel construction of private quantum channels [8,9] It is an important aspect of our work that, by virtue of an explicit model, these saturated lower bounds translate into bounds on the physical size of a SOR. In the sense that we do not require perfect control in either the states prepared by the SOR or the timing of the process, and further recurrent, in the sense that, for large system dimension d, continuous time versions of our noisy processes mapinffiffitffiain a state close to the desired final state for times τ ∝ d, at which point the system recurs to the initial state
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