Catalytic quantum randomness

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 $\sqrt{d}$ 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 re-used. 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.

arXiv comment18 pages, 6 figures; Fixed typo in expander theorem;

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Cited by in corpus (24)

• Von Neumann entropy from unitarity
• Catalysis of entanglement and other quantum resources
• Correlation in Catalysts Enables Arbitrary Manipulation of Quantum Coherence
• Entropy and reversible catalysis
• Randomness cost of masking quantum information and the information conservation law
• Quantum chaos, equilibration and control in extremely short spin chains
• By-passing fluctuation theorems
• Amplifying asymmetry with correlated catalysts
• The Min-entropy as a Resource for One-Shot Private State Transfer, Quantum Masking and State Transition
• Catalysis in Action via Elementary Thermal Operations
• Randomness for quantum channels:Genericity of catalysis and quantum advantage of uniformness
• Classical analogue of quantum superdense coding and communication advantage of a single quantum system
• Quantum one-time tables for unconditionally secure qubit-commitment
• Faithfulness and sensitivity for ancilla-assisted process tomography
• Delocalized and Dynamical Catalytic Randomness and Information Flow
• Catalytic transformations with finite-size environments: applications to cooling and thermometry
• Unlocking inaccessible performance of the quantum refrigerator with catalysts
• Correlational Resource Theory of Catalytic Quantum Randomness under Conservation Law
• Mixing of quantum states under Markovian dissipation and coherent control
• Experimental Catalytic Amplification of Asymmetry
• Hacking Quantum Networks: Extraction and Installation of Quantum Data
• Finite-size catalysis in quantum resource theories
• Two coupled qubits under the influence of a minimal, phase-sensitive environment
• Thermal operations from informational equilibrium