Energy-constrained discrimination of unitaries, quantum speed limits and a Gaussian Solovay-Kitaev theorem

Abstract

We investigate the energy-constrained (EC) diamond norm distance between unitary channels acting on possibly infinite-dimensional quantum systems, and establish a number of results. Firstly, we prove that optimal EC discrimination between two unitary channels does not require the use of any entanglement. Extending a result by Acín, we also show that a finite number of parallel queries suffices to achieve zero error discrimination even in this EC setting. Secondly, we employ EC diamond norms to study a novel type of quantum speed limits, which apply to pairs of quantum dynamical semigroups. We expect these results to be relevant for benchmarking internal dynamics of quantum devices. Thirdly, we establish a version of the Solovay--Kitaev theorem that applies to the group of Gaussian unitaries over a finite number of modes, with the approximation error being measured with respect to the EC diamond norm relative to the photon number Hamiltonian.

24 pages, no figures. In v2 we added the estimates in Eq. (3) and (4), and improved Thm 6. In v3, which is close to the published version, we improved the overall presentation (e.g. of the proof of Thm 3 in the SM)

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• Dynamics of Loschmidt echoes and fidelity decay
• The Quantum Chernoff Bound
• Quantum speed limits in open system dynamics
• The Chernoff lower bound for symmetric quantum hypothesis testing
• Monitoring derivation of the quantum linear Boltzmann equation
• Energy-constrained diamond norm with applications to the uniform continuity of continuous variable channel capacities
• Entanglement in channel discrimination with restricted measurements

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