The Wehrl entropy has Gaussian optimizers
arXiv:1703.02552 · doi:10.1007/s11005-017-0994-3
Abstract
We determine the minimum Wehrl entropy among the quantum states with a given von Neumann entropy, and prove that it is achieved by thermal Gaussian states. This result determines the relation between the von Neumann and the Wehrl entropies. The key idea is proving that the quantum-classical channel that associates to a quantum state its Husimi Q representation is asymptotically equivalent to the Gaussian quantum-limited amplifier with infinite amplification parameter. This equivalence also permits to determine the p->q norms of the aforementioned quantum-classical channel in the two particular cases of one mode and p=q, and prove that they are achieved by thermal Gaussian states. The same equivalence permits to prove that the Husimi Q representation of a one-mode passive state (i.e. a state diagonal in the Fock basis with eigenvalues decreasing as the energy increases) majorizes the Husimi Q representation of any other one-mode state with the same spectrum, i.e. it maximizes any convex functional.
Proof extended to multimode states
References in corpus (2)
Cited by in corpus (11)
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- Gaussian optimizers for entropic inequalities in quantum information
- Dissipative evolution of quantum Gaussian states
- Energy upper bound for structurally stable N-passive states
- Phase space density limitation in laser cooling without spontaneous emission
- Formal relation between Pegg-Barnett and Paul quantum phase frameworks
- Uncertainty relations with quantum memory for the Wehrl entropy
- Relating the Glauber-Sudarshan, Wigner and Husimi quasiprobability distributions operationally through the quantum limited amplifier and attenuator channels
- Spectral stabilizability