The Wehrl entropy has Gaussian optimizers

TY - JOUR

T1 - The Wehrl entropy has Gaussian optimizers

AU - de Palma, Giacomo

PY - 2018/1/1

Y1 - 2018/1/1

N2 - 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 with 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.

AB - 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 with 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.

KW - Husimi Q representation

KW - Quantum Gaussian states

KW - Schatten norms

KW - Von Neumann entropy

KW - Wehrl entropy

UR - http://www.scopus.com/inward/record.url?scp=85028983061&partnerID=8YFLogxK

U2 - 10.1007/s11005-017-0994-3

DO - 10.1007/s11005-017-0994-3

M3 - Journal article

AN - SCOPUS:85028983061

SN - 0921-3767

VL - 108

SP - 97

EP - 116

JO - Mathematical Physics Studies

JF - Mathematical Physics Studies

IS - 1

ER -