We consider families of tight upper bounds on the difference $S(\rho)-S(\sigma)$ with a rank/energy constraint imposed on the state $\rho$ , which are valid provided that the state $\rho$ partially majorizes the state $\sigma$ and is close to the state $\sigma$ with respect to the trace norm. The upper bounds within these families depend on the parameter $m$ of partial majorization. The upper bounds corresponding to $m=0$ coincide with the (unconditional) optimal semicontinuity bounds for the von Neumann entropy with a rank/energy constraint obtained in [1] and [2]. State-dependent improvements of these semicontinuity bounds are proposed and analysed numerically. The notion of $\varepsilon$ -sufficient majorization dimension of the set of states with bounded energy is introduced and analysed. Classical versions of the above results formulated in terms of probability distributions and Shannon entropy are also considered. Bibliography: 20 titles.We consider families of tight upper bounds on the difference $S(\rho)-S(\sigma)$ with a rank/energy constraint imposed on the state $\rho$, which are valid provided that the state $\rho$ partially majorizes the state $\sigma$ and is close to the state $\sigma$ with respect to the trace norm. The upper bounds within these families depend on the parameter $m$ of partial majorization. The upper bounds corresponding to $m=0$ coincide with the (unconditional) optimal semicontinuity bounds for the von Neumann entropy with a rank/energy constraint obtained in [1] and [2]. State-dependent improvements of these semicontinuity bounds are proposed and analysed numerically. The notion of $\varepsilon$-sufficient majorization dimension of the set of states with bounded energy is introduced and analysed. Classical versions of the above results formulated in terms of probability distributions and Shannon entropy are also considered. Bibliography: 20 titles.

arXiv:2504.08098Sbornik MathematicsJan 1, 20260 citations

A variation on von Neumann Entropy and a result of Varadarajan

Glenn D. Appleby

Expositiones MathematicaeApr 25, 20241 citation

Majorization-based entanglement criterion for fermion systems using the von Neumann entropy

Y. Akbari-Kourbolagh, E. Rezazadeh-Dizaji

The European Physical Journal DSep 1, 20240 citations

Multimode Gaussian optimizers for the Wehrl entropy and quantum Gaussian channels

Giacomo De Palma, Dario Trevisan, Vittorio Giovannetti

We prove in the multimode scenario a fundamental relation between the Wehrl and the von Neumann entropy, stating that the minimum Wehrl entropy among all the quantum states with a given von Neumann entropy is achieved by thermal Gaussian states. We also prove that thermal Gaussian input states minimize the output von Neumann entropy of multimode quantum Gaussian attenuators, amplifiers and phase-contravariant channels among all the input states diagonal in some product basis and with a given entropy. This result constitutes a major step towards the proof of the same property for generic input states, which is still an open conjecture. This conjecture is necessary to determine the maximum communication rates for the triple trade-off coding and broadcast communication with the Gaussian quantum-limited attenuator. Finally, we prove that the tensor product of n identical geometric input probability distributions minimizes the output Shannon entropy of the n-mode thinning among all the input probability distributions with a given entropy.We prove in the multimode scenario a fundamental relation between the Wehrl and the von Neumann entropy, stating that the minimum Wehrl entropy among all the quantum states with a given von Neumann entropy is achieved by thermal Gaussian states. We also prove that thermal Gaussian input states minimize the output von Neumann entropy of multimode quantum Gaussian attenuators, amplifiers and phase-contravariant channels among all the input states diagonal in some product basis and with a given entropy. This result constitutes a major step towards the proof of the same property for generic input states, which is still an open conjecture. This conjecture is necessary to determine the maximum communication rates for the triple trade-off coding and broadcast communication with the Gaussian quantum-limited attenuator. Finally, we prove that the tensor product of n identical geometric input probability distributions minimizes the output Shannon entropy of the n-mode thinning among all the input probability distributions with a given entropy.

arXiv:1705.00499May 1, 20173 citations

7 von Neumann entropy

Mou-Hsiung Chang

De Gruyter eBooksAug 8, 20220 citations

We consider families of tight upper bounds on the difference

S(\rho)-S(\sigma)

with a rank/energy constraint imposed on the state

\rho

, which are valid provided that the state

\rho

partially majorizes the state

\sigma

and is close to the state

\sigma

with respect to the trace norm. The upper bounds within these families depend on the parameter

m

of partial majorization. The upper bounds corresponding to

m=0

coincide with the (unconditional) optimal semicontinuity bounds for the von Neumann entropy with a rank/energy constraint obtained in [1] and [2]. State-dependent improvements of these semicontinuity bounds are proposed and analysed numerically. The notion of

\varepsilon

-sufficient majorization dimension of the set of states with bounded energy is introduced and analysed. Classical versions of the above results formulated in terms of probability distributions and Shannon entropy are also considered. Bibliography: 20 titles.