6 research outputs found
New lower bounds to the output entropy of multi-mode quantum Gaussian channels
We prove that quantum thermal Gaussian input states minimize the output
entropy of the multi-mode quantum Gaussian attenuators and amplifiers that are
entanglement breaking and of the multi-mode quantum Gaussian phase
contravariant channels among all the input states with a given entropy. This is
the first time that this property is proven for a multi-mode channel without
restrictions on the input states. A striking consequence of this result is a
new lower bound on the output entropy of all the multi-mode quantum Gaussian
attenuators and amplifiers in terms of the input entropy. We apply this bound
to determine new upper bounds to the communication rates in two different
scenarios. The first is classical communication to two receivers with the
quantum degraded Gaussian broadcast channel. The second is the simultaneous
classical communication, quantum communication and entanglement generation or
the simultaneous public classical communication, private classical
communication and quantum key distribution with the Gaussian quantum-limited
attenuator
Uncertainty relations with quantum memory for the Wehrl entropy
We prove two new fundamental uncertainty relations with quantum memory for
the Wehrl entropy. The first relation applies to the bipartite memory scenario.
It determines the minimum conditional Wehrl entropy among all the quantum
states with a given conditional von Neumann entropy and proves that this
minimum is asymptotically achieved by a suitable sequence of quantum Gaussian
states. The second relation applies to the tripartite memory scenario. It
determines the minimum of the sum of the Wehrl entropy of a quantum state
conditioned on the first memory quantum system with the Wehrl entropy of the
same state conditioned on the second memory quantum system and proves that also
this minimum is asymptotically achieved by a suitable sequence of quantum
Gaussian states. The Wehrl entropy of a quantum state is the Shannon
differential entropy of the outcome of a heterodyne measurement performed on
the state. The heterodyne measurement is one of the main measurements in
quantum optics and lies at the basis of one of the most promising protocols for
quantum key distribution. These fundamental entropic uncertainty relations will
be a valuable tool in quantum information and will, for example, find
application in security proofs of quantum key distribution protocols in the
asymptotic regime and in entanglement witnessing in quantum optics
The Entropy Power Inequality with quantum conditioning
The conditional entropy power inequality is a fundamental inequality in
information theory, stating that the conditional entropy of the sum of two
conditionally independent vector-valued random variables each with an assigned
conditional entropy is minimum when the random variables are Gaussian. We prove
the conditional entropy power inequality in the scenario where the conditioning
system is quantum. The proof is based on the heat semigroup and on a
generalization of the Stam inequality in the presence of quantum conditioning.
The entropy power inequality with quantum conditioning will be a key tool of
quantum information, with applications in distributed source coding protocols
with the assistance of quantum entanglement
Linear growth of the entanglement entropy for quadratic Hamiltonians and arbitrary initial states
We prove that the entanglement entropy of any pure initial state of a bipartite bosonic quantum system grows linearly in time with respect to the dynamics induced by any unstable quadratic Hamiltonian. The growth rate does not depend on the initial state and is equal to the sum of certain Lyapunov exponents of the corresponding classical dynamics. This paper generalizes the findings of [Bianchi et al., JHEP 2018, 25 (2018)], which proves the same result in the special case of Gaussian initial states. Our proof is based on a recent generalization of the strong subadditivity of the von Neumann entropy for bosonic quantum systems [De Palma et al., arXiv:2105.05627]. This technique allows us to extend our result to generic mixed initial states, with the squashed entanglement providing the right generalization of the entanglement entropy. We discuss several applications of our results to physical systems with (weakly) interacting Hamiltonians and periodically driven quantum systems, including certain quantum field theory models
Gaussian optimizers for entropic inequalities in quantum information
We survey the state of the art for the proof of the quantum Gaussian
optimizer conjectures of quantum information theory. These fundamental
conjectures state that quantum Gaussian input states are the solution to
several optimization problems involving quantum Gaussian channels. These
problems are the quantum counterpart of three fundamental results of functional
analysis and probability: the Entropy Power Inequality, the sharp Young's
inequality for convolutions, and the theorem "Gaussian kernels have only
Gaussian maximizers." Quantum Gaussian channels play a key role in quantum
communication theory: they are the quantum counterpart of Gaussian integral
kernels and provide the mathematical model for the propagation of
electromagnetic waves in the quantum regime. The quantum Gaussian optimizer
conjectures are needed to determine the maximum communication rates over
optical fibers and free space. The restriction of the quantum-limited Gaussian
attenuator to input states diagonal in the Fock basis coincides with the
thinning, which is the analog of the rescaling for positive integer random
variables. Quantum Gaussian channels provide then a bridge between functional
analysis and discrete probability