2,520 research outputs found
The Wehrl entropy has Gaussian optimizers
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.Comment: Proof extended to multimode state
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
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
The conditional entropy power inequality for quantum additive noise channels
We prove the quantum conditional Entropy Power Inequality for quantum
additive noise channels. This inequality lower bounds the quantum conditional
entropy of the output of an additive noise channel in terms of the quantum
conditional entropies of the input state and the noise when they are
conditionally independent given the memory. We also show that this conditional
Entropy Power Inequality is optimal in the sense that we can achieve equality
asymptotically by choosing a suitable sequence of Gaussian input states. We
apply the conditional Entropy Power Inequality to find an array of
information-theoretic inequalities for conditional entropies which are the
analogues of inequalities which have already been established in the
unconditioned setting. Furthermore, we give a simple proof of the convergence
rate of the quantum Ornstein-Uhlenbeck semigroup based on Entropy Power
Inequalities.Comment: 26 pages; updated to match published versio
The One-Mode Quantum-Limited Gaussian Attenuator and Amplifier Have Gaussian Maximizers
We determine the p->q norms of the Gaussian one-mode quantum-limited
attenuator and amplifier and prove that they are achieved by Gaussian states,
extending to noncommutative probability the seminal theorem "Gaussian kernels
have only Gaussian maximizers" (Lieb in Invent Math 102(1):179-208, 1990). The
quantum-limited attenuator and amplifier are the building blocks of quantum
Gaussian channels, which play a key role in quantum communication theory since
they model in the quantum regime the attenuation and the noise affecting any
electromagnetic signal. Our result is crucial to prove the longstanding
conjecture stating that Gaussian input states minimize the output entropy of
one-mode phase-covariant quantum Gaussian channels for fixed input entropy. Our
proof technique is based on a new noncommutative logarithmic Sobolev
inequality, and it can be used to determine the p->q norms of any quantum
semigroup.Comment: Annales Henri Poincar\'e (2018
Experiments testing macroscopic quantum superpositions must be slow
We consider a thought experiment where the preparation of a macroscopically
massive or charged particle in a quantum superposition and the associated
dynamics of a distant test particle apparently allow for superluminal
communication. We give a solution to the paradox which is based on the
following fundamental principle: any local experiment, discriminating a
coherent superposition from an incoherent statistical mixture, necessarily
requires a minimum time proportional to the mass (or charge) of the system. For
a charged particle, we consider two examples of such experiments, and show that
they are both consistent with the previous limitation. In the first, the
measurement requires to accelerate the charge, that can entangle with the
emitted photons. In the second, the limitation can be ascribed to the quantum
vacuum fluctuations of the electromagnetic field. On the other hand, when
applied to massive particles our result provides an indirect evidence for the
existence of gravitational vacuum fluctuations and for the possibility of
entangling a particle with quantum gravitational radiation.Comment: 12 pages, 1 figur
Classical capacity of Gaussian thermal memory channels
The classical capacity of phase-invariant Gaussian channels has been recently
determined under the assumption that such channels are memoryless. In this work
we generalize this result by deriving the classical capacity of a model of
quantum memory channel, in which the output states depend on the previous input
states. In particular we extend the analysis of [C. Lupo, et al., PRL and PRA
(2010)] from quantum limited channels to thermal attenuators and thermal
amplifiers. Our result applies in many situations in which the physical
communication channel is affected by nonzero memory and by thermal noise.Comment: 14 pages, 8 figure
Passive states optimize the output of bosonic Gaussian quantum channels
An ordering between the quantum states emerging from a single mode
gauge-covariant bosonic Gaussian channel is proven. Specifically, we show that
within the set of input density matrices with the same given spectrum, the
element passive with respect to the Fock basis (i.e. diagonal with decreasing
eigenvalues) produces an output which majorizes all the other outputs emerging
from the same set. When applied to pure input states, our finding includes as a
special case the result of A. Mari, et al., Nat. Comm. 5, 3826 (2014) which
implies that the output associated to the vacuum majorizes the others
Gaussian states minimize the output entropy of one-mode quantum Gaussian channels
We prove the longstanding conjecture stating that Gaussian thermal input
states minimize the output von Neumann entropy of one-mode phase-covariant
quantum Gaussian channels among all the input states with a given entropy.
Phase-covariant quantum Gaussian channels model the attenuation and the noise
that affect any electromagnetic signal in the quantum regime. Our result is
crucial to prove the converse theorems for both the triple trade-off region and
the capacity region for broadcast communication of the Gaussian quantum-limited
amplifier. Our result extends to the quantum regime the Entropy Power
Inequality that plays a key role in classical information theory. Our proof
exploits a completely new technique based on the recent determination of the
p->q norms of the quantum-limited amplifier [De Palma et al.,
arXiv:1610.09967]. This technique can be applied to any quantum channel
Gaussian States Minimize the Output Entropy of the One-Mode Quantum Attenuator
We prove that Gaussian thermal input states minimize the output von Neumann
entropy of the one-mode Gaussian quantum-limited attenuator for fixed input
entropy. The Gaussian quantum-limited attenuator models the attenuation of an
electromagnetic signal in the quantum regime. The Shannon entropy of an
attenuated real-valued classical signal is a simple function of the entropy of
the original signal. A striking consequence of energy quantization is that the
output von Neumann entropy of the quantum-limited attenuator is no more a
function of the input entropy alone. The proof starts from the majorization
result of De Palma et al., IEEE Trans. Inf. Theory 62, 2895 (2016), and is
based on a new isoperimetric inequality. Our result implies that geometric
input probability distributions minimize the output Shannon entropy of the
thinning for fixed input entropy. Moreover, our result opens the way to the
multimode generalization, that permits to determine both the triple trade-off
region of the Gaussian quantum-limited attenuator and the classical capacity
region of the Gaussian degraded quantum broadcast channel
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