21 research outputs found
Continuous-variable entropic uncertainty relations
Uncertainty relations are central to quantum physics. While they were
originally formulated in terms of variances, they have later been successfully
expressed with entropies following the advent of Shannon information theory.
Here, we review recent results on entropic uncertainty relations involving
continuous variables, such as position and momentum . This includes the
generalization to arbitrary (not necessarily canonically-conjugate) variables
as well as entropic uncertainty relations that take - correlations into
account and admit all Gaussian pure states as minimum uncertainty states. We
emphasize that these continuous-variable uncertainty relations can be
conveniently reformulated in terms of entropy power, a central quantity in the
information-theoretic description of random signals, which makes a bridge with
variance-based uncertainty relations. In this review, we take the quantum
optics viewpoint and consider uncertainties on the amplitude and phase
quadratures of the electromagnetic field, which are isomorphic to and ,
but the formalism applies to all such variables (and linear combinations
thereof) regardless of their physical meaning. Then, in the second part of this
paper, we move on to new results and introduce a tighter entropic uncertainty
relation for two arbitrary vectors of intercommuting continuous variables that
take correlations into account. It is proven conditionally on reasonable
assumptions. Finally, we present some conjectures for new entropic uncertainty
relations involving more than two continuous variables.Comment: Review paper, 42 pages, 1 figure. We corrected some minor errors in
V
Multidimensional entropic uncertainty relation based on a commutator matrix in position and momentum spaces
The uncertainty relation for continuous variables due to Byalinicki-Birula
and Mycielski expresses the complementarity between two -uples of
canonically conjugate variables and in terms of Shannon differential entropy. Here, we consider the
generalization to variables that are not canonically conjugate and derive an
entropic uncertainty relation expressing the balance between any two
-variable Gaussian projective measurements. The bound on entropies is
expressed in terms of the determinant of a matrix of commutators between the
measured variables. This uncertainty relation also captures the complementarity
between any two incompatible linear canonical transforms, the bound being
written in terms of the corresponding symplectic matrices in phase space.
Finally, we extend this uncertainty relation to R\'enyi entropies and also
prove a covariance-based uncertainty relation which generalizes Robertson
relation.Comment: 8 pages, 1 figur
Using Graph Theory to Derive Inequalities for the Bell Numbers
The Bell numbers count the number of different ways to partition a set of
elements while the graphical Bell numbers count the number of non-equivalent
partitions of the vertex set of a graph into stable sets. This relation between
graph theory and integer sequences has motivated us to study properties on the
average number of colors in the non-equivalent colorings of a graph to discover
new non trivial inequalities for the Bell numbers. Example are given to
illustrate our approach
Entropy-power uncertainty relations : towards a tight inequality for all Gaussian pure states
We show that a proper expression of the uncertainty relation for a pair of
canonically-conjugate continuous variables relies on entropy power, a standard
notion in Shannon information theory for real-valued signals. The resulting
entropy-power uncertainty relation is equivalent to the entropic formulation of
the uncertainty relation due to Bialynicki-Birula and Mycielski, but can be
further extended to rotated variables. Hence, based on a reasonable assumption,
we give a partial proof of a tighter form of the entropy-power uncertainty
relation taking correlations into account and provide extensive numerical
evidence of its validity. Interestingly, it implies the generalized
(rotation-invariant) Schr\"odinger-Robertson uncertainty relation exactly as
the original entropy-power uncertainty relation implies Heisenberg relation. It
is saturated for all Gaussian pure states, in contrast with hitherto known
entropic formulations of the uncertainty principle.Comment: 15 pages, 5 figures, the new version includes the n-mode cas
Detection of non-Gaussian entangled states with an improved continuous-variable separability criterion
Currently available separability criteria for continuous-variable states are
generally based on the covariance matrix of quadrature operators. The
well-known separability criterion of Duan et al. [Phys. Rev. Lett. 84, 2722
(2000)] and Simon [Phys. Rev. Lett. 84, 2726 (2000)] , for example, gives a
necessary and sufficient condition for a two-mode Gaussian state to be
separable, but leaves many entangled non-Gaussian states undetected. Here, we
introduce an improvement of this criterion that enables a stronger entanglement
detection. The improved condition is based on the knowledge of an additional
parameter, namely the degree of Gaussianity, and exploits a connection with
Gaussianity-bounded uncertainty relations [Phys. Rev. A 86, 030102 (2012)]. We
exhibit families of non-Gaussian entangled states whose entanglement remains
undetected by the Duan-Simon criterion.Comment: Revised presentation, results unchanged. 10 pages, 6 figure
Complex-valued Wigner entropy of a quantum state
It is common knowledge that the Wigner function of a quantum state may admit
negative values, so that it cannot be viewed as a genuine probability density.
Here, we examine the difficulty in finding an entropy-like functional in phase
space that extends to negative Wigner functions and then advocate the merits of
defining a complex-valued entropy associated with any Wigner function. This
quantity, which we call the complex Wigner entropy, is defined via the analytic
continuation of Shannon's differential entropy of the Wigner function in the
complex plane. We show that the complex Wigner entropy enjoys interesting
properties, especially its real and imaginary parts are both invariant under
Gaussian unitaries (displacements, rotations, and squeezing in phase space).
Its real part is physically relevant when considering the evolution of the
Wigner function under a Gaussian convolution, while its imaginary part is
simply proportional to the negative volume of the Wigner function. Finally, we
define the complex-valued Fisher information of any Wigner function, which is
linked (via an extended de Bruijn's identity) to the time derivative of the
complex Wigner entropy when the state undergoes Gaussian additive noise.
Overall, it is anticipated that the complex plane yields a proper framework for
analyzing the entropic properties of quasiprobability distributions in phase
space.Comment: 14 pages + 10 pages of Appendi
Relating the Entanglement and Optical Nonclassicality of Multimode States of a Bosonic Quantum Field
The quantum nature of the state of a bosonic quantum field manifests itself
in its entanglement, coherence, or optical nonclassicality which are each known
to be resources for quantum computing or metrology. We provide quantitative and
computable bounds relating entanglement measures with optical nonclassicality
measures. These bounds imply that strongly entangled states must necessarily be
strongly optically nonclassical. As an application, we infer strong bounds on
the entanglement that can be produced with an optically nonclassical state
impinging on a beam splitter. For Gaussian states, we analyze the link between
the logarithmic negativity and a specific nonclassicality witness called
"quadrature coherence scale".Comment: 13 pages, 2 figures, change of notation in v
Quadrature coherence scale driven fast decoherence of bosonic quantum field states
International audienceWe introduce, for each state of a bosonic quantum field, its quadrature coherence scale (QCS), a measure of the range of its quadrature coherences. Under coupling to a thermal bath, the purity and QCS are shown to decrease on a time scale inversely proportional to the QCS squared. The states most fragile to decoherence are therefore those with quadrature coherences far from the diagonal. We further show a large QCS is difficult to measure since it induces small scale variations in the state's Wigner function. These two observations imply a large QCS constitutes a mark of "macroscopic coherence". Finally, we link the QCS to optical classicality: optical classical states have a small QCS and a large QCS implies strong optical nonclassicality
Higher Order Nonclassicality from Nonlinear Coherent States for Models with Quadratic Spectrum
Harmonic oscillator coherent states are well known to be the analogue of classical states. On the other hand, nonlinear and generalised coherent states may possess nonclassical properties. In this article, we study the nonclassical behaviour of nonlinear coherent states for generalised classes of models corresponding to the generalised ladder operators. A comparative analysis among them indicates that the models with quadratic spectrum are more nonclassical than the others. Our central result is further underpinned by the comparison of the degree of nonclassicality of squeezed states of the corresponding models