In the space of mixed states the Schrödinger-Robertson uncertainty relation holds though it can never be saturated.
Two tight extensions of this relation in the space of mixed states exist; one proposed by Dodonov
and Man'ko, where the lower limit on the uncertainty depends on the purity of the state, and another where
the uncertainty is bounded by the von Neumann entropy of the state proposed by Bastiaans. Driven by the
needs that have emerged in the field of quantum information, in a recent work we have extended the puritybounded
uncertainty relation by adding an additional parameter characterizing the state, namely its degree of
non-Gaussianity. In this work we alternatively present a extension of the entropy-bounded uncertainty relation.
The common points and differences between the two extensions of the uncertainty relation help us to draw more
general conclusions concerning the bounds on the non-Gaussianity of mixed states.
We study the transmission of classical information via optical Gaussian channels with a classical additive noise
under the physical assumption of a finite input energy including the energy of classical signal (modulation)
and the energy spent on squeezing the quantum states carrying information. Multiple uses of a certain class of
memory channels with correlated noise is equivalent to one use of parallel independent channels generally with
a phase-dependent noise. The calculation of the channels capacity implies finding the optimal distribution of
the input energy between the channels. Above a certain input energy threshold, the optimal energy distribution
is given by a solution known in the case of classical channels as water-filling. Below the threshold, the optimal
distribution of the input energy depends on the noise spectrum and on the input energy level, so that the channels
fall into three different classes: the first class corresponds to very noisy channels excluded from information
transmission, the second class is composed of channels in which only one quadrature (q or p) is modulated
and the third class corresponds to the water-filling solution. Although the non-modulated quadrature in the
channels of the second class is not used for information transmission, a part of the input energy is used for the
squeezing the quantum state which is a purely quantum effect. We present a complete solution to this problem
for one mode and analyze the influence of the noise phase dependence on the capacity. Contrary to our intuition,
in the highly phase-dependent noise limit, there exists a universal value of the capacity which neither depends
on the input energy nor on the value of noise temperature. In addition, similarly to the case of lossy channels
for weak thermal contribution of the noise, there exists an optimal squeezing of the noise, which maximizes the
capacity.
Jérôme Lodewyck, Matthieu Bloch, Raúl García-Patrón, Simon Fossier, Evgueni Karpov, Eleni Diamanti, Thierry Debuisschert, Nicolas Cerf, Rosa Tualle-Brouri, Steven McLaughlin, Philippe Grangier
We report on both theoretical and experimental aspects of a fully implemented quantum key distribution device
with coherent states. This system features a final key rate of more than 2 kb/s over 25 km of optical fiber. It
comprises all required elements for field operation: a compact optical setup, a fast secret bit extraction using
efficient LDPC codes, privacy amplification algorithms and a classical channel software. Both hardware and
software are operated in real time.
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