Hi all,
tomorrow we'll have a special talk from Nilanjana Datta of University of
Cambridge. 3pm in E41.1. Title and Abstract below.
-joe
---------
Title: Entropy power inequalities for qudits
Abstract:
Shannon’s entropy power inequality (EPI) can be viewed as a statement of
concavity of an entropic function of a continuous random variable under a
scaled addition rule:
f(√a X + √(1-a) Y) ≥ a f(X) + (1 − a)f(Y) ∀ a ∈ [0, 1].
Here, X and Y are continuous random variables and the function f is either
the differential entropy or, for a = 1/2, the entropy power. Koenig and
Smith obtained quantum analogues of these inequalities for
continuous-variable quantum systems, where X and Y are replaced by bosonic
fields and the addition rule is the action of a beamsplitter with
transmissivity a on those fields. We similarly establish a class of EPI
analogues for d-level quantum systems (i.e. qudits). The underlying
addition rule for which these inequalities hold is given by a quantum
channel that depends on the parameter a ∈ [0, 1] and acts like a
finite-dimensional analogue of a beamsplitter with transmissivity a,
converting a two-qudit product state into a single qudit state. We refer to
this channel as a partial swap channel because of the particular way its
output interpolates between the states of the two qudits in the input as a
is changed from zero to one. We obtain analogues of Shannon’s EPI, not only
for the von Neumann entropy and the entropy power for the output of such
channels, but for a much larger class of functions as well. This class
includes the Renyi entropies and the subentropy. We also prove a qudit
analogue of the entropy photon number inequality (EPnI). Finally, for the
subclass of partial swap channels for which one of the qudit states in the
input is fixed, our EPIs and EPnI yield lower bounds on the minimum output
entropy and upper bounds on the Holevo capacity. This is joint work with
Koenraad Audenaert and Maris Ozols.