Theoretical Statistics and Mathematics Unit, ISI Delhi

From particle counting to Gaussian tomography

by K. R. Parthasarathy and Ritabrata Sengupta

The momentum and position observables in an $n$-mode
boson Fock space $\Gamma(\mathbb{C}^n)$ have the whole real
line $\mathbb{R}$ as their spectrum. But the total number
operator $N$ has a discrete spectrum
$\mathbb{Z}_+=\{0,1,2,\cdots\}$. An $n$-mode Gaussian state
in $\Gamma(\mathbb{C}^n)$ is completely determined by the
mean values of momentum and position observables and their
covariance matrix which together constitute a family of
$n(2n+3)$ real parameters. Starting with $N$ and its unitary
conjugates by the Weyl displacement operators and operators
from a representation of the symplectic group $Sp(2n)$ in
$\Gamma(\mathbb{C}^n)$ we construct $n(2n+3)$ observables
with spectrum $\mathbb{Z}_+$ but whose expectation values in
a Gaussian state determine all its mean and covariance
parameters. Thus measurements of discrete-valued
observables enable the tomography of the underlying Gaussian
state and it can be done by
using 5 one mode and 4 two mode Gaussian symplectic gates
in single and pair mode wires of $\Gamma(\mathbb{C}^n)
= \Gamma(\mathbb{C})^{\otimes n}$. Thus the tomography
protocol admits a simple description in a language similar
to circuits in quantum computation theory. Such a Gaussian
tomography applied to outputs of a Gaussian channel with
coherent input states permit a tomography of the channel
parameters. However, in our procedure the number of counting
measurements exceeds the number of channel parameters
slightly. Presently, it is not clear whether a more
efficient method exists for reducing this tomographic
complexity.
As a byproduct of our approach an elementary derivation
of the probability generating function of $N$ in a Gaussian
state is given. In many cases the distribution turns out to
be infinitely divisible and its underlying L\'evy measure
can be obtained. However, we are unable to derive the exact
distribution in all cases. Whether this property of infinite
divisibility holds in general is left as an open problem.

isid/ms/2015/04 [fulltext]

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