Theoretical Statistics and Mathematics Unit, ISI Delhi

On the Kolmogorov - Wiener - Masani spectrum of a multi-mode weakly stationary quantum process

by K. R. Parthasarathy and Ritabrata Sengupta

We introduce the notion of a $k$-mode weakly stationary
quantum process ${\varrho}$ based on the canonical
Schr\"odinger pairs of position and momentum observables in
copies of $L^2(\mathbb{R}^k)$, indexed by an additive
abelian group $D$ of countable cardinality. Such observables
admit an autocovariance map $\widetilde{K}$ from $D$
into the space of real $2k \times 2k$ matrices. The map
$\widetilde{K}$ on the discrete group $D$ admits a
spectral representation as the Fourier transform of a
$2k \times 2k$ complex Hermitain matrix-valued totally
finite measure $\Phi$ on the compact character group
$\widehat{D}$, called the Kolmogorov-Wiener-Masani (KWM)
\emph{spectrum} of the process ${\varrho}$. Necessary and
sufficient conditions on a $2k \times 2k$ complex Hermitian
matrix-valued measure $\Phi$ on $\widehat{D}$ to be the KWM
spectrum of a process ${\varrho}$ are obtained. This
enables the construction of examples. Our theorem reveals
the dramatic influence of the uncertainty relations among
the position and momentum observables on the KWM spectrum of
the process ${\varrho}$. In particular, KWM spectrum
cannot admit a gap of positive Haar measure in $\widehat{D}$.
The relationship between the number of photons in a
particular mode at any site of the process and its KWM
spectrum needs further investigation.

isid/ms/2016/05 [fulltext]

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