\section{Introduction}

    $\delta$ Scuti stars are a class of pulsating stars located
    on the Hertzsprung-Russell Diagram on or
    around the Main Sequence and intersecting the instability strip.
    They are 1.5 - 2.5 $M_{\odot}$ stars pulsating often in one main
    dominant oscillation mode or many lower amplitude pulsation modes.
    They have been interesting targets seismologically, because
    the oscillation amplitudes often reach tenths of magnitudes, and
    we
    understand  their stellar structure relatively well, so seismology can
    allow us to probe details of the microphysics such as energy
    transport mechanisms, convective core overshoot, as well as
    other less well-developed theories such as rapid rotation.

    The main setbacks that $\delta$ Scuti seismology face (as well as
    other pulsating stars) are 1) the fundamental stellar parameters are
    not well-enough constrained to allow the few pulsation modes to
    probe the structure, and 2) rapid rotation causes each of the
    degrees $l$ to split into $2l+1$ $m$-modes, making mode-identification
    a difficult task \cite{gou05}.
    Indeed these problems are not exclusive nor exhaustive.
    This has prompted authors to look towards objects where at least one
    of these problems can be eliminated \cite{lb02, ah04, mac06, cos07}.

    Observing stars where the number of free parameters is constrained,
    such as in open clusters or multiple systems
    is a possibility for overcoming these obstacles.

    In order to use seismology to probe the interior of a star,
    the parameters of the star need to be known quite well,
    for example, the mass should be known to 1-2\% \cite{cre07}.
    The observables from a binary system provide strict constraints
    on the parameters of the component stars.
    If the binary is an eclipsing and spectroscopic system, the
    absolute values of the masses and radii can be extracted to 1-2\% (e.g.
    \cite{rib99,las00}).

    The objective of this study is to find out if the uncertainties
    in the stellar parameters can be reduced,

    so that seismology can
    be applied to those stars that exhibit one or few pulsation modes.
    We look at the particular case of a pulsating star in an eclipsing
    binary system and compare the parameter uncertainties with those of an
    isolated star.
    This study quantifies how well the stellar parameters can be extracted
    in various hypothetical systems.

    \section{Methods}

    The mathematical basis of this study lies in the application of
    Singular Value Decomposition (SVD) techniques to physical models.
    This technique has been used in previous studies, such as \cite{bro94,
      mm05, cre07}
    and it is being applied to many other areas of astrophysics and
    science, because of its powerful diagnostic properties.
    Intricate details of this mathematical technique are elaborated upon in
    the above mentioned publications as well as in \cite{pre92}.
    Here we give the basic equations to enable an understanding of this
    work.

    \subsection{Singular Value Decomposition}
    SVD is the decomposition of any $M\times N$  matrix {\bf D} into
    3 components ${\bf U}$, ${\bf V^T}$
    and ${\bf W}$ given by ${\bf D} = {\bf U W V^T}$.
    ${\bf V^T}$ is the transpose of ${\bf V}$ which is an
    $N \times N$ orthogonal matrix that contains the
    {\it input} basis vectors for ${\bf D}$, or the vectors
    associated with the parameter space.
    ${\bf U}$ is an $M \times N$ orthogonal matrix that contains the
    {\it output} basis vectors for ${\bf D}$, or the vectors
    associated with the observable space.
    ${\bf W}$ is a diagonal matrix that contains the
    {\it singular values} of ${\bf D}$.

    The key element to our work is the description of the matrix {\bf D}.
    Here we define {\bf D} to be a matrix whose elements consist
    of the partial derivatives of each
    of the observables with respect to each of the parameters of the system,
    in function of the expected measurement errors on each
    of the observables:
    \begin{equation}
      D_{ij} = \frac{\partial B_i}{\partial P_j} \epsilon_i^{-1}.
      \label{eqn:designmatrix}
      \end{equation}
    Here $B_i$ are each of the $i = 1, 2, ...M$
    observables of the system, with measurement or expected
    errors $\epsilon_i$,
    and $P_j$ are each of the $j = 1,2,...,N$ free parameters of the
    system (see section \ref{sec:obspar}
    for discussion on the observables and the
    parameters).

    By writing the design matrix in function of the measurement errors,
    we provide a quantitative description of the information content
    of each of the observables for determining the stellar parameters
    and their uncertainties.

    Supposing that we are looking for the true solution ${\bf P}_{\rm R}$
of the system.
    By starting from an initial close guess of the solution ${\bf P}_0$,
    SVD can be used as an inversion technique by calculating a set
    of parameter corrections ${\bf \delta P}$
    that minimizes some goodness-of-fit function:
    ${\bf \delta P = V\bar{W}^{-1}U^T \delta B}$,
    where ${\bf \delta B}$ are the differences
    between the set of actual observations {\bf O}
    and the calculated observables
    ${\bf B}_{\rm 0}$ given the initial parameters ${\bf P}_{\rm 0}$.
    ${\bf \bar{W}}$ is a modification of the matrix {\bf W} such that
    the inverse of the values below a certain threshold are set to 0.

    The formal errors are comprised of the sum of all of
    the ${\bf V}_k/w_k$, where each
    ${\bf V}_k/w_k$ describes the direction and magnitude to move
    each parameter, so that the true solution ${\bf P}_{\rm R}$ and formal
    uncertainties can be given by
    \begin{equation}
      {\bf P}_{\rm R} = {\bf P}_{\rm 0} +
      {\bf V\bar{W}^{-1}U^T \delta B} \left ( \pm \frac{{\bf V}_1}{w_1}
      \pm  \frac{{\bf V}_2}{w_2} \pm ...
      \pm \frac{{\bf V}_N}{w_N} \right).
      \label{eqn:dscuti_er}
    \end{equation}
    The {\it covariance matrix} {\bf C} consequently comes
    in a very neat and compact
    form:
    \begin{equation}
      C_{jl} = \sum_{k=1}^N \frac{V_{jk}V_{lk}}{w_{k}^2},
      \label{eqn:covariance}
    \end{equation}
    and the square
    roots of the  diagonal elements of the covariance matrix are the
    theoretical parameter uncertainties:
    \begin{equation}
      \sigma_j^2 =  \sum_{k=1}^N \left ( \frac{V_{jk}}{w_k} \right )^{2}.
      \label{eqn:uncertainties}
    \end{equation}

    \subsection{Observables, Parameters \& Models\label{sec:obspar}}

    We describe a single $\delta$ Scuti star by a set of parameters or
    ingredients.  The ingredients for the stellar model are
    mass $M$, age $\tau$,
    rotational velocity $v$, initial
    Hydrogen and heavy metals mass fraction $X$ and $Z$ where $X+Y+Z = 1$ and
    $Y$ is initial Helium mass fraction,
    and mixing-length parameter $\alpha$ where applicable\footnote{For masses
    larger than about 1.5 $M_{\odot}$ the outer convective layer is
    relatively thin, so
    the observables of the star are not very
    sensitive to the value of the mixing-length parameter.}.
    The distance to the object $d$ is also included as a parameter.

    For a binary system, the additional parameters are the system properties:
    separation of components $a$, eccentricity of orbit $e$,
    longitude of periastron $\omega$ and inclination of orbit $i$.
    Fortunately, both stellar components in a binary system
    share the parameters $\tau$, $X$ and $Z$,

    so then the individual stars differ mainly by
    $M$ and $v$.
    The parameters of the binary system of this study are given in
    Table \ref{tab:parameters}.

    The observables are the measurable quantities of the system.
    These include things such as radius,
    effective temperature, gravity, metallicity and parallax
    for a single star system.
    For a binary system, the observables include
    effective temperature ratio,
    relative radii, radial velocities and orbital period.
    The Aarhus STellar Evolution Code \cite{chr82} is used to calculate the
    stellar evolution models.
    This code uses the stellar parameters as the input ingredients,
    and returns
    a set of global stellar properties such as radius and effective
    temperature,
    as well as the interior profiles of the star such as mass,
    density and pressure.
    Oscillation frequencies for this rapidly rotating star are calculated
    using MagRot \cite{gt90, bt06}.
    Using the global stellar properties and the distance to the star,
    SDSS \cite{yor00}
    magnitudes and colours in various filters are evaluated using
    the Basel model atmospheres \cite{lej97}.
    Most of the binary observables are calculated analytically using the
    combined stellar and binary parameters.
    In this work we refer to the non-seismic data as
    the {\it classical} observables.
    Most of the classical observables of the binary system are given in
    Table~\ref{tab:parameters}.

    A clear distinction should be made between the input parameters of the
    system and output measurable quantities, the observables.
    So to discriminate between the errors in both parameters and observables,
    we shall denote the (derived) parameter uncertainties by $\sigma$,
    while $\epsilon$ is reserved for the
    observable errors.

    Both luminosity and effective temperature can be observables, but
    in Section \ref{sec:lt} the derived uncertainties of these properties are
    discussed.
    This refers specifically to the calculated error boxes in the
    luminosity-temperature (L-T) diagram, and here their uncertainties
    will also be denoted by $\sigma$.

    \begin{table}
      \centering
      \caption{\label{tab:parameters}
	System Parameters \& Observables}
      \begin{tabular}{@{}l*{15}{l}}
	\br
	Parameter  & Value ($P_j$) & & & Observable & Value ($O_i$)
	& $\epsilon_i$ & & \\
	\mr
	$M_A$ & 1.8 $M_{\odot}$ & & &$R_A$ & 1.95 $R_{\odot}$ & 0.02 \\
	$M_B$ & 1.7 $M_{\odot}$ & & &$R_B$ & 1.81 $R_{\odot}$ & 0.02 \\
	$\tau$ & 0.7 Gyr & & & $T_B/T_A$ & 0.97 & 0.05\\
	$X$ & 0.700 & & & $T_{\rm eff}$ & 6965 (K)  & 100 \\
	$Z$ & 0.035 & & & [M/H] & 0.31 (dex) & 0.05\\
	$v_A$ & 100.0 km s$^{-1}$  & & & $v_A \sin i$ & 99.7 km s$^{-1}$
	& 2.5 \\
	$v_B$ & 80.0 km s$^{-1}$ & & & $v_B \sin i$ & 59.8 km s$^{-1}$& 2.5 \\
	$d$ & 200 pc  & & & $\pi$ & 5.0 (mas) & 0.5\\
	$a$ & 0.15 AU  & & &$\Pi$ & 0.031 (yrs) & 0.00001\\
	$i$ & 85.6 $^{\circ}$ & & & $i$  & 85.6 $^{\circ}$ & 0.05\\
	$e$ & 0.0  & & & $M_A \sin^3 i $ & 1.78  $M_{\odot}$& 0.06\\
	$\omega$ & 0.0  & & & $M_B \sin^3 i$ & 1.69  $M_{\odot}$& 0.05\\
	\br
      \end{tabular}
    \end{table}

    \section{Results}

    The theoretical uncertainties in each of the
    parameters of $M_A$, $\tau$, $X$ and $Z$ are calculated as a function
    of error in radius $\epsilon_{R_{\star}}$
    and as a function of error in colour
    $\epsilon_{(i-z)}$,
    using Equation \ref{eqn:uncertainties} coupled with the
    observable errors given
    in Table \ref{tab:parameters}.
    Consequently the theoretical uncertainties in luminosity and effective
    temperature are calculated using Equation \ref{eqn:dscuti_er},
    for three different
    values of $\epsilon_{R_{\star}}$ and three different values
    of effective temperature error $\epsilon_{T_{\rm eff}}$.
    The results are shown in Figures \ref{fig1} and \ref{fig2} below.

    \subsection{Parameter Uncertainties}

    Figure \ref{fig1} shows the theoretical uncertainties ($\sigma$)
    in $M_A$, $\tau$, $X$ and $Z$
    of a pulsating star
    as a function of error in the radius ({\it left panels}) and
    as a function of error in the photometric colours ({\it right panels}).
    Note that the left panels do {\it not} include photometric information,
    i.e. no colours
    nor magnitudes.
    The dashed lines show the results for a single star (the observables
    are radius, effective temperature, gravity, metallicity, ...)
    while the solid lines show the results for a component of a binary
    system (observables are those of the single star and the binary
    observables).
    The lines with the diamonds include one identified mode as well as the
    classical observables.
    The results for the other stellar parameters are not shown, because
    the four aforementioned parameters and $v_A$ are responsible for
    determining
    the model structure of the pulsating star.
    $\sigma (v_A)$ is
    is usually independent of
    $\epsilon_{R_{\star}}$ and $\epsilon_{(i-z)}$; it is determined mainly
    by the observables $v \sin i$ and $i$ from a combination of
    spectroscopy and the
    photometric light curve.

    \begin{figure}
      \begin{center}
	\includegraphics[width = 0.45\textwidth, height = 0.75\textheight]
			{fnrad_sgdl.eps}
	\includegraphics[width = 0.45\textwidth, height = 0.75\textheight]
			{fncol_sgdl.eps}
      \end{center}
      \caption{\label{fig1} Theoretical uncertainties ($\sigma$) in mass $M$,
	age $\tau$, initial hydrogen $X$ and metal $Z$ content as a function
	of observable error.
	The left panel shows the uncertainties as a function of observable
	radius error, here no photometric information has been included.
	The right panel shows the uncertainties as a function of observable
	error in photometric colour.
	The dashed and continuous lines show the results for the single
	star and the binary system respectively,
	those with diamonds show the results when an identified mode
	is included in the set of observables.}
    \end{figure}

    For the single star without an identified mode ({\it dashed lines no
    diamonds}), the parameter uncertainties remain at a large
    constant value as a function of radius ({\it left panels}) but do decrease
    slightly with improved photometric data ({\it right panels}).
    Only when seismic data are included for the single star system
    ({\it dashed lines, diamonds}) {\it and} the observable
    errors are small,  the parameters are constrained to a usable amount.

    By comparing the solid lines with and without diamonds in Figure \ref{fig1},
    it can be seen that
    the addition of one identified mode
    makes almost no difference to the parameter uncertainties
    for the binary system.
    This implies that there is enough information provided by the
    binary constraints to sufficiently determine the stellar parameters.
    In this sense, the identified mode is {\it redundant} information,
    and thus can be used maybe to test the interior of the star.

    Including photometric information ({\it right panels}) provides
    an interesting result:
    the information provided
    by the single star system can supersede that of the binary system for $\tau$ and $Z$ .
    This is because
    the colours are uncontaminated by a component star.
    This only happens at very small measurement errors, and only
    when an identified mode is included for the single star.

    \subsection{Luminosity-Temperature Error Box \label{sec:lt}}

    The correlation matrices come in a compact form when using SVD.
    This then allows a calculation of the theoretical uncertainties
    in both effective temperature and luminosity (L-T)
    (Equation \ref{eqn:dscuti_er}).
    Figure \ref{fig2} shows the theoretical error box in effective
    temperature and luminosity for a single star system ({\it dashed lines})
    and a binary system ({\it solid lines}).
    The observables do not include photometric information,
    and for the single star an identified mode is included\footnote{Figure
      \ref{fig1} shows that the parameters are
    not constrained for the single star if the identified mode
    is not included.},
    while
    for the binary system no seismic data is included.

    \begin{figure}
      \begin{center}
	\includegraphics[width = 1.\textwidth, height = 0.3\textheight]
			{rad_sgdl_lt.eps}
      \end{center}
      \caption{\label{fig2} The theoretical error boxes for luminosity and
	effective temperature.  The dashed lines represent the results for the
	single star while the continous lines represent the results for the
	binary system.
	The left panel shows the results while reducing the error in radius,
	and the right panel shows the results while reducing the error
	in effective temperature.}
    \end{figure}

    \subsubsection{Single Star}

    Observe how the error box reduces significantly while reducing the error
    in the radius observable ({\it left panel}).
    The uncertainty in $T_{\rm eff}$ also reduces slightly.
    The right panel also shows that by reducing
    the errors in the observable $T_{\rm eff}$, an expected
    corresponding reduction in the uncertainties in $T_{\rm eff}$ is noted.
    The $\epsilon_{T_{\rm eff}}$ of 200, 100, and 50 K, produces a
    $\sigma (T_{\rm eff})$ of 250, 110, and 50 K.
    The fact that these uncertainties are reproduced also gives
    confidence in this method.
    $\sigma(L_{\star})$ changes slightly as a function of
    $\epsilon_{T_{\rm eff}}$, its value is determined mostly by
    the error in the radius observable (2\%).
    Looking back to the left panel, we see that
    interpolating between 1\% and 3\% $\epsilon_{R_{\star}}$
    produces a $\sigma(L_{\star})$ = 0.5
    L$_{\odot}$ for $\epsilon_{R_{\star}}$ = 2\%.
    This is the value that is shown in the right panel.

    \subsubsection{Binary System}

    For the binary system ({\it solid lines}), {\it no} identified mode is
    included.
    The error box for the binary system does not reduce while
    reducing the errors
    in the radius,
    because of the small
    uncertainties in these parameters.
    However, the error box does reduce when the error in effective temperature
    is reduced, reproducing accurately the input $\epsilon_{T_{\rm eff}}$
    of $\sigma(T_{\rm eff})$ = 200, 100, and 50 K.
    $\sigma (L_{\star})$ does not decrease in either panel,
    because the mass is well-determined for the binary system
    and provides this narrow constraint on $L_{\star}$.

    In all cases, note that
    the constraints provided by the binary system without an identified
    mode are more
    effective than those from the single star when an identified mode is
    included.

    \section{Conclusions}

    This study investigated whether the uncertainties in the stellar
    parameters of a pulsating component in an eclipsing binary
    system were sufficient so that an observed pulsation mode could
    be used to test the physics of the stellar interiors.
    Additionally we studied the information content of a pulsating
    star in a single star system
    to quantify how much is gained in terms of precision in parameters
    and size of the L-T error box by observing the
    star in a detached eclipsing binary system.
    The conclusions are summarized as follows:

    \begin{itemize}
    \item A single star system without an identified mode remains
      poorly understood
      when observables such as the radius or colours are poorly measured.
      The parameter uncertainties are too large to correctly place
      the star in the L-T diagram.
    \item A binary system {\it without} seismic information
      provides better constraints than the single star system
      when an oscillation
      mode has been identified.
      \item Reducing the size of some observable errors has little or no impact
	on the parameter determinations for the binary system,
	because these parameters are already well constrained.

    \item The tight constraints provided by the binary system for the stellar
      parameters reduces the size of the error box in the L-T
      diagram significantly.

      \item By carefully constraining the parameters of the star, just as
	an eclipsing binary system allows us to do,
	an accurate estimate of the stellar model under study can be obtained.
	This allows the redundant observables (like an oscillation mode)
	to be used exclusively to test the physics of the interior
	of a star.
    \end{itemize}

\section*{References}


\section{Introduction}
Uncertainty relations place fundamental limits on the precision achievable in  measuring non-commuting observables. The original idea of uncertainty relation was first introduced  by  Heisenberg~\cite{heis}  for position and momentum observables $Q$, $P$. Subsequently,  a  mathematically precise version (with $\hbar=1$)
\be
\label{pqunc}
(\Delta Q)^2 \, (\Delta P)^2 \geq \frac{1}{4}
\ee
was  formulated by Kennard~\cite{kennard}. Here  $(\Delta \Gamma)^2 = \langle \Gamma^2  \rangle - \langle \Gamma  \rangle^2$ denotes the variance of the observable $\Gamma=Q \ {\rm or}\  P$ and  the bracket $\langle \cdots \rangle=\mbox{Tr}\,[\rho\cdots ]$ corresponds to the expectation value in a quantum state $\rho$.

Weyl~\cite{weyl} and Robertson~\cite{robt} extended the uncertainty relation  (\ref{pqunc}) for any arbitrary pair of physical observables $A_1$, $A_2$:
\be
\label{unc}
(\Delta A_1)^2 (\Delta A_2)^2 \geq \frac{1}{4}\left\vert
\langle\,[A_1,\,A_2 ]\,\rangle \right\vert^2
\ee
which is commonly referred to as the Heisenberg-Robertson uncertainty relation in the literature. Here $[A_1,\,A_2 ]=A_1A_2-A_2A_1$ denotes the commutator of the observables $A_1$ and $A_2$. The uncertainty relation (\ref{unc}) imposes restrictions on the product of variances
$(\Delta A_1)^2$ and $(\Delta A_2)^2$  --   essentially limiting the capability towards precise prediction of the  measurement results of non-commuting observables. In general the Heisenberg-Robertson approach of placing limits on the uncertainties of incompatible observables in the given quantum state $\rho$  sets the conventional framework for deriving preparatory uncertainty relations~\cite{busch_werner,note1}.

Apart from their fundamental interest, uncertainty relations play a significant role in the field of quantum information processing, with several applications such as entanglement detection~\cite{hf,hfbound, ogunhe},  quantum cryptography~\cite{koshi, berta, hangi, tom, branciard, aru_hsk, coles}, quantum metrology~\cite{reid2011}  and   foundational tests  of quantum theory~\cite{scully,reid}. Motivated by their applicability,  there has  been an ongoing interest in  reformulating uncertainty relations expressing trade-off of more than two incompatible observables, formalized in terms of  variances~\cite{hf,ogunhe,akp1,rivas,varbase,akp2,chen,shabbir,xiao,bagchi,ma,song,bsanders,maconne19, busch2019,zheng,zukow}, or via information entropies~\cite{ hirschman, beckner, bial1, deutch, partovi, bial2, kraus, mu, sw, bial3}. Recently several experimental tests have been carried out to verify different forms of uncertainty relations~\cite{expt_wang, expt2017, chen_expt, expt2019}.

In this paper we investigate variance based LSUR for local angular momentum operators of a bipartite quantum system,  proposed  by Hofmann and Takeuchi~\cite{hf}, violation of which witnesses entanglement. One of the main advantages of employing LSUR is the fact that it is possible to detect entanglement without a complete knowledge of the quantum state  and  it suffices to determine experimental friendly  variances of local angular momentum observables for this purpose. We show that the angular momentum  LSUR, which places lower bound on the set of all bipartite separable states,  gets  violated if and only if the covariance matrix of the two-qubit reduced system of the $N$-qubit permutation symmetric state is negative. Since it has been shown ~\cite{ijmp06,pla07,prl07} that the covariance matrix negativity
  serves as a necessary and sufficient for entanglement in a two-qubit symmetric system, our result   establishes a one-to-one connection between entanglement and violation of the  LSUR. Furthermore, we show that  negative covariance matrix of the two-qubit reduced system of a $N$-qubit symmetric state  leads to a variance based test of pairwise entanglement.

\section{Sum Uncertainty Relation}

It may be noted that the term  $\langle\,[A_1,\,A_2 ]\,\rangle={\rm Tr}\left(\rho\,[A_1,\,A_2] \right) $ appearing in the right hand side of the Heisenberg-Robertson uncertainty relation
 (\ref{unc})  vanishes in some specific quantum states $\rho$.  In such cases, one ends up with a  trivial relation $(\Delta A_1)^2 (\Delta A_2)^2 \geq 0$ for the product of variances of non-commuting observables $A_1, \, A_2$.
 Moreover,  variance   vanishes in the eigenstate of one of the observables. In such cases the Heisenberg-Robertson uncertainty relation (\ref{unc}) fails to capture the intrinsic indeterminacy of non-commuting observables.  To overcome such issues it is preferable to  employ uncertainty relations placing non-trivial bounds on the sum of variances $(\Delta A_1)^2 + (\Delta A_2)^2$. In fact, a lower bound for the sum of variances may be found  by using  the inequality
 $\sum_{\alpha=1}^{m} a_\alpha/m \geq  \left(\prod_{\alpha}a_\alpha\right)^{1/m}$ between the airthmetic mean and the geometirc mean of real non-negative numbers $a_\alpha, \alpha=1,2,\ldots, m$. Choosing $a_1=(\Delta A_1)^2,\  a_2=(\Delta A_2)^2$ and using (\ref{unc}) one obtains a variance based sum uncertainty relation
\be
\label{sur0}
(\Delta A_1)^2 + (\Delta A_2)^2\geq \left\vert \langle\,[A_1,\,A_2 ]\,\rangle \right\vert.
\ee

However the sum uncertainty relation (\ref{sur0}) is a byproduct of the Heisenberg-Robertson inequality (\ref{unc}) and thus it is  non-informative for some quantum states $\rho$ in which one of the variances and/or  $\langle\,[A_1,\,A_2 ]\,\rangle$ vanish.

\subsection{Sum uncertainty relations for angular momentum operators}

Hofmann and Takeuchi~\cite{hf}  proposed that  a non-trivial lower bound ${\cal U}>0$ must exist for the sum of variances of a set $\{A_\alpha\}$, $\alpha=1,2,\ldots $ of
non-commuting observables, as they do not share any simultaneous eigenstate:
\be
\label{sur1}
\sum_{\alpha}\,(\Delta A_\alpha)^2\geq {\cal U}.
\ee
The lower bound ${\cal U}$ in the sum uncertainty relation (\ref{sur1}) corresponds to the absolute  minimum value $\left[\sum_{\alpha}\, (\Delta A_\alpha)^2\right]_{\rm min}$ for any quantum state $\rho$. While it is, in general, tough to determine  ${\cal U}$ for any arbitrary set  $\{A_{\alpha}\}$ of non-commuting observables, there are some important physical examples  in finite dimensional Hilbert spaces, where the limiting value $U$ can be readily identified. For example, in the case of a spin-$j$ quantum system the components of angular momentum operators
$\{J_1, J_2, J_3\}$ satisfy the following conditions:
\begin{eqnarray*}
 \left\langle \left(J^2_1+J^2_2+J^2_3\right)\right\rangle&=&j(j+1) \\
  \langle J_1\rangle^2+\langle J_2\rangle^2+\langle J_3\rangle^2 &\leq& j^2.
  \end{eqnarray*}
Thus, one obtains a variance based  sum uncertainty relation for the components $(J_1, J_2, J_3)$ of angular momentum operator~\cite{hf}:
\be
\label{sur}
(\Delta J_1)^2+(\Delta J_2)^2+(\Delta J_3)^2 \geq j,
\ee
imposing a limit on the measurement precision of more than one of the angular momentum componets.

In the specific  example of $j=1/2$ (i.e., for a qubit), we have $J_\alpha=\sigma_\alpha/2, \alpha=1,2,3$, where $\sigma_\alpha$ denote the Pauli matrices.  One then obtains the sum uncertainty relation for qubits~\cite{hf}
\be
\label{paulisur}
(\Delta \sigma_1)^2+(\Delta \sigma_2)^2+(\Delta \sigma_3)^2 \geq 2.
\ee
Given an arbitrary single qubit density matrix $\rho$, expressed in the standard basis $\{\vert 0\rangle, \vert 1\rangle\}$,
\begin{eqnarray}
\rho&=&\frac{1}{2}\left(I+\sum_{\alpha=1}^3\,  s_\alpha \sigma_\alpha\right), \\
s_\alpha&=&{\rm Tr}[\rho\, \sigma_\alpha]; \ \ \ \sum_{\alpha} s_\alpha^2\leq 1, \nonumber
\end{eqnarray}
one obtains $(\Delta \sigma_1)^2+(\Delta \sigma_2)^2+(\Delta \sigma_3)^2=3-\sum_{\alpha} s_\alpha^2\geq 2$, in accordance with the sum uncertainty relation (\ref{paulisur}). Equality sign holds when $\sum_\alpha\, s_\alpha^2=1$ i.e., for pure states of qubit.

\subsection{Local sum uncertainty relations for bipartite systems}

Let us consider angular momentum operators $J_{A\alpha}, \ J_{B\alpha}$ acting on the Hilbert spaces ${\cal H}_A$, ${\cal H}_B$ of dimensions $d_A=(2j_A+1)$, $d_B=(2j_B+1)$ respectively. They satisfy the sum uncertainty relations
\be
\label{main}
\sum_{\alpha=1}^3\, (\Delta J_{A\alpha})^2 \geq j_A, \ \  \sum_{\alpha=1}^3\, (\Delta J_{B\alpha})^2 \geq j_B.
\ee
Then,  the  LSUR~\cite{hf}
\begin{eqnarray}
\label{lsur}
\sum_{\alpha=1}^3\, \left[\Delta \left(J_{A\alpha}+J_{B\alpha}\right)\right]^2\geq j_A+j_B
\end{eqnarray}
is necessarily satisfied by the set of all bipartite separable states $\rho^{(\rm sep)}_{AB}=\sum_{k}\, p_k\,\left(\rho_{Ak}\otimes \rho_{Bk}\right)$ in the Hilbert space ${\cal H}_A\otimes {\cal H}_B$.  Violation of the LSUR (\ref{lsur})  serves as  a  clear signature of entanglement. This is readily seen by considering a bipartite system with $j_A=j_B=j$ prepared in a  spin singlet state
$$\vert\Psi^{\rm singlet}_{AB}\rangle=\frac{1}{\sqrt{2j+1}}\sum_{m=-j}^j\, (-1)^{j-m}\, \vert j, m\rangle_A \otimes \vert j,- m\rangle_B$$
in which one obtains
$$\left(J_{A\alpha}+J_{B\alpha}\right)\vert\Psi^{\rm singlet}_{AB}\rangle\equiv 0,\ \  \alpha=1,\,2,\,3,$$ leading to $$\sum_{\alpha=1}^3\, \left[\Delta \left(J_{A\alpha}+J_{B\alpha}\right)\right]^2=0.$$  In other words, the LSUR (\ref{lsur}) gets violated in  a  spin singlet state $\vert\Psi^{\rm singlet}_{AB}\rangle$, thus highlighting  the entanglement property that local observables  of the subsystem $A$ can be   determined by performing measurements on the other subsystem $B$. More specifically, violation of LSUR signifies in general that correlations between subsystems in an entangled state can be determined with enhanced precision than those in a separable state.

It is  of interest to explore if violation of local sum uncertainty relations is both necessary and sufficient to detect  entanglement in some special classes of bipartite quantum systems. To this end, Hoffmann and Takeuchi~\cite{hf} considered the following two-qubit LSUR
\be
\label{lsur_qubit}
\sum_{\alpha=1}^3\,\left[\,\Delta(\sigma_\alpha \otimes I+I \otimes \sigma_\alpha)\right]^2\geq 4,
\ee
obtained by substituting $j_A=j_B=1/2$ and $J_{A\alpha}=(\sigma_\alpha\otimes I)/2$, $J_{B\alpha}=(I\otimes \sigma_\alpha)/2,\, \alpha=1,\, 2,\, 3$ in (\ref{lsur}).
In the one-parameter family of two-qubit Werner states,
\be
\label{werner}
\rho^{\rm Werner}_{AB}=\frac{(1-x)}{4}(I_2\otimes I_2)+x\, \vert\Psi\rangle_{AB}\langle \Psi\vert,
\ee
where $\vert\Psi\rangle_{AB}=\frac{1}{\sqrt{2}}\, (\vert 0_A\,1_B\rangle-\vert 1_A\,0_B\rangle$ and $0\leq x\leq 1$. It is readily seen that
 $$[\Delta(\sigma_\alpha \otimes I+I \otimes \sigma_\alpha)]^2=2\, (1-x), \ \ \alpha=1,2,3.$$
 Thus, the left hand side of LSUR (\ref{lsur_qubit}) is given by
\be
\label{WerLSUR}
\sum_{\alpha=1}^3\,\left[\,\Delta(\sigma_\alpha \otimes I+I \otimes \sigma_\alpha)\right]^2=6\, (1-x).
\ee
in the Werner class of two-qubit states, which are known to be entangled for $x> 1/3$.  From (\ref{WerLSUR}) it is evident that the LSUR (\ref{lsur_qubit}) is violated in the parameter range  $1/3<x\leq 1$. Thus,  violation of the two-qubit LSUR (\ref{lsur_qubit}) is both necessary and sufficient for detecting entanglement in the one-parameter family of two-qubit Werner states.

In the next section  we discuss violation of LSUR  in permutation symmetric $N$-qubit states.

\section{Violation of LSUR by permutation symmetric $N$-qubit states}
\label{sec2I}
Permutation symmetric $N$-qubit states draw  attention due to their experimental feasibility and  for the mathematical simplicity offered by them~\cite{ijmp06,pla07,prl07,sym01,sym05,aups,sym06,braun,markham,arus,lamata,symnew}. This class of states belong to the $d=2j+1=N+1$ dimensional subspace of the $2^N$ dimensional Hilbert space, which  corresponds to the maximum value $j=N/2$ of angular momentum of $N$-qubit system. Invariance under exchange of particles labeled by $\alpha$, $\beta$ in a multiparty state $\rho^{\rm sym}$ is exhibited by the property
\be
\Pi_{\alpha\beta}\rho^{\rm sym}=\rho^{\rm sym}\Pi_{\alpha\beta}=\rho^{\rm sym}
\ee
where  $ \Pi_{\alpha\beta}$ denotes the permutation operator corresponding to the interchange of labels $\alpha,\, \beta$.

We focus our attention on permutation symmetric $N$-qubit states ${\rho_{AB}^{\rm sym}}$ with even $N$, i.e., $N=2n,\ n=$integer (which corresponds to the value $j=N/2=n$ associated with  the set $\{J^{\rm total}_{\alpha}=J_\alpha\otimes I_2^{\otimes n}+I_2^{\otimes n}\otimes J_\alpha,\ \alpha=1,2,3\}$ of collective angular momentum operators  of the $N$-qubit system) and explore the LSUR  (\ref{lsur}) for bipartite divisions $A$, $B$  characterized respectively by  $j_A=j_B=n/2$. The angular momentum operators of the bipartitions $A$ and $B$ are explicitly given by
\begin{eqnarray}
\label{jajb}
J_{A\alpha}=J_\alpha\otimes I_2^{\otimes n}& =&\frac{1}{2}\, \sum_{k=1}^{n}\,\sigma_{k\alpha}\otimes I_2^{\otimes n}  \nonumber \\
J_{B\alpha}= I_2^{\otimes n} \otimes J_\alpha &=& I_2^{\otimes n}\otimes \frac{1}{2}\,\sum_{k=1}^{n}\,  \sigma_{k\alpha},
\end{eqnarray}
where $I_2^{\otimes n}=I_2\otimes I_2\otimes\ldots \otimes I_2$ denotes $n$-th order tensor product of $2\times 2$ identity matrix  $I_2$  and
\begin{equation}
\label{sigman}
\sigma_{k\alpha}=I_2\otimes I_2\otimes\ldots\otimes \sigma_\alpha\otimes I_2\otimes \ldots \otimes I_2
\end{equation}
with $\sigma_\alpha,\ \alpha=1,2,3$ appearing at the position $k\leq n$ in the $n^{\rm th}$ order tensor product. Using angular momentum algebra it readily follows that
\begin{eqnarray}
\label{nnp1A}
\sum_{\alpha=1}^3\, \left\langle\, J^2_{A\alpha}\right\rangle_{\rho_{AB}^{\rm sym}}=j_A(j_A+1)=\frac{n(n+2)}{4} \\
\label{nnp1B}
\sum_{\alpha=1}^3\, \left\langle\, J^2_{B\alpha}\right\rangle_{\rho_{AB}^{\rm sym}}=j_B(j_B+1)=\frac{n(n+2)}{4}.
\end{eqnarray}
Furthermore, the expectation values $\left\langle J_{A\alpha}\right\rangle_{\rho_{AB}^{\rm sym}}$, $\left\langle  J_{B\alpha}\,\right\rangle_{\rho_{AB}^{\rm sym}}$, $\left\langle J_{A\alpha}\,J_{B\alpha}\right\rangle_{\rho_{AB}^{\rm sym}}$ evaluated in the $N$-qubit symmetric state
 ${\rho_{AB}^{\rm sym}}$ can be expressed in terms of two-qubit averages as follows:
\begin{eqnarray}
\label{s1}
\left\langle J_{A\alpha}\right\rangle_{\rho_{AB}^{\rm sym}}&=& \left\langle J_{\alpha}\otimes\,I_2^{\otimes n}\right\rangle_{\rho_{AB}^{\rm sym}} \nonumber \\
 & =&\frac{1}{2}\, \sum_{k=1}^n\, \left\langle\,\sigma_{k\,\alpha} \otimes I_2^{\otimes{n}}\,\right\rangle_{\rho_{AB}^{\rm sym}}\nonumber  \\
&=&\frac{n}{2}\,  \langle\,\sigma_{\alpha}\otimes I_2 \rangle_{\varrho^{\rm sym}} \\
\left\langle  J_{B\alpha}\,\right\rangle_{\rho_{AB}^{\rm sym}}&=& \left\langle I_2^{\otimes n}\otimes J_{\alpha}\,\right\rangle_{\rho_{AB}^{\rm sym}}\nonumber \\
&=&\frac{1}{2}\, \sum_{l=1}^n\,  \left\langle\,  I_2^{\otimes{n}}\, \otimes \sigma_{l\,\alpha}  \right\rangle_{\rho_{AB}^{\rm sym}}\nonumber \\
\label{s2}
&=&\frac{n}{2}\,  \langle\,I_2\otimes\sigma_{\alpha}\rangle_{\varrho^{\rm sym}},\\

\label{scor}
\left\langle J_{A\alpha}\,J_{B\alpha}\right\rangle_{\rho_{AB}^{\rm sym}}&=& \left\langle J_{\alpha}\,\otimes J_{\beta}\right\rangle_{\rho_{AB}^{\rm sym}} \nonumber \\
& =&  \frac{1}{4}\, \sum_{k,l=1}^n\, \left\langle\,\sigma_{k\,\alpha} \otimes\,  \sigma_{l\,\alpha}\right\rangle_{\rho_{AB}^{\rm sym}},  \nonumber \\
&=&  \frac{n^2}{4}\,  \left\langle\,\sigma_{\alpha} \otimes\,  \sigma_{\alpha}\right\rangle_{\varrho^{\rm sym}}
\end{eqnarray}
Here it may be noticed  that
\begin{eqnarray*}
 \left\langle\,\sigma_{k\,\alpha} \otimes I_2^{\otimes{n}}\,\right\rangle_{\rho_{AB}^{\rm sym}}&=& \langle\,\sigma_{\alpha}\otimes I_2 \rangle_{\varrho^{\rm sym}} \\
 \left\langle\,  I_2^{\otimes{n}}\, \otimes \sigma_{k\,\alpha}  \right\rangle_{\rho_{AB}^{\rm sym}}&=& \langle\,I_2\otimes\sigma_{\alpha}\rangle_{\varrho^{\rm sym}}, \\
\left\langle\,\sigma_{k\,\alpha} \otimes\,  \sigma_{l\,\alpha}\right\rangle_{\rho_{AB}^{\rm sym}}&=&\left\langle\,\sigma_{\alpha} \otimes\,  \sigma_{\alpha}\right\rangle_{\varrho^{\rm sym}
\end{eqnarray*}
irrespective of the qubit labels $k,l=1,2,\ldots n$ (because the system is symmetric under interchange of constituent qubits)~\cite{ijmp06,pla07, prl07}, with   $\varrho^{\rm sym}$ denoting  the two-qubit reduced state of {\em any random pair} $(k,l)$ of qubits,  drawn from the $n$-qubit partitions  $A$ and $B$ of the  $N=2n$-qubit state ${\rho_{AB}^{\rm sym}}$.

Expressing the two-qubit symmetric density matrix
\begin{eqnarray}
\label{varrho}
\varrho^{\rm sym}&=&\frac{1}{4}\left(I_2\otimes I_2+\sum_{\alpha=1}^3 \, (\sigma_\alpha\otimes I_2+I_2\otimes \sigma_\alpha)\, s_\alpha\right. \nonumber \\
&& \ \ \ \  \left.+\sum_{\alpha,\beta=1}^3 \, (\sigma_\alpha\otimes \sigma_\beta)\, t_{\alpha\,\beta}\right)
\end{eqnarray}
in terms of its 8 real state parameters~\cite{ijmp06,pla07,prl07}
\begin{eqnarray}
\label{salpha0}
s_\alpha&=&\langle\,\sigma_{\alpha}\otimes I_2 \rangle_{\varrho^{\rm sym}}=\langle\,I_2\otimes\sigma_{\alpha}\rangle_{\varrho^{\rm sym}}  \\
\label{tab0}
t_{\alpha\,\beta}&=&\left\langle\,\sigma_{\alpha} \otimes\,  \sigma_{\beta}\right\rangle_{\varrho^{\rm sym}}=\left\langle\,\sigma_{\beta} \otimes\,  \sigma_{\alpha}\right\rangle_{\varrho^{\rm sym}}\\
&=&t_{\beta\,\alpha},  \ \ \ \ \ \  t_{11}+t_{22}+t_{33}=1,  \nonumber
\end{eqnarray}
leads to the following identifications for the expectation values $\left\langle J_{A\alpha}\right\rangle_{\rho_{AB}^{\rm sym}}$, $\left\langle  J_{B\alpha}\,\right\rangle_{\rho_{AB}^{\rm sym}}$, $\left\langle J_{A\alpha}\,J_{B\alpha}\right\rangle_{\rho_{AB}^{\rm sym}}$ (see (\ref{s1}), (\ref{s2}), (\ref{scor})):
\begin{eqnarray}
\label{salpha1}
\left\langle J_{A\alpha}\right\rangle_{\rho_{AB}^{\rm sym}}&=&\frac{n}{2}\, s_\alpha
= \left\langle  J_{B\alpha}\,\right\rangle_{\rho_{AB}^{\rm sym}} \\
\label{t}
\left\langle J_{A\alpha}\,J_{B\alpha}\right\rangle_{\rho_{AB}^{\rm sym}}&=& \frac{n^2}{4}\, t_{\alpha\alpha
\end{eqnarray}

It may be noted that under {\em identical} local unitary transformation    $\varrho^{\rm sym}\rightarrow U^{\otimes 2}\varrho^{\rm sym}\left(U^{\dag}\right)^{\otimes 2}$,\   $U\in SU(2)$, the qubit orientation vector $s=(s_1,s_2,s_3)^T$ (see (\ref{salpha1})) and the $3\times 3$ real symmetric two-qubit correlation matrix $T=(t_{\alpha\,\beta}),\ \alpha,\beta=1,2,3$ (see (\ref{t})) transform  as~\cite{pla07}
\begin{eqnarray}
 s\rightarrow s'&=&R\, s \nonumber \\
 T\rightarrow T'&=& R\,T\,R^T
\end{eqnarray}
where $R\in SO(3)$ denotes the  $3\times 3$ rotation matrix.

Let us consider the angular momentum operators of the partitions $A$ and $B$ of the even $N$-qubit density matrix $\rho_{AB}^{\rm sym}$  (See (\ref{jajb}))
\[
J_{A\alpha}=J_\alpha\otimes I_2^{\otimes n} =\frac{1}{2}\, \sum_{k=1}^{n}\,\sigma_{k\alpha}\otimes I_2^{\otimes n},
\]
\begin{eqnarray}
\label{jb}
J'_{B\alpha}&=&I_2^{\otimes n} \otimes \left\{ \left[U^\dagger(\hat{a},\theta)\,\right]^{\otimes n}\,J_\alpha \left[U(\hat{a},\theta)\right]^{\otimes n}\right\} \nonumber \\
&=& I_2^{\otimes n}\otimes \frac{1}{2}\,\sum_{k=1}^{n}\, \left\{\left[U^\dagger(\hat{a},\theta)\,\right]^{\otimes n} \sigma_{k\alpha}\,\left[U(\hat{a},\theta)\right]^{\otimes n}\right\}\nonumber \\
&=&I_2^{\otimes n}\otimes \frac{1}{2}\,\sum_{k=1}^{n}\,\left\{\sum_{\beta=1}^{3}\, R_{\alpha\beta}(\hat{a},\theta)\,  \sigma_{k\beta}\right\} \nonumber \\
&=&\sum_{\beta=1}^3 R_{\alpha\beta}(\hat{a},\theta)\, J_{B\beta}
\end{eqnarray}
where $U(\hat{a},\,\theta)=\exp \left(\frac{-i(\sigma\cdot{\hat a})\theta}{2}\right)\in SU(2)$  denotes local unitary operation and
 $R_{\alpha\beta}(\hat{a},\theta)$, $\alpha,\, \beta=1,\,2,\,3$ denote the elements of the corresponding $3\times 3$ proper orthogonal rotation matrix  $R(\hat a,\, \theta) \in SO(3)$,  with $\hat{a},\ \theta$ being the  axis, angle parameters.

In a symmetric even $N$-qubit state, the angular momentum operators $J_{A\alpha}$, $J'_{B\alpha}$, $\alpha=1,\,2,\,3$  obey the following sum uncertainty relations (see (\ref{main})):
\begin{eqnarray}
& &\sum_{\alpha=1}^3 (\Delta J_{A\alpha})^2 \geq \frac{n}{2}, \ \ \ \ \  \sum_{\alpha=1}^3 (\Delta J'_{B\alpha})^2\geq \frac{n}{2},
\end{eqnarray}
where $n=N/2$. Then the ensuing  LSUR (See (\ref{lsur}))
\be
\label{lsursym}
\sum_{\alpha=1}^3 \left[\Delta  \left(J_{A\alpha}+ J'_{B\alpha}\right)\right]^2\geq n.
\ee
is satisfied by the set of all separable symmetric states.

Violation of the LSUR (\ref{lsursym}) i.e.,
$\sum_{\alpha=1}^3 \left[\Delta  \left(J_{A\alpha}+ J'_{B\alpha}\right)\right]^2~<~n$  reveals that the bipartite state
$\rho^{\rm sym}_{AB}$ is  entangled.

We define
\begin{eqnarray}
\label{chin}
&&\chi(\hat{a},\theta)=\frac{1}{n^2}\left( \sum_{\alpha=1}^3\left\{ \Delta \left[ J_{A\alpha}+ J'_{B\alpha}\right]\right\}^2-n\right) \nonumber \\
                    &&\ \ =\frac{1}{n^2}\left(  \sum_{\alpha=1}^3\left\{\Delta \left[J_{A\alpha}+\sum_{\beta=1}^3\, R_{\alpha\beta}(\hat{a},\theta)\,J_{B\beta}\right]\right\}^2-n\right) \nonumber \\
\end{eqnarray}
and observe that  $\chi(\hat{a},\theta)<0 \Longrightarrow$ the LSUR (\ref{lsursym}) is violated. This in turn implies that  $\rho^{\rm sym}_{AB}$ is entangled.

We now proceed to prove the following Lemma.

\begin{lem}
	\label{chiexp}
 In an even $N$-qubit permutation symmetric state $\rho_{AB}^{\rm sym}$ it is seen that
\be
\label{lemF}
\chi(\hat{a},\theta)=\frac{1}{2}\left(\,1- s_0^2+{\rm Tr}\, [R(\hat{a},\theta)\,C]\right)
\ee
where   $C=T-ss^T$ is the  $3\times 3$ real symmetric {\em covariance matrix}~\cite{pla07,prl07} defined in terms of the two-qubit correlation matrix $T=(t_{\alpha\beta}),\ \alpha,\beta=1,2,3,  {\rm Tr}\,[T]=1$ (see (\ref{tab0})) and the qubit orientation vector $s=(s_1, s_2,s_3)^T$ (see (\ref{salpha0})). The squared magnitude of the qubit orientation vector is denoted by $s_0^2=s^T s=s_1^2+s_2^2+s_3^2$.
\end{lem}
\begin{pf}
  Let us express
\begin{eqnarray*}
\left[\Delta (J_{A\alpha}+J'_{B\alpha})\right]^2&=&\left[\Delta (J_{A\alpha}\right]^2+\left[\Delta (J'_{B\alpha}\right]^2 \\
 &+& 2\langle J_{A\alpha} J'_{B\alpha} \rangle - 2\langle J_{A\alpha}\rangle\, \langle J'_{B\alpha} \rangle
\end{eqnarray*}
and simplify the left hand side of the  LSUR (\ref{lsursym}) by substituting  (\ref{s1})--(\ref{scor}), (\ref{salpha0}),  (\ref{tab0}) to obtain
\begin{eqnarray}
\label{final}
\sum_{\alpha=1}^3\, \left[\Delta (J_{A\alpha}+J'_{B\alpha})\right]^2&=&\frac{n(n+2)}{2}-\frac{n^2}{2}\, s_0^2  \nonumber \\
& & +\frac{n^2}{2}\, {\rm Tr}\,
\left[ R(\hat{a},\theta) C\right].
\end{eqnarray}
It readily follows that  $\chi(\hat{a},\theta)$ defined in (\ref{chin}) reduces to the simple form given by (\ref{lemF}).   \hskip 1.3in $\square$
\end{pf}
Our main result is presented in the form of the following theorem:
\begin{thm}
	Permutation symmetric even $N$-qubit state violates the angular momentum LSUR (\ref{lsursym}) if the associated two-qubit covariance matrix $C=T-ss^T$ is negative.
\end{thm}
\begin{pf}
	Violation of the LSUR (\ref{lsursym}) is ensured whenever 	$\chi(\hat{a},\theta)<0.$
for any values of the axis-angle parameters    $a~=~(a_1,a_2,a_3)^T$, $a^Ta=a_1^2+a_2^2+a_3^2=1$, $0\leq \theta\leq 2\pi.$
Substituting the explicit form~\cite{kns} for the elements of the rotation matrix  $R(\hat{a},\theta)$ i.e.,
	\begin{eqnarray}
	\label{rot}
	R_{\alpha\beta}&=&\cos \theta\, \delta_{\alpha\beta}+(1-\cos \theta)a_\alpha a_\beta-\sin \theta\,\epsilon_{\alpha\beta\gamma}\,a_\gamma, \nonumber \\
	& &\hskip 1in \alpha,\,\beta,\,\gamma=1,\,2,\,3
	\end{eqnarray}
in terms of the axis-angle parameters  and using the property
$c_{\alpha\beta}=c_{\beta\alpha}$
of the covariance matrix $C=T-s^Ts$,
 we obtain
\begin{eqnarray}
\label{rotC}
 {\rm Tr}\, [R(\hat{a},\theta)\,C]&=& \cos \theta\, {\rm Tr}\, [C] + (1-\cos \theta)\, a^T\,C\,a \nonumber\\
 &=& \cos\theta\, (1-s_0^2) + (1-\cos \theta)\, a^TCa.   \nonumber
 \end{eqnarray}
 where we have substituted
 \be
 {\rm Tr}\, [C]=(1-s_0^2).
 \ee
 Furthermore, choosing  the rotation axis  $a=(a_1,a_2,a_3)^T$ to be the eigenvector of the two-qubit covariance matrix $C$, corresponding to its least eigenvalue $c_L$,   and setting  the rotation angle to be $\theta=\pi$ in (\ref{rotC}) we get
  \begin{eqnarray}
  \label{rotCf}
  {\rm Tr}\, [R(\hat{a},\theta=\pi)\,C]&=& -(1-s_0^2) + 2\, c_L.
  \end{eqnarray}
 Substituting (\ref{rotCf}) in (\ref{lemF}) leads to
 \begin{equation}
\chi(\hat{a},\theta=\pi)= c_L.
 \end{equation}
It is thus evident that the LSUR (\ref{lsursym}) is violated, whenever $c_L<0$,  thus proving the Theorem.  \hskip 0.7in $\square$
\end{pf}

In Ref.~\cite{pla07} it has been established by some of us that a two-qubit symmetric state $\varrho^{\rm sym}$ is entangled (negative under partial transpose) if and only if its associated covariance matrix $C=T-ss^T$ is negative. Thus the above Theorem draws attenton to the fact that entanglement in the two-qubit reduced state $\varrho^{\rm sym}$ of the whole symmetric even $N$-qubit system $\rho_{AB}^{\rm sym}$ reflects itself in the violation of the LSUR (\ref{lsursym}).

Substituting (\ref{rotCf}) in (\ref{final}), the left hand side of LSUR, one obtains
\begin{eqnarray}
\label{lsurlhs}
\sum_{\alpha=1}^3\, \left[\Delta (J_{A\alpha}+J'_{B\alpha})\right]^2=n(1+n\, c_L).
\end{eqnarray}
Thus
\be
\label{lsurc}
\sum_{\alpha=1}^3\, \left[\Delta (J_{A\alpha}+J'_{B\alpha})\right]^2\geq n \Rightarrow (1+n\, c_L)\geq 1
\ee
where  LSUR (\ref{lsursym}) is expressed in terms of the least eigenvalue of the two-qubit covariance matrix $C$  . As long as $c_L\geq 0$ (which happens to be the case for  separable symmetric states) the LSUR (\ref{lsurc})  is obeyed. In the next section we discuss two specific physical examples of $N$-qubit symmetric states which violate (\ref{lsurc}).

\section{Examples of $N$-qubit symmetric states violating the angular momentum LSUR }
\subsection{Symmetric multiqubit state generated by one-axis twisting Hamiltonian}
Kitagawa and Ueda~\cite{ku} had proposed a non-linear Hamiltonian $$\hat{H}=\chi J^2_1,$$ referred to as the {\em   one-axis twisting Hamiltonian} for generating multiqubit spin squeezed states, where $J_1$ denotes one of the components of the collective angular momentum operator of the $N$-qubit system.

Dynamical evolution of an initially {\em spin-down} state $\vert j,\,-j\rangle$ of the \break $N=2j$ qubit system,  governed by the one-axis twisting  Hamiltonian, results in~\cite{ku}:
\begin{eqnarray}
\label{oath}
\vert\Psi_{\rm {KU}}\rangle&=&\exp(-iHt)\vert j,\,-j\rangle, \ \ j=\frac{N}{2}.
\end{eqnarray}

In Ref. \cite{ijmp06,aups}, authored by some of us, the mean spin vector $s=(s_1,s_2,s_3)^T$ and the correlation matrix $T$ of a random pair of qubits drawn from the state $\vert\Psi_{\rm {KU}}\rangle$ were explicitly given: The mean spin vector is given by
\be
{s}^T=\left(0,\,0,\,-\cos^{(N-1)}(\chi\,t)\right)
\ee
and the the non-zero elements of the $3\times 3$ real symmetric matrix $T$ take the form,
\begin{eqnarray}
& & t_{11}=t_{13}=t_{23}=0, \ \  \ t_{12}=\cos^{(N-2)}(\chi\,t)\,\sin(\chi\,t) \ \nonumber\\
& & t_{22}=\frac{1}{2}\left[1- \cos^{(N-2)}(2\chi\,t) \right],\ \  t_{33}=1-t_{22}
\end{eqnarray}
We then construct the covariance matrix $C=T-{s}\, {{s}}^T$ and evaluate its eigenvalues as a function of the number $N$ of qubits  and the dimensionless dynamical parameter $\chi\,t$. Fig.~1 illustrates the behaviour of the left hand side of the LSUR (\ref{lsurc})  with respect to $\chi\,t$ for different choices of $n=N/2$.

\begin{figure}[h]
	\label{1}
	\begin{center}
		\includegraphics*[width=3in,keepaspectratio]{KU.eps}
		\caption{(Color online) The left hand side of the LSUR (\ref{lsurlhs}) i.e., $\sum_{\alpha=1}^3\, \left[\Delta (J_{A\alpha}+J'_{B\alpha})\right]^2=n(1+n\, c_L)$ in the  $N$-qubit symmetric state $\vert\Psi_{\rm {KU}}\rangle$ given in (\ref{oath}), as a function of the dimensionless dynamical parameter $\chi t$ for different choices of  $n=N/2$.  Violation of LSUR (\ref{lsurc}) is clearly seen as  $1+ n c_L< 1$.  This in turn highlights the pairwise entanglement in the symmetric $N$-qubit state (\ref{oath}) where bipartite divisions are characterized by the collective angular momenta  $j_A=j_B=n/2$.}
	\end{center}
\end{figure}

\subsection{One parameter family of W-class $N$ qubit states}
We consider the one-parameter   $N$-qubit  symmetric state of the W-class~\cite{adsum}:
\begin{eqnarray}
\label{wclass}
\vert \Psi\rangle_{\rm W}&=&\, a\,\vert 0_1,0_2,\ldots, 0_N\rangle+\sqrt{1-a^2}\,\vert W\rangle,  \nonumber\\
&& \hskip 1in 0<a<1,
\end{eqnarray}
where
\begin{eqnarray}
\vert W\rangle &=&\frac{1}{\sqrt{N}}\, \left(\vert 1_1,0_2,\ldots\,0_N\rangle + \vert 0_1,1_2,0_3,\ldots\,0_N\rangle +\right. \nonumber \\
&& \hskip 0.4in \left.\ldots\ldots +\vert 0_1,0_2,\ldots\,1_N\rangle \right)
\end{eqnarray}
denotes the symmetric $N$-qubit  W state.

Reduced two-qubit density matrix $\varrho^{\rm sym}={\rm Tr}_{N-2}\, [\,\vert \Psi\rangle_{\rm W}\langle \Psi\vert\, ]$ obtained by tracing any of the $N-2$ qubits is found to be ~\cite{adsum}
\be
\label{2wclass}
\varrho^{\rm sym}=\frac{1}{A+2D}\left(\begin{array}{cccc} A & B &B & 0\\
	B & D &D & 0\\  B & D &D & 0\\
	0 & 0 & 0 & 0        \end{array}\right)
\ee
where
\begin{eqnarray}
A&=&\frac{N^2a^2+(N-2)(1-a^2)}{N^2\,a^2+N(1-a^2)},\ B=\frac{a\sqrt{1-a^2}}{1+a^2(N-1)},\  \nonumber \\
D&=& \frac{1-a^2}{N^2\,a^2+N(1-a^2)}.\
\end{eqnarray}
The covariance matrix elements $c_{\alpha\beta}=t_{\alpha\beta}-s_{\alpha}s_{\beta}$ are readily evaluated in the two-qubit symmetric state $\varrho^{\rm sym}={\rm Tr}_{N-2}\, [\,\vert \Psi\rangle_{\rm W}\langle \Psi\vert\, ]$ and
the associated $3\times 3$ matrix $C=T-ss^T$ takes the form
\be
\label{cwclass}
C=\frac{1}{A+2D}\left(\begin{array}{ccc} 2D-4B^2 & 0 & 2B(1-D) \\
	0 & 2D & 0 \\  2B(1-D) & 0 & A-2D-A^2
\end{array}\right)
\ee
We have plotted  the left hand side of the LSUR (\ref{lsurc}) as a function of the parameter $a$, for different choices of the angular momenta  $j_A=j_B=n/2$ in Fig.~2. Violation of the LSUR i.e., $(1+n\, c_L) \leq 1$ is manifestly seen in Fig.~2, thus highlighting the pairwise entanglement in the bipartite division (characterized by collective angular momenta $j_A=j_B=n/2$) of the W-class $N$-qubit symmetric state (\ref{wclass}).
\begin{figure}[h]
	\label{1}
	\begin{center}
		\includegraphics*[width=3in,keepaspectratio]{Wclass.eps}
		\caption{(Color online) Plot of  $(1+n\, c_L)$ i.e., the left hand side (\ref{lsurc}) of the LSUR, in the W-class $N$-qubit symmetric state (\ref{wclass}),   as a function of the parameter $a$ for different values of  $n=N/2$. $c_L$ denotes the minimum eigenvalue of the covariance matrix $C$ (see (\ref{cwclass})) associated with the W-class state.}
	\end{center}
	\end{figure}

\section{Conclusion}
In this paper, it is shown that violation of local sum uncertainty relations (LSUR) in angular momentum operators is a necessary and sufficient condition for entanglement in two-qubit permutation symmetric states.
The angular momentum LSUR for permutation symmetric even $N$-qubit states are shown to be violated when the least eigenvalue of the covariance matrix of the two-qubit reduced state is negative. The entanglement in the two-qubit reduced system is thus shown to imply entanglement between bipartite divisions of the even $N$-qubit symmetric state. Our illustration of the result in two important classes of multiqubit states helps in discerning the bipartite entanglement through the parameters of two-qubit marginal.

\section*{Acknowledgements}  HSK acknowledges the support of NCN through grant SHENG (2018/30/Q/ST2/00625). Sudha and ARU are supported by the Department of Science and Technology, India (Project No. DST/ICPS/QUST/Theme-2/2019).


\section{#1}\setcounter{equation}{0}}

\newcommand{\leavevmode\hbox{\rm{\small1\kern-3.8pt\normalsize1}}}{\leavevmode\hbox{\rm{\small1\kern-3.8pt\normalsize1}}}

\def{\hbox{$\sqcup$}\llap{\hbox{$\sqcap$}}}{{\hbox{$\sqcup$}\llap{\hbox{$\sqcap$}}}}

\begin{document}
\addtolength{\baselineskip}{1.5mm}
\renewcommand{\thefootnote}{\fnsymbol{footnote}}

\begin{flushright}
DAMTP--1998--160\\
gr-qc/9812032\\
\end{flushright}
\smallskip

\begin{center}
{\huge{Bounds on negative energy densities in static \\[2mm]
        space-times}}\\[12mm]
{Christopher J. Fewster\footnote{Electronic address: {\tt cjf3@york.ac.uk}}}\\[6mm]
{\sl Department of Mathematics, University of York, Heslington,
York YO10 5DD, England}\\[8mm]

{Edward Teo\footnote{Electronic address: {\tt E.Teo@damtp.cam.ac.uk}}}\\[6mm]
{\sl Department of Applied Mathematics and Theoretical Physics,
University of Cambridge,\\
Silver Street,
Cambridge CB3 9EW,
England\\[3mm]
Department of Physics, National University of Singapore,
Singapore 119260}
\end{center}
\vspace{0.8cm}

\centerline{\bf Abstract}\bigskip
\noindent
Certain exotic phenomena in general relativity, such as backward time
travel, appear to require the presence of matter with negative energy.
While quantum fields are a possible source of negative energy densities,
there are lower bounds---known as quantum inequalities---that constrain
their duration and magnitude. In this paper, we derive new quantum
inequalities for scalar fields in static space-times, as measured by
static observers with a choice of sampling function. Unlike those
previously derived by Pfenning and Ford, our results do not assume any
specific sampling function. We then calculate these bounds in static
three- and four-dimensional Robertson--Walker universes, the de Sitter
universe, and the Schwarzschild black hole. In each case, the new
inequality is stronger than that of Pfenning and Ford for their
particular choice of sampling function.

\vfill\eject

\setcounter{footnote}{0}
\renewcommand{\thefootnote}{\arabic{footnote}}

\sect{Introduction}

In recent years, there has been much interest in various exotic solutions
of general relativity---such as traversable wormholes \cite{MT,MTY}, the
Alcubierre ``warp drive'' \cite{alcubierre}, and the Krasnikov ``tube''
\cite{kransnikov}---that permit hyperfast or backward time travel. However,
these space-times without exception require the presence of matter which
possess {\it negative\/} energy densities \cite{FR-worm,PF-warp,ER,olum},
and hence violate the standard energy conditions.

Now, it is well-known that quantum field theory, unlike classical physics,
allows energy density to be unboundedly negative at a point in space-time
\cite{epstein}. Should the theory place no restrictions on this negative
energy, quantum fields could be used to produce gross macroscopic effects
such as those mentioned above, or even a violation of cosmic censorship
or the second law of thermodynamics. It is therefore important
to have a quantitative handle on the permitted amount of negative energy
in a neighbourhood of a space-time point.

Ford and Roman \cite{FRa,FR} have found inequalities which constrain the
duration and magnitude of negative energy densities for quantised free,
real scalar fields in Minkowski space. They show that a static observer,
who samples the energy density by time-averaging it against the Lorentzian
function
\begin{equation}
\label{Lorentzian}
f(t)={t_0\over\pi}{1\over t^2+t_0^2}\,,
\end{equation}
obtains a result which is bounded from below by a negative quantity
depending inversely on the characteristic timescale $t_0$. For example,
in the case of a massless scalar field in four dimensions, the
renormalised energy density in any quantum state satisfies
\begin{equation}
\rho\geq-{3\over32\pi^2t_0^4}\,.
\end{equation}
This means the more negative the energy density that is present in an
interval, the shorter the duration of this interval must be. Thus, this
``quantum inequality''---in a way reminiscent of the uncertainty principle
of quantum mechanics\footnote{However, the derivation of the
quantum inequalities does not depend on any putative time-energy
uncertainty principle.}---serves to limit any large-scale, long-time
occurrence of negative energy. In the infinite sampling time limit
$t_0\rightarrow\infty$, it reduces to the usual averaged weak energy
condition (for quantum fields \cite{PFb,pfenning}).

Eveson and one of the present authors~\cite{fewster} have recently presented
a different derivation of the quantum inequalities for a massive scalar
field in $n$-dimensional Minkowski space (with $n\geq2$). The method used is
straightforward---involving only the canonical commutation relations and
the convolution theorem of Fourier analysis---and has the virtue of being valid
for any smooth, non-negative and even sampling function decaying sufficiently
quickly at infinity. Furthermore, the resulting bounds turn out to
be stronger than those obtained by Ford and Roman~\cite{FRa,FR} when the
Lorentzian sampling function is applied.

In the present paper, we extend this method to derive quantum inequalities
for scalar fields in generally curved but static space-times using
arbitrary smooth, non-negative (although not necessarily even, as assumed
in \cite{fewster}) sampling functions of sufficiently rapid decay.
We obtain a lower bound on the averaged normal-ordered energy density in
the Fock space built on the static vacuum in terms of the appropriate
mode functions. Since the normal-ordered energy density in a given state
is the difference between the renormalised energy density in this state
and the (generally nonzero and potentially negative) renormalised energy
density of the static vacuum, our bound also constrains the renormalised
energy density (cf.~\cite{PFb}).

We apply our bound to
several examples where the bound can be explicitly evaluated, namely the
three- and four-dimensional Robertson--Walker universes, the de Sitter
universe, and the Schwarzschild black hole. In all these cases, we obtain
bounds which are up to an order of magnitude stronger than those previously
derived by Pfenning and Ford \cite{PFb,pfenning,PFa} for the specific
sampling function they used.

\sect{Derivation of the quantum inequality}
\label{QI}

We shall consider $n+1$-dimensional space-times that are globally static,
with time-like Killing vector $\partial_t$. The metric of such a space-time
takes the general form
\begin{equation}
{\rm d}s^2=-|g_{tt}({\bf x})|{\rm d}t^2+
g_{ij}({\bf x}){\rm d}x^i{\rm d}x^j,
\end{equation}
where ${\bf x}=(x^1,x^2\dots,x^n)$ and $i,j=1,2,\dots,n$. The equation of a
free, real scalar field $\phi$ of mass $\mu\geq0$ in this space-time is
\begin{equation}
-{1\over|g_{tt}|}\partial_t^2\phi+\nabla^i\nabla_i\phi
-\mu^2\phi=0\,.
\end{equation}
Suppose it admits a complete, orthonormal set of positive frequency
solutions. We write these mode functions as
\begin{equation}
f_\lambda(t,{\bf x})=U_\lambda({\bf x}){\rm e}^{-i\omega_\lambda t},
\end{equation}
where $\lambda$ denotes the set of quantum numbers needed to specify the
mode (which may be continuous or discrete). A general quantum scalar field
can then be expanded as
\begin{equation}
\phi=\sum_\lambda(a_\lambda f_\lambda+a_\lambda^\dagger
f_\lambda^\ast)\,,
\end{equation}
in terms of creation and annihilation operators $a_\lambda^\dagger$,
$a_\lambda$ obeying the canonical commutation relations
\begin{equation}
\label{ccr}
[a_\lambda,a_{\lambda^\prime}^\dagger]=\delta_{\lambda
\lambda^\prime}\leavevmode\hbox{\rm{\small1\kern-3.8pt\normalsize1}}\,,\qquad
[a_\lambda,a_{\lambda^\prime}]=[a_\lambda^\dagger,
a_{\lambda^\prime}^\dagger]=0\,,
\end{equation}
and which generate the Fock space built on the static vacuum state
$|0\rangle$.
We shall be interested in the energy density of $\phi$
along the world-line $x^\mu(t)=(t,{\bf x_0})$ of a {\em static\/} observer,
with ${\bf x_0}$ kept fixed. If the field is in a normalised quantum state
$|\psi\rangle$, the normal-ordered energy density as measured
by such an observer at time $t$ is \cite{PFb,pfenning}
\begin{eqnarray}
\langle\,:T_{\mu\nu}u^\mu u^\nu:\,\rangle&=&{\rm Re}\,
\sum_{\lambda,\lambda^\prime}\bigg\{{\omega_\lambda\omega_{\lambda^\prime}
\over|g_{tt}|}\Big[U_\lambda^\ast U_{\lambda^\prime}\langle a_\lambda^\dagger
a_{\lambda^\prime}\rangle{\rm e}^{i(\omega_\lambda-\omega_{\lambda^\prime})t}
-U_\lambda U_{\lambda^\prime}\langle a_\lambda a_{\lambda^\prime}\rangle
{\rm e}^{-i(\omega_\lambda+\omega_{\lambda^\prime})t}\Big]\cr
&&\hskip.5in+\Big[\nabla^iU_\lambda^\ast \nabla_i
U_{\lambda^\prime}\langle a_\lambda^\dagger a_{\lambda^\prime}\rangle
{\rm e}^{i(\omega_\lambda-\omega_{\lambda^\prime})t}
+\nabla^iU_\lambda\nabla_iU_{\lambda^\prime}\langle a_\lambda
a_{\lambda^\prime}\rangle
{\rm e}^{-i(\omega_\lambda+\omega_{\lambda^\prime})t}\Big]\cr
&&\hskip.5in+m^2\Big[U_\lambda^\ast
U_{\lambda^\prime}\langle a_\lambda^\dagger a_{\lambda^\prime}\rangle{\rm
e}^{i(\omega_\lambda-\omega_{\lambda^\prime})t}+U_\lambda U_{\lambda^\prime}
\langle a_\lambda a_{\lambda^\prime}\rangle
{\rm e}^{-i(\omega_\lambda+\omega_{\lambda^\prime})t}\Big]\bigg\}\,,
\label{Tmunuexp}
\end{eqnarray}
where $u^\mu=\big(|g_{tt}|^{-1/2},{\bf 0}\big)$ is
the observer's four-velocity, and $U_\lambda$ and its derivatives are
evaluated at ${\bf x_0}$. We have also written
$\langle\,\cdot\,\rangle\equiv\langle\psi|\cdot|\psi\rangle$
for brevity. Recall that the normal-ordered energy density is the difference between
the renormalised energy density in the two states $|\psi\rangle$ and
$|0\rangle$.

We now define a weighted energy density
\begin{equation}
\rho = \int_{-\infty}^\infty{\rm d}t\,\langle\,:T_{\mu\nu}u^\mu u^\nu
:\,\rangle\, f(t)\,,
\end{equation}
where $f$ is any smooth, non-negative function decaying at least as fast as
${\rm O}(t^{-2})$ at infinity, and normalised to have unit integral.
Ford and coworkers~\cite{FRa,FR,PFb,pfenning,PFa} employ
the Lorentzian function~(\ref{Lorentzian}), whose specific properties
play a key r\^ole in their arguments [in particular, the Fourier transform
of~(\ref{Lorentzian}) is simply the function $\exp(-|\omega|t_0)$]; we
emphasise that our arguments apply to general $f$. Substituting from
Eq.~(\ref{Tmunuexp}), the weighted energy density
measured by the observer is
\begin{eqnarray}
\rho &=&{\rm Re}\,
\sum_{\lambda,\lambda^\prime}\bigg\{{\omega_\lambda\omega_{\lambda^\prime}
\over|g_{tt}|}\Big[U_\lambda^\ast U_{\lambda^\prime}\langle a_\lambda^\dagger
a_{\lambda^\prime}\rangle\widehat{f}(\omega_{\lambda'}-\omega_\lambda)
-U_\lambda U_{\lambda^\prime}\langle a_\lambda a_{\lambda^\prime}\rangle
\widehat{f}(\omega_\lambda+\omega_{\lambda'})\Big]\cr
&&\hskip.5in+\Big[\nabla^iU_\lambda^\ast \nabla_i
U_{\lambda^\prime}\langle a_\lambda^\dagger a_{\lambda^\prime}\rangle
\widehat{f}(\omega_{\lambda'}-\omega_\lambda)
+\nabla^iU_\lambda\nabla_iU_{\lambda^\prime}\langle a_\lambda
a_{\lambda^\prime}\rangle
\widehat{f}(\omega_\lambda+\omega_{\lambda'})
\Big]\cr
&&\hskip.5in+m^2\Big[U_\lambda^\ast
U_{\lambda^\prime}\langle a_\lambda^\dagger a_{\lambda^\prime}\rangle
\widehat{f}(\omega_{\lambda'}-\omega_\lambda)
+U_\lambda U_{\lambda^\prime}
\langle a_\lambda a_{\lambda^\prime}\rangle
\widehat{f}(\omega_\lambda+\omega_{\lambda'})\Big]\bigg\}\,,
\end{eqnarray}
where we define the Fourier transform of $f$ by
\begin{equation}
\widehat{f}(\omega)=\int_{-\infty}^\infty{\rm d}t\,f(t){\rm
e}^{-i\omega t}\,.
\end{equation}

By applying the inequality (\ref{inequality}), proved in the Appendix,
to each of the cases
$q_\lambda={\omega_\lambda\over|g_{tt}|^{1/2}}U_\lambda$, $\nabla_iU_\lambda$
and $mU_\lambda$, we obtain the following manifestly negative lower bound for $\rho$:
\begin{equation}
\rho\geq-{1\over2\pi}\int_0^\infty{\rm d}\omega\,\sum_\lambda
\bigg({\omega_\lambda^2\over|g_{tt}|}U_\lambda^\ast U_\lambda
+\nabla^iU_\lambda^\ast\nabla_iU_\lambda+m^2U_\lambda^\ast U_\lambda\bigg)
\left|\widehat{f^{1/2}}(\omega+\omega_\lambda)\right|^2.
\end{equation}
Using the field equation satisfied by the spatial mode function
\cite{PFb,pfenning}:
\begin{equation}
\nabla^i\nabla_iU_\lambda+\bigg({\omega_\lambda^2\over|g_{tt}|}
-m^2\bigg)U_\lambda=0\,,
\end{equation}
this inequality can be rewritten as
\begin{equation}
\label{QIb}
\rho\geq-{1\over\pi}\int_0^\infty{\rm d}\omega\,\sum_\lambda
\bigg({\omega_\lambda^2\over|g_{tt}|}+{1\over4}\nabla^i\nabla_i\bigg)
|U_\lambda|^2\left|\widehat{f^{1/2}}(\omega+\omega_\lambda)\right|^2.
\end{equation}
This is the desired quantum inequality, which is valid for general
sampling functions $f(t)$, subject to the above-stated conditions.
Another useful form of it can be obtained by
introducing the new variable $u=\omega+\omega_\lambda$:
\begin{equation}
\label{QIc}
\rho\geq-{1\over\pi}\int_{\omega_{\rm min}}^\infty{\rm d}u\,
\left|\widehat{f^{1/2}}(u)\right|^2\sum_{\lambda~{\rm s.t.}~\omega_\lambda\leq u}
\bigg({\omega_\lambda^2\over|g_{tt}|}+{1\over4}\nabla^i\nabla_i\bigg)
\,|U_\lambda|^2,
\end{equation}
with $\omega_{\rm min}\equiv\min_\lambda\omega_\lambda$.

To simplify it any further would require a specific choice of $f(t)$. For
example, with the even sampling function
\begin{equation}
\label{mysamp}
f(t)={2\over\pi}{t_0^3\over(t^2+t_0^2)^2}\,,
\end{equation}
that is peaked at $t=0$, we have
\begin{equation}
\left|\widehat{f^{1/2}}(\omega)\right|^2=2\pi t_0{\rm e}^{-2|\omega|t_0}.
\end{equation}
In this case, the quantum inequality can be expressed in terms of the
Euclidean Green's function
\begin{equation}
G_{\rm E}(t,{\bf x};t^\prime,{\bf x}^\prime)=\sum_\lambda
U_\lambda^\ast({\bf x})U_\lambda({\bf x}^\prime){\rm e}^{\omega_\lambda
(t-t^\prime)},
\end{equation}
quite compactly as
\begin{equation}
\rho\geq-\hbox{$1\over4$}{\hbox{$\sqcup$}\llap{\hbox{$\sqcap$}}}_{\rm E}G_{\rm E}(-t_0,{\bf x};
t_0,{\bf x})\,,
\end{equation}
where ${\hbox{$\sqcup$}\llap{\hbox{$\sqcap$}}}_{\rm E}\equiv{1\over|g_{tt}|}\partial_{t_0}^2+\nabla^i\nabla_i$
is the Euclidean wave operator. This bound is, in
fact, identical to one that was derived in \cite{PFb,pfenning} assuming the
Lorentzian sampling function (\ref{Lorentzian}). But because (\ref{mysamp})
is a more sharply peaked function [half the area under the Lorentzian function
lies within $|t|<t_0$, while this figure is
${1\over2}+{1\over\pi}\simeq0.82$ for
(\ref{mysamp})], this is a first indication that the
inequality derived here is a stronger result.

Finally, we record the fact that for the Lorentzian function,
\begin{equation}
\label{FT}
\left|\widehat{f^{1/2}}(\omega)\right|^2=\frac{4t_0}{\pi}
K_0(t_0|\omega|)^2,
\end{equation}
where $K_0(x)$ is the modified Bessel function of zeroth order.
In the rest of this paper, we shall consider the quantum inequality
in specific examples of globally static space-times where the left-hand
side of (\ref{QIb}) or (\ref{QIc}) can be explicitly evaluated.
As these examples have been considered previously by Pfenning and Ford
\cite{PFb,pfenning,PFa}, we shall at times be brief and refer to their
papers for more details. For the most part we will closely follow their
notation and conventions.

\sect{Minkowski space}

We begin with a review of the quantum inequality in $n+1$-dimensional
Minkowski space, the case that was treated in \cite{fewster}. The mode
functions for a free scalar field of mass $\mu$ are
\begin{equation}
U_{\bf k}({\bf x})={1\over[(2\pi)^n
2\omega_{\bf k}]^{1/2}}{\rm e}^{i{\bf k}\cdot{\bf x}},\qquad
\omega_{\bf k}=\sqrt{|{\bf k}|^2+\mu^2}\,,
\end{equation}
with each component of the $n$-dimensional (spatial) momentum covector ${\bf k}$
satisfying $-\infty<k_i<\infty$. The quantum inequality (\ref{QIb}) becomes
\begin{eqnarray}
\label{Mbound}
\rho&\geq&-{1\over2\pi}\int_0^\infty{\rm d}\omega
\int{{\rm d}^n{\bf k}\over(2\pi)^n}\,\omega_{\bf k}
\left|\widehat{f^{1/2}}(\omega+\omega_{\bf k})\right|^2\cr
&=&-\frac{C_n}{2\pi}\int_0^\infty{\rm d}\omega\int_\mu^\infty{\rm d}\omega'\,
{\omega'}^2({\omega'}^2-\mu^2)^{n/2-1}
\left|\widehat{f^{1/2}}(\omega+\omega')\right|^2,
\end{eqnarray}
where $C_n$ is equal to the area of the unit $n-1$-sphere divided by
$(2\pi)^n$, i.e.,
\begin{equation}
C_n\equiv\frac{1}{2^{n-1}\pi^{n/2}\Gamma(\frac{1}{2}n)}\,.
\end{equation}
If we make the change of variables $u=\omega+\omega'$ and $v=\omega'$,
the quantum inequality (\ref{Mbound}) can be rewritten as
\begin{equation}
\label{Mbound2}
\rho\ge-\frac{C_n}{2\pi(n+1)}\int_\mu^\infty{\rm d}u\,
\left|\widehat{f^{1/2}}(u)\right|^2 u^{n+1}Q_n\bigg(\frac{u}{\mu}\bigg),
\end{equation}
where the functions $Q_n(x)$ are defined by
\begin{equation}
Q_n(x) = (n+1)x^{-(n+1)}\int_1^x{\rm d}y\, y^2(y^2-1)^{n/2-1}.
\end{equation}

There are several special cases in which this bound can be evaluated
analytically \cite{fewster}, notably massless fields in two and
four dimensions with the sampling function
(\ref{Lorentzian}). In the former, the bound is four times stronger
than that derived by Ford and Roman \cite{FR}, but $1{1\over2}$ times weaker
than the optimal one of Flanagan \cite{flanagan}. In the latter case, the
present bound is $9\over64$ of Ford and Roman's result.

\sect{Three-dimensional closed universe}
\label{IIIDCU}

The line element for the static, three-dimensional closed universe is
\begin{equation}
{\rm d}s^2=-{\rm d}t^2+a^2({\rm d}\theta^2+\sin^2\theta\,{\rm d}\varphi^2)\,,
\end{equation}
where $a$ is the radius of the two-sphere at each constant time-slice,
and the angular variables take values $0\leq\theta\leq\pi$,
$0\leq\varphi<2\pi$ (and will do so for all the space-times considered in
this paper). We
consider the massive scalar field equation on this background with a
coupling of strength $\xi$ to the scalar curvature $R=2/a^2$:
\begin{equation}
{\hbox{$\sqcup$}\llap{\hbox{$\sqcap$}}}\phi-(\mu^2+\xi R)\phi=0\,,
\end{equation}
whose mode-function solutions are given in terms of the usual spherical
harmonics $Y_{lm}(\theta,\varphi)$ by~\cite{pfenning,PFa}
\begin{equation}
U_{lm}({\bf x})={1\over(2\omega_la^2)^{1/2}}Y_{lm}(\theta,\varphi)\,,
\end{equation}
for $l=0,1,2,\dots$ and $m=-l,-l+1,\dots,l$, with
\begin{equation}
\omega_l=a^{-1}\sqrt{l(l+1)+2\xi+(a\mu)^2}\, .
\end{equation}
The $Y_{lm}(\theta,\varphi)$ obey the sum rule
\begin{equation}
\label{sumrule}
\sum_{m=-l}^{l}\big|Y_{lm}(\theta,\varphi)\big|^2={2l+1\over4\pi}\,,
\end{equation}
which can be used in (\ref{QIb}) and (\ref{QIc}) to show that
\begin{eqnarray}
\label{3Dbound}
\rho&\geq&-{1\over8\pi^2a^2}\int_0^\infty{\rm d}\omega\sum_{l=0}^\infty\,
(2l+1)\omega_l\left|\widehat{f^{1/2}}(\omega+\omega_l)\right|^2\cr
&=&-{1\over8\pi^2a^2}\int_{\omega_0}^\infty{\rm d}u\,
\left|\widehat{f^{1/2}}(u)\right|^2
\sum_{l=0}^{N(u)}\,(2l+1)\omega_l\,.
\end{eqnarray}
Here, $N(u)\equiv\max\{n\in{\bf Z}:\,\omega_n\leq u\}$, i.e.,
\begin{equation}
N(u)=\left\lfloor{\sqrt{1-4[2\xi+(a\mu)^2-(au)^2]}-1\over2}
\right\rfloor,
\end{equation}
where $\lfloor x \rfloor$ denotes the integer part of $x$.

While the bound in (\ref{3Dbound}) can be readily evaluated using
numerical techniques, it may be worthwhile to first simplify it
analytically as much as possible. This may be useful if one should
want to draw conclusions about its general properties. In particular,
we shall present a general strategy for approximating finite summations
like that in (\ref{3Dbound}).

\begin{figure}[t]
\begin{center}
\epsfxsize=6.in
\epsffile{QI-fig1.eps}
\caption{Graph of the QI bound for the 3D
closed universe [dashed line], and that obtained by Pfenning and Ford
[solid line], against $\mu$.}
\label{fig2}
\end{center}
\end{figure}

The summation in (\ref{3Dbound}) can be evaluated using the
trapezoidal rule of numerical integration (e.g., see Eq.~(3.6.1)
of \cite{Hildebrand}):
\begin{equation}
\label{trapez}
\sum_{n=0}^Ng(n)=\int_0^N{\rm d}x\,g(x)+{1\over2}\left[g(0)+g(N)\right]
+{N\over12}g''(\zeta)\,,
\end{equation}
for some $\zeta\in(0,N)$. In the present case, $g(x)=(2x+1)\sqrt{x(x+1)+
2\xi+(a\mu)^2}$, and the integral in (\ref{trapez}) can be evaluated
analytically. Furthermore, $g''(\zeta)$ is non-decreasing in
the interval in question, so its occurrence in (\ref{3Dbound}) can be
replaced by $g''(N)$, at the expense of weakening the bound slightly.
We obtain the final inequality
\begin{equation}
\label{3Dbound1}
\rho\geq-{1\over8\pi^2a^3}\int_{\omega_0}^\infty{\rm d}u\,
\left|\widehat{f^{1/2}}(u)\right|^2
\left\{\int_0^{N(u)}{\rm d}x\,g(x)+{1\over2}\left[g(0)+g(N(u))\right]
+{N(u)\over12}g''(N(u))\right\},
\end{equation}
with
\begin{eqnarray}
\int{\rm d}x\,g(x)&=&{2\over3}\left[x(x+1)+2\xi+(a\mu)^2\right]^{3/2},\cr
g''(x)&=&{3(2x+1)\over\sqrt{x(x+1)+2\xi+(a\mu)^2}}-{1\over4}
{(2x+1)^3\over\big[x(x+1)+2\xi+(a\mu)^2\big]^{3/2}}\,.
\end{eqnarray}

The graph of the bound in (\ref{3Dbound}) is plotted against mass
in Fig.~\ref{fig2}, for $a=1$ and $\xi=0$. As usual, the sampling
function $f(t)$ is taken to be the Lorentzian function
(\ref{Lorentzian}), with $t_0=1$. When plotted on the same scale, that
of (\ref{3Dbound1}) is almost indistinguishable from the former graph.
For comparison, the corresponding bound obtained by Pfenning and Ford
\cite{pfenning,PFa} is also plotted in  Fig.~\ref{fig2}. It is clear that
our bound is stronger for all values of mass.

\sect{Four-dimensional Robertson--Walker universe}
\label{IVDRWU}

We shall first consider the case of the open universe, before proceeding
to the closed universe. The line element is
\begin{equation}
{\rm d}s^2=-{\rm d}t^2+a^2\left[{\rm d}\chi^2+\sinh^2\chi\,
\big({\rm d}\theta^2+\sin^2\theta\,{\rm d}\varphi^2\big)\right],
\end{equation}
where $a$ characterises the scale of the spatial section, and
$0\leq\chi<\infty$.
The mode functions for a scalar field of mass $\mu$ are \cite{parker}
\begin{eqnarray}
U_{qlm}({\bf x})&=&{1\over(2\omega_qa^3)^{1/2}}
\Pi^{(-)}_{ql}(\chi)Y_{lm}(\theta,\varphi)\,,\cr
\omega_q&=&\sqrt{{q^2+1\over a^2}+\mu^2}\,,
\end{eqnarray}
with $0<q<\infty$ and $l,m$ as usual. The functions $\Pi^-_{ql}(\chi)$ satisfy
\begin{equation}
\Pi^{(-)}_{ql}(\chi)\propto\sinh^l\chi\,
\bigg({{\rm d}\over{\rm d}\cosh\chi}\bigg)^{l+1}\cos q\chi\,,
\end{equation}
and obey the sum rule
\begin{equation}
\sum_{l,m}\,\left|\Pi^{(-)}_{ql}(\chi)Y_{lm}
(\theta,\varphi)\right|^2 ={q^2\over 2\pi^2}\,.
\end{equation}
The right-hand side does not depend on the angular variables, as is
expected of a system with isotropic symmetry. Hence, the quantum inequality
(\ref{QIb}) becomes
\begin{equation}
\label{Cbound}
\rho\geq-{1\over4\pi^3a^3}\int_0^\infty{\rm d}\omega
\int_0^\infty{\rm d}q\,\omega_qq^2
\left|\widehat{f^{1/2}}(\omega+\omega_q)\right|^2.
\end{equation}

Note that this bound is identical in form to that in (four-dimensional)
Minkowski space. Both (\ref{Mbound}) and (\ref{Cbound}) can, in fact, be
written as
\begin{eqnarray}
\rho&\geq&-{1\over4\pi^3}\int_0^\infty{\rm d}\omega
\int_C^\infty{\rm d}\omega'\,\omega'^2\sqrt{\omega'^2-C^2}
\left|\widehat{f^{1/2}}(\omega+\omega')\right|^2\cr
&=&-\frac{1}{16\pi^3}\int_C^\infty{\rm d}u\,\left|\widehat{f^{1/2}}(u)\right|^2
u^4 Q_3\left(\frac{u}{C}\right),
\end{eqnarray}
where
\begin{equation}
C\equiv\sqrt{{\epsilon\over a^2}+\mu^2},\qquad\epsilon=
\cases{0&Minkowski space;\cr1&open universe,}
\end{equation}
and an explicit expression (and graph) for $Q_3(x)$ can be found in
\cite{fewster}. The Minkowski space result is obviously recovered in
the limit of infinite $a$. Furthermore, since $Q_3(x)$ is an increasing
function on $[1,\infty)$, the bound for general $a$ is tighter than that
in Minkowski space for all sampling functions $f(t)$. Pfenning and Ford
\cite{pfenning,PFa} also noted this for their particular choice of $f(t)$.

We now turn to the closed or Einstein universe, with line element
\begin{equation}
{\rm d}s^2=-{\rm d}t^2+a^2\left[{\rm d}\chi^2+\sin^2\chi\,
\big({\rm d}\theta^2+\sin^2\theta\,{\rm d}\varphi^2\big)\right],
\end{equation}
where $0\leq\chi\leq\pi$. The mode functions are \cite{parker}
\begin{eqnarray}
U_{nlm}({\bf x})&=&{1\over(2\omega_na^3)^{1/2}}
\Pi^{(+)}_{nl}(\chi)\, Y_{lm}(\theta,\varphi)\,,\cr
\omega_n&=&\sqrt{{n(n+2)\over a^2}+\mu^2}\,,
\end{eqnarray}
with $n=0,1,2,\dots$, $l=0,1,\dots,n$, $m=-l,-l+1\ldots,l$, and
\begin{equation}
\Pi^{(+)}_{nl}(\chi)\propto\sin^l\chi\,
\bigg({{\rm d}\over{\rm d}\cos\chi}\bigg)^{l+1}\cosh (n+1)\chi\,.
\end{equation}
Using the sum rule
\begin{equation}
\sum_{l,m}\,\left|\Pi^{(+)}_{nl}(\chi)Y_{lm}
(\theta,\varphi)\right|^2={(n+1)^2\over 2\pi^2}\,,
\end{equation}
we obtain the quantum inequality
\begin{eqnarray}
\label{RWbound}
\rho&\geq&-{1\over4\pi^3a^3}\int_0^\infty{\rm d}\omega
\sum_{n=0}^\infty\,\omega_n(n+1)^2
\left|\widehat{f^{1/2}}(\omega+\omega_n)\right|^2\cr
&=&-{1\over4\pi^3a^3}\int_\mu^\infty{\rm d}u\,
\left|\widehat{f^{1/2}}(u)\right|^2
\sum_{n=0}^{N(u)}\,\omega_n(n+1)^2,
\end{eqnarray}
with
\begin{equation}
N(u)\equiv\Big\lfloor\sqrt{(au)^2-(a\mu)^2+1}-1\Big\rfloor\,.
\end{equation}

\begin{figure}[t]
\begin{center}
\epsfxsize=6.in
\epsffile{QI-fig2.eps}
\caption{Graphs of the QI bounds for the 4D
closed universe with $a\mu=1$: (\ref{Boundmua1}) using a dashed line,
(\ref{Boundmua2}) using a dotted line, and that obtained by Pfenning and
Ford [solid line].}
\label{fig3}
\end{center}
\end{figure}

An obvious special case to investigate is $a\mu=1$, in which
$\omega_n=\mu(n+1)$. The sum in (\ref{RWbound}) may then be evaluated exactly,
to give
\begin{equation}
\label{Boundmua1}
\rho \ge -\frac{1}{16\pi^3a^4}\int_\mu^\infty{\rm d}u\,
\left|\widehat{f^{1/2}}(u)\right|^2(N(u)+1)^2(N(u)+2)^2.
\end{equation}
This bound may be weakened slightly, by replacing $N(u)=\lfloor
au-1\rfloor$ with the larger quantity $au-1$, to give
\begin{equation}
\label{Boundmua2}
\rho \ge -\frac{1}{16\pi^3}\int_{\mu}^\infty{\rm d}u\,
\left|\widehat{f^{1/2}}(u)\right|^2
\bigg( u^4+ \frac{2u^3}{a} + \frac{u^2}{a^2}\bigg)\,.
\end{equation}
It clearly differs from the massless Minkowski bound by ${\rm O}(1/a)$ terms.
The bounds in (\ref{Boundmua1}) and (\ref{Boundmua2}) are plotted in
Fig.~\ref{fig3} against mass. The difference between these two graphs can
be further minimised using the approximations below, but at the expense of
having a more complicated expression for the bound. The corresponding bound
derived in \cite{pfenning,PFa} is also plotted on the same graph.

\begin{figure}[t]
\begin{center}
\epsfxsize=6.in
\epsffile{QI-fig3.eps}
\caption{Graphs of the QI bounds for the 4D closed universe:
(\ref{Boundgen1}) using a solid line, and its approximation
(\ref{Boundgen2}) using a dashed line.}
\label{fig4}
\end{center}
\end{figure}

\begin{figure}[t]
\begin{center}
\epsfxsize=6.in
\epsffile{QI-fig4.eps}
\caption{Graphs of the QI bounds for the 4D closed universe:
(\ref{Boundgen1}) using a dashed line, and that of Pfenning and Ford
[solid line].}
\label{fig5}
\end{center}
\end{figure}

Returning to the general case, we note that (\ref{RWbound}) can be
written as
\begin{equation}
\label{Boundgen1}
\rho\geq-{1\over4\pi^3a^4}\int_\mu^\infty{\rm d}u\,
\left|\widehat{f^{1/2}}(u)\right|^2
\sum_{n=1}^{N'}\,n^2\sqrt{n^2+(a\mu)^2-1}\,,
\end{equation}
where $N'\equiv\lfloor a\sqrt{u^2-\mu^2}\rfloor$. The finite sum can
again be approximated analytically using the trapezoidal rule, now in the form:
\begin{equation}
\sum_{n=1}^Ng(n)=\int_1^N{\rm d}x\,g(x)+{1\over2}\left[g(1)+g(N)\right]
+{N-1\over12}g''(\zeta)\,,
\end{equation}
for some $\zeta\in(1,N)$. From the fact that the second derivative
of $g(x)=x^2\sqrt{x^2+(a\mu)^2-1}$ is non-decreasing in this interval,
we obtain the inequality
\begin{equation}
\label{Boundgen2}
\rho\geq-{1\over4\pi^3a^4}\int_\mu^\infty{\rm d}u\,
\left|\widehat{f^{1/2}}(u)\right|^2
\left\{\int_1^{N'}{\rm d}x\,g(x)+{1\over2}\left[g(1)+g(N')\right]
+{N'-1\over12}g''(N')
\right\}\,,
\end{equation}
with
\begin{eqnarray}
\int{\rm d}x\,g(x)&=&{1\over4}x\big[x^2+(a\mu)^2-1\big]^{3/2}-{1\over8}
\big[(a\mu)^2-1\big]x\sqrt{x^2+(a\mu)^2-1}\cr
&&\qquad-{1\over8}\big[(a\mu)^2-1\big]^2\ln\Big(x+\sqrt{x^2+(a\mu)^2-1}\Big)\,,
\cr
g''(x)&=&2\sqrt{x^2+(a\mu)^2-1}+{5x^2\over\sqrt{x^2+(a\mu)^2-1}}
-{x^4\over\big[x^2+(a\mu)^2-1\big]^{3/2}}\,.
\end{eqnarray}

The bound in (\ref{Boundgen1}) and its approximation in (\ref{Boundgen2})
are plotted in Fig.~\ref{fig4} against $\mu$, for $a=1$. As can be seen,
the approximation is only very slightly weaker than the exact bound.
Also plotted in Fig.~\ref{fig5} is the bound obtained in \cite{pfenning,PFa},
for comparison with (\ref{Boundgen1}).

\sect{de Sitter space-time}

A convenient static parametrisation of the de Sitter universe is
\begin{equation}
{\rm d}s^2 = -\bigg(1-{r^2\over\alpha^2}\bigg)\,{\rm d}t^2
+\bigg(1-{r^2\over\alpha^2}\bigg)^{-1}{\rm d}r^2+r^2({\rm d}\theta^2
+\sin^2\theta\,{\rm d}\varphi^2)\,,
\end{equation}
with $0\le r\le \alpha$. The surface $r=\alpha$ is the particle horizon
for an observer located at the origin. In this representation, the mode
functions for a scalar field with mass $\mu$ and energy $\omega$ are
\begin{equation}
U_{klm}({\bf x})={1\over(4\pi\alpha^2k)^{1/2}}
f_{kl}(z)Y_{lm}(\theta,\varphi)\,,
\end{equation}
where we denote $z\equiv r/\alpha$ and $k\equiv\alpha\omega$. The latter
continuously parametrises the mode function from zero to infinity, while
$l$ and $m$ are as in Sec.~\ref{IIIDCU}. The radial function can then be
solved in terms of the hypergeometric function $F(a,b;c;z)$ as
\cite{higuchi}
\begin{equation}
f_{kl}(z)={\Gamma(b^+_l)\Gamma(b^-_l)\over\Gamma(l+{3\over 2})
\Gamma(ik)}z^l(1-{z^2})^{ik/2}F\bigg(b^+_l,b^-_l;l+{3\over 2};z^2\bigg)\,,
\end{equation}
with
\begin{equation}
b^\pm_l\equiv{1\over 2}\bigg(l+{3\over2}+ik
\pm\sqrt{{9\over4}-\alpha^2\mu^2}\bigg)\,.
\end{equation}

Using the sum rule (\ref{sumrule}) in the quantum inequality (\ref{QIb}),
we have for an observer at the origin,
\begin{eqnarray}
\rho&\geq&-{1\over64\pi^3\alpha^4}\int_0^\infty{\rm d}\omega
\int_0^\infty{\rm d}k\sum_{l=0}^\infty\,{2l+1\over k}\bigg|{\Gamma(b^+_l)
\Gamma(b^-_l)\over\Gamma(l+{3\over 2})\Gamma(ik)}\bigg|^2\cr&&\quad
\lim_{z\rightarrow0}
\bigg\{{4k^2\over1-z^2}+{1\over z^2}\partial_z\big[z^2(1-z^2)\partial_z\big]
\bigg\}z^{2l}\bigg|F\bigg(b^+_l,b^-_l;l+{3\over 2};z^2\bigg)\bigg|^2
\left|\widehat{f^{1/2}}(\omega+k/\alpha)\right|^2.~~~~~~~
\end{eqnarray}
In fact, only the $l=0$ and $l=1$ terms contribute (cf. Eqs.~(4.126)
and~(4.127) of~\cite{pfenning}), and the expression may
be simplified to give
\begin{eqnarray}
\label{deSitter}
\rho&\geq&-{1\over8\pi^5\alpha^4}\int_0^\infty{\rm d}\omega
\int_0^\infty{\rm d}k\,\sinh(\pi k)\Big\{(k^2+\alpha^2\mu^2)\big|
\Gamma(b^+_0)\Gamma(b^-_0)\big|^2+4\big|\Gamma(b^+_1)\Gamma(b^-_1)\big|^2
\Big\}\cr&&\hskip1.65in
\times\left|\widehat{f^{1/2}}(\omega+k/\alpha)\right|^2.
\end{eqnarray}

\begin{figure}[t]
\begin{center}
\epsfxsize=6.in
\epsffile{QI-fig5.eps}
\caption{Graphs of the QI bounds for de Sitter
space-time: (\ref{deSitter}) using a dashed line, and that of Pfenning
and Ford [solid line].}
\label{fig6}
\end{center}
\end{figure}

As was noted in \cite{PFb,pfenning}, there are two cases for which the gamma
functions in (\ref{deSitter}) can be evaluated analytically, namely when
$\mu=0$ and $\sqrt{2}/\alpha$. Assuming the Lorentzian sampling function
(\ref{Lorentzian}) and using (\ref{FT}), we obtain, for the massless case,
\begin{equation}
\rho\geq-{t_0\over2\pi^4\alpha^2}\int_0^\infty{\rm d}\omega
\int_0^\infty{\rm d}\omega^\prime\,(5\omega^\prime+2\alpha^2\omega^\prime{}^3)
K_0\big(t_0(\omega+\omega^\prime)\big)^2.
\end{equation}
Defining the new variables $u=\omega+\omega^\prime$ and $v=\omega^\prime$,
the bound becomes
\begin{equation}
-{t_0\over2\pi^4\alpha^2}\int_0^\infty{\rm d}u\,K_0(t_0u)^2
\int_0^u{\rm d}v\,(5v+2\alpha^2v^3)\,.
\end{equation}
This can be explicitly evaluated using the integral
\begin{equation}
\label{integral}
\int_0^\infty{\rm d}u\,u^{\alpha-1}K_0(t_0u)^2={2^{\alpha-3}
\over t_0^\alpha\Gamma(\alpha)}\Gamma\Big({\alpha\over2}\Big)^4,
\end{equation}
to obtain
\begin{equation}
\label{mydS}
\rho\geq-{3\over32\pi^2t_0^4}{9\over64}\bigg[1+{16\over9}{5\over3}
\Big({t_0\over\alpha}\Big)^2\bigg]\,.
\end{equation}
This bound is at least four times stronger than that obtained in
\cite{PFb,pfenning}:
\begin{equation}
\label{forddS}
\rho\geq-{3\over32\pi^2t_0^4}\bigg[1+{5\over3}
\Big({t_0\over\alpha}\Big)^2\bigg]\,.
\end{equation}
In the limit $\alpha\rightarrow\infty$ or $t_0\rightarrow0$, we expect
to recover the results for Minkowski space. Indeed, the bound in (\ref{mydS})
is then $9\over64$ that in (\ref{forddS}), as was observed in \cite{fewster}.

When $\mu=\sqrt{2}/\alpha$, we similarly obtain the quantum inequality
\begin{equation}
\label{mydS2}
\rho\geq-{3\over32\pi^2t_0^4}{9\over64}\bigg[1+{16\over9}
\Big({t_0\over\alpha}\Big)^2\bigg]\,,
\end{equation}
in contrast to that derived in \cite{PFb,pfenning}:
\begin{equation}
\rho\geq-{3\over32\pi^2t_0^4}\bigg[1+
\Big({t_0\over\alpha}\Big)^2\bigg]\,.
\end{equation}
The bound in (\ref{deSitter}) and that derived in \cite{PFb,pfenning} are
plotted for general $\mu$, and $\alpha=1$, in Fig.~\ref{fig6}.

We have, in fact, proved that for general $\mu$, the de Sitter bound
(\ref{deSitter}) differs from the Minkowski space bound~(\ref{Mbound2})
by terms no greater than order $\alpha^{-1/2}$ as $\alpha\to\infty$, and
so our results for these cases agree in this limit.
This estimate involves bounds on the integrand in Eq.~(\ref{deSitter})
which are uniform in $k$ and $\omega$. The proof, which we omit, is
accordingly somewhat technical. It is unclear whether the argument can
be strengthened to show that the deviation is in fact
${\rm O}(\alpha^{-2})$ in general, as it is for the specific
cases considered in~(\ref{mydS}) and~(\ref{mydS2}).

\sect{Schwarzschild space-time}

As the final example, we shall examine the quantum inequalities in a
black hole space-time. The line element for the Schwarzschild black hole
of mass $M$ is
\begin{equation}
{\rm d}s^2=-\left(1-{2M\over r}\right){\rm d}t^2
+\left(1-{2M\over r}\right)^{-1}{\rm d}r^2
+r^2({\rm d}\theta^2 + \sin^2\theta\,{\rm d}\varphi^2)\,.
\end{equation}
For simplicity, we shall only consider a massless scalar field in
this space-time. The mode
functions, in the region exterior to the horizon $r>2M$, take the form
\cite{deWitt}
\begin{eqnarray}
\stackrel{\rightarrow}{U}_{\omega lm}({\bf x})&=&
{1\over(4\pi\omega)^{1/2}}\stackrel{\rightarrow}{R}_l(\omega|r)
Y_{lm}(\theta,\varphi)\,,\cr
\stackrel{\leftarrow}{U}_{\omega l m}({\bf x})&=&{1\over(4\pi\omega)^{1/2}}
\stackrel{\leftarrow}{R}_l(\omega|r)Y_{lm}(\theta,\varphi)\,,
\end{eqnarray}
where, as usual, $\omega$ is the energy of the field and
$Y_{lm}(\theta,\varphi)$ are the spherical harmonics.
$\stackrel{\rightarrow}{R}_l(\omega|r)$ and
$\stackrel{\leftarrow}{R}_l(\omega|r)$
are the outgoing and ingoing solutions to the radial part of the wave
equation, respectively. Although this equation cannot be solved
analytically, the asymptotic forms of the solutions are known near the horizon
and at infinity.

Again, using the sum rule (\ref{sumrule}), we see that the quantum inequality
(\ref{QIb}) becomes
\begin{eqnarray}
\label{Schwarz}
\rho&\geq&-{1\over16\pi^3}\int_0^\infty{\rm d}\omega
\int_0^\infty{{\rm d}\omega^\prime\over\omega^\prime}\sum_{l=0}^\infty\,
(2l+1)\,
\bigg\{{\omega^\prime{}^2\over1-{2M\over r}}+{1\over4r^2}\partial_r\Big[
r^2(1-2M/r)\partial_r\Big]\bigg\}\cr
&&\hskip1.65in\times\,\bigg[\Big|\stackrel{\rightarrow}{R}_l(\omega^\prime|r)\Big|^2+
\Big|\stackrel{\leftarrow}{R}_l(\omega^\prime|r)\Big|^2\bigg]
\left|\widehat{f^{1/2}}(\omega
+\omega^\prime)\right|^2.
\end{eqnarray}
In writing this, we are assuming that the mode functions are defined to
have positive frequency with respect to the time-like Killing vector
$\partial_t$. This is the Boulware vacuum.
Now, in the two regions where $\stackrel{\rightarrow}{R}_l(\omega|r)$
and $\stackrel{\leftarrow}{R}_l(\omega|r)$ are known
explicitly, we have \cite{candelas}
\begin{equation}
\sum_{l=0}^\infty\,(2l+1)\Big|\stackrel{\rightarrow}{R}_l(\omega|r)\Big|^2
\simeq\cases{4\omega^2(1-2M/r)^{-1},&$r\rightarrow2M$,\cr
\displaystyle{{1\over r^2}\sum_{l=0}^\infty}(2l+1)\big|B_l(\omega)\big|^2,
&$r\rightarrow\infty$,}
\end{equation}
and
\begin{equation}
\sum_{l=0}^\infty\,(2l+1)\Big|\stackrel{\leftarrow}{R}_l(\omega|r)\Big|^2
\simeq\cases{
\displaystyle{{1\over4M^2}
\sum_{l=0}^\infty}(2l+1)\big|B_l(\omega)\big|^2,&$r\rightarrow2M$,\cr
4\omega^2,&$r\rightarrow\infty$.}
\end{equation}
If we further assume the low-energy condition $2M\omega\ll1$, then
\cite{jensen}
\begin{equation}
\label{B_l}
B_l(\omega)\simeq{(l!)^3\over(2l+1)!(2l)!}(-4iM\omega)^{l+1}.
\end{equation}
These results can be substituted into (\ref{Schwarz}), and the bound
explicitly evaluated using the integral (\ref{integral}). However, the
maximum value of $l$ for which the expansion in (\ref{Schwarz}) is valid
depends on the order of the leading terms which have been dropped in
$B_l(\omega)$. If (\ref{B_l}) is exact to ${\rm O}\left[(M\omega)^{l+2}
\right]$, then only the $l=0$ terms should be retained, as the $B_1$
contribution would be smaller than the corrections to $B_0$
\cite{PFb,pfenning}.

Near the horizon, the quantum inequality can be expressed in terms
of the observer's proper time:
\begin{equation}
\tau_0=\bigg(1-{2M\over r}\bigg)^{1\over2}t_0\,,
\end{equation}
as
\begin{equation}
\label{Sbounda}
\rho\geq-{3\over32\pi^2\tau_0^4}\bigg\{
{1\over24}\bigg({2M\tau_0\over r^2}\bigg)^2\bigg(1-{2M\over r}\bigg)^{-1}
+{9\over64}\bigg[1+\bigg(1-{2M\over r}\bigg)\bigg]+\cdots\bigg\}\,,
\end{equation}
where the ellipsis denotes higher-order terms that have been dropped.
This is to be compared with the result derived in \cite{PFb,pfenning}:
\begin{equation}
\label{Sboundb}
\rho\geq-{3\over32\pi^2\tau_0^4}\bigg\{
{1\over6}\bigg({2M\tau_0\over r^2}\bigg)^2\bigg(1-{2M\over r}\bigg)^{-1}
+1+\bigg(1-{2M\over r}\bigg)+\cdots\bigg\}\,.
\end{equation}
The bound in (\ref{Sbounda}) is between $9\over64$ and $1\over4$ that in
(\ref{Sboundb}), at least in the present approximation. Note that in
either case, the bound becomes arbitrarily negative near the horizon
of the black hole.

On the other hand, the quantum inequality for an observer near infinity becomes
\begin{equation}
\rho\geq-{3\over32\pi^2\tau_0^4}{9\over64}\bigg\{1-{2M\over r}
+\bigg({2M\over r}\bigg)^2\bigg[1+{16\over9}{1\over3}\bigg(
{\tau_0\over r}\bigg)^2\bigg]-\bigg({2M\over r}\bigg)^3
\bigg[1+{16\over9}\bigg({\tau_0\over r}\bigg)^2\bigg]+\cdots\bigg\}\,,
\end{equation}
while the corresponding inequality obtained in \cite{PFb,pfenning} is
\begin{equation}
\rho\geq-{3\over32\pi^2\tau_0^4}\bigg\{1-{2M\over r}+\bigg({2M\over r}\bigg)^2
\bigg[1+{1\over3}\bigg({\tau_0\over r}\bigg)^2\bigg]-\bigg({2M\over r}\bigg)^3
\bigg[1+\bigg({\tau_0\over r}\bigg)^2\bigg]+\cdots\bigg\}\,.
\end{equation}
Again, the former bound is between $9\over64$ and $1\over4$ the latter.
It gives the correct Minkowski space result in the limit $r\rightarrow\infty$
or $M\rightarrow0$.

\sect{Concluding remarks}

In summary, we have derived new quantum inequalities (\ref{QIb}) or
(\ref{QIc}) on the normal-ordered averaged energy density in static
space-times, that are valid for quite general
sampling functions. They were then applied to several standard examples
using the Lorentzian sampling function. (Of course, other
space-times could readily be considered, such as Rindler
space, flat space with perfectly reflecting mirrors, and other black holes
\cite{PFb,pfenning}.) The resulting bounds are stronger than previous results,
and would lead to even tighter constraints on the
various exotic space-times mentioned at the beginning of the paper.
Before we conclude, a few comments are in order.

An important question is whether our quantum inequalities are optimal.
This could, for example, be proved by finding a quantum state that
saturates the bound, which would necessarily belong to the kernel of all
the operators ${\cal O}^\pm(\omega)$ in (\ref{O}). However, it is known
that our bound, when applied to a massless scalar field in two-dimensional
Minkowski space, is $1{1\over2}$ times weaker than the optimal value
obtained by Flanagan \cite{flanagan}. Unfortunately, his
derivation relies on some special features of two-dimensional massless
field theory, and does not appear to generalise to other more realistic cases.

An interesting application of our quantum inequality would be to
the static Morris--Thorne-type wormholes \cite{MT}. Ford and Roman have
applied the flat-space version of their quantum inequalities to this case,
and have found that they constrain the size of such wormholes \cite{FR-worm}.
They justified this procedure by making the sampling timescale much shorter
than the minimum characteristic curvature scale, so that space-time appears
locally flat. However, it would be desirable to verify this calculation
using the full curved space results; this should not be too difficult
once the form of the scalar field mode functions in the wormhole space-time
have been determined.

\section*{Acknowledgment}

CJF thanks Simon Eveson for useful discussions concerning the use of the
trapezoidal rule in Secs.~\ref{IIIDCU} and~\ref{IVDRWU}.