% On the concentration of quantum states in phase space
% by R.F. Werner 
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\REF HLP \HLP \par  
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\REF vNe \vNeum \par  
\REF OCW \OConnell  \par  
\REF RS \RSimon \par  
\REF Rie \Riesz \par  
\REF Sim \Simon    \par  
\REF Tak \Takahashi \par  
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%title page: 
\line{}
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{\BF \baselineskip=25pt
\centerline{ On the Concentration of }
\centerline{ Quantum States in Phase Space}
}
\vskip 1.0cm
\centerline{ June 16, 1993 }
\vskip1cm
\centerline{
\bf R.F. Werner
\footnote{$\11$}%
{{\sl FB Physik, Universit\"at Osnabr\"uck,
            Postfach 4469, D-4500 Osnabr\"uck, Germany.
}}%
$\1,$\footnote{$\1{2}$}
{{ \sl Electronic mail:\quad 
   \tt reinwer@dosuni1.rz.Uni-Osnabrueck.DE}}
}
\vskip 1.0cm

{\baselineskip=12pt
\midinsert\narrower\narrower\noindent
{\bf Abstract.}\
Let $E(x)$, for $x$ in a $2d$-dimensional phase space, be an 
irreducible Weyl system, and $\Phi:\Rl\1+\to\Rl\1+$ a convex function 
with $\Phi(0)=0$. We discuss the maximum of 
$\int dx\ \Phi\bigl(\vert\langle\phi,E(x)\psi\rangle\vert\12\bigr)$ 
with respect to unit vectors $\phi,\psi$. 
When $\Phi(t)=t\1p$ with $1<p<\infty$ the maximum is attained if and 
only if $\phi$ and $\psi$ are coherent states with respect to the 
same quadratic form. We show that this statement is not correct for 
more general convex functions $\Phi$. 
\endinsert
}
\vskip20 pt \vfil\eject

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% body:  
\noindent 
The study of non-relativistic quantum mechanics in terms of an 
associated classical phase space is an old subject. Perhaps the most 
widely known map taking quantum states to functions on phase space 
is the Wigner transformation \cite\Wigner. However, since the Wigner 
function of a state need not be integrable, it often represents a
``probability'' density, in which an infinite positive probability 
is cancelled by an infinite negative probability to give formally the 
normalization to unity. These infinities make the Wigner function 
practically useless in studies in which the statistical 
interpretation of Quantum Theory is taken seriously, or, on the 
technical side, whenever norm estimates of states or observables are 
desired. 
 
There is a well-known alternative to the Wigner function avoiding 
these difficulties, which is variously known as the 
``Husimi-function''\cite\Takahashi, a ``phase space 
observable''\cite{\Davies,\Holevo}, a ``Berezin upper/lower 
symbol''\cite{\Simon}, or ``convolution with a coherent 
state''\cite\QHA. One price to pay in all these approaches is that 
while the Wigner function has an intrinsic characterization in terms 
of the Weyl operators alone \cite\OConnell, these positive 
distribution functions depend on the choice of a ``reference state'', 
usually a coherent state. (In the latter case the frequency $\omega$ 
of the oscillator Hamiltonian $(P\12+\omega\12Q\12)/2$ of which the coherent 
reference state is the ground state is a free parameter). 
 
Apart from this arbitrariness these positive distributions have the 
unpleasant feature of introducing additional ``uncertainty'': 
they have larger variances for both position and momentum than the 
corresponding quantum state (When it makes sense, the Wigner 
function reproduces the quantum variances). The least increase in 
variance is guaranteed by constructing the positive distribution 
with coherent (minimal uncertainty) states, and this is indeed the 
only option usually considered in the literature. However, the 
variance is only a very crude measure of the spread of a probability 
distribution in phase space, and for many purposes other measures of 
``concentration in phase space'' are relevant. Folklore would 
suggest that, whatever the notion of concentration, the most 
concentrated classical densities associated with any quantum state 
are those of a coherent state, where the positive density is also 
constructed with a coherent reference state. 
 
In this note we give a positive and a negative result in this 
direction. The positive result is a strengthening of a result by 
Lieb \cite\Lieb, and makes the above statement precise when the 
measure of concentration is an $\L p$ norm, or the Shannon entropy 
of the classical density. This result, plus an analogy with a 
well-known inequality for the rearrangement of convolutions had led 
the present author to conjecture a much more general inequality. The 
conjecture was presented at the Nottingham Quantum Probability 
workshop. In the meantime, however, it has turned out that it fails in 
general: there are measures of concentration of quantum states in 
phase space which are {\it not} maximized by coherent states. 
 
We now fix the basic notations and conventions for phase space 
quantum mechanics. For a more complete exposition the reader is 
referred to \cite{\QHA,\PHU}. 
By a {\it phase space} we will mean a $2d<\infty$ dimensional real 
vector space $X$ with a non-degenerate antisymmetric bilinear form 
$\sigma:X\times X\to\Rl$. 
The connection between quantum mechanics and the classical phase 
space structure is made by an {\it irreducible Weyl system}, i.e.\ 
an irreducible set of unitary operators $\bigl(\wy(x)\bigr)_{x\in 
X}$ on a Hilbert space $\H$ such that $t\mapsto E(tx)$ is strongly 
continuous, and 
$$ \wy(x)\wy(y)=e\1{i\sigma(x,y)/2}\ \wy(x+y) 
\quad.\eqno(1)$$ 
By von Neumann's Uniqueness Theorem \cite\vNeum\ such a system is 
uniquely defined up to unitary equivalence, and we may take, without 
loss of generality, 
$$\eqalign{     \H &= \L2(\Rl\1d,d\xi) \cr 
   \sigma(x_1,x_2) &= p_1\cdot q_2- p_2\cdot q_1 
                     \qquad\hbox{with}\quad 
                     x_i=(p_i,q_i)\in\Rl\1d\times\Rl\1d \cr 
 \bigl(\wy(p,q)\psi\bigr)(\xi) 
          &= \exp({{\textstyle{i\over2}}}p\cdot q + i p\cdot\xi )\ 
                  \psi(\xi+q) 
\quad.\cr}\eqno(2)$$ 
Von Neumann's Theorem also implies that symplectic linear 
transformations of $X$ are implemented by unitary operators. We will 
only need the phase space inversion $x\mapsto-x$, given by the 
unitary operator $(U_-\psi)(\xi)=\psi(-\xi)$. It will be convenient 
to denote the automorphisms of phase space translation and 
inversion of classical and quantum observables by the same letter 
$\alpha$: 
$$\hbox to 250pt{\hfil\vbox{
    \halign{\hfil$#$&=$#$\hfil\hskip40pt&
           \hfil$#$&=$#$\hfil\cr
%
       \alpha_x(A)& \wy(x)A\wy(x)\1*   &
    (\alpha_xf)(y)& f(y-x)           \cr 
       \alpha_-(A)& U_-AU_-\1*         & 
    (\alpha_-f)(y)& f(-y) 
\qquad.\cr}}\hfil}  
\eqno(3)$$

A crucial formula of the theory is the square integrability of the 
Weyl system in the sense that 
$$ \int dx \abs{\bra \psi, \wy(x)\phi>}\12 
        =\norm{\psi}\12 \norm{\phi}\12 
\quad,\eqno(4)$$ 
where ``$dx$'' is the Lebesgue measure on phase space, normalized 
using Planck's constant ${\rm h}=2\pi$, i.e.\ 
$dx=(2\pi)\1{-d}dp_1\cdots dp_d\,dq_1\cdots dq_d
   =(2\pi)\1{-d}d\1dp\,d\1dq$. 
 
The correspondence between classical states and observables, given 
by functions on $X$, and quantum states and observables, given 
by operators on $\H$, is best expressed in terms of an extended 
{\it convolution} operation ``\cvl'' such that the convolution of two 
operators gives a function, and the convolution of an operator and a 
function is an operator \cite\QHA. Let $\Tcl(\H)$ 
denote the trace class. Then, for $f,g\in\L1(X,dx)$, and 
$A,B\in\Tcl(\H)$,  we define
$$\eqalign{ 
       (f\cvl g)(x) &= \int dy\ f(y)\ g(x-y)    \cr 
    f\cvl A=A\cvl f &= \int dy\ f(y)\ \alpha_y(A) \cr 
       (A\cvl B)(x) &= \tr\bigl(A(\alpha_x\alpha_-B)\bigr) 
\quad.\cr}\eqno(5)$$ 
Note that the second an third lines follow from the first by 
substituting the appropriate action of $\alpha_x$ and $\alpha_-$, 
and replacing the integral by the trace if necessary. Note that 
$A\cvl B$ is an integrable function by virtue of equation (4) 
 
These definitions make the space $\R1=\L1(X,dx)\oplus\Tcl(\H)$ into a 
$\Ir_2$-graded commutative Banach algebra. Its Gel'fand transform is 
the extension of the Fourier transform on $\L1(X,dx)$, which is best 
written in the form 
$$\eqalign{ 
   \bigl(\FouWey f\bigr)(x)&= \int\!\!dy\ e\1{i\sigma(x,y)} f(y) \cr 
   \bigl(\FouWey T\bigr)(x)&= \tr \bigl(E(x)T\bigr) 
\quad,}\eqno(6)$$ 
for $f\in\L1(X,dx)$, $T\in\Tcl(\H)$. 
The definitions (5) of convolution make sense also when one of the
factors is in $\R1$, and the other is in
$\R\infty:=\L\infty(X,dx)\oplus\B(\H)$. As in the classical case we
may apply interpolation theory to extend the definitions to suitable
spaces $\R p:=\L p(X,dx)\oplus\Tcl\1p(\H)$, and Young's inequality 
holds (for the definition of $\Tcl\1p$, see section IX.4 in 
\cite\RSimon). For the Fourier transform we have the Hausdorff-Young 
inequality, the Riemann-Lebesgue Lemma, and ``twisted'' versions of 
Bochner's Theorem, and the formulas relating products in $\R1$ and 
convolutions of Fourier transforms \cite\QHA. A fundamental result 
is the extension of Wiener's Approximation Theorem, which states 
that the phase space translates $\alpha_x(T)$ of a trace class 
operator $T\in\Tcl(\H)$ are norm dense in $\Tcl(\H)$ if and only if 
$\FouWey T$ has no zeros. As a consequence the ideal theory of the 
Banach algebra $\R1$, including all questions of harmonic synthesis 
(relating ideals to the set of points where all Fourier transforms 
of elements of the ideal vanish) is completely reduced to the 
classical case. Another useful consequence is a one-to-one 
correspondence of phase space translation invariant subspaces of 
$\L\infty(X,dx)$ and $\B(\H)$, respectively \cite{\QHA,\PHU}. Under 
this correspondence the continuous functions vanishing at infinity 
are associated with the compact operators, the CCR algebra is 
associated with the almost periodic functions, and so on. One 
obtains simple proofs for some operator theoretic theorems by using 
correspondence for ``reduction to the classical case''. Typical 
examples are the Riemann- Lebesgue Lemma for the inverse Fourier 
transform (``$\intdx f(x)E(x)$ is a compact operator for 
$f\in\L1(X,dx)$''), and the theorem that an operator $A\in\B(\H)$, 
which is strongly continuous for $\alpha_x$, is in the norm closed 
subspace generated by the $E(x)$ (i.e.\ in the CCR-algebra) if and 
only if the orbit $\set{\alpha_x(A)\stt x\in X}$ is norm-precompact.
 
We have cited these results to give support to the intuition that 
the operation ``$\cvl$'' indeed deserves the name ``convolution'', 
and that it is reasonable to expect quantum analogues of results in 
classical harmonic analysis. On the other hand, the results cited 
above do not refer to the ordering of $\L1(X,dx)$ and $\Tcl(\H)$. 
But it is precisely the difference in these order structures which 
determines the geometry of the respective state spaces, and hence 
make all the difference between quantum and classical theories.
Of course, the convolution of positive objects is positive. In fact,
the convolutions (5) may be characterized by their order properties:
any normal operator from $\L\infty(X,dx)$ to $\B(\H)$, or in the
opposite direction, which takes positive elements into positive
elements, and intertwines the actions $\alpha_x$, is of the form
``convolution with a fixed density matrix'' (positive trace class
operator of trace $1$) \cite{\QHA,\CLQ}. This is the general form of 
the positive classical distribution functions mentioned in the 
introduction. 
 
The operator of convolution with a density matrix is ``doubly 
stochastic'' in the sense that it preserves $\idty$, and maps the 
trace into the integral, and conversely. As a corollary one has the 
Berezin-Lieb inequalities \cite{\Simon,\QHA} 
$$ \intdx \PHI{f(x)} 
      \geq \tr\PHI{ f\cvl T_1} 
      \geq \intdx \PHI{(T_2\cvl T_1\cvl f)(x)} 
\quad,\eqno(7)$$ 
where $\Phi$ is any positive convex function on $\Rl\1+$ with 
$\Phi(0)=0$, and $T_i$ are density matrices. This is saying that 
convolution with $T_i$ produces functions/operators which are less 
and less concentrated in phase space. If one wants to get sharp 
estimates of quantum operators $f\cvl T_1$ from the Berezin-Lieb 
inequalities, one would like to have the difference between the two 
sides as small as possible, which requires $T_1\cvl T_2$ to be 
concentrated in phase space as sharply as possible. Since 
$\abs{(T_1\cvl T_2)(x)}\leq1$ for all $x$, $\delta$-function-like 
concentration is impossible. For studying quantum-classical 
correspondence we therefore need to learn more about the set of 
functions 
$$ \DD:= \Set\Big{T_1\cvl T_2  \stt \hbox{$T_i$ density matrices}\ } 
      \subset\L1(X,dx) 
\quad.\eqno(8)$$ 
Unfortunately, very little is known about this set. 
 
The aim of the present note is to describe this set better with regard to 
``concentration in phase space''. We can make this notion 
precise by setting up a theory of ``stochastic majorization'' in the 
context of semi-finite W*-algebras with distinguished faithful trace. 
The basic order relation of such a theory ``$A\mxd B$'', read as 
``$A$ is more mixed than $B$'', is defined for $A,B$ positive 
elements, not necessarily in the same W*-algebra. One of the 
equivalent ways of defining it is that 
$\tr\Phi(A)\leq \tr\Phi(B)$, holds for{\it  all} positive convex functions 
$\Phi$ vanishing at $0$. This is not quite the same as the 
Alberti-Uhlmann ordering \cite\Uhla, since it does not trivialize in 
the abelian case, and depends explicitly on the trace chosen in each 
algebra. The Berezin-Lieb inequalities can be written as 
$T_2\cvl T_1\cvl f\mxd T_1\cvl f\mxd f$, and are hence an example of 
a comparison of elements of different algebras in this ordering. 
When both $A\mxd B$ and $B\mxd A$, we say that $A$ and $B$ are {\it 
rearrangements} of each other. This is equivalent to saying that 
they have the same distribution function with respect to the 
respective traces. 
 
In phase space the natural trace is given by the $\int dx$. 
Therefore, we would like to find density matrices $T_i$ such that 
$$  \intdx \PHI{(T_2\cvl T_1)(x)} 
     \buildrel!\over=\max 
\quad.\eqno(9)$$ 
A more detailed version of this problem asks for the rearrangements 
of $T_1$ and $T_2$ making such an integral maximal for a given 
$\Phi$. If we find rearrangements $T_1'$ and $T_2'$ of $T_1$ and 
$T_2$ making this integral maximal for all $\Phi$ simultaneously, we 
have found a smallest element of the set $T_1'\cvl T_2'$ in the 
ordering $\mxd$.
 
It may seem unreasonable that such least mixed elements should 
exist. However, a classic result by F.~Riesz \cite\Riesz, presented 
also in the classic book by Hardy, Littlewood, and P\`olya 
\cite{\HLP, {\bf Theorem 379}}, says that for the classical 
convolution of functions they do exist: the maximizing $T_i'$ are 
the symmetrically decreasing rearrangements of the given functions 
$T_i$. In order to state a quantum analogue of this result, we have 
to say what ``symmetrically decreasing'' means for an operator (we 
assume $d=1$ for simplicity). A symmetrically decreasing function is 
a decreasing function of $h=(p\12+q\12)/2$. Then we shall call a trace 
class operator $T$ {\it symmetrically decreasing}, if it has the 
same eigenvectors as $H=(P\12+Q\12)/2$, where $P,Q$ are momentum and 
position operators in the standard representation (2), and if the 
eigenvalues of $A$ are decreasing with respect to the eigenvalues of 
$H$. It is clear that every positive trace class operator $T$ has a 
unique decreasing rearrangement $\symdec T$ in this sense, computed 
by forgetting its eigenbasis, and arranging the eigenvalues along 
the spectrum of $H$. There is a simple kind of rearrangement 
inequalities in \cite\HLP\ stating that the integral of a product of 
two functions becomes maximal, when they are rearranged to be 
monotonic with respect to each other (e.g.\ both symmetrically 
decreasing). Such statements carry over trivially to the 
rearrangement of operators. However, the result of Riesz is much 
deeper. Its quantum analogue would be the affirmative answer to the 
following Question, and was conjectured to be true by the author at 
the QP workshop in Nottingham.
 
\iproclaim Question 1. 
Let $T_1,T_2\in\Tcl(\H)$, or let 
$T_1\in\Tcl(\H)$ and $T_2\in\L1(X,dx)$ be positive, 
and let $\symdec{T_i}$ denote the symmetrically decreasing 
rearrangement of $T_i$. 
Is it always the case that 
$$   T_1\cvl T_2\mxd \symdec{T_1}\cvl\symdec{T_2} 
\quad?$$ 
\eproclaim 
 
\noindent 
One can show, using the generating function of the Laguerre 
polynomials, that the convolution of symmetrically decreasing objects 
(functions or operators) is symmetrically decreasing. This readily 
implies (as in \cite\HLP) that if the above question has an 
affirmative answer for projection operators $T_1,T_2$, this holds for 
general $T_i$, and also for an arbitrary number $n\geq2$ of factors. 
 
Of course, the most interesting case is to find least mixed elements
of the set $\DD$ of probability densities, without constraint on the
rearrangement classes. This amounts to maximizing (9) over all density
matrices $T_1, T_2$. It is clear from the convexity of $\Phi$ that
this maximum will be attained when both density matrices are pure. The
only symmetrically decreasing one-dimensional projection is the ground
state projection of $H$, which represents a coherent state. Since pure
states are a rearrangement class by itself, we arrive at the following
special version of Question 1. Since it makes no reference to the 
symmetrically decreasing rearrangement of general operators, we can 
state it in any number $d$ of degrees of freedom. 
 
\iproclaim Question 2. 
Let $\phi,\psi\in\H$ be unit vectors, and let $\Phi$ be a positive 
convex function vanishing at zero. Let $\chi$ be a coherent unit 
vector. Is it true that 
$$    \intdx \PHI{\vert\langle\phi, 
               \wy(x)\psi\rangle\vert\12} 
     \leq\intdx \PHI{\vert\langle\chi, 
               \wy(x)\chi\rangle\vert\12} 
     =\int_0\1\infty\!\!\! dt\ {t\1{d-1}\over(d-1)!}\ 
          \PHI{e\1{-t}} 
\quad?$$ 
\eproclaim 
 
Apart from the classical analogy the main piece of supporting evidence
for conjecturing the above Questions to have affirmative answers was
the following Theorem, which is a slight extension of a result of Lieb
\cite\Lieb, who proved it under the additional assumption that one of
the two vectors $\phi,\psi$ is already coherent.
 
\iproclaim Theorem. 
The answer to Question 2 is ``yes'', when $\Phi(t)=t\1s$ with 
$1<s<\infty$. 
\eproclaim 
 
\proof:
The proof follows closely the arguments in \cite\Lieb, who considers
only the special case in which $\phi$ is already assumed to be
Gaussian. The main idea is to utilize the best constants in the Young
and Hausdorff-Young inequalities \cite{\Beckner,\Brascamp} in $\Rl\1n$,
i.e.\
$$\eqalign{ 
   \norm{f\cvl g}_r 
       &\leq \bigl(C(p)C(q)/C(r)\bigr)\1d\ \norm{f}_p\,\norm{g}_q  \cr 
&\qquad \hbox{for}\quad1+r\1{-1}=p\1{-1}+q\1{-1} 
\quad;\quad  1\leq p,q\leq\infty\cr 
   \norm{\Fourier f \,}_s 
       &\leq \bigl(C(s')\,(2\pi)\1{1/s}\bigr)\1d  \norm{f}_{s'} \cr 
&\qquad \hbox{for}\quad s\1{-1}=1-(s')\1{-1} 
\quad;\quad  2\leq s\leq\infty \cr 
\hbox{where}\qquad\qquad 
   \ln C(s)&= {\ln s\over 2s}  - {\ln s'\over 2s'} 
\quad. }$$ 
 
We introduce the functions 
$$ f_q(\xi)=\Bar{\psi(\xi)}\phi(\xi+q) 
\quad,$$ 
for $\xi,q\in\Rl\1d$. Then by definition (2) of the Weyl operators 
$\wy(x)$, the left hand side of the inequality becomes 
$$\eqalign{ 
 I(s)&= \int {d\1dq\,d\1dp\over (2\pi)\1d}\  \abs{\Fourier f_q(p)}\1{2s} 
      = \int {d\1dq\over (2\pi)\1d}\  \norm{\Fourier f_q}\1{2s}_{2s} 
\cr 
     &\leq C(s')\1{2ds}\ \int\!\! d\1dq\ \norm{f_q}_{s'}\1{2s} 
\quad,}$$ 
where $(s')\1{-1}=1-(2s)\1{-1}$. On the other hand, 
$$ \norm{f_q}_{s'}\1{s'} 
    =\int\!\!d\xi\ \abs{\psi(\xi)}\1{s'}\,\abs{\phi(\xi+q)}\1{s'} 
    = \bigl(\abs\psi\1{s'}\cvl\abs\phi\1{s'}\bigr)(-q) 
\quad. $$ 
Hence, with $r=2s/s'=2s-1$, and $p=q=2/(1+r\1{-1})=(2s-1)/s$: 
$$\eqalign{ 
  I(s)&\leq C(s')\1{2ds}\ 
         \norm{\abs\psi\1{s'}\cvl\abs\phi\1{s'}}_r\1r \cr 
      &\leq C(s')\1{2ds}\ \bigl(C(p)\12(C(r)\bigr)\1{dr} 
         \norm{\abs\psi\1{s'}}_p\1r \norm{\abs\phi\1{s'}}_p\1r 
\quad.\cr}$$ 
Since $ps'=2$, the norms on the right hand side are equal to $1$. 
The logarithm of the constant is $\ln C=-d \ln s$. 
Since 
$$\int_0\1\infty\!\!\!dt\ {t\1{d-1}\over(d-1)!}\ e\1{-ts}=s\1{-d}
\quad,$$ 
this proves the Theorem. 
\QED 
 
 
This result can be used to obtain bounds on other quantities. On the 
basis of Lieb's result this was done by Uhlmann \cite\Uhlb. Here we 
get correspondingly stronger results: for example, let $N$ be a 
measurable set of phase space measure $m$, and let $T_1,T_2$ be 
density matrices. Then 
$$ \int_N\!\!dx\ \bigl(T_1\cvl T_2\bigr)(x)
       \leq m\,\exp\Bigl({-d\over e}m\1{1/d}\Bigr)
\quad,\eqno(10)$$
as long as $m\leq e\1d$. The proof is immediately reduced to the case 
of pure states $T_i$ by taking convex combinations. Then the Theorem 
gives a bound on the $p$-norm of $T_1\cvl T_2$, and we use this 
together with $\norm{\chi_N}_q=m\1{1/q}$ for $p\1{-1}+q\1{-1}=1$, 
and optimizing with respect to $p$. It is interesting to compare 
this with the bound that an affirmative answer to Question 2 would 
have guaranteed: to compute this we would only have to take $N$ as a 
ball of the appropriate radius, and to take $T_1,T_2$ as coherent. 
Taking $d=1$ for simplicity, we would get the improvement 
$$ \int_N\!\!dx\ \bigl(T_1\cvl T_2\bigr)(x)
      \leq 1-e\1{-m}  \leq  m e\1{-m/e}
\quad.\eqno(11)$$
In fact, this improved bound would be equivalent to Question 2. 

 
The first indication that the answer to Question 2 (and {\it a 
fortiori} the answer to Question 1) is ``no'', came from an 
analysis of the expression in Question 2, when both $\phi$ and 
$\psi$ are close to the same coherent state $\chi$. We briefly 
indicate the argument without giving the details of the 
computations. Let us consider differentiable curves 
$t\mapsto\phi_t,\psi_t$ of unit vectors, with 
$\phi_0=\psi_0=\chi$. For our purposes we can write 
$$ \phi_t= \chi+t\dot\phi+{\textstyle{1\over2}}t\12\ \ddot\phi
               +\Order(t\13) 
\quad,\eqno(12)$$ 
and similarly for $\psi$. The normalization condition then means 
that $\Re\bra\chi,\dot\phi>=0$, and 
$\Re\bra\chi,\ddot\phi>+\bra\dot\phi,\dot\phi>=0$. 
Let $\Phi$ be a fixed positive convex function with $\Phi(0)=0$, 
which we assume to be twice differentiable. Then, setting 
$\rho_t(x)=\vert\langle\phi_t, 
               \wy(x)\psi_t\rangle\vert\12$, 
and $\rho=\rho_0(x)$, our aim is to decide whether 
$I(t)=\intdx \Phi({\rho_t(x)})$ has a local maximum at $t=0$. 
Hence we expand $I$ to second order in $t$.  
The first derivative of $I$ vanishes because $\chi$ 
is an eigenvector of the operator 
$$A=\intdx \Phi'\bigl(\rho(x)\bigr)\  
         \wy(x)\vert\chi\rangle\langle\chi\vert \wy(x)\1*
   = \Phi'(\rho)\cvl P_0 
\quad,\eqno(13)$$ 
where $P_0=\vert\chi\rangle\langle\chi\vert$ is the ground state 
projection of the harmonic oscillator. For the vanishing of the first 
order it would be sufficient that $\phi_0$ and $\psi_0$ are both 
oscillator eigenvectors. Using the equation for the vanishing of the 
first order, one can eliminate the $\ddot\phi$ and $\ddot\psi$ from 
$\ddot I$. Moreover, in many of the terms making up $\ddot I$ the 
angular integration gives zero. In particular, all terms vanish in 
which only one derivative appears, or in which all 
$\dot\phi,\dot\psi$ appear in the antilinear (resp. linear) 
arguments of the scalar products. 
Then the second order becomes
$$\eqalign{ 
     \ddot I(0)&= -2\Set\Big{\bra\dot\phi,B\dot\phi> 
                            +\bra\dot\psi,B\dot\psi> 
                            +2\Re\bra\dot\psi,C\dot\phi>} 
\quad,\cr 
\hbox{where}\hskip30pt 
     B&= \bigl({\textstyle \intdx \rho\,\Phi'(\rho)}\bigr)\idty 
           - \bigl(\Phi'(\rho)+\rho\Phi''(\rho)\bigr) \cvl P_0 
\quad,\cr 
\hbox{and}\hskip30pt 
     C&=  -\intdx \sqrt\rho\,\Phi'(\rho)\ \wy(x) 
          -\intdx \rho\,\Phi''(\rho)\ \wy(x)P_0\wy(x) 
\quad.}\eqno(14)$$ 
(The omission of a star on the last factor in $C$ is not a misprint). 
One readily verifies that both $B$ and $C$ are diagonal in the 
oscillator eigenbasis, and the definiteness of the second derivative 
becomes equivalent to 
$$ B_n\geq\abs{C_n} 
\quad,\eqno(15)$$ 
where $B_n$ and $C_n$ are the $n\th$ eigenvalues of $B$ and $C$. 
We get 
$$\eqalign{ 
     B_n &= \int_0\1\infty\!\!\! dt\ e\1{-2t}\Phi''\bigl(e\1{-t}\bigr)\ 
            \set{\sum_{k=1}\1{n-1}{t\1k\over k!}}  \cr 
     C_n &= \int_0\1\infty\!\!\! dt\ e\1{-2t}\Phi''\bigl(e\1{-t}\bigr)\ 
               \Set\Big{\Lag_n(t)-\Lag_{n-1}(t)-(-1)\1n
                        {t\1n\over n!} }
\quad,}\eqno(16)$$
where $\Lag_n$ denotes the $n\th$ Laguerre polynomial. Here we have 
performed various partial integrations so that only $\Phi''$ appears 
rather than $\Phi$ and $\Phi'$.
Hence the weight in both integrals is an essentially arbitrary
positive function. If we insert $\Phi(t)=t\1p$, we get 
$$ B_n=1-p\1{-(n-1)}\quad \geq \quad 
       p\1{-n}(p-1)\abs{(p-1)\1{n-1}+(-1)\1n}=\abs{C_n}
\quad,\eqno(17)$$
which is clear from the Theorem. 
On the other hand, we can choose for $\Phi$ ( a smooth 
approximation of ) $\Phi(\rho)=(\rho-\lambda)_+$, so that the 
integrand has a delta function $\delta(e\1{-t}-\lambda)$. Hence it 
suffices to discuss the condition (15) for the expressions in braces 
in equation (16), pointwise for each $t$. Since $\dot\phi$ and 
$\dot\psi$ are orthogonal to $\chi$, they do not appear in $\ddot 
I$. If $\phi_t=\wy(tx)\chi$, we get $\dot\phi$ proportional to the 
first oscillator function. But the value of the integral is constant 
under such shifts. Hence along such paths $\ddot I=0$, and we must 
have $B_1=C_1=0$. Further we have $B_2=t=-C_2$. But 
$$ B_3=t+{1\over2}t\12 
      \quad\not\geq\quad 
         \abs{-t+t\12}=\abs{C_3} 
\quad,\eqno(18)$$ 
and in fact the inequality (15) fails for all $n\geq3$ and sufficiently 
large $t$. Hence we have proven that the answer to Questions 1 and 2 
is ``no'' in general. 

Of course, this proof is rather indirect, and an explicit 
counterexample would be preferable. We can take the above 
computation as a guideline, and set 
$$\eqalign{
      \phi&=\cos\gamma\  \chi_0 + \sin\gamma\  \chi_3 \cr
      \psi&=\cos\gamma\  \chi_0 - \sin\gamma\  \chi_3 
\quad,\cr}\eqno(19)$$
where $\chi_n$ is the $n\th$ oscillator eigenvector.  
Then with 
$\gamma\approx 0.0606 $, % 0.06059556000549346
and 
$\lambda\approx 0.0039$, % 0.003915610151391912
we get 
$$ \intdx \Bigl(\vert\langle\phi, \wy(x)\psi\rangle\vert\12 
           -\lambda\Bigr)_+
\approx 6.11 *10\1{-5}   % 0.0000611532
 + \intdx \Bigl(\vert\langle\chi_0, \wy(x)\chi_0\rangle\vert\12 
           -\lambda\Bigr)_+
\quad, \eqno(20)$$ 
so the difference is indeed positive. 
This computation has to be done with some care, since we are looking 
for a relatively small difference. It is simplified by observing 
that for small (resp.\ large) radial coordinates the density 
$\vert\langle\phi, \wy(x)\psi\rangle\vert\12$ is larger (resp. 
smaller) than $\lambda$ for all angles, so that these parts can be 
integrated analytically. The remaining integral is then done in 
polar coordinates, where the integration over the angle can also be 
done analytically, leaving an expression involving an $\arcsin$. 

It is clear that this counterexample is not optimal. It would be 
interesting to maximize the above integral with respect to 
$\phi,\psi$ for any fixed value of $\lambda$. It would also be 
interesting to see whether there is also a counterexample when one 
of the two vectors is coherent, corresponding to the special case of 
the Theorem originally treated by Lieb. Another open problem is to 
determine the best constants of the Young and Hausdorff-Young 
inequalities for the convolutions (5) in analogy with \cite\Beckner. 
Are they attained for Gaussians, as in the classical case?

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \vfill\eject
\let\REF\doref 
\ACKNOW 
The author is indebted to Roland Franzius for an independent 
verification of the numerical counterexample at the end of the 
paper. He also acknowledges financial support from the DFG (Bonn).

 
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\bye