}\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?$$
\eproclaim
Apart from the classical analogy the main piece of supporting evidence
for conjecturing the above Questions to have affirmative answers was
the following Theorem, which is a slight extension of a result of Lieb
\cite\Lieb, who proved it under the additional assumption that one of
the two vectors $\phi,\psi$ is already coherent.
\iproclaim Theorem.
The answer to Question 2 is ``yes'', when $\Phi(t)=t\1s$ with
$1~~=0$, and
$\Re\bra\chi,\ddot\phi>+\bra\dot\phi,\dot\phi>=0$.
Let $\Phi$ be a fixed positive convex function with $\Phi(0)=0$,
which we assume to be twice differentiable. Then, setting
$\rho_t(x)=\vert\langle\phi_t,
\wy(x)\psi_t\rangle\vert\12$,
and $\rho=\rho_0(x)$, our aim is to decide whether
$I(t)=\intdx \Phi({\rho_t(x)})$ has a local maximum at $t=0$.
Hence we expand $I$ to second order in $t$.
The first derivative of $I$ vanishes because $\chi$
is an eigenvector of the operator
$$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).
\REF AU \Uhla \Bref
P.M. Alberti, A. Uhlmann
"Stochasticity and partial order"
D. Reidel, Dordrecht 1982
\REF Bec \Beckner \Jref
W. Beckner
"Inequalities in Fourier analysis"
Ann.Math. @102(1975) 159--182
\REF BL \Brascamp \Jref
H.J. Brascamp, E.H. Lieb
"Best constants in Young's inequality, its converse, and its
generalization to more than three functions"
Advan.Math. @20(1976) 151--173
\REF Dau \Daubech \Jref
I. Daubechies
"Continuity statements and counterintuitive examples in
connection with Weyl quantization"
J.Math.Phys. @24(1983) 1453--1461
\REF Dav \Davies \Bref
E.B. Davies
"Quantum theory of open systems"
Academic Press, London 1976
\REF HLP \HLP \Bref
G. Hardy, J.E. Littlewood, G. P\'olya
"Inequalities"
Cambridge University Press, Cambridge 1934
\REF Hol \Holevo \Bref
A.S. Holevo
"Probabilistic and statistical aspects of quantum theory"
North Holland, Amsterdam 1982
\REF Lie \Lieb \Jref
E.H. Lieb
"Proof of an entropy conjecture by Wehrl"
Commun.Math.Phys. @62(1978) 35--41
\REF vNe \vNeum \Jref
J. von Neumann
"Die Eindeutigkeit der Schr\"odingerschen Operatoren"
Math.Ann. @104(1931) 570--578
\REF OCW \OConnell \Jref
R.F. O'Connell, E.P. Wigner
"Quantum-Mechanical distribution functions:
conditions for uniqueness"
Phys.Lett. @83A (1981) 145--148
\REF RS \RSimon \Bref
M. Reed, B. Simon
"Methods of Modern Mathematical Physics, vol.II:
Fourier analysis and self-adjointness"
London: Academic Press, 1975
\REF Rie \Riesz \Jref
F. Riesz
"Sur une in\'egalit\'e int\'egrale"
J.London.Math.Soc. @5(1930) 162--168
\REF Sim \Simon \Jref
B. Simon
"The classical limit of quantum partition of functions"
\hfill\break
Commun.Math.Phys. @71(1980) 247--276
\REF Tak \Takahashi \Jref
K. Takahashi
"Wigner and Husimi functions in quantum mechanics"
J.Phys.Soc.Jap. @55(1986) 762--779
\REF Uhl \Uhlb \Jref
A. Uhlmann
"Remarks on the relation between quantum and classical entropy as
proposed by A.~Wehrl"
Rep.Math.Phys. @18(1980) 177--182
\REF We1 \QHA \Jref
R.F. Werner
"Quantum harmonic analysis on phase space"
J.Math.Phys. @25 (1984) 1404--1411
\REF We2 \PHU \Jref
R.F. Werner
"Physical uniformities on the state space of
non-relativistic quantum mechanics"
Found.Phys. @13 (1983) 859--881
\REF We3 \CLQ \Gref
R.F. Werner
"The classical limit of quantum mechanics"
In preparation
\REF Wig \Wigner \Jref
E.P. Wigner
"On the Quantum Correction for Thermodynamic Equilibrium"
\hfill\break
Phys.Rev. @40 (1932) 749
\bye
~~