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\begin{titlepage}
\begin{flushright}
UWThPh-1996-27\\
%\today
\end{flushright}
\vspace{2cm}
\begin{center}
{\Large \bf Why Schr\"odinger's Cat is most likely to be either
alive or dead}\\[50pt]
H. Narnhofer, W. Thirring \\
Institut f\"ur Theoretische Physik \\ Universit\"at Wien\\
Boltzmanngasse 5, A-1090 Wien
\vfill
{\bf Abstract} \\
\end{center}
With appropriate assumptions on the time evolution a sequence of events
makes a mixed state pure on the classical quantities.
\vfill
\end{titlepage}
\section{Introduction}
Ever since Schr\"odinger's seminal 1935 paper \cite{S} quantum physicists
are wondering why we always find states where classical observables have
definite values but never mixtures of them. Mixed states like the
canonical state seem to be the rule, we live in a $3^0$K universe,
but we observe always pure ones on macroscopic observables. The latter
can be characterized mathematically as the center of the observable
algebra and states which are pure over the center are called factor
states. This means they assign a definite $c$-number to each element
of the center, they become dispersion free.
The first hint about
what distinguishes physically the factor states is found in
\cite{NR} where it is shown that the canonical (= KMS) states which
are also factor states enjoy the following property:
If one disturbes the Hamiltonian $H$ by any sequence of perturbations
$h_n$ such that in the limit $n \ra \infty$ $H + h_n$ generate the
same time evolution as $H$ then the associated canonical state converges
to the one associated to $H$ exactly if the latter is a factor. In one
direction this theorem is obvious since if we take the
$h_n \neq c {\bf 1}$ from the center this will not change the time
evolution but it does change the canonical state. The converse of this
statement is not so trivial.
However this kind of stability is not
exactly what one wants: it does not explain a dynamical purification of
the state. Ideally one should show that the unavoidable interactions
with the surrounding drive the state into a factor state. That this is
so if the interactions are measurements and one accepts the usual
reduction of the state postulate was shown by one of the authors
\cite{N}.
In this letter we want first to give detailed derivations of this
result within the many histories \cite{G,GH,O} interpretation of quantum
mechanics which gives the precise formulation of the reduction
postulate. Secondly we will show by a certain model for the measuring
device that the same result can be derived by a unitary time evolution
with a time dependent Hamiltonian. We consider the simplest case of the
mixture of two factor states and show that for almost each history the
ratio of the two contributions (or its inverse) decreases exponentially
with the number of outside interactions. We say interactions for measurements
since our model does not contain an observer (or his friend) who takes note
of the result. The observables being measured are a set of complete
orthogonal projectors and replace the classical partitioning of
phase space. Thus we are dealing with the quantum generalization
of symbolic dynamics where the trajectory is replaced by a list of the
gates through which the trajectory went at a discrete sequence of times. For
some dynamical systems this list if it becomes infinitely long coincides
with that of a single trajectory only, corresponding to a state with
maximal purification (= sharp initial conditions). Our results cannot be
derived for general quantum systems but only for infinite systems where
inequivalent representation and a center of the algebra can occur.
Furthermore they must be chaotic in as much as we need decaying time
correlations. This can be proved for the so-called K-systems, both
classical and quantum mechanical.
\section{Purification by measurement}
In the many histories interpretation of quantum mechanics one assigns
probabilities $W$ to histories consisting of a sequence of events.
The latter are represented by projection operators $P_k(t) \in \A$,
$k = 1 \ldots r$, where for the time dependence the Heisenberg
representation is used and $\A$ is the algebra of observables. For
finite quantum systems a state is given by a density matrix $\rho$ and
the key formula for the probabilities is
\beqa
W(\Un{\alpha}) &=& \mbox{tr } P_{\alpha_n}(t_n) \ldots P_{\alpha_1}(t_1)
\rho P_{\alpha_1}(t_1) \ldots P_{\alpha_n}(t_n) \no \\
&=& \mbox{tr } \sqrt{\rho} \; P_{\alpha_1}(t_1) \ldots P_{\alpha_n}(t_n)
P_{\alpha_{n-1}}(t_{n-1}) \ldots P_{\alpha_1}(t_1) \sqrt{\rho}
\eeqa
where $\Un{\alpha} = (\alpha_1 \ldots \alpha_n)$,
$\alpha_i \in \{1 \ldots r\}$. If the $P_k$ are a complete set of
orthogonal projections, $P_k P_{k'} = \delta_{kk'} P_k$,
$\sum_{k=1}^r P_k = 1$ the probabilities are correctly normalized
\beq
W(\Un{\alpha}) \geq 0, \qquad \sum_{\Un{\alpha}}
W(\Un{\alpha}) = 1.
\eeq
For the infinte systems we are concerned with, the density matrix $\rho$
will not exist but the notion of a state as positive linear functional
$\omega$ as generalization of tr~$\sqrt{\rho} \; A \; \sqrt{\rho} =
\omega(A)$ carries over. For these systems in physical situations the
time correlations are expected to decay for an extremal time invariant
state $\omega$ such that (in \cite{T,NT2} this is proved for the
so-called K-systems)
\beq
W(\Un{\alpha}) \ra \prod_i \omega(P_{\alpha_i}) \qquad \mbox{if }
t_{i+1} - t_i \ra \infty \; \forall \; i.
\eeq
Extremal invariant means that $\omega$ is not a combination
\beq
\mu \omega_1 + (1 - \mu) \omega_2, \qquad 0 < \mu < 1,
\eeq
of two other invariant states and the factorization (3) is lost by
convex combinations.
Nevertheless consistency of histories is preserved since it can be
expressed in shorthand by $\omega(P_{\Un{\alpha}'} P_{\Un{\alpha}}) =
\delta_{\Un{\alpha}',\Un{\alpha}}$. It holds for every extremal
invariant state and is preserved by convex combinations.
Extremal invariance requires that $\A$ does not contain an
element constant in time $z \neq c {\bf 1}$ since with it one could
decompose $\omega$ into
$$
\omega_1(A) = \frac{\omega(z^*Az)}{\omega(z^*z)} \qquad \mbox{and}
\qquad \omega_2(A) = \frac{\omega(\sqrt{1-z^*z}\;A\;\sqrt{1-z^*z})}
{\omega(1 - z^*z)}
$$
(we may assume $z^*z < 1$ by replacing $z$ by $z/\|z\|$).
For infinite quantum systems with suitable interactions constant elements
belong to the center which is the classical part of the system. On the
other hand, for equilibrium (KMS) states all elements of the center
are constant.
The extremal invariant states are the ones where $z$ is represented by
a $c$-number, thus to the classical quantities one assigns a definite
value. In the example of the spin chain to be studied in Section~3
the classical quantities are the mean magnetization
$$
\vec m = \lim_{N \ra \infty} \frac{1}{2N} \sum_{i = -N}^N \vec \sigma_i
$$
and classically pure states are the ones where all spins except
a negligible number of them point in the same direction $\vec m$. For the mixed
states of the form (4) the histories $W$ are also convex combinations
\beq
W_\omega(\Un{\alpha}) = \mu W_{\omega_1}(\Un{\alpha}) +
(1 - \mu) W_{\omega_2}(\Un{\alpha})
\eeq
and what we will show that for long histories $n \ra \infty$,
$t_{i+1} - t_i \ra \infty$ they purify in the sense that in (5) either
$W_{\omega_1}$ or $W_{\omega_2}$ dominates such that
$W_{\omega_i}/W_{\omega_j} < \ve$ for arbitrarily small $\ve$.
Which one dominates depends on the history. Of course one cannot dominate
over the other for all histories since
$$
\sum_{\Un{\alpha}} W_{\omega_1}(\Un{\alpha}) =
\sum_{\Un{\alpha}} W_{\omega_2}(\Un{\alpha}) = 1.
$$
We shall elaborate only on the simplest nontrivial case with two states
$\omega_{1,2}$ and two projectors $P$, $1-P$. The generalization to several
states or propositions is easy and shows the same features. Denote
$\omega_{1,2}(P) = p_{1,2}$, then for a history with $\ell$ projectors
$P$ and $n-\ell$ projectors $(1-P)$, $0 \leq \ell \leq n$, we get from
(3)
\beq
W(\Un{\alpha}) = W(n,\ell) = \mu p_1^\ell(1-p_1)^{n-\ell} +
(1 - \mu) p_2^\ell(1 - p_2)^{n-\ell}.
\eeq
The amount of mixing is given by the ratio
$$
R = \frac{\mu}{1-\mu} \left( \frac{p_1}{p_2}\right)^\ell
\left( \frac{1-p_1}{1-p_2}\right)^{n-\ell}
$$
of the two contributions. The mixing is noticeable if for some small
number $\ve > 0$ we have $\ve < R < 1/\ve$ or
\beq
\frac{\ve(1 - \mu)}{\mu} < a^\ell b^{n-\ell} <
\frac{1 - \mu}{\ve \mu}
\eeq
where $a := p_1/p_2$ and $b := (1 - p_1)/(1 - p_2)$.
For definiteness we may assume $a > 1$, thus $b < 1$.
If we scale $\ell$ with $n$, $\ell = n \lambda$, $0 \leq \lambda \leq 1$
the condition (7) becomes
\beq
\frac{1}{\ln a/b} \left( \ln \frac{1}{b} + \frac{1}{n} \ln
\frac{1 - \mu}{\mu} - \frac{1}{n} \ln \frac{1}{\ve} \right) <
\lambda < \frac{1}{\ln a/b} \left( \ln \frac{1}{b} + \frac{1}{n}
\ln \frac{1- \mu}{\mu} + \frac{1}{n} \ln \frac{1}{\ve} \right).
\eeq
Thus for the mixed histories $\lambda$ is in an interval of length
$2/(n \ln a/b) \ln 1/\ve$. We are interested in long histories,
$n \ra \infty$, for which this is a small number and want now to
calculate which fraction of the histories are mixed. Since at each
event the histories can take two courses there are $2^n$ histories
altogether. Their probabilities (6) depend only on $n$ and $\ell$
and there are $n!/(\ell !(n - \ell)!)$ histories with this
probability. For $n \ra \infty$ we calculate with Stirlings"s formula
the density $d$ of histories with $\lambda = (1 + \delta)/2$ to be
\beq
d(\delta) = \sqrt{\frac{n}{\pi}}\; e^{-n\delta^2}, \qquad
\int_{-1}^1 d\delta \; d(\delta) = 1 + O(e^{-n}).
\eeq
Then (8) tells us that for $n \ra \infty$ the fraction of histories more
mixed than $\ve$ is less than
$$
\frac{4}{\sqrt{\pi n}} \; \frac{\ln 1/\ve}{\ln a/b} \;
e^{-n \delta_m^2}
$$
where
$$
\delta_m = \left| \frac{\ln 1/b - \ln a}{\ln 1/b + \ln a}\right|
$$
is the minimum of $|\delta|$ in the interval defined by (8). We have
$\delta_m > 0$ and therefore exponential decrease unless $a = 1/b$ which
happens if $p_1 = p_2$ or $1-p_2$. In the first case $\omega_1 = \omega_2$
for the algebra generated by $P$ and $1-P$ and there is no mixing.
In the second case $\omega_1 = \omega_2$ combined with a reflection and
in this special case the fraction decreases only as $1/\sqrt{n}$. In
any case for $n \ra \infty$ most histories are pure in the sense that
within $\ve$ only states where the classical quantities have a definite
value contribute.
Since the $\ell$-dependence of $W$ goes with $(p/(1-p))^\ell$ for the
most probable history we have $\ell = 0$ or $n$ depending on whether
$p <$ or $> 1/2$. Thus most likely the answer to each experiment
is always the most probable one.
\section{The spin chain}
In this section we shall illustrate the abstract development of
Section~2 by a standard model of a measuring apparatus. It is the
simplest nontrivial physical example which shows these
features. Consider an infinite spin chain $\A = \{ \vec \sigma_i\}$,
the index $i$ ranging over the integers. As discrete evolution we take
the shift $\sigma_i \ra \sigma_{i+1}$. For the benefit of people working
on the interpretation of quantum mechanics who prefer to think in terms
of wave functions we shall use now the Schr\"odinger representation.
For each direction $\vec n_i$, $\vec n_i^2 = 1$ there is a vector
$| \vec n_i\rangle_i$ in Hilbert space such that $\vec \sigma_i$ points
in this direction, $\vec \sigma_i \cdot \vec n_i |\vec n_i\rangle_i =
| \vec n_i\rangle_i$. For all spins together the corresponding vector
is the tensor product $\bigotimes_i |\vec n_i\rangle_i$. If there were
$N$ spins these vectors spanned a $2^N$-dimensional space but for
$N = \infty$ the space is huge (nonseparable). We shall work in
separable subspaces of this monster which are obtained like the Fock
space by letting $\A$ act on a reference vector
$|\Omega_1\rangle : \Ha_1 = \A |\Omega_1\rangle$.
For $|\Omega_1\rangle$ we chose a polarized state where all spins
point in the same direction
\beq
|\vec n\rangle : |\Omega_1\rangle = \bigotimes_{i = -\infty}^\infty
| \vec n\rangle_i.
\eeq
In $\Ha_1$ we get an irreducible representation $\pi_1$ of $\A$.
Of course, for individual spins $\pi_1(\sigma_i)$ acts like $\sigma_i$
but weak limits like the mean magnetization
\beq
\vec M = \lim_{N \ra \infty} \frac{1}{2N + 1} \sum_{i = -N}^N
\pi_1(\vec \sigma_i) = \vec n {\bf 1}
\eeq
({\bf 1} the unit operator) depend on the representation. Had we
considered another reference state $|\Omega_2\rangle =
\bigotimes_{i = -\infty}^\infty |m\rangle_i$ the magnetization would turn
out to be $\vec m$. Thus the two representations are not unitarily
equivalent, $U^{-1} \pi_1(\sigma_i) U = \pi_2(\sigma_i)$ would imply
$$
U^{-1} \vec n \cdot {\bf 1} U = \vec m \cdot {\bf 1}
$$
which is impossible since $U$ cannot change the unity {\bf 1}.
The $| \Omega_{1,2}\rangle$ define states
$\omega_{1,2}(\cdot) = \langle \Omega_{1,2}|\cdot |\Omega_{1,2}\rangle$
and the mixed state $\mu \omega_1 + (1-\mu)\omega_2$ is obtained by a
vector in the orthogonal sum of $\pi_1$ and $\pi_2$
\beq
|\Omega_S\rangle = \sqrt{\mu}\; |\Omega_1\rangle \oplus \sqrt{1-\mu}\;
|\Omega_2\rangle \in \Ha_1 \oplus \Ha_2 =: \Ha_S, \qquad
\pi = \pi_1 \oplus \pi_2,
\eeq
$\langle \Omega| \pi(\vec \sigma_i)|\Omega\rangle = \mu \vec n +
(1-\mu)\vec m$. This representation is reducible, there are two
``superselection sectors'' \cite{L}, the magnetization
$\vec M$ is in the center and is not a multiple of unity
\beq
\lim_{N \ra \infty} \frac{1}{2N + 1} \sum_{i = -N}^N \pi(\vec \sigma_i)
= \mu \vec n {\bf 1}_1 \oplus (1-\mu) \vec m \cdot {\bf 1}_2.
\eeq
If we identify with a poetic license $\vec M$ with Schr\"odinger's cat
and $\vec n$ means alive and $\vec m$ means dead then the vectors of
$\Ha_1$ ($\Ha_2$) represent the cat
alive (dead) whereas in
$|\Omega\rangle$ (with $\mu = 1/2$) it is half dead and half alive.
We shall now show how by a succession of measurements $|\Omega\rangle$
purifies in the sense that the dominant components turn into either
$\Ha_1$ or $\Ha_2$.
First we have to construct for the measuring device a classical system
which can store the information contained in $\A$. Since a measurement
of $\sigma_i$ can have only two outcomes pointers with 2 positions
suffice. Since also for classical systems the Hilbert space
description is useful we represent that state of the device measuring
$\sigma_i$ by a two-dimensional vector $\left( \ba{c} u_i \\ d_i \ea
\right)$, $u$ and $d$ meaning pointer up or down. The measuring array
for all spins is again an infinite tensor product
$\bigotimes_i \left( \ba{c} u_i \\ d_i \ea \right)_i \in \Ha_A$
and we start with an $|\Omega_A\rangle$ where all $u_i$ are zero.
For the time evolution we take a shift $U$ and then an instantaneous
measurement of a direction $\vec s$ of a spin. The corresponding
proposition is the projector
$$
P_k = \frac{1}{2} (1 + \vec \sigma_k \cdot \vec s) = |s\rangle_k \;
{}_k\langle s|.
$$
If the answer is one we have the pointer unchanged, if the answer is zero
we turn the pointer up. This turning of the pointer is affected by an
operator $\tau$,
$$
\tau_k \left( \ba{c} u \\ d \ea \right)_k = \left( \ba{c} d \\ u \ea
\right)_k.
$$
Thus the effect of measuring $\vec \sigma_1$ is
$V_1 = P_1 + (1-P_1)\tau_1$ or written out in full as operators in
$\Ha = \Ha_S \otimes \Ha_A$
\beq
V_1 = (\pi_1 (P_1) \oplus \pi_2(P_1) \otimes {\bf 1} +
(1 - \pi_1(P_1)) \oplus (1 - \pi_2(P_1)) \otimes \tau_1).
\eeq
Note that $V_1$ is unitary and in $\Ha_S \otimes \Ha_A$ there is no
reduction of the full state vector. The time evolution $U$ between the
measurements shifts by one unit
$U \pi_{1,2}(P_k) = \pi_{1,2}(P_{k+1})U$, $U \tau_k = \tau_{k+1}U$,
so that the full time evolution of
$|\Omega\rangle = |\Omega_S\rangle \otimes |\Omega_A\rangle$
is after $n$ time units
\beq
|\Omega(n)\rangle = V_1 U V_1 U \ldots V_1 U |\Omega\rangle =
V_1 V_2 \ldots V_n |\Omega\rangle
\eeq
since $U^k |\Omega\rangle = |\Omega\rangle$. The results of the
measurements are encoded in the $\Ha_A$-part of $|\Omega(n)\rangle$ and
so we decompose it in an orthogonal basis of $\Ha_A$
\beq
| \Omega(n)\rangle = \sum_{\alpha_i = 0}^1 v(\Un{\alpha})
\tau_1^{\alpha_1} \tau_2^{\alpha_2} \ldots \tau_n^{\alpha_n}
| \Omega_A\rangle .
\eeq
Wherever $\alpha_i = 0$ the corresponding spin was in direction
$\vec s$, for $\alpha_i = 1$ we had $- \vec s$. If we have such a
situation the system is left with a wave function $v(\Un{\alpha})$
which has a component in $\Ha_1$ and one in $\Ha_2$:
\beq
v(\Un{\alpha}) = v_1(\Un{\alpha}) \oplus v_2(\Un{\alpha}).
\eeq
To calculate the length of the $v_{1,2}(\Un{\alpha})$ we have to use
$P_1 |n\rangle_1 = |s\rangle_1 \langle s|n\rangle$,
$(1 - P_1) |n\rangle = |-s\rangle \langle -s|n\rangle$.
If we introduce $| \langle s|n\rangle|^2 = p_1$, thus
$| \langle -s|n\rangle |^2 = 1 - p_1$ and similarly
$| \langle s|m\rangle |^2 = p_2$, $|\langle -s|m\rangle |^2 = 1 - p_2$
we find that if $\Un{\alpha}$ contains $\ell$ zeros and $n - \ell$
ones we have
\beq
\| v_1(\Un{\alpha})\|^2 = \mu p_1^\ell (1-p_1)^{n-\ell}, \qquad
\| v_2(\Un{\alpha})\|^2 = (1 - \mu) p_2^\ell (1 - p_2)^{n-\ell}.
\eeq
Thus with $W(\Un{\alpha}) = \|v(\Un{\alpha})\|^2$ we arrive at exactly the
expression (6) and the decomposition (16) displays the $2^n$ histories.
Remember that we had a unitary time evolution, there was no collapse of
the wave function after each measurement, only at the end we were
reading the configuration of classical pointers.
\paragraph{Remarks}
\begin{enumerate}
\item In terms of convergence of states our result can be expressed as follows.
The vector $|\Omega\rangle$ gives for $\mu \neq 0,1$ a mixed state over
the algebras of system $\otimes$ apparatus since the system is represented
reducibly. It evolves by the unitary evolution into the vector
$|\Omega(n)\rangle$ such that the state
$\omega_n(\cdot) = \langle \Omega(n)| \cdot |\Omega(n)\rangle$
over the system $\otimes$ apparatus stays pure for $\mu = 0$ or 1,
otherwise it stays mixed. Reading the pointer in a position $\Un{\alpha}$
changes the state to
$\omega_{n,\Un{\alpha}}(\cdot) = \omega_n (P_{\Un{\alpha}} \cdot)
W_\omega(\Un{\alpha})^{-1}$
where $P_\alpha$ projects onto the vector
$\tau_1^{\alpha_1} \ldots \tau_n^{\alpha_n} |\Omega_A\rangle$.
For $n \ra \infty$ (and making the history $\Un{\alpha}$ infinite)
this converges weakly to a pure state.
$\lim_{n \ra \infty} \omega_n = \sum_{\Un{\alpha}} \omega_{n,\Un{\alpha}}
W(\Un{\alpha})$ is a mixed state even for $\mu = 0$ or 1. This is in
accordance with the result of K. Hepp who observed that in a similar
situation of an infinite quantum system weak limits of pure states may
be mixed \cite{H}.
\item That the states $\omega_{1,2}$ in the example were pure is irrelevant,
for asymptotic abelian systems they only have to be extremely invariant
to possess the required cluster properties. If we restrict ourselves
to canonical temperature states then only at phase transition points
one has to be aware that they themselves are mixtures of states with the
same temperature but different values of a central element, corresponding
to the fact that we know that at a transition point space clustering
becomes critical (on time clustering no rigorous results are available).
\item When we talk about macroscopic quantum systems we mean many
degrees of freedom and not just large in size. So to the proposal of
Leggett et al. \cite{LG} our considerations are not applicable.
What we really need is that the relaxation time of the system is
shorter than the time between measurements.
\end{enumerate}
To summarize we have first to emphasize that we stay within the
Copenhagen interpretation of quantum mechanics.
The many history interpretation is a systematic formulation of the
reduction of the wave function postulate. We consider it as a simple
description of the essential effect of a measurement without going
into the details of a complicated mechanism (compare \cite{BJ,P}).
For one simple model of a measuring apparatus we have shown that
actually the unitary evolution of the joint system leads to the same
result. Our goal is to see what the special properties of large
quantum systems imply for the histories.
To get the optimal knowledge of the state of a finite quantum system
one has to repeat the experiment on other members of an ensemble of
equally prepared systems to gather enough statistics. For our infinite
system one can also redo after an appropriate relaxation time the
experiment on the same system and get the same result provided the
initial state was pure on the macroscopic part. In all quantum
systems even in an optimally refined state some quantities will remain
fluctuating. This remains true for infinite quantum systems but we have
seen that by repeated measurements the classical observables will
assume definite values. Thus f.i. below the phase transition domains
of a magnet will be magnetized in a definite direction. Even if nobody
looks at it there will be enough ``events'' (= interactions with the
surroundings) to purify the state over the classical part. However,
a quantum mechanically pure state over all the microscopic observables
will not be obtainable because for these systems all observable
projections are infinite dimensional, onedimensional projections in
Hilbert space will not belong to the algebra of observables.
\newpage
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\end{document}