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\begin{document}
\title{Statistics of Measurement of Non-commuting Quantum Variables}
\classification{03.65.Ta, 03.65.Wj, 03.67.Lx, 73.23.-b}
\keywords{quantum measurement, qubit, detector, linear amplifier}
\author{ Hongduo Wei and  Yuli V. Nazarov }
{address=
{Kavli Institute of NanoScience, Delft University of
Technology, 2628 CR Delft, The Netherlands}}

\begin{abstract}
We address continuous weak linear quantum measurement and argue that
it is best understood in terms of statistics of the outcomes of the
linear detectors measuring a quantum system, for example, a qubit.
We develop a proper formalism to evaluate the statistics of
such measurement. Generally, we are able to evaluate the joint
probability distribution of the detector outcomes and the qubit
variables.
We  concentrate on two setups.
The application of our method to the setup where a single 
pseudospin component is measured gives a comphrehensive picture
of quantum non-demolition measurement. 
More interesting setup consists of a qubit and three
independent detectors that simultaneously monitor three
non-commuting operator variables, those corresponding to three
pseudo-spin components of the qubit. When analyzing the distribution in the limit of big
values of the outcomes, we reveal a high degree of correspondence
between the three outcomes and three components of the qubit
pseudo-spin after the measurement. This enables a high-fidelity
monitoring of all three components. We discuss the relation between
the monitoring described and the algorithms of quantum information
theory that use the results of the partial measurement.
The formalism is based on
Feynman-Vernon approach, roots in the theory of full counting
statistics, and boils down to a Bloch-Redfield equation augmented
with counting fields.
\end{abstract}
\maketitle
\section{Introduction}
The theory of quantum measurement, being a foundation of quantum
physics, is  attracting more and more attention
\cite{quantum-measure}.
Intrinsic paradoxes  {\cite{Leggett}} are definitely a main reason for studying
quantum measurements. More motivation comes
from the practical needs to understand the
real solid-state based devices \cite{solid-state,qubit-review}
developed for quantum computing \cite{quantum-computing}.
Measurements in solid-state setups may provide access to extra
variables that facilitate the read out of the quantum information stored
in the elementary
two-level quantum systems (qubits). The concept of continuous weak
linear measurement (CWLM), where the interaction between the
detector and the measured system is explicit and sufficiently weak,
has been recently elaborated in context of the solid state quantum
computing \cite{ Korotkov1, Averin, Clerk, Jordan, Korotkov}.
%change
CWLM provides a universal description of the measurement process and is
based on general linear response theory
\cite{point}. It applies to a large class of linear
detectors: From common amplifiers to more exotic on-chip detectors
such as quantum point contact \cite{QPC}, superconducting SET
transistors \cite{SET}, generic mesoscopic conductors \cite{MES},
%and in the experiment setup by manipulation of ballistic motion of
%individual
fluxons in a Josephson transmission line to measure a
flux qubit \cite{ARS,ARD}.

It is an important feature of CWLM that the (quantum) information is
transferred from a quantum system  being measured --- a qubit --- to
other degrees of freedom: those of the detector. The outcome of the
measurement is thus represented by the detector degrees of freedom
rather than those of the qubit. We will address both the statistics
of the outcomes and joint statistics of the outcomes and the qubit
degrees of freedom.

We stress the difference between the detector outcomes
and the outcomes of a projective measurement of a qubit.
In distinction from the result of a projective measurement,
the detector outcome is not discrete, since the detector output
(for instance, voltage or current) is a continuous variable.
The outcomes do not even have to correlate with the state of the qubit
if the detector is uncoupled. Further, the detector variables are subject to noise
not related to the qubit. Owing to the feedback of the detector at the qubit,
this noise affects the qubit too.


In comparison with the text-book projective measurement that
instantly provides a result and projects the system onto the state
corresponding to the result, the CWLM takes time both to accumulate
the information and to distort the qubit. The time $\tau_{m}$
required to obtain a sufficiently accurate measurement result is
called "measurement time" and is a characteristic of a CWLM setup.
It is not a duration of an individual measurement in this setup: the
latter may vary. The distortion is due to the inevitable  back
action of the detector and is characterized by the dephasing rate
$\Gamma_d$. It has been shown \cite{Korotkov1,Averin,Clerk} that for
an optimized
--- quantum limited --- detector $\tau_{m}\Gamma_d = 1/2$ while the
"measurement time" $\tau_m$ greatly exceeds $1/2\Gamma_d$ for less
optimal detectors.

In the context of quantum information theory, CWLM may be understood
as an interaction of the qubit with infinitely many ancillary qubits
representing the detector degrees of freedom. Each ancilla is
brought to weakly interact with the qubit for a short time and is
subsequently measured. Owing to the interaction, the quantum state
of the ancillae is entangled with the state of the qubit. The
detector output is proportional to the sum of the measurement
results of a large set of ancillae.  This allows to transfer quantum
information from the qubit to the detector without formal projective
measurement of the qubit. Therefore the peculiarities of the CWLM
can be understood in the framework of a projective measurement,
although a more complicated one involving the detector degrees of
freedom. The CWLM can be thus seen as a build-up of an entanglement
between the qubit and the detector. An outcome of an individual
CWLM is the detector output accumulated during the time interval of
a certain duration $\tau_d$. Any CWLM can be described as a
generalized quantum measurement, that involves qubit and detector
degrees of freedom.

The outcome randomly varies from measurement to measurement. We
argue here that studying statistics of the measurement outcomes of a
CWLM is the best way to understand and characterize such a
measurement.  This is especially important for the simultaneous
measurement of non-commuting variables (say, $A$ and $B$) we
concentrate on in this work. In this case, the text-book projective
measurement can not help to predict the statistics of the results:
it would depend on the order of measurements of $A$ and $B$. This
property of the measurements in non-commuting bases enables most
quantum cryptography \cite{crypt} algorithms and has been
extensively elucidated in  Ref. \cite{Busch}.

One can straightforwardly realize in experiment a
CWLM of a quantum system where $A$ and $B$ are measured simultaneously.
 If $A$ and $B$ commute,
the statistics of the outcomes of sufficiently long CWLM corresponds
to the predictions of projective measurement scheme (see Sec. III).
The projective measurement scheme loses its predictive power if $A$
and $B$ do not commute. The reason is that the order of measurement
of $A$ and $B$ is not determined in the course of a continuous
measurement. The statistics of CWLM outputs thus can not be
straightforwardly conjectured and has to be evaluated from the
quantum mechanical treatment of the whole system consisting of the
qubit and the detectors.

In a sharp contrast to the case of commuting variables,
the most probable outcome of a sufficiently
long CWLM of non-commuting variables does not depend
on the qubit state. Therefore it provides no information about the qubit.
The information is however hidden in the statistics of random outcomes.
Recently, the
simultaneous acquisition of two non-communing observables was
investigated in the framework of CWLM {\cite{Jordan}}, and  the
correlation of the random output of two detectors was found to be informative.
Not only noise, but the whole full counting statistics (FCS) of the
non-commuting measurements has been recently addressed for an
example of many spins traversing the detectors \cite{Lorenzo}.

\section{Results in short} 
We develop the necessary
formalism for the measurement statistics. 
Our approach stems from the FCS theory of
electron transfers \cite{Levitov} in the extended Keldysh formalism
\cite{Nazarov}, which has been recently discussed
\cite{Makhlin-Lesovik} in the context of the quantum measurement. At
first step, we obtain a Feynman-Vernon action to describe the
fluctuations of the input and output variables of the detector(s).
In the relevant limit, the action is local in time. So at the second
step we reduce the path integral to the solution of a differential
equation that appears to be a Bloch-Redfield equation augmented with
the counting field. We exemplify the formalism
addressing a relatively simple case of quantum non-demolition (QND)
measurement \cite{QND}. We evaluate the distribution of the outcomes
for a single detector and
 understand  the statistics of  a recently proposed  quantum un-demolition
 measurement\cite{Korotkov}.
The main results concern the
statistics of measurement of non-commuting variables for the case of
three independent detectors measuring the three components of the
qubit pseudo-spin. We find the statistical correspondence between
the three outcomes and three wavefunction components after the
measurement. The correspondence is characterized by a fidelity that
generally increases with the magnitude of the outcomes reaching the
ideal value $1$ in the limit of large magnitudes. Since very large
outcomes are statistically rare and require long waiting times, this
result could be of a purely theoretical value. To prove the
opposite, we have evaluated the fidelity at moderate magnitudes of
outcomes and measurement durations $\tau_d$ and we were able to
demonstrate the fidelity of $0.95$ for $\tau_d \simeq 7 \tau_{m}$.
We term this "quantum monitoring". Ideally, the result of the
quantum monitoring is a pure state of the qubit and three numbers
(detector outputs) giving the polarization of the state. The same
result can be also achieved by preparing the qubit state of the
known polarization, for instance, by a projective measurement along
a certain axis. The difference is that in the case of preparation
the polarization axis is known to the observer in advance, while in
the case of monitoring it is not so: both the three numbers and the
state emerge from dynamics of the quantum system that encompasses
the qubit and the detectors. A fuller account of the results can be found in \cite{Wei}.


%We discuss the relation between the quantum monitoring proposed and
%the quantum algorithms that use the results of partial measurements
%that we summarize in Sec. V.  We evaluate the detector action in the
%Appendix A. We prove in the Appendix B that our approach correspond
%to a Lindblad scheme for a system consisting of the detectors and
%the qubit.
\section{Method}
We start with a single detector setup. The Hamiltonian reads as
follows:
\begin{subequations}
\begin{eqnarray}
 %\nonumber to remove numbering (before each equation)
  H &=& H_q+H_{int}+H_{d} \,,\label{Hamiltonian}\\
  H_{q} &=& \sum\limits_{i=1}^{3} H_i \hat{\sigma}_i; \;
  H_{int} = \hat{\sigma}_3 \hat{Q} \label{Hamiltonian-int} \,.
 \end{eqnarray}
\end{subequations}
Here, $H_q$  is the Hamiltonian of the qubit in terms of Pauli
matrices $\hat{\sigma}_{i}$ ($i=1$, $2$, $3$) corresponding to three
components of the qubit pseudo-spin.
 $H_{int}$ gives the interaction between the detector and the third component of
the pseudo-spin of the qubit, $\hat{Q}$ being the detector {\it
input} variable.  $H_{d}$ is the Hamiltonian of the detector which
we do not specify assuming linear dynamics of the detector
variables. Our goal is the statistics of the detector output
variable $\hat{V}$. Following \cite{Nazarov-2} we introduce a
counting field $\chi(t)$ coupled to the output variable $\hat{V}$
and use Keldysh scheme with different Hamiltonians $H^{\pm}$ on two
parts of the Keldysh contour\cite{Keldysh}, $ H^{\pm}=H\pm
\hbar\chi(t) \hat{V}/2$. $\pm$  corresponding to the upper/lower
branch of the  contour. The counting field $\chi(t)$ plays a role of
the variable in the probability-generating function of the detector
outcomes $V(t)$, this function is given by:
\begin{equation}\label{action}
%e^{-S(\{\chi(t)\})}
Z(\{\chi(t)\}) =\mathrm{Tr}\bigl( \overrightarrow{\mathrm{T}} e^{
\frac{-i}{\hbar} \int dtH^{+}} \hat{\mathrm{R}}(0)
\,\overleftarrow{\mathrm{T}}e^{ \frac{i}{\hbar}\int    dt H^{-}}
\bigr) \,.
\end{equation}
$\mathrm{Tr}(\cdots)$ implying the trace over  both  detector and
qubit variables. Here, $\overrightarrow{
\mathrm{T}}$($\overleftarrow{\mathrm{T}}$) denotes time (reversed)
ordering in evolution exponents and $\hat{\mathrm{R}}(0)$ is the
initial density matrix of the qubit and detector. Next we employ the
path integral representation for the probability-generating function
\cite{Nazarov-2}. The integral is over the detector variables,
\begin{eqnarray}\label{Z}
% \nonumber to remove numbering (before each equation)
%   e^{-\frac{1}{\hbar} S(\{\chi(t)\})}
 Z(\{\chi(t)\})  &=& \int\mathcal{D}{\bar X}^{+} \mathcal{D} {\bar X}^{-}
   e^{A_{d}(\{\bar{X}^{+}(t), \bar{X}^{-}(t),\bar{\chi}(t) \})}  \nonumber \\
   & &\times \mathrm{Tr}_{\rm qubit}
   \bigl(\overrightarrow{\mathrm{T}} e^{-\frac{i}{\hbar}\int dt
 \,(H_q + \hat{\sigma}_3 \hat{Q}^{+}(t)) }\hat{\mathrm{\rho}}(0)\nonumber \\
   & & \times \overleftarrow{\mathrm{T}} e^{\frac{i}{\hbar}\int dt \,(H_q +
\hat{\sigma}_3 \hat{Q}^{-}(t))
   }
   \bigr)\,.
\end{eqnarray}
Here, $\bar{X}^{\pm}(t)$ are two-dimensional vectors of the detector
variables $\bar{X}^{\pm}(t)=(Q^{\pm}(t), V^{\pm}(t))^T$,
$\mathcal{D}{\bar X}^{\pm} = \prod\limits_t \mathcal{D}{\bar
X}^{\pm}(t)$, $\hat{\mathrm{\rho}}(0)$ is the qubit density matrix.
The action $A_d$ is bilinear in $\bar{X}^\pm$ to describe linear
dynamics of the detector. Following common assumptions about CWLM,
\cite{Averin,Clerk} we assume instant detector responses and white
noises to arrive at
\begin{eqnarray}\label{newaction1}
A_d &=& \int dt [ -\frac{1}{2}\bar{x}^{T}(t)
(\check{a}^{-1})^{T}\check{S}\check{a}^{-1}\bar{x}(t) \nonumber \\
&&+i\bar{X}^{T}(t)\check{a}^{-1}\bar{x}(t)+i
\bar{\chi}^{T}(t)\bar{X}(t) ]\,,
\end{eqnarray}
where we switch to the "quantum" ($\bar{x}$)
 and "classical"($\bar{X}$)
variables defined as follows: ${\bar x}=({\bar X}^{+}-{\bar
X}^{-})/\hbar$, ${\bar X}=({\bar X}^{+}+{\bar X}^{-})/2$. Here,
$\bar{\chi}=(0,\,\chi)^{T}$, the $2\times 2$ matrices $\check{a}$,
$\check{S}$ give the response functions and noises of the detector
respectively. $S_{11}$ is the noise of the input variable
responsible for the backaction of the detector and decoherence of
the qubit; $S_{22}$ is the output noise and $S_{12}=S_{21}$ presents
the correlation of these two noises. $a_{12}$ determines the
detector response on the qubit pseudo-spin, $\langle \hat{V} \rangle
= a_{12} \langle \hat{\sigma}_3\rangle$. Other response functions
$a_{21},a_{22},a_{11}$ are respectively related to reverse gain,
output and input impedances of the detector and are not of immediate
interest for us. The detector is characterized with the dephasing
rate $\Gamma_d=2 S_{11}/\hbar^2$ and the "measurement time" $\tau_m=
S_{22}/a_{12}^2$.\cite{Makhlin-2, Averin} The Cauchy-Schwartz
inequality $4 S_{11}S_{22}-4S_{12}^2\geq \hbar^2 a_{12}^2$
\cite{Averin, Clerk} guarantees $\tau_m \Gamma_d \ge 1/2$.

It is important for further advance that the action
(\ref{newaction1}) is local in time. In this case, the path integral
in (\ref{Z}) can be taken at each time slice separately. The result
of integration is expressed in terms of the solution of a
local-in-time evolution equation, which is a familiar Bloch-Riedel
equation for the density matrix modified by the counting
field.\cite{Romito} It reads:
\begin{eqnarray}
      % \nonumber to remove numbering (before each equation)
      \frac{\partial{\hat{\rho}}}{\partial t} &=&
      -\frac{i}{\hbar} [\hat{H}_q,\hat{\rho}
      ]+\frac{\chi^2}{2}S_{22}\hat{\rho}
+ \frac{i a_{12}\chi}{2}(\hat{\rho} \hat{\sigma}_z + \hat{\sigma}_z
\hat{\rho}) \nonumber \\
        & &-\frac{S_{12}}{\hbar}\chi(\hat{\rho} \hat{\sigma}_z - \hat{\sigma}_z \hat{\rho})
        -\frac{S_{11}}{\hbar^2}(\hat{\rho}-  \hat{\sigma}_z \hat{\rho}\hat{\sigma}_z
        ) \,.
\label{master-eq}
\end{eqnarray}
The locality in time is a relevant but strong assumption which in
fact corresponds to a  {\it classical} detector (indeed, the action
(\ref{newaction1}) does not contain any $\hbar$.) This is why we do
not have to worry about possible quantum uncertainties of the
detector output that could complicate the interpretation of the
statistics. \cite{Nazarov-2} The scheme described can be easily
extended to more qubits and/or detectors: One just adds extra
(counting) fields for detectors and extra Pauli matrices for qubits.
The case of interest for us is the simultaneous CWLM of three
pseudo-spin projections. The coupling term becomes
\begin{equation}
\label{three-detectors}
 H=\hat{\sigma}_1 \hat{Q}_1 +\hat{\sigma}_2 \hat{Q}_2+\hat{\sigma}_3 \hat{Q}_3 \,.
\end{equation}
 $\hat{Q}_k$ ($k=1$, $2$, $3$) being the input fields of the three detectors
. Three counting fields $\chi_k$  are coupled to the corresponding
output variables $V_{k}$  of the three detectors. We assume for
simplicity that the detectors are independent and identical each
described by the action (\ref{newaction1}). The corresponding
Bloch-Reidel equation reads
\begin{eqnarray}
      % \nonumber to remove numbering (before each equation)
      \frac{\partial{\hat{\rho}}}{\partial t} &=&
      -\frac{i}{\hbar} [\hat{H}_q,\hat{\rho}
      ]+\frac{\chi^2}{2}S_{22}\hat{\rho}
+ \frac{i a_{12}}{2}\sum_{k=1}^{3}\chi_k
[\hat{\sigma}_k,  \hat{\rho}]_{+}\nonumber \\
        & &+\frac{S_{12}}{\hbar}\sum_{k=1}^{3} \chi_k[ \hat{\sigma}_k,
        \hat{\rho}]
        -\frac{S_{11}}{\hbar^2}(3\hat{\rho}-\sum_{k=1}^{3}\hat{\sigma}_k
    \hat{\rho}\hat{\sigma}_k)\,,
\label{master-eq2}
\end{eqnarray}
where $\chi=\sqrt{\chi_1^2+\chi_2^2+\chi_3^2}$.
\begin{figure}[htb]
  
  \centerline{\includegraphics[width=0.5\textwidth]{QND.eps}}
 % 
 \caption{Quantum Non-demolition measurent: Two successive measurements. 
 In each pair of the curves,
 the solid one gives the distribution of outcome of the first
 measurement while the dashed one gives the distribution for
 the second measurement {\it provided} 
 the first measurement gave $v_1=-1$.
 Lower (upper) pair of curves corresponds to long, $\tau_{1,2}=2$
 (short, $\tau_{1,2}=0.3$)
 measurements. The long measurement is repetitive, the short one is not.
 }
\end{figure}
\begin{figure}
\centerline{\includegraphics[width=0.5\textwidth]{sigma1.eps}}
\caption{Quantum "undemolition" measurement:
$\sigma_1(v, \tau)$ characterizes the dephasing of
 the superposition after a time $\tau$($\tau=1$ for plots)
 {\it provided} the detector outcome is $v$.
 A quantum-limited detector ($C=C_{12}=0$, upper curve)
 allows for the quantum un-demolition measurement ($\sigma_1=1$)
 at $v=0$. This does not work for a worse detector
 ($C=C_{12}=1$, lower curve).}
\end{figure}

\section{Single variable}
Let us first illustrate the method with one-detector QND measurement
recently realized for superconducting qubits.\cite{QNM-hans}
 To satisfy non-demolishing condition \cite{QND}, we should set
$\hat{H}_q = \epsilon \hat{\sigma}_3$. In this case, $H_q$ is
canceled by transformation to the rotating frame, $\rho(t) \to
e^{i\hat{H}_qt/\hbar}\rho(t)e^{-i\hat{H}_qt/\hbar}$. Let us perform
two measurements that immediately follow each other. During the
first measurement, the detector output is collected in the time
interval $(0,t_1)$ so the measurement outcome is $V_1=\int_0^{t_1}
dt V(t)/t_1$. Similarly, for the second measurement
$V_2=\int_{t_1}^{t_1+t_2} dt V(t)/t_2$. The statistics of the two
outcomes is given by a piece-wise constant  $\chi(t)
=\chi_1(\chi_2)$ during the first(second) time interval and
$\chi(t)=0$ otherwise. We parameterize $\hat{\rho}$ as follows:
$\hat{\rho}(t)=(1+\hat{\sigma}_3)\rho_{+}(t)/2+(1-\hat{\sigma}_{3})\rho_{-}(t)/2
+\hat{\sigma}_{1}\rho_{1}(t)+\hat{\sigma}_{2}\rho_{2}(t)$. Eq.
(\ref{master-eq}) gives two decoupled pairs of equations for
$\rho_{\pm}$ and $\rho_{1,2}$ respectively. Solving for
$\rho_{\pm}(t)$ with initial conditions $\rho_{\pm}(0)$ (assuming
$\rho_{1,2}(0)=0$) and transforming the generating function gives a
very simple probability distribution of two outcomes:
\begin{equation}
P(v_1,v_2) = \sum\limits_{\pm}
\frac{\sqrt{\tau_1\tau_2}}{2\pi}\rho_{\pm}(0)
e^{-\frac{(v_1\mp1)^2\tau_1}{2}} e^{-\frac{(v_2\mp1)^2\tau_2}{2}}\,,
\end{equation}
where we switched to the dimensionless times $\tau = t/\tau_m$, and
outputs $v = V/a_{12}$. This result is in fact classical: It does
not depend on the dephasing rate. Initially, the qubit comes either
in the state $+$ or $-$ (with probabilities $\rho_{\pm}(0)$). The
state persists during the measurements, the outcome of each
measurement is distributed normally around $\pm 1$ with the standard
deviation $\sqrt{\tau_{1,2}}$. The repeatability of the measurements
is illustrated in Fig. 1(a).

To illustrate the quantum aspect, let us set the initial wave
function to a superposition: $\hat\rho(0)=\hat{\sigma}_1$, and
evaluate the corresponding projection of the pseudo-spin {\it after}
the measurement in time interval $(0,\tau)$ that gives the outcome
$v$. In addition to equations for $\rho_{\pm}$, we have to solve two
equations for $\rho_{1,2}$,
\begin{equation}
\frac{\partial \rho_{1,2}}{\partial t} = \mp\frac{2i S_{21}\chi}
{\hbar} \rho_{2,1}-(\Gamma_d +\frac{\chi^2}{2}S_{22})\rho_{1,2}\,.
\end{equation}
that do contain the dephasing rate. The quantity of interest is
obtained by transforming the generating functions
$\rho_1(\tau,\chi)$, ${\rm Tr}\left(\hat\rho(\tau,\chi)\right)$ and
reads (Fig. 2(b)):
\begin{eqnarray}
\sigma_1(v,\tau) &=&\frac{\cos (C_{12}v\tau) }  {\cosh(v\tau)}
e^{-\frac{C}{2}\tau}\,,
\end{eqnarray}
where we introduce dimensionless constants $C \equiv
4(S_{11}S_{22}-S_{12}^2)/(\hbar a_{12})^2 -1$ and $C_{12} =2
S_{12}/\hbar a_{12}$. Generally, $\sigma_1(v,\tau)$ quickly decays
with increasing $\tau$. This indicates the dephasing of the
superposition by the measurement. Remarkably enough, for a
quantum-limited detector ($C=C_{12}=0$) and for a special value of
the measurement outcome $v=0$ the dephasing is absent and the wave
function retains the initial value. This was called "quantum
un-demolition measurement" in \cite{Korotkov}. Physical meaning of
this is that the phase shift between the states $\pm$, acquired from
the detector, $2\int_0^{\tau}\,dt Q(t)/\hbar$,
 is zero at this (rather unprobable, \cite{Korotkov}) value of the outcome.
  We stress that the strict
correspondence between the phase shift and outcome does not hold for
a general detector, so that $\sigma_1(v=0, \tau) = \exp(-C\tau/2)$
decreases with the time of measurement.

\section{Non-commuting variables}
These simple examples prove the use of the statistical approach.
Thus encouraged, we turn to the statistics of the CWLM of
non-commuting variables. We assume that the Hamiltonian $\epsilon
\sigma_3$ is removed by transforming to the rotating frame. This
presumes that the signal from $\sigma_{1,2}$ is collected at
frequencies $\epsilon/\hbar$ rather than at zero frequency as the
signal from $\sigma_3$ is $V_{1,2} = \int_0^{\tau}\,
dt(\cos(\epsilon t/\hbar) V_{1,2}(t) \mp \sin(\epsilon t/\hbar)
V_{2,1}(t))/\tau$. Without the term $H_q$, the Eq.
(\ref{master-eq2}) is readily solved in proper basis in pseudo-spin
space: One of the Pauli matrices is defined as $\hat{\sigma}_{\chi}=
(\chi_1 \hat{\sigma}_1 +\chi_2 \hat{\sigma}_2+\chi_3
\hat{\sigma}_3)/\chi$, while two others are orthogonal to it.

We stress that the CWLM we are about to describe is hardly a
measurement of the {\it initial} state of the qubit. In contrast to
QND where the dephasing is limited to $1,2$ components, the
detectors randomly rotate the pseudo-spin in all three directions.
The quantum information about initial state is quickly lost at the
time scale of $1/\Gamma_d$, that is, before a statistically reliable
measurement result can be accumulated. To this end, the initial
condition hardly matters and we choose the unpolarized density
matrix $\hat\rho(0) = \frac{1}{2}\hat 1$. Albeit we will see that
this CWLM can be rather informative.

Let us first discuss the distribution of the detector outputs.
In the limit of long time of measurement $\tau \gg 1$,
the log of the generating function reads:
\begin{equation}\label{S}
-\log Z =\tau\left( C_d -\sqrt{C^2_d-\chi^2}+\frac{\chi^2}{2}\right)
\,.
\end{equation}
where $\chi_i$ has been made dimensionless $\chi_i S_{22}/a_{12} \to
\chi_i $ as to give the cumulants of dimensionless outputs $v_i$.
Here, $C_d \equiv \Gamma_d\tau_m  = (C+1+C_{12}^2)/2\geq 1/2$. The
distribution is isotropic in three outputs depending on $v\equiv
\sqrt{v_1^2+v_2^2+v_3^2}$ only and in this limit is obtained by the
saddle-point method. The distribution is concentrated at zero and is
essentially non-Gaussian at $v \simeq 1$. The presence of the qubit
exponentially enhances probabilities of such outcomes.(Fig. 2(a)) As
to low cumulants, the presence of the qubit enhances output noises
of each detector by the factor $1+1/C_d$. Most importantly, it gives
rise to non-zero fourth cumulant $\langle\langle v^2_i
v^2_j\rangle\rangle = -(1+2\delta_{ij})/(C_d\tau)^3$--- correlation
of noise {\it variations} in formally independent detectors.
\begin{figure}[htb]
%\begin{center}
  
    \centerline{\includegraphics[width=0.5\textwidth]{s-v.eps}}
  
\caption{Statistics of CWLM of non-commuting variables. Logarithm of the
outcome distribution is a function of $v\equiv
\sqrt{v_1^2+v_2^2+v_3^2}$. The curves from the top to the bottom:
quantum-limited detector ($C_d=1/2$), worse detector ($C_d=2$),
detector not connected to the qubit. At big values of
outputs $v \gg 1$, the probability of a big output is exponentially enhanced by the presence of the qubit. }
%\end{center}
\end{figure}
\begin{figure}
 \centerline{\includegraphics[width=0.5\textwidth]{fvt.eps}}
 \caption{ Fidelity of quantum
monitoring $f$ versus measurement time $\tau$ for $v$ ranging from
$4$ to $1.5$ (from upper to lower curves in each subfigure)
Bullets (triangles) at each curve indicate the value of
$\tau$ at which the probability to get the outcome
 $>v$ is 10\% (50 \%).
(b) The quantum-limited
detector.  (c) A worse detector
($C_d=2$).}
\end{figure}

Let us discuss the correlation of the detector outputs and the
pseudo-spin {\it after} such measurement. We characterize this with
a fidelity $f(v)$, inner product of the normalized vector of the
outcomes and averaged pseudo-spin at {\it given} outcome $v$,
$f=\sum_i\langle \sigma_i \rangle v_i/v $.
 The fidelity is $1$ if the values of the outputs precisely give
all three pseudo-spin components. From the saddle-point solution we
obtain that $f$ does not depend on $\tau$ in the limit $\tau \gg 1$,
and at large values of the outcomes $v\gg 1$ reaches the ideal value
$f \approx 1 - C_d/v$. This, quite unexpectedly, enables an
efficient {\it quantum monitoring} of non-commuting variables.
Roughly, one continuously measures the system and waits for
sufficiently high values of the outputs. When this is achieved, the
state of the qubit is known with any accuracy desired.

The better the accuracy desired $a_{des}\equiv 1-f \ll 1$, the
bigger outputs are required, $v \gg C_d/a_{des}$, so the typical
waiting time grows exponentially, $\mathrm{log}(t_w) \simeq
a_{des}^{-2}$. To prove that the monitoring is practical, we have to
show that a reasonably high fidelity can be achieved in a reasonably
short time. We evaluate and plot (Fig. 2(b) and 2(c) ) $f(v)$ of a
single measurement of duration $\tau$ versus $\tau$. We see from the
plots that $f=0.95$ is achieved for a quantum-limited detector at
$v=4$ and $\tau=0.7$. At these parameters, $10\%$ of the
measurements are successful, i.e. give the output $v >4$. We
conclude that the $5\%$ accuracy is typically achieved in time
interval $\simeq 7 \tau_m$.

To conclude, we present the statistical approach to the CWLM  and
illustrate the use of it. We propose a scheme for an efficient
monitoring of non-commuting variables and prove its feasibility.
\begin{theacknowledgments}
H.W. acknowledges support of NanoNed (project DSC.7023). Y.N.
appreciates the participation in 2006 Aspen Summer Program where he
got the impetus to this work.
\end{theacknowledgments}
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\end{document}