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quantum zeno effect , proposed by misra and sudarshan @xcite in 1977 , is believed to exist uniquely in the quantum world due to informatical description of quantum states and the projective measurement in quantum mechanics ( for a review , see ref .
while its classical correspondences , such as the fletcher s paradox , becomes an antinomy in newtonian mechanics .
the basic reason , from our judgement , is the absence of the projective measurement in classical physics .
however , though newtonian mechanics excludes the projective measurement , the later does exist in the classical world . in a recent work on this issue sjgumario , one of us ( gu ) touched the possibility of the classical zeno and anti - zeno effects .
gu used a scenario of super mario s prison break to show that a classical state " might not decay if it is observed continuously .
nevertheless , the scenario involves too much subjective freedoms coming from super mario s intelligent feedback , such that it is , though reasonable in everyday life , far beyond the objective laws of the classical world .
this consideration leads to the main motivation of the present work to seek a possibility of the classical zeno effect without involving any subjective judgement .
we consider here such a scenario of transferring a classical state , for instance a digitized image , through a classical channel with a noise surrounding ( see fig .
[ sketch.eps ] ) .
the state , if there is no assisted equipment , decays inevitably with the distance due to the surroundings interferences , and finally loses all information . in order to suppress the decay rate or
even ensure zero decay , we need to check the information of state within a certain distance . instead of the intelligent judgement made by super mario @xcite
, here we use a machine ( repeater ) , which has nothing to do with living beings , to examine and try to recover the state .
we show that if there are enough number of repeaters , the transferred state then might decay much more slowly , or even never decay , then a classical zeno effect occurs .
this work is organized as follows . in section
[ sec : class ] , we introduce the basic formulas for describing the classical transportation . in section [ sec : example ] , we take the emblem of the chinese university of hong kong ( cuhk ) as an explicit example to show how the classical zeno effect occurs during a classical transportation . a further discussion and a brief summary are given in section [ sec : dis ] and [ sec : sum ] respectively .
through a classical channel .
up : @xmath0 is transferred through the channel without any repeater and decays with the distance .
low : a repeater t is installed to check the completeness of the state , then @xmath0 might decay more slowly . , width=302 ]
to begin with , we consider a classical @xmath1-bit state describe by @xmath2 with @xmath3 , which in computer science can be saved physically by for instance in a flash disk or a perforated paper tape .
now we want to transfer the state through a classical channel , which is subject to a noise background of temperature @xmath4 . during the transferring process
, the state carried by each bit is protected via an energy gap @xmath5 .
so if the temperature is zero , there is no background inference , and the state can be transferred over an infinite distance without loss of information .
however , the thermal fluctuation of the surrounding might flip the state of bit . once the bit is flipped one or more times ,
we assume it loses the correlation to the original global state .
the transition probability of each bit through a unit distance @xmath6 is determined by @xmath7where @xmath8 denotes uncorrelated state .
then if the bit is transferred over a distance of @xmath9 ( in unit of @xmath6 ) , the surviving probability becomes @xmath10where @xmath11 .
therefore , the classical state will decay exponentially to a complete random state with the transferred distance .
explicitly , in the basis of @xmath12 , the initial state of the bit is @xmath13while the final state becomes@xmath14the distance between these two states can be measured by the fidelity @xmath15then at the very beginning , i.e. small @xmath9 @xmath16which decays algebraically .
we notice that the leading term of the fidelity here is linear instead of quadratic in quantum fidelity .
nevertheless , the algebraic decay of the fidelity ensure that we have chance to correct the lost information via a repeater .
in this section , we give a simple example to illustrate how such a process can be realized in practice .
we assume that alice wants to transfer a picture , i.e , the cuhk emblem , to bob who is at a long distance via a classical channel .
the emblem is prepared in a bmp file with @xmath17 resolution and 256 colors . for simplicity , we assume alice and bob have already known all other structural information , such as 256 elemental colors and the size of the picture , except the image data , which can be expressed as a @xmath17 matrix @xmath18 @xmath19with each element @xmath20 ( @xmath21,j\in \lbrack 1,200]$ ] ) being an 8-bit integer . during the transportation ,
each element of the matrix @xmath18 is subject to environmental interference .
one of elements might be changed to a random 8-bit integer with the probability @xmath22 with @xmath23 over the unit distance @xmath6 .
clearly , the lower the temperature , the more stable the data . meanwhile , as the transferred distance increase , the emblem s information will inevitably lose .
we use the classical monte carlo method to simulate such a process .
the leftmost column of fig .
[ cuhk.eps ] shows the emblem at the various distance .
we can see that the figure will finally be blurred .    in order
to suppress the decay rate , alice and bob need to install repeaters on the channel they use .
the repeater s first function is to check the integrability of data .
secondly , if possible , it can partially recover the integrability or require the previous repeater to resend the data .
for this purpose , they add row - column check code to the end of the matrix elements .
@xmath24the row and column check codes are defined as@xmath25respectively .
so if one of elements @xmath20 or @xmath26 is changed , the corresponding row and column check code do not match . since the eq .
( [ eq : checkcode ] ) are simple summation , in this case , the repeater is able to recover the lost information , that is@xmath27or@xmath28 if one of elements in @xmath26 is lost , the repeater recalculate @xmath29 from the matrix @xmath18 . _ for simplicity _ ,
if there are two or more elements are lost , we assume that the repeater is neither able to recover the integrability , nor requires the previous repeater to resend the picture again , but recalculate eq .
( [ eq : checkcode ] ) to ensure the integrability of data for next repeater .
we show our results in fig .
cuhk.eps . in the second column ,
the distance between two repeaters is @xmath30 .
the emblem keep much more information than the first column .
moreover , if we add more repeaters to the channel , the decay rate will be suppressed further .
especially if @xmath31 , the emblem at @xmath32 is almost the same as the original one .
we then call such a phenomenon as a repeater - assisted classical zeno effect in random processes .
precisely , we can introduce the fidelity to describe the honesty of the state to the original one . in this case
, the fidelity can be defined as @xmath33 the results are shown in fig .
[ fidelity.eps ] , from which we can see that the decay rate can be significantly suppressed if enough number of repeaters are installed . on the other hand , if there are two or more elements that have been found to be lost and the repeater can ask the previous repeater to resend the data again , then the classical state will never decay .
( in unit of 20000 steps ) for various @xmath34 . , width=302 ]
now the game is over .
however , we find ourselves in an embarrassed situation
. does such an issue still belong to physics ?
the answer seems to be yes since there is no any interferences from living beings and the  repeater "- like device needs energy only .
nevertheless , a digitized state and its decay clearly neither belong to quantum mechanics nor newtonian mechanics .
such a puzzle keeps a challenge for us and is not able to be answered easily in physical science .    on the other hand ,
the classical zeno effect is very popular phenomenon in the world . besides the case of super mario s prison break @xcite , the classical zeno effect exists in many fields , such as the medical field and the economical field .
take the former as an example : a human being usually is in a metastable state .
some small deceases can introduce some fluctuations around the equilibrium point . in these cases , the body s immune response and a suitable cure
will recover the metastable .
however , there exists a class of vital deceases , such as cancer , that will finally drag the state of human beings from the metastable point and never back if the state is far from the equilibrium point .
necessary cures clearly can slow down or ever stop such a process .
however , to stop such a process , the most important is to find and cure the decease as early as possible , then the probability to recover the metastable state is very high .
in summary , we touched such a possibility of classical zeno effect in classical stochastic processes , as illustrated by a scenario of transferring a classical state through a noise channel .
we show explicitly that the zeno effect is no longer a patent of quantum mechanics , but exists in classical stochastic processes .
we gave an example of transferring a bmp file through a noise channel , as simulated by classical monte carlo method .
the example manifests that the decay rate of the bmp file will be significantly suppressed , even never decay , if there are enough number of repeaters .
as a classical state , for instance a digitized image , is transferred through a classical channel , it decays inevitably with the distance due to the surroundings interferences . 
 however , if there are enough number of repeaters , which can both check and recover the state s information continuously , the state s decay rate will be significantly suppressed , then a classical zeno effect might occur . 
 such a physical process is purely classical and without any interferences of living beings , therefore , it manifests that the zeno effect is no longer a patent of quantum mechanics , but does exist in classical stochastic processes .