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the behavior of quantum - mechanical systems within relativistic settings is a subject of ongoing research , with the unruh effect being a prime aspect of these explorations @xcite .
questions raised by relativistic quantum information go beyond the problem of relative particle content and hold a promise of unveiling general properties of quantum mechanics when exposed to non - inertial motions and strong gravitational fields .
this includes analysis of detection of entanglement by non - inertial observers @xcite , extraction of entanglement from the vacuum @xcite , generation of entanglement due to motion @xcite and improvement of relativistic metrological protocols @xcite .
it also involves discussion of universal decoherence due to the gravitation dilation @xcite , fundamental limits to building ideal clocks @xcite , experimental simulation closed time - like curves @xcite and entangling power of the expanding universe @xcite , proposals for verification of the space - time topology @xcite and relativistic protocols within circuit quantum electrodynamics architecture @xcite .
here we would like to discuss one of the quantum features , namely the nonclassicality of quantum states , and ask how its operationally defined measure is influenced by general motions .
nonclassicality of quantum states can be linked to a variety of phenomena observed experimentally @xcite , such as photon antibunching @xcite , sub - poissonian photon statistics @xcite , squeezing @xcite , and can be considered a resource @xcite .
there is a number of different approaches to this concept @xcite .
however , no unique definition exists .
we are motivated by the approach taken in the field of quantum optics where the criterion for nonclassicality is frequently based on the properties of the @xmath0-representation .
what we are interested in , is verifying whether such a criterion , defined operationally , is sensitive to the motion of the probe detecting nonclassicality .
we show that indeed such sensitivity occurs and the universality of the operational criterion is further limited by a presence of relative motions .
even the minkowski vacuum state , which is considered classical from the perspective of an inertial frame , can yield non - classical properties when viewed by an accelerated observer , associated with a different basis @xcite .
the purpose of this work is to show explicitly that operational measures of nonclassicality such as the one studied in this paper , are actually observer - dependent . in order to study such effects
let us first review the necessary formalism behind the @xmath0-representation .
any state of a harmonic oscillator can be represented in the basis of coherent states , the so - called diagonal basis : @xmath1 where @xmath2 is the glauber - sudarshan diagonal @xmath0-representation , which is a probability quasi - distribution function @xcite .
coherent state represents the closest approximation of a classical oscillator , with the minimum uncertainty and oscillating expectation value of the position and the momentum @xcite .
therefore a quantum state @xmath3 , which can be represented as a statistical mixture of coherent states , can be considered classical @xcite .
this occurs when the corresponding @xmath0-representation @xmath2 satisfies the properties of a probability distribution function and is a positive definite function .
however , when @xmath2 takes on negative values or is highly singular , this statement does not hold anymore .
it is possible to derive the @xmath0-representation from the full quantum state tomography @xcite .
however , in general this procedure is highly sensitive to experimental errors and thus not practical .
for that reason , the solution is to search for measures that can tune in to the properties of @xmath0-representation without the necessity of using state tomography . in this body of work
we are interested in the quantum - optical approach that leads to an operationally defined witness of nonclassicality .
we follow the proposal presented in @xcite where the authors introduced a witness based on a two - level probe interacting with a harmonic oscillator . similarly to that we probe a state of a quantum field with a two - level detector and infer the presence of the aforementioned nonclassicality of this state from the detector s readings .
our analysis reveals how different motions of the detector affect the performance of the witness based on the @xmath0-representation .
the paper is structured as follows : in sec .
[ s2 ] we recall the formalism of the witness of nonclassicality , in sec .  [ s3 ] we discuss the impact of relative motion of the detector and the measured system on the performance of the witness and in sec .  [ s4 ] we give a summary and an outlook . throughout the paper
we use natural units with @xmath4 .
let us start by briefly reviewing the operational witness of nonclassicality based on the approach presented in @xcite .
we are interested in the operational evaluation of nonclassicality of an unknown initial state of a quantum system .
we consider a qubit detector which interacts with a harmonic oscillator .
after the interaction , by analyzing the state of the detector we are able to verify the presence of nonclassicality in the initial state of the harmonic oscillator . + the system is assumed to start in a product state @xmath5 , corresponding to the qubit and the harmonic oscillator , and evolve according to a unitary transformation @xmath6 generated by a hamiltonian : @xmath7 in the above equation @xmath8 denote the usual bosonic annihilation and creation operators , @xmath9 are the pauli operators , @xmath10 , @xmath11 are the frequencies of the respective subsystems , and @xmath12 is the coupling constant between them .
we choose to parametrize the initial state in the following manner : @xmath13 where we have used the diagonal representation for the harmonic oscillator s density operator @xmath14 and the eigenbasis of @xmath15 for the qubit s operator @xmath16 .
the full density operator evolves according to the equation @xmath17 and the state of the detector at any time @xmath18 is given by the reduced density operator @xmath19 . since @xmath20=0 $ ] , the only non - trivially evolving elements of the qubit density matrix , @xmath21 , are the off - diagonal ones : @xmath22 where @xmath23 denote @xmath24 eigenstates with corresponding eigenvalues @xmath25 .
an explicit calculation of the evolution of @xmath26 yields the following result : @xmath27 is the _ _ witness function _ _ as the witness function . ] .
if the underlying @xmath0-representation is positive , then one can write the _ witness inequality _ ( or the _ classicality bound _ ) : latexmath:[\ ] ] and confirm that @xmath156 satisfies the relation .
based on this observation we can calculate the time evolution operator to obtain : @xmath157 in the above @xmath158 denotes the displacement operator and we have introduced : @xmath159 the above formulae can be used for evaluating the time evolution operator @xmath160 discussed in the core body of the manuscript .
recall that the @xmath0-representation for coherent and thermal states take the following form ( here we consider one mode only and omit the label @xmath108 ) : @xmath161 where @xmath162 is the average number of excitations .
one can insert these formulae into the first line of equation ( [ witness - moving ] ) and evaluate the absolute value of the witness function . for the coherent state
the corresponding absolute value of the witness function is equal to one , whereas for the thermal state we obtain : @xmath163 in both cases we obtain a function which does not exceed unity .
here we evaluate the expression for @xmath104 that appears in eq .  .
let us repeat the underlying integral : @xmath164 and the notation @xmath114 . for further convenience
we introduce @xmath165 and continue the calculation to obtain :      this expression reveals a resonant behavior with a peak at @xmath167 , namely if we take the limit @xmath168 , we obtain a finite @xmath169 whose envelope increases linearly with @xmath48 .
it allows us to identify the critical velocity given in eq .  .
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we show that the operational definition of nonclassicality of a quantum state depends on the motion of the observer . 
 we use the relativistic unruh - dewitt detector model to witness nonclassicality of the probed field state . 
 it turns out that the witness based on the properties of the @xmath0-representation of the quantum state depends on the trajectory of the detector . 
 inertial and non - inertial motion of the device have qualitatively different impact on the performance of the witness .