Gaussification through decoherence
We investigate the loss of nonclassicality and non-Gaussianity of a single-mode state of the radiation field in contact with a thermal reservoir. The damped density matrix for a Fock-diagonal input is written using the Weyl expansion of the density operator. Analysis of the evolution of the quasiprobability densities reveals the existence of two successive characteristic times of the reservoir which are sufficient to assure the positivity of the Wigner function and, respectively, of the representation. We examine the time evolution of non-Gaussianity using three recently introduced distance-type measures. They are based on the Hilbert-Schmidt metric, the relative entropy, and the Bures metric. Specifically, for an -photon-added thermal state, we obtain a compact analytic formula of the time-dependent density matrix that is used to evaluate and compare the three non-Gaussianity measures. We find a good consistency of these measures on the sets of damped states. The explicit damped quasiprobability densities are shown to support our general findings regarding the loss of negativities of Wigner and functions during decoherence. Finally, we point out that Gaussification of the attenuated field mode is accompanied by a nonmonotonic evolution of the von Neumann entropy of its state conditioned by the initial value of the mean photon number.
pacs:03.67.-a, 42.50.Dv, 42.50.Ex, 03.65.Yz
In quantum optics, a nonclassical state is defined as having either a negative or a highly singular representation (more singular than Dirac’s ). Otherwise, we term it as a classical state. Cahill (1) and later on Hillery (2) proved that the only pure states that are classical are the coherent ones: all other classical states are mixtures. Accordingly, a pure state whose Wigner function has also negative values is neither classical nor Gaussian. On the other hand, Hudson’s theorem (3) implies that the Wigner function of a non-Gaussian pure state is not pointwise non-negative. Thus, for pure states, negativity of the Wigner function could be interpreted as an indicator of nonclassicality as prominent as the negativity of the representation. Moreover, signatures of nonclassicality were identified by means of the negativity of the Wigner function for some mixed non-Gaussian states as well (4). Interest in the non-Gaussian states has interestingly emerged in quantum information processing. Very recently it was shown that continuous-variable states and operations represented by non-negative (Gaussian or non-Gaussian) Wigner functions can be efficiently simulated on a classical computer (5); (6). This result is a continuous-variable generalization of the Gottesman-Knill theorem for qubits (7). At the same time, the negativity of the discrete Wigner function defined for odd-dimensional discrete-variable systems was identified to be a quantum computational resource (8); (5). This means that the states with negative Wigner functions cannot be efficiently simulated on a classical computer or, otherwise said, a quantum computer outperforms a classical one. This property was extended to the infinite-dimensional case (5) and examples of useful resource states for quantum computation were given (6). Some time ago, it was also realized that non-Gaussian resources and operations could be more efficient in some quantum protocols such as teleportation (9); (10); (11) and cloning (12) or even indispensable in entanglement distillation (13). To quantify the non-Gaussianity as a resource in such cases, some distance-type measures of this property were proposed (14); (15). However, in order to avoid the complications encountered in extremization procedures of the previously defined distance-type degrees of entanglement (16), another pattern was followed in the case of non-Gaussianity degrees (14); (15). Let us denote by the Gaussian state having the same average displacement and covariance matrix as the given state . This state was reasonably used as a reference one in defining a degree of non-Gaussianity as the distance between the given state and the Gaussian state . In Refs.(14); (15); (17), Genoni et al. employed the Hilbert-Schmidt metric to define a corresponding non-Gaussianity measure,
and the relative entropy to introduce the entropic measure of non-Gaussianity:
On the one hand, Eq. (1) gives an easily computable expression. On the other hand, despite its not being a true distance, the relative entropy is acceptable and used as a measure of distinguishability between two quantum states. It is remarkable that the relative entropy (2) reduces to a difference of von Neumann entropies:
where is the von Neumann entropy of the state . General properties of the non-Gaussianity measures (1) and (3) were discussed in detail in Refs.(14); (15); (17). Another approach to quantifying non-Gaussianity is based on the function and leads to a measure expressed by the difference between the Wehrl entropies of the Gaussian state and the given non-Gaussian state (18). It is also worth noting that the non-Gaussianity measures (14); (15) were used to check on the relation between non-Gaussianity and the possibility of extending Hudson’s theorem to mixed states (19); (20). In the two-mode case, the relation between non-Gaussianity and entanglement both quantified by the relative entropy was analyzed for photon-added and photon-subtracted two-mode squeezed vacuum states (21). From the experimental part, non-Gaussianity in terms of relative entropy (15) was measured for single-photon-added coherent states (22). An experimental study on the same measure was recently reported for phase-averaged coherent states (23).
The aim of the present paper is threefold. First, we reinforce the Bures distance to the associate Gaussian state as a measure of non-Gaussianity of an arbitrary one-mode state . We have recently introduced it on general grounds in Ref.(24) by using this well-known metric related to the fidelity between two quantum states (25). Our motivation to use this measure was given by the valuable distinguishability properties of the fidelity described in Refs.(26); (27). Application of this measure for -photon-added thermal states showed us its good agreement with previously defined distance-type measures (14); (15); (17). Note that fidelity-based metrics have proven to be fruitful in quantum optics and quantum information as measures of nonclassicality (28) and entanglement (29); (30); (31). Second, we use the three above-mentioned distance-type measures to investigate the loss of non-Gaussianity for -photon-added thermal states of a field coupled to a heat bath. To this end, we give a new derivation of the Fock-basis solution of the quantum optical master equation by employing very conveniently the Weyl expansion of the density operator. Third, we write the damped quasiprobability densities and examine their evolution towards the equilibrium thermal state imposed by the reservoir. This evolution is marked by the loss of negativity of both Wigner and functions. Collaterally, we notice the nonmonotonic evolution of the von Neumann entropy during decoherence as depending on the mean photon number of the input state.
The plan of the paper is as follows. In Sec. II we recapitulate the principal features of the Bures degree of non-Gaussianity arising from some general properties of the fidelity. We here concentrate on an easily workable case, namely, that of mixed states having diagonal density matrix in the Fock basis. A comparison between various degrees of non-Gaussianity is further intended. Therefore, we consider in Sec. III a set of states generated as solutions of the quantum optical master equation (32). The density matrix of the damped field mode and the quasiprobability densities at the moment are derived for an arbitrary input state which is diagonal in the Fock basis. The evolution of the quasiprobability densities allows us the formulation of some general properties. Section IV is dedicated to a specific damped state which is important in applications: An -photon-added thermal state evolving under the influence of a thermal reservoir. With this explicit input density matrix at the moment and exploiting some results of Sec. III, we here find the damped quasiprobability distributions and density matrix in a closed analytic form. Further, the time evolutions of the Hilbert-Schmidt, entropic, and Bures measures of non-Gaussianity are analyzed and compared. Special attention is paid to the entropy production as another tool to examine Gaussification of single-mode states due to the field-reservoir interaction. Section V summarizes our results focused on the consistency of the analyzed measures of non-Gaussianity and the principal features of nonclassicality decay considered in the present paper.
Ii Bures measure of non-Gaussianity
Let us consider two arbitrary states, and , of a given quantum system. According to Uhlmann, when both states are mixed, a good measure of the closeness between their properties is the maximal quantum-mechanical transition probability between their purifications in an enlarged Hilbert space (25); (26). This is an extended notion of transition probability between quantum states which is now called fidelity (26) and has the intrinsic expression (25)
In Ref.(24) we have defined a fidelity-based degree of non-Gaussianity
As seen in Eq. (5), the fidelity is tightly related to the Bures metric introduced in Ref.(33) on mathematical grounds. Several general properties of fidelity are listed below together with the features they transfer to our definition (5).
(P1) and if and only if . This property implies
(P3) When at least one of the states is pure, then the inequality is saturated: Accordingly, for any pure state Eq. (5) reads
Unlike , which has the same value for all pure states possessing the same covariance matrix, is state dependent.
holds, and therefore
It follows that, in the one-mode case, does not depend on one-mode squeezing and displacement operations.
(P5) (monotonicity under any trace-preserving, convex-linear, and completely positive map ). This means that fidelity does not decrease strictly under any trace-preserving quantum operation, including an arbitrary nonselective measurement. Therefore, by virtue of definition (5), the Bures degree of non-Gaussianity does not increase under any such quantum operation.
(P6) (multiplicativity). Let us consider a bipartite product state . If is a Gaussian state, we get
(P7) For commuting density operators, Eq. (4) simplifies to
Properties (6), (9), and (10) of are shared by the non-Gaussianity measures (1) and (3) as well (17). In general, the Hilbert-Schmidt measure (1) is easier to compute than or and one could wonder why these last two were in fact considered? The answer is given by the monotonicity property under any trace-preserving quantum operation shared by the relative entropy and the fidelity, unlike the Hilbert-Schmidt distance (27). One of our goals in this work is to notice the role the monotonicity plays in the evolution of non-Gaussianity under damping (see Sec. IV).
In the following we concentrate on an interesting computable case. The Gaussian reference state of a Fock-diagonal non-Gaussian one,
is a thermal state with the same mean photon occupancy :
where is the purity of the state , and
Here we have employed the von Neumann entropy of a Gaussian state. In this special case, we notice the commutation relation , which allows one the use of Eq. (11) to get
An important example of states having the density operator of the form (12) is the class of phase-averaged (or phase-randomized) coherent states. A complete theoretical and experimental characterization of such states including measurements of the degrees of non-Gaussianity (LABEL:hs2), (LABEL:re2), and (16) was recently reported in Ref.(34). Note finally that various excitations on a thermal state of the type have the density operator of the type (12). Here and are the amplitude operators of the field mode.
Iii Master equation of one-mode field damping
Loss of nonclassical properties of the single-mode field states (Gaussian and non-Gaussian as well) during the interaction with a dissipative environment has been intensively studied in the last decades (35); (36); (37); (38); (39); (40). Quite recently, more general properties of damped non-Gaussian states such as the evolution of mixing measured by linear-entropy production (41); (42); (43), or the evolution of various measures of nonclassicality (44) were investigated by employing the quantum optical master equation in the interaction picture (32):
In Eq. (17), is the reduced density operator of the field, is the coupling constant between field and bath, and stands for the mean occupancy of the reservoir. This popular master equation is of the Lindblad type and thus preserves both the positivity and normalization of the density operator. It has the clear physical significance of describing decoherence of a field mode coupled to a heat bath. For a comprehensive list of references regarding solutions arising from Eq. (17) for the expectation values of field operators, characteristic functions, and quasiprobability densities, we refer the reader to some recent work of Dodonov (45); (46).
iii.1 Damped density matrix
We here intend to write the evolving damped-mode state generated by this master equation from an input non-Gaussian one. Then we will compare the time evolution of the above-discussed degrees of non-Gaussianity. To go on this programme, we need a workable solution of Eq. (17) in the Fock basis. In what follows our main tool is the Weyl expansion of the density operator:
In Eq. (18), is the characteristic function (CF) of the state , defined as the expectation value of the displacement operator ,
The mean photon number in the damped mode is
where denotes the thermal mean occupancy in the field mode at time and is the initial mean photon number.
We expect that the damping master equation (17), which is phase insensitive, preserves the diagonal form of an evolving state whose initial density matrix is Fock-diagonal. Indeed, the CF of an input Fock-diagonal state is
where is the photon-number distribution and is a Laguerre polynomial of degree . Equation (20) gives further
The density matrix of the damped field state is obtained from the Weyl expansion (18) after here inserting the CF (23). Use is also made of the matrix elements of the displacement operator in the Fock basis. After a simple calculation using the polar coordinates and an obvious change of variable in the integral (18) we are left with the following series expansion:
The integral in the above equation can be routinely performed (48) yielding the formula
where is a Gauss hypergeometric function, Eq. (50). Equation (25) is the general solution for the damped density matrix as function of the input one in the Fock-diagonal case. Let us take the limit in Eq. (25). The result is a thermal state with the Bose-Einstein mean occupancy . We thus deal with a Gaussification process, namely, evolution under this master equation eventually destroys the non-Gaussianity and also the nonclassicality properties of any input state. A special case of Eq. (25) arises for the thermal contact with a zero-temperature bath (), i.e., for the field coupling to the vacuum. The evolving state (25) then becomes:
Equation (26) describes both dissipation by contact with a zero-temperature reservoir and photon counting for which the exponential should be simply replaced by the quantum efficiency of the detector (38).
iii.2 Damped quasiprobability densities
We recall the most useful -ordered CFs (49):
Their corresponding normalized Fourier transforms
can be interpreted as quasiprobability densities and are important tools for understanding the nonclassical features of quantum states. Specifically, they are: for , (Glauber’s function), , (Wigner function), and , (Husimi function).
We first perform the integration over the polar angle, then we are left with a known integral over whose evaluation (48) leads us to the following double summation:
In Eq. (30) we have introduced a bath-dependent parameter whose relevance for the evolution of the quasiprobability densities will arise further:
Now, the sum over is of the type (52) and leads to a rather simple result:
Any Laguerre polynomial displays, via Eqs. (53) and (51), its positivity for negative values of the argument . Equation (32) presents two main advantages. First, it makes possible a simultaneous evaluation of all quasiprobability densities . Second, its structure implies a general statement regarding the positivity of the quasiprobability densities independent of the initial state . Indeed, at any all the quasiprobability densities [Eq. (32)] are positive for positive . This allows us to write the ultimate times at which any quasiprobability distribution displaying initially some negativies becomes positive due to the field interaction with the thermal bath.
(i) . As expected from its definition as the average value of the density operator in a coherent state, the function is always positive.
(ii) is positive for
This is the time at which the Wigner function completely loses any negativity.
(iii) is positive for
Beyond the time , the Glauber-Sudarshan representation exists as a genuine probability density. The threshold times (33) and (34) are constants of the bath valid for any input Fock-diagonal state. Note also that , that is, the Wigner function is more fragile than the function under damping. Let us now remark that the characteristic time (34) was previously obtained for particular input states such as even coherent states in Refs.(43); (44) and Fock states in (42). The limit time (33) was found in Refs.(42); (43) in describing the mixing process during the damping of an input pure state. As an onset of positivity of the Wigner function for damped photon-added thermal states, the time was written in Refs.(50); (51). In the case of a zero-temperature reservoir, , we get an unique state-independent time for disappearance of any negativity of the Wigner function, while, according to Eq. (34), the representation could have negative domains at any time.
What happens at earlier times ? We cannot make general predictions using our Eq. (32). The behavior of the distribution functions and will depend on the initial state .
Iv A case study: damped photon-added thermal states
According to a pertinent remark of Lee (52), adding photons to a classical state implies removal of the vacuum state from its Fock expansion. A nonclassical output is obtained in every case. Addition of photons to a classical Gaussian state, such as a coherent or a thermal one, generates a non-classical output which is no longer Gaussian. We can say that photon-added states are both non-classical and non-Gaussian. An -photon-added thermal state (53); (54) is an interesting example of a Fock-diagonal state whose non-classicality was recently investigated experimentally (55); (56); (57). The density operator of the state obtained by adding photons to a thermal one is
The spectral decomposition of a given thermal state is of the type (13), with a mean photon number denoted . The nonclassicality of the state (35) was first discussed by Agarwal and Tara (53) in terms of its nonpositive representation and Mandel’s -parameter. The non-Gaussianity of the state (35) was recently evaluated in Ref.(18) by employing the Wehrl-entropy measure and found to coincide with the non-Gaussianity of the number state , being thus independent of the thermal mean occupancy . We employ here Eq. (35) to write down the density matrix:
The photon-number distribution of the associate thermal state
is obtained by using the shared mean photon occupancy . The non-Gaussianity degrees (LABEL:hs2), (LABEL:re2), and (16) for the state (35) were carefully analyzed in our work (24). We have there adopted as a reasonable criterion for their appropriateness a monotonic behavior with respect to the average photon number of the state. In the case of Hilbert-Schmidt degree (LABEL:hs2) we were able to give an analytic result, while for the fidelity-based degree and the relative-entropy measure we performed numerical evaluations. We found that they depend on the thermal mean occupancy unlike the Wehrl-entropy measure (18). Our evaluations have also shown a consistent relation between these three non-Gaussianity measures. Their behavior under damping will be analyzed in the following by extending the results reported in Ref.(24).
iv.1 Decay of negativities
The damped photon-added thermal states offer a good test bed to examine the evolution of the initial quasiprobability distributions with negativities towards positive representations following our general findings in Sec. III B. We evaluate first the time development of quasiprobability densities by inserting the density matrix (36) into Eq. (32). By applying Eq. (58) we finally get after minor rearrangements:
where we have denoted
We can see that the distributions and have a similar functional form which guarantees the existence of some negativity regions. Indeed, the Laguerre polynomial has precisely distinct positive roots. According to Eq. (53), , so that for . On the contrary, a polynomial is negative on a finite number of subintervals of : this number is either , if is odd, or , if is even. Therefore, a product of the type , occurring in both expressions (41) and (42) presents negative values. Hence the -photon-added thermal states are nonclassical for any values of the parameters and .
As regards the distributions of the damped field, we remark that the state-dependent quantity is positive at any time. Therefore, the discussion on the characteristic times at which the distributions and lose their initial negativities as presented in Sec. III is equally valid here. There is no alteration of the limit times [Eq. (34)] and [Eq. (33)] produced by the specific input state [Eq. (35)] we are dealing with. The photon-added thermal states with their limit case , that is, a number state , represent a perfect example of nonclassicality lost during Gaussification by damping. They develop in time through three distinct stages (regimes) of decoherence which can be related to their efficiency in continuous-variable quantum computation (5); (6); (8) as follows.
(1) For , the damped states are nonclassical from both points of view of quantum optics (negative function) and quantum information processing (negative function) and could be used for quantum computational speed-up.
(2) For , the damped states have positive Wigner function, while their representation still displays negativities (region of bound universal states (8)). According to the recent results in Refs.(5); (6); (8), such states admit an efficient classical simulation due to the positivity of their Wigner distribution, in spite of not having a well-behaved representation.
(3) For , the damped states are classical and can be efficiently simulated on a classical computer.
The discussion simplifies when dealing with a zero-temperature bath. We have and . This case was recently tackled in Ref.(6) for . The fidelity between the damped states and the vacuum was there considered as a measure of the closeness to the convex hull of Gaussian states. This was found to be small for , namely, for the set of bound universal states created by dissipation. We shall further reconsider this problem for an arbitrary thermal reservoir by using the more rigourous measures of non-Gaussianity presented in Secs. I and II.
iv.2 Measures of non-Gaussianity
Insertion of the input photon-number distribution (36) into Eq. (25) gives the Fock-diagonal density matrix of a damped photon-added thermal state. Technically we have to evaluate the sum of a power series involving Gauss hypergeometric functions which is discussed in the Appendix. We apply Eq. (57) and get
The reference Gaussian state associated with this density matrix is a thermal one whose mean occupancy equals the average photon number of the damped field state. According to Eq. (21), this is
We have used the time-dependent density matrix (LABEL:dpa) and the mean occupancy (44) to evaluate numerically two of the distance-type measures of non-Gaussianity we are interested in: and via Eqs. (16) and (LABEL:re2), respectively. Our results are displayed in Figs. 1 and 2, where the time evolutions of these non-Gaussianity measures are presented for weak damping () and for a large thermal mean occupancy of the reservoir (), respectively. On both figures we have marked the three regions delimitated by the bath parameters and . In Fig. 1 the two regions of non-classicality are large enough ( and ). They contain states with higher non-Gaussianity, while the third (classical) region displays states slowly evolving to the Gaussian equilibrium state of the reservoir. Figure 2 shows that in a noisy bath the nonclassicality rapidly dissappears ( and ), while non-Gaussianity is still large at the classicality threshold . The functions and are monotonically decreasing, in accordance with the corresponding property discussed in Sec. II. Both Figs. 1 and 2 display a very good agreement between the two measures. They show consistency by inducing the same ordering of non-Gaussianity when considering specific sets of damped states continuously generated by the field-reservoir interaction.
The last degree of non-Gaussianity we are interested in is evaluated using Eq. (LABEL:hs2). Fortunately, as in the input-state case (24), we are able to give nice analytic expressions for both the scalar product and the purity of the damped state. We first write
where the expectation value is given in Eq. (44). By inserting Eq. (LABEL:dpa) for the damped density matrix into Eq. (45), we eventually get a sum of hypergeometric polynomials of the type (55). After a little algebra we obtain the formula
Now, the purity of the damped state,
is proportional to the sum of a power series whose coefficients are squared Gauss hypergeometric polynomials. We apply the summation formula (60) and find after some minor rearrangements:
Note that we can express the right-hand side of Eq. (LABEL:purity1) in terms of a Legendre polynomial via Eq. (54). The equilibrium values of both the scalar product (46) and the purity (LABEL:purity1) are equal to the purity of the thermal state eventually imposed by the reservoir, namely, . The interesting case of a number state is given by setting in the above formulas. To compute the Hilbert-Schmidt degree of non-Gaussianity, we have to substitute into Eq. (LABEL:hs2) the overlap (46), the purity (LABEL:purity1) of the damped state, and the purity of the reference Gaussian state,
In Fig. 3 we point out the time evolution of the Hilbert-Schmidt degree of non-Gaussianity for the same sets of states as in Figs. 1 and 2.
The agreement between the Hilbert-Schmidt degree of non-Gaussianity and the other two distance-type measures we discuss here is more accurate during the interaction with noisier reservoirs, as shown in Figs. 2 and 3. All plots showing the time development of and in Figs. 1 and 2 have a similar aspect which indicates a good consistency of these two measures coming from their monotonicity under the damping map. We expect they induce the same ordering of non-Gaussianity. This property is not shared by the Hilbert-Schmidt degree of non-Gaussianity and we have to remark the different concavity of the plots in the left-hand side of Fig. 3. For weak damping this measure could not induce the same ordering of the non-Gaussianity as the relative entropy or Bures distance. In Fig.4 (left) we have plotted the entropic measure as a function of the fidelity-based degree for the set of damped states with and different values of . The two measures show here a good consistency by preserving the same ordering of non-Gaussianity. Also plotted in Fig.4 is as a function of and for the same sets of states as in the left panel.
We notice that both of them have the same concavity but they lack consistency for some values of the parameters. Indeed, as shown by Figs. 1-3, at a given time the all three measures are increasing with the number . This is fairly displayed by the left plot in Fig.4 but not entirely by the other two. We see that the non-Gaussianity of the states with appears to be larger than in the case of .
iv.3 Gaussification and entropy production
The monotonic decay of all non-Gaussianity degrees under damping does not tell us too much about the loss of nonclassicality of the states (35). Decreasing of negativity regions of quasiprobability densities seems to be a faithful indicator of nonclassicality decay as shown on very general grounds in Sec. III. Some time ago, another interesting evolution was found for two functions related to the von Neumann entropy, namely, the linear entropy and the 2-entropy (42); (43). In Ref.(38) it was shown that a nonclassical Gaussian state under damping displays a maximum in its linear entropy provided that its initial mean occupancy exceeds the mean occupancy of the reservoir . See also a more recent analysis in Ref.(58). For some pure nonclassical input states, a number state in Ref.(42) and an even coherent state in Ref.(43), it was found analytically that a maximum in the evolution of the -entropy exists under the same mean-occupancy condition. It appears to us that the existence of a maximum in the evolution of the entropy under damping is primarily conditioned by the nonclassicality (expressed in terms of a not well-behaved representation) of the input state. For instance, the entropy of a classical Gaussian state does not exhibit a maximum under damping in any case. But this statement was not proven for mixed non-Gaussian states so far.
The mixed states we are dealing with in this paper are both nonclassical and non-Gaussian. We thus find it interesting to check on their entropic evolution under damping because we can take advantage of having evaluated the von Neumann entropy for both the time-dependent state and the reference Gaussian one . In the latter case, the field-reservoir interaction preserves the feature of being a thermal state. Its mean photon number [Eq. (44)] varies monotonically from the input value to the reservoir mean occupancy . Therefore, the von Neumann entropy is a monotonic function of time between the corresponding imposed limits at and . Good examples of this monotonic evolution are shown on the left-hand side of Fig.5 for a noisy reservoir with .
The input mean occupancy for the states with and various are placed in Table I together with the corresponding values of the entropy for the reference Gaussian states and the given ones . To calculate the latter ones, we made use of the density matrix (LABEL:dpa). Notice that in all the chosen cases the entropy of the input is smaller than that of the reservoir. The von Neumann entropy of the damped state is plotted on the right-hand side of Fig. 5 for the same parameters.
By contrast to its monotonic evolution for (cases ), we point out the existence of a short-time maximum of the von Neumann entropy when (cases