An Entropic Measure of Nonclassicality of Single Mode Quantum Light
Soumyakanti Bose S. N. Bose National Centre for Basic Sciences Block-JD, Sector-III, Salt Lake, Kolkata 700106
India.
(Dated: November 15, 2017)
Nonclassical states of a quantized light are described in terms of Glauber-Sudarshan P function which is not a genuine classical probability function. Despite several attempts, defining a uniform measure of nonclassicality (NC) for the single mode quantum states of light is yet an open task. Here, we propose a measure of NC of single mode quantum states of light, in terms of its Wehrl entropy, that require no numerical minimization. It exploits classical like Q distribution which could be calculated easily as well be retrieved experimentally in optical heterodyne detection. We show that, alongwith the simple states which are generated under single NC-inducing operations, for the broader class of states, generated under multiple NC-inducing operations, our measure quantifies the NC consistently, in contrast to the other measures. The work, presented in this paper, becomes important in describing NC of quantum processes including open quantum systems as well as in understanding the role of single mode NC as a resource in several information processing protocols.
Keywords: 03.67.-a, 42.50.-p

arXiv:1701.00064v2 [quant-ph] 11 Nov 2017

I. INTRODUCTION
Quantum states of light exhibit several intriguing features such as photon antibunching, sub-Poissonian distribution, oscillatory number distribution etc. [1]. States revealing such characters possess phase-space distributions beyond the scope of classical probability theory. For any quantum state of light , if the phase-space GlauberSudarshan P distribution [2] behaves like a classical probability distribution, i.e., positive semidefinite or singular no more than a delta function,  is said to be classical [3], otherwise nonclassical. These nonclassical states could be generated by various nonclassicality (NC)-inducing operations such as photon excitation [4], quadrature squeezing [5], kerr squeezing [6] etc., applied on the classical states.
Several attempts have been made to quantify [7­11] the NC of single mode quantum state of light, described in terms of the distance in Hilbert space [7, 8], phasespace sigularity/negativity [9, 10] as well as the normal ordered operator values [11]. Recently, people [12] have shown that the distance based [8] as well as the phasespace based [9, 10] measures fail to capture the NC of single mode quantum states, generated under multiple NC-inducing operations, reasonably. On the other hand, the operational approach by Gehrke et. al. [11], also finds all the photon number states maximally nonclassical and a squeezed state maximally nonclassical at a moderate squeezing strength. People have also proposed measure of single mode NC in terms of the entanglement monotones [13]. However, recent results also show that the single mode NC, defined in terms of entanglement monotones, depends upon the specific choice of entanglement potential [14]. Besides, a BS converts the NC of input states into output entanglement partially [15]. This necessitates the search for a consistent measure of NC of the single mode quantum optical states.

Here, we propose a measure of NC for a single mode quantum optical state in terms of its Wehrl entropy [16] exploiting the Q distribution [1], the 'classical' like distribution [17]. For Schrodinger kittens [18], we find that the odd superposition is more nonclassical than the even superposition. However, for macroscopic cat states, both the superposition states become equally nonclassical. In the case of single mode quantum states, generated under multiple NC-inducing operations, we obtain NC in line of the concerned BS generated entanglement [12]. On the other hand, in the case of mixed states, it is noteworthy that for squeezed thermal state, we detect NC for all non-vanishing squeezing strengths unlike reported earlier [13].
This paper is organized as follows. In Sec. II, we introduce the measure of NC of single mode quantum optical states. In Sec. III, we consider some generic examples of pure and mixed states of a quantized electromagnetic field. We discuss in Sec. IV the key results and relevance of this measure to the contemporary works. In Sec. V, we conclude with the relevance of the work and the future perspectives.

II. WEHRL ENTROPIC MEASURE OF NC AND ITS PROPERTIES

IIA. Measure of NC

We measure the NC of any quantum state nc as the absolute difference between its Wehrl entropy and that of its nearest classical state,

Nw(nc)

=

inf
cl

|Hw(nc)

-

Hw (cl )|,

(1)

where the infimum has been considered over the set of all classical states.

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Since the coherent states | are the only classical pure states [3], in the case of pure states, we consider the set of coherent states (| ) as the classical reference. It is well known that the minimum Wehrl entropy of any quantum state of light is unity [20] and it attains the minimum for a coherent state | . As a consequence, NC (Eq. 1) for all pure states reduces to,

Nw(| nc) = Hw(| nc) - 1

(2)

On the other hand, in any optical experiment with decoherence, quantum states, by loosing all the quantum correlations, end up at the thermal state th(n¯) which is a classical mixed state. Consequently, for mixed states, we consider the thermal states as the set of classical reference states. For a nonclassical mixed state nc, its nearest thermal state can be chosen by using the theorem 1.

Theorem 1 The nearest thermal state to any quantum
state nc is the thermal state present in the Gaussian counterpart of nc, gnc.
proof: It has been shown [21] that the minimal relative distance of any Gaussian state, G, from a nonGaussian state, , is the relative distance between  and its Gaussian counterpart, g, the Gaussian state formed with the first and second order moments same as the  itself. Speaking mathematically,

S(||G) = S(||g) + S(g||G)

(3)

yielding the minimal value of S(||G) (= S(||g)) with the choice of G = g. S(||) denotes the quan-
tum relative entropy between the states  and  [22]. Now, for  = nc and G = th, Eq. (3) becomes S(nc||th) = S(nc||th) + S(gnc||th) where the particular setting th = gnc is not permissible, in general. Consequently, infth S(nc||th) is obtained by minimizing S(gnc||th) over the set of all th.
S(gnc||th) can further be written as S(gnc||th) = -T r[(gnc - th) ln th] + S(th) - S(gnc). Since th = gnc, the minimization of S(gnc||th) essentially boils down to the minimization of S(th)-S(gnc). It is well known that any single mode Gaussian state so as the gnc can be written as a displaced squeezed thermal state [23] for which
the Von-Neumann entropy is given by the Von-Neumann
entropy of the corresponding thermal state [24]. As a consequence, infth S(th) - S(gnc) yields the thermal state same as that in the decomposition of gnc [23] which concludes the proof.
Furthermore, the Wehrl entropy of a thermal state
th(n) has an analytic form, Hw(th(n)) = 1 + log[n + 1].
Thus, we obtain NC (Eq. 1) of mixed states in the form,

Nw(nc) = |Hw(nc) - Hw(th(n))|

= |Hw(nc) - 1 - log[n + 1]|

(4)

where n denotes the thermal state in the decomposition
gnc [23]. Next, we illustrate our measure of NC for certain single mode pure and mixed quantum states of light.

IIB. Some Properties of Nw

A. Invariance under displacement :

Let's consider a nonclassical state light nc. Its NC

is given by Eq. 1. Under the action of a phase-space

displacement nc  nc = bution changes as Qnc ()

D()ncD(), its Q distri-



Q
nc

()

=

Qnc (

- ).

Since the displacement works as rigid translation in phase

Wehrl entropy remain unchanged under such transforma-

tion [16, 20], e.g. pure states | nc

Hw 

(nc |

)  Hw (nc) = Hw(nc), nc = D()| nc, Nw(|

for nc )

all 

Nw (| nc) = Nw(| nc).

In the case of a mixed state nc, nearest classical state

is the thermal part of its gaussian counterpart of nc. It

is well known that, the Von-Neumann entropy of a gaus-

sian state is independent of its displacement [24]. Conse-

quently, Nw(nc) remains invariant under displacement. Thus, in general, D() :    , Nw()  Nw () =
Nw().

B. Invariance under passive rotation :

Let's consider a passive rotation TU : nc  nc.

Under this transformation Q distribution changes as

Qnc ()



Q
nc

()

=

Qnc (U -1)

=

Qnc (),

where

U denotes a rotation in phase-space. Consequently,

Hw(nc) remains unchanged. As quite straightforwardly, for pure states Nw remains conserved under such transformation. On the other hand, under such rotation

in phase-space, Von-Neumann entropy does not change.

This leads to the fact that under such rotation, the choice

of nearest classical state doesn't change (the same way as

discussed in the case of "invariance under displacement")

resulting in the invariance of Nw(nc).

Hence, Nw() 

we can Nw ()

write, in = Nw(),

general, TU : nc  nc, where TU is a passive ro-

tation.

III. NC of Some Pure and Mixed Quantum States of Light
A. Examples of Pure States
As examples of nonclassical pure states of light, generated under single NC-inducing operation, we consider photon number state, quadrature squeezed coherent state and photon added coherent state. We also consider the superposition of coherent states as examples of nonclassical pure states obtained by quantum superposition. In this context, we further study the quantum optical states, generated under multiple NC-inducing operations, in particular, photon added squeezed vacuum state and squeezed number state.
Photon Number State : A photon number state |m is obtained by applying photon excitation ( am ) on
m!

3

NW
Nw

the vacuum. For |m , we obtain analytic expression for NC as,

Nw(|m ) = m + log m! - m(m + 1)

(5)

We plot the Nw(|m ) for different values of m in Fig. 1(a). For small m( 4) we observe a rapid increase in
Nw(|m ). With further increase in m, Nw(|m ) saturates for very high m. As m increases, the monotonic increase
in Nw(|m ) falls in line of the increasing negativity in the Wigner distribution [10].

1.5
1
0.5 (a)
5 10 15 m

0.4

0.3

0.2

0.1 (b)

0.4

0.8

r

FIG. 1: (Color Online) Plot of Nw for a: Photon number state and b: Squeezed coherent state.

Squeezed Coherent State : A squeezed coherent

state, |sc = S()| , is generated under quadrature

squeezing

(S()

=

exp(

 a2 -  a2 2

)),

applied

on

a

coherent

state | , where  = rei; r and  being the squeezing

strength and the squeezing angle respectively. For |sc we obtain a logarithmic NC as

Nw(|sc ) = log µr

(6)

where µr = cosh r. Evidently, the Nw(|sc ) is independent of  since it
only sets the direction of squeezing rather than the de-

gree of squeezing. Moreover, the Nw(|sc ) is independent of the coherent displacement , e.g., Nw(S()| ) =
Nw(S()|0 ). This is because, |sc could be written as |sc = S()| = D()S()|0 , where,  = µ - ei,  = sinh r and the Wehrl entropy is independent of dis-

placement. Fig. 1(b) shows the dependence of Nw(|sc ) upon r. We observe an initial slow and then rapid in-

crease in Nw(|sc ) with increase in r. However, for very high r it saturates asymptotically (not shown in the fig-

ure).

Photon Added Coherent State : An m-photon

added coherent state (PAC), |pac

=

1 Nm

am

|

,

is

ob-

tained by applying photon excitation on a coherent state,

where, Nm = m!Lm(-||2). For the sake of simplicity we consider real displacement, e.g.,  = R. In Fig. 2(a) we

plot the dependence of Nw(|pac ) on R for different m values. With increase in m, Nw(|pac ) increases monotonically while as R increases it decreases. For sufficiently

high R (>> 1), Nw(|pac ) becomes almost independent of m.

Coherent Superposition States : We further study

the even (|+ ) an the odd (|- ) superposition of co-

herent states |±

= 1

(| ± | -  ).

2(1±e-2||2 )

For

1.2

0.6

0.8

0.4

0.4

(a)

(b) 0.2

0.8 1.6 2.4 R

0.8 1.6 2.4 R

FIG. 2: (Color Online) Plot of dependence of NC on R for (a) |pac for m = 1 (black solid line), 2 (yellow dashed line), 3 (green dotted line), 4 (blue dashed dotted line) and 5 (red dashed double dotted line) and (b) |± with |+ (black solid line) and |- (red dashed line).

the sake of simplicity we consider real displacement,

e.g.,  = R. We show the dependence of NC on R

for |± in Fig. 2(b). It is noteworthy that for small R( 1.0), |- is more nonclassical than |+ ; however,

for large R( 1.5) both |± contains equal amount of NC. This, we expect at high R due to the increase in

the distance between the coherent amplitudes that ef-

fectively leads to the same mixed state superposition

(limr

±



1 2

(|R

R| + | - R

-R|).

Photon Added Squeezed Vacuum State and

Squeezed Number State: We have also considered the

single mode quantum optical states generated under suc-

cessive application of multiple NC-inducing operations,

in particular, photon excitation and quadrature squeez-

ing. The ordered application of these operations on vac-

uum lead to the states known as photon added squeezed

vacuum state (PAS) and squeezed number state (SNS).

These are given as,

|pas

=  1 amS(r)|0 Nm

|sns = S(r)|m ,

(7)

where, Nm = m!µmPn(µ), µ = cosh r and Pn(x) is the nth order Legendre polynomial.
In Fig. 3 we have plotted the dependence of Nw on r for PAS and SNS. In the case of PAS [Fig. 3(a)], we observe that Nw is non-monotonic on both r and m. For m = 1 it increases monotonically with r. However, for all m  2, as r increases it first decreases and then increases. In addition to that, we also notice that for a moderate r (0.30  r  0.60), Nw for higher m becomes smaller than the lower m. It becomes prominent with increase in m. For very high value of r ( 0.80), Nw becomes predominantly dependent on r.
In the case of SNS [Fig. 3(b)] we observe a monotonic dependence of Nw on both r and m. For m = 1, both SNS and PAS yield similar NC; however for m  2 they are different. This is due to the fact that aS(r)|0 = S(r)|1 and for m  2, states are very much different as discussed in [12].

4

Nw

1.2

1.4

1.1

0.8 0.8

(a)

(b)

0.25 0.5 0.75 r

0.25 0.5 0.75 r

FIG. 3: (Color Online) Plot of Nw vs r for m = 1 (black solid line), 2 (yellow dashed line), 3 (green dotted line), 4 (blue dashed dotted line) and 5 (red dashed double dotted line) for (a) PAS and (b) SNS.

B. Examples of Mixed States

We now test the validity our formalism for nonclassi-

cal mixed states, namely, photon excited and quadrature

squeezed thermal state th(n¯). Photon Added Thermal State : For

an m-photon added thermal state (m-PATS),

m-PATS

=

1 (1+n)m

m!

am

th

(n)am

,

its Gaussian

cNowu(ntmer-pPaArTt S) gm=-P|HATwS(mis-aPATthSe)r-maHlws(tagmte-PrAeTsSu)l|tiwnghicinh

is nothing but the non-Gaussianity (NG) of m-PATS

[25]. Furthermore, it has been shown that NG of

m-PATS is equal to the NG of |m [25]. Thus, we obtain an analytic form for NC of m-PATS as

Nw(m-PATS) = log[m + 1] - m - log[m!] + m(m + 1), (8)
where,  is the di-Gamma function.
In Fig. 4(a) we plot Nw(m-PATS) for different m. With increase in m, Nw(m-PATS) increases monotonically and saturates at very high m. It is noteworthy that,
Nw(m-PATS) is completely independent of thermal state parameter n.

Nw

0.9
0.6
0.3
(a)
5 10 15
m

0.4
r

0.3
0.2 0.1
(b)
0.8

FIG. 4: (Color Online) Plot of (a) Nw(m-PATS) with m and (b) Nw(ST) with r for n = 1 (black solid line), 2 (yellow dashed line), 3 (green dotted line), 4 (blue dashed dotted line) and 5 (red dashed double dotted line).

Squeezed Thermal State : A squeezed thermal state, ST = S(r)th(n)S(r), is a Gaussian mixed state for which the nearest thermal (classical) state has the
same average number of thermal photons as present is

ST. Consequently, Nw(ST) attains the analytic form,

Nw(ST)

=

|

1 2

ln[µ2(1

+

2n)

+

n2]

-

ln[1

+

n]|

(9)

As is explicit from the Eq. 9, in the lim r  0,
Nw(ST)  | ln[1 + n] - ln[1 + n]| = 0, describing the thermal state. On the other hand, in the lim n  0,
Nw(ST)  ln[µ] which indicates the NC of squeezed vacuum state (Eq. 6). In Fig. 4(b) we show the de-
pendence of Nw(ST) on r for several n. Evidently, Nw(ST) increases with increase in r as well as increasing n. We detect the effect of squeezing for all values of
r in contrast to [13] that reads ST nonclassical only if r  rc(= log[1 + 2n]).

IV. DISCUSSION
The Wehrl entropic measure of NC, for both pure and mixed states, requires no optimization over the set of classical states. In addition to that, it remains invariant under phase-space displacement and rotation. In the case of nonclassical states of light, generated under single NC-inducing operation, Nw quantifies the NC efficiently. It successfully distinguishes between the even and odd Schrodinger kittens (when coherent amplitude is small) as well as shows that both the states are maceoscopically equally nonclassical, irrespective of the parity, as observed in terms of the Wigner negativity [10]. Our measure of NC also sheds light on the relative competition between the NC-inducing operations in the case of quantum optical states which are generated under multiple NC-inducing operations as predicted in [12]. It consistently quantifies the non-monotonic NC for the PAS and the SNS.
In the case of photon excited thermal states, NC depends only on the number of photon excitation. On the other hand, for Gaussian squeezed mixed states, it depends on the average thermal photon. We show that, for ST, n~ = n as considered in [8].
In contrast to the use of phase-space singularity and negativity [9, 10, 26, 27], our measure of NC is defined in terms of the classical like distribution that can be easily computed as well as be retrieved experimentally in heterodyne detection [28]. Current formalism, can be extended to the finite dimensional quantum systems [29] alongwith macroscopic optomechanical systems [30] by using the description of Q-function in finite-dimension [31] and thus sets a framework for studying the convertion of NC into entanglement by the action of BS, in general context [32].

V. CONCLUSION
In summary, we define a measure of NC for single mode quantum optical states in terms of Wehrl entropy. We show that the our measure quantifies the NC of both

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pure and mixed quantum optical states, generated under the action of single as well as multiple NC-inducing operations, efficiently.
Present work plays an important role in analyzing the NC of quantum optical states under quantum processes [33, 34]. It also becomes important in understanding the quantum-classical transition in open quantum systems [35] alongwith the role of NC of quantum states in several information tasks processing such as entanglement distillation [36], entanglement distribution [37, 38], quantum computation[39, 40] etc.
Author acknowledges M. Sanjay Kumar and S. Dutta for numerous discussions and critical insight on improving the manuscript.
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