arXiv:1709.05445v1 [quant-ph] 16 Sep 2017
Quantum coherence and nonclassical
correlations of photonic qubits carrying
orbital angular momentum through
atmospheric turbulence
Mei-Song Wei1, Jicheng Wang1, Yixin Zhang1, Qi-Liang He2and Zheng-Da Hu1,∗
1Jiangsu Provincial Research Center of Light Industrial Optoelectronic Engineering and Technology, School of Science, Jiangnan University, Wuxi 214122, China
2School of Physics and Electronics, Guizhou Normal University, Guiyang 550001, China
Abstract: We investigate the decay properties of the quantum coherence and nonclassical correlations of two photonic qubits, which are partially entangled in their orbital angular momenta, through Kolmogorov turbulent atmosphere. It is found that the decay of quantum coherence and quantum discord may be qualitatively different from that of entanglement when the initial state of two photons is not maximally entangled. We derive two universal decay laws for quantum coherence and quantum discord, respectively, and show that the decay of quantum coherence is more robust than nonclassical correlations.
© 2017 Optical Society of America
OCIS codes: (010.1300) Atmospheric propagation; (010.1330) Atmospheric turbulence; (270.5585) Quantum information and processing.
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1. Introduction
the turbulent atmosphere, which may lead to random phase aberrations on a propagating optical beam [9]. A large amount of efforts, both theoretical [10–14] and experimental [15–18], have been devoted to exploring the impacts of atmospheric turbulence on the propagation of photons carrying OAM and protecting OAM photons from decoherence in turbulent atmosphere.
In QIS, it is typically critical to understand the behaviors of nonclassically correlated photons traveling in the turbulent atmosphere since the quantumness contained in the encoded states are usually fragile and can be easily destroyed. Quantum entanglement is a fundamental quantum resource in QIS which has been insensitively studied [19]. Recently, the entanglement decay of photonic OAM qubit states in turbulent atmosphere has been reported [20–22], which only focuses on the ideal case of Bell state with maximal entanglement. However, entanglement may not be the unique resource which can be utilized in QIS. There exist other resources such as quantum discord [23] and quantum coherence [24,25] attracting much attention. Recently, a measure of nonclassical correlations, termed as local quantum uncertainty (LQU), has been proposed as a genuine measure of quantum discord and is exactly computable for any bipartite quantum state [26]. The LQU is defined as quantification of the minimal quantum uncertainty achievable on a single local measurement, based on the skew information [27]. Following to the ideas of LQU, the authors in Ref. [26] put forward an observable measure for quantum coherence, which is termed as local quantum coherence (LQC) [25]. This measure of quantum coherence is defined by the skew information, which is experimentally friendly and depen-dent on the local observableKmeasured. It may be interesting to study the effect of turbulent atmosphere on these resources other than entanglement.
In this paper, we investigate the effects of atmospheric turbulence on the LQC and LQU of two photonic qubits of partially entangled in their OAM. The evolutions for LQC and LQU of the two-qubit state are discussed by introducing the the phase correlation length of an OAM beam, which may be qualitatively different from that of entanglement when the initial state of two photons is not maximally entangled. We derive two universal decay laws for quantum coherence and quantum discord, respectively, and show that the decay of quantum coherence is more robust than nonclassical correlations.
This paper is organized as follows. In Sec.2, we derive the photonic OAM state influenced by the turbulent atmosphere and discuss the evolution of the entanglement of the initial extended Werner-like state. In Sec.3, the evolutions of the LQU and LQC as well as their decay laws are discussed. Conclusions are presented in Sec.4.
2. Partially entangled OAM state through atmospheric turbulence
In this work, we use two Laguerre-Gaussian (LG) beams [28] to generate a twin-photon state [8]. As is shown in Fig.1, two correlated LG beams, generated by the source, propa-gate through the turbulent atmosphere and then are received by the two detectors. The beams carry pairs of photons that may not be maximally entangled in their OAM due to impurity and imperfection, and the non-maximal entanglement is encoded by LG modes with the opposite azimuthal quantum numberl[8].
We assume that the input LG modes have a beam waistω0, a radial quantum numberp0=0,
and azimuthal quantum numbersl0and−l0. The photon pair is initially prepared in an extended
Werner-like state defined as
ρ(0)=1−γ
4 I+γ|Ψ0ihΨ0|, (1) where 0≤γ≤1 denotes the purity of the initial state and|Ψ0iis the Bell-like state given by
|Ψ0i=cos
θ
2
|l0,−l0i+eiφsin
θ
2
Fig. 1. A source produces pairs of OAM-entangled photons propagating through turbulent atmosphere and received by two detectors.
with 0≤θ≤π and 0≤φ≤2π. It is worth noting that the quantum state (1) recovers the Bell state considered in Ref. [29] for purityγ=1 andθ=π/2. The extended Werner-like states play an important role in many applications of QIS [30,31].
The action of the turbulent atmosphere on the photons can be treated as a linear mapΛ, in terms of which the received state at the detectors reads as
ρ= (Λ1⊗Λ2)ρ(0), (3)
whereΛ1andΛ2are the actions of the atmospheric turbulence on the individual photon state.
The density matrix of the photonic state (3) can be constructed as [32]
ρ=
∑
ii′,j j′
ρ|i jihi′j′||i jii′j′
=
∑
ii′,j j′
ρii′,j j′|i ji
i′j′
=
∑
ii′,j j′
∑
ll′,mm′
Λll′
1ii′Λmm ′
2j j′ρll(0′),mm′|i ji
i′j′
=
∑
ii′,j j′ll′
∑
,mm′Λll′
1ii′Λmm ′
2j j′ρ|(lm0)ihl′m′||i ji
i′j′. (4)
Here, we letΛ1=Λ2=Λdue to the same effect of turbulent atmosphere on the photons. The
elementsΛl0,l0′ l,l′ =∑pΛ
0l0,0l′0
pl,pl′ of the linear mapΛis given by [29]
Λl0,l0′ l,±l′ =
δl0−l′
0,l∓l 2π
Z ∞
0
drRpl0(r)R∗pl0(r)r
Z 2π
0
dϑe−iϑ[l±l−(l0+l0′)]/2e−0.5Dφ(2r|sin(ϑ/2)|), (5)
where
Rpl0(r) =2
p!
(|l0+p|) 1/2
1
ω0
r√2
ω0 !|l0|
L|l0| l
2r2 ω0
exp
−r
2
ω2 0
(6)
is the radial part of LG beam atz=0 [10] with generalized Laguerre polynomials
L|ll0|(x) =
p
∑
m=0
(−1)m (|l0|+p)! (p−m)!(|l0|+m)!m!
xm. (7)
Here, we considerDφ=6.88(r/r0)5/3, which is the phase structure function of the Kolmogorov
model of turbulence and
r0= 0.423C2nk2L
−3/5
is the Fried parameter, whereCn2is the index-of-refraction structure constant,Lis the propaga-tion distance, andkis the optical wave number [33].
The density matrix of the input state (1) in the basis{|l0,l0i,|l0,−l0i,|−l0,l0i,|−l0,−l0i}
can be written as an X form
ρ(0)=
According to Eq. (4), the density matrix of the output state (3) can also be expressed in the X form as
In this sense, the output OAM state can be treated as two-qubit.
Then, we can express the entanglement by Wootters’ concurrence [34] for the X state
From Eqs. (12) and (13), we can derive the analytical form for the concurrence as
C(ρ) =max
(
0, a
2sinθ−2ab
γ (a+b)2 −
1−γ
2
)
. (15)
For convenience, one can introduce the phase correlation lengthξ(l0)which is defined as the
average distance between the points in the LG beam cross-section that have a phase difference ofπ/2. The phase correlation length can be expressed as [29]
ξ(l0) =sin
π
2|l0|
ω
0
2
Γ(|l0|+3/2)
Γ(|l0|+1)
, (16)
withΓ(x)the Gamma function .
Then, we plot the concurrenceC(ρ)as a function of the ratiox=ξ(l0)/r0and the initial
state parameterθin Fig.2(a). We can see that the concurrence decays with the increase of the correlation length and decreases fast to zero with non-asymptotical vanishing, the phenomenon of which is termed as entanglement sudden death (ESD) [35,36]. Moreover, we plot the concur-rence as a function of the ratioξ(l0)/r0and the purity parameterγin Fig.2(b). It is found that
the concurrence is zero when the initial state is unentangled for 0≤γ≤1/(1+2 sinθ), since no entanglement can be created between the two photons under independent atmospheric tur-bulences for initially disentangled states. In the next part, we will investigate the properties of quantum coherence and quantum correlation via LQC and LQU, which may exhibit qualitative difference from that of entanglement via concurrence.
(a) (b)
Fig. 2. (a) Concurrence as function ofξ(l0)/r0 and θ. (b) Concurrence as function of ξ(l0)/r0 andγ. The parameters are chosen as p=0,l0=1,ω0=1, (a)γ=1 and (b) θ=π/2.
3. Quantum coherence and quantum correlation of OAM state in atmospheric turbu-lence
We can quantify the degree of quantum coherence and quantum correlation via LQC [25] and LQU [26], which are both defined by the skew information [27]. For a bipartite systemAB sub-ject to a local measurement on the subsystemA, quantum uncertainty will yield if the subsystem
Ahas quantum coherence. Then, the LQC of an bipartite quantum stateρABis given by [27]
LQC(ρAB) =F(ρAB,KA⊗IB) =− 1 2Tr
[√ρAB,KA⊗IB]2
whereKAis a local observable on systemAandIBis the identity operator of subsystemB. Here, we choose the local observableKAto be the third Pauli operatorσz.
Another measurement we use to express quantum correlation is LQU, which is the mini-mal quantum uncertainty achievable on a single local measurement. LQU is also the minimini-mal optimization of the LQC over all possible local observables, which can be written as
LQU(ρAB) =min {KA}
F(ρAB,KA⊗IB). (18)
The LQU has a simple expression for a 2×dquantum system as [26]
UA(ρAB) =1−λmax{WAB}, (19)
whereλmaxis the maximal eigenvalue of the 3×3 matrixWABwith the elements
(WAB)i j=Tr
√
ρAB(σiA⊗IB)√ρAB σjA⊗IB, (20)
whereσiA(i=x,y,z)are the Pauli matrixes of subsystemA.
Then we plot the LQC and LQU as functions ofξ(l0)/r0andθin Fig.3(a) and Fig.3(b). We
can see that the LQC and LQU decay fast with the increase of the ratioξ(l0)/r0whenθis close
toπ/2 and then decrease slowly in a non-vanishing manner even when theξ(l0)/r0is large
enough. Therefore, we cannot see the phenomenon of sudden vanishing as for the entanglement (ESD). The LQC and LQU as functions ofξ(l0)/r0andγ are also displayed in Fig.3(c) and
Fig.3(d). When the purityγ of the initial state is close to zero, the quantum coherence and quantum correlation of the two qubits are very weak but still be non-vanishing, which is quite different from the entanglement shown in Fig.2(b). Whenγ is close to 1, the LQC and LQU seem to decay in a nearly exponential manner with the increase of ξ(l0)/r0, which is also
contrast to that of entanglement with sudden vanishing in Fig.2(b).
In order to see the evolution in turbulent atmosphere more clearly, the LQC and LQU as functions of ξ(l0)/r0 for differentθ are also shown in Fig. 4. We can clearly see the LQC
and LQU decay more and more slowly with the increase of the ratioξ(l0)/r0and the speed
become almost the same whenθis close toπ/2. For certain initial state, for instanceθ=π/3 shown in Fig.4(b), the decay rate of LQU may be suddenly changed when theξ(l0)/r0is still
small, which is termed as the sudden change phenomenon [37,38] as for discord-like quantum correlations.
Moreover, we would like to explore the precise decay laws of quantum coherence and quan-tum correlation for different values of phase correlation length which is determined by the azimuthal quantum numberl0as shown in Eq. (16). The decays of LQC and LQC as functions
ofξ(l0)/r0for different values ofl0is shown in Fig.5(a) and Fig.5(b). It is shown that both
the LQC and LQU decay fastest atl0=1. When the azimuthal quantum numberl0increases,
the decays of LQC and LQU are slowed down and finally collapse onto two universal curves, i.e., LQC(ρ)≈ f(x) = 0.185
(a) (b)
(c) (d)
Fig. 3. (a,b) LQC and LQU as function ofξ(l0)/r0 andθ and (c,d) LQC and LQU as function ofξ(l0)/r0andγ. The parameters are chosen asp=0,l0=1,ω0=1, (a,b)γ=1 and (c,d)θ=π/2.
θ=π/6
θ=π/4
θ=π/3
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.0
0.2 0.4 0.6 0.8 1.0
ξ(l0)/r0
LQC
(a)
θ=π/6
θ=π/4
θ=π/3
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
0.0 0.2 0.4 0.6 0.8 1.0
ξ(l0)/r0
LQU
(b)
Fig. 4. (a) LQC as function ofξ(l0)/r0and (b) LQU as function ofξ(l0)/r0. The parame-ters are chosen asp=0,l0=1,ω0=1 andγ=1.
4. Conclusions
ƽƽƽ
OAM qubits, generated from a source, are initially prepared in an extended Werner-like state (partially entangled), the decay effects of the turbulent atmosphere are explored for the output state received by the detectors. It is shown that the concurrence, LQC and LQU all decay as the increase of the ratio of phase correlation length and Fried parameter but with different phenom-ena. The concurrence decays suddenly to zero with the so-called entanglement sudden death (ESD), while the LQC and LQU decay asymptotically. For certain initial state, the LQU may demonstrate an extra sudden change phenomenon when the ratio of phase correlation length to Fried parameter is not large. Moreover, we derive two precise decay laws of quantum co-herence and quantum correlation for different values of phase correlation length (the azimuthal quantum number). As the azimuthal quantum number becomes large, two different universal decay laws emerge for the LQC and LQU, respectively. The decay of LQU is universally in an exact exponential manner similar to that of entanglement already reported in Ref. [29] but with asymptotic vanishing. By contrast, the decay of LQC is merely polynomial, which illustrates that the LQC can be more robust against atmospheric turbulence.
Acknowledgments
