Quantum Physics authors titles recent submissions
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 He2 and
Zheng-Da Hu1,∗
1 Jiangsu
Provincial Research Center of Light Industrial Optoelectronic Engineering and
Technology, School of Science, Jiangnan University, Wuxi 214122, China
2 School
of Physics and Electronics, Guizhou Normal University, Guiyang 550001, China
∗
huyuanda1112@jiangnan.edu.cn
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 fact that photons can carry orbital angular momentum (OAM) to encode quantum states
makes them very useful for quantum information science (QIS) [1–8]. However, the decay of
quantumness will be unavoidable when the encoded photons with OAM transmit through turbulent atmosphere. Photons’ wave front will be distorted due to refractive index fluctuations of
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 dependent on the local observable K measured. 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, propagate 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 number l [8].
We assume that the input LG modes have a beam waist ω0 , a radial quantum number p0 = 0,
and azimuthal quantum numbers l0 and −l0 . The photon pair is initially prepared in an extended
Werner-like state defined as
1−γ
I + γ |Ψ0 i hΨ0 | ,
ρ (0) =
(1)
4
where 0 ≤ γ ≤ 1 denotes the purity of the initial state and |Ψ0 i is the Bell-like state given by
θ
θ
|Ψ0 i = cos
|l0 , −l0 i + eiφ sin
|−l0 , l0 i ,
(2)
2
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 purity γ = 1 and θ = 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 map Λ, in
terms of which the received state at the detectors reads as
ρ = (Λ1 ⊗ Λ2 ) ρ (0) ,
(3)
where Λ1 and Λ2 are 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 = ∑ ρ|i jihi′ j′ | |i ji i′ j′
ii′ , j j ′
=
′
′ ′
i j
′
(0)
′
′
(0)
ρii′ , j j′ |i ji
∑
′
′
ii , j j
=
∑ ′ Λll1ii′ Λmm
∑
2 j j ′ ρll ′ ,mm′ |i ji
′
′ ′
ii , j j ll ,mm
=
′ ′
i j
∑ ′ Λll1ii′ Λmm
∑
2 j j ′ ρ|lmihl ′ m′ | |i ji
′
′ ′
ii , j j ll ,mm
′ ′
i j .
(4)
Here, we let Λ1 = Λ2 = Λ due to the same effect of turbulent atmosphere on the photons. The
l ,l ′
0l ,0l ′
δl0 −l0′ ,l∓l Z
∞
0 0
elements Λl,l0 ′ 0 = ∑ p Λ pl,pl
′ of the linear map Λ is given by [29]
l0 ,l0′
Λl,±l
′
=
2π
0
drR pl0 (r) R∗pl0 (r) r
Z 2π
0
′
d ϑ e−iϑ [l±l−(l0 +l0 )]/2 e−0.5Dφ (2r|sin(ϑ /2)|) ,
(5)
where
p!
R pl0 (r) = 2
(|l0 + p|)
1/2
1
ω0
√ !|l0 |
2
2
r
r 2
|l | 2r
Ll 0
exp − 2
ω0
ω0
ω0
(6)
is the radial part of LG beam at z = 0 [10] with generalized Laguerre polynomials
|l |
Ll 0 (x) =
p
(|l0 | + p)!
∑ (−1)m (p − m)! (|l0 | + m)!m! xm .
(7)
m=0
Here, we consider Dφ = 6.88 (r/r0 )5/3 , which is the phase structure function of the Kolmogorov
model of turbulence and
−3/5
r0 = 0.423Cn2k2 L
(8)
is the Fried parameter, where Cn2 is the index-of-refraction structure constant, L is the propagation distance, and k is the optical wave number [33].
The density matrix of the input state (1) in the basis {|l0 , l0 i , |l0 , −l0 i , |−l0 , l0 i , |−l0 , −l0 i}
can be written as an X form
(0)
(0)
ρ11
ρ14
(0)
(0)
ρ22 ρ23
(9)
ρ (0) =
,
(0)
(0)
ρ32 ρ33
(0)
(0)
ρ41
ρ44
where
θ
1 − γ (0) 1 − γ
,
, ρ22 =
+ γ cos2
4
4
2
1−γ
1−γ
θ
(0)
(0)
ρ33 =
, ρ44 =
+ γ sin2
,
4
2
4
1 − γ γ −iφ
(0)
(0)
+ e sin θ ,
ρ14 =0, ρ23 =
4
2
(0)
(0)∗ (0)
(0)∗
ρ41 =ρ14 , ρ32 = ρ23 .
(0)
ρ11 =
(10)
According to Eq. (4), the density matrix of the output state (3) can also be expressed in the
X form as
ρ11
ρ14
ρ22 ρ23
,
ρ =
(11)
ρ32 ρ33
ρ41
ρ44
with
(0)
(0)
(0)
(0)
(0)
(0)
(0)
(0)
(0)
(0)
(0)
(0)
(0)
(0)
(0)
(0)
ρ11 =a2 ρ11 + abρ22 + abρ33 + b2 ρ44 ,
ρ22 =abρ11 + a2ρ22 + b2ρ33 + abρ44 ,
ρ33 =abρ11 + b2ρ22 + a2ρ33 + abρ44 ,
ρ44 =b2 ρ11 + abρ22 + abρ33 + a2 ρ44 ,
(0)
(0)
(0)
(0)
ρ14 =a2 ρ14 , ρ41 = a2 ρ41 ,
ρ23 =a2 ρ23 , ρ23 = a2 ρ32 ,
(12)
where
l ,l
−l ,−l
−l ,l
l ,−l
a =Λl00 ,l00 = Λ−l00 ,−l00 = Λ−l00 ,l00 = Λl00 ,−l00 ,
−l ,−l0
b =Λl0 ,l00
l ,l
0 0
.
= Λ−l
0 ,−l0
(13)
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
√
√
C (ρ ) = max {0, |ρ14 | − ρ22 ρ33 , |ρ23 | − ρ11 ρ44 } .
(14)
From Eqs. (12) and (13), we can derive the analytical form for the concurrence as
)
(
a2 sin θ − 2ab γ 1 − γ
.
C (ρ ) = max 0,
−
2
(a + b)2
(15)
For convenience, one can introduce the phase correlation length ξ (l0 ) which is defined as the
average distance between the points in the LG beam cross-section that have a phase difference
of π /2. The phase correlation length can be expressed as [29]
π
ω0 Γ (|l0 | + 3/2)
ξ (l0 ) = sin
,
(16)
2 |l0 | 2 Γ (|l0 | + 1)
with Γ (x) the Gamma function .
Then, we plot the concurrence C (ρ ) as a function of the ratio x = ξ (l0 ) /r0 and the initial
state parameter θ in 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 concurrence as a function of the ratio ξ (l0 ) /r0 and the purity parameter γ in Fig. 2(b). It is found that
the concurrence is zero when the initial state is unentangled for 0 ≤ γ ≤ 1/ (1 + 2 sin θ ), since
no entanglement can be created between the two photons under independent atmospheric turbulences 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 of ξ (l0 )/r0 and θ . (b) Concurrence as function of
ξ (l0 )/r0 and γ . 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 turbulence
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 system AB subject to a local measurement on the subsystem A, quantum uncertainty will yield if the subsystem
A has quantum coherence. Then, the LQC of an bipartite quantum state ρAB is given by [27]
1 √
LQC(ρAB ) = F (ρAB, KA ⊗ IB) = − Tr [ ρAB , KA ⊗ IB]2 ,
2
(17)
where KA is a local observable on system A and IB is the identity operator of subsystem B. Here,
we choose the local observable KA to be the third Pauli operator σ z .
Another measurement we use to express quantum correlation is LQU, which is the minimal quantum uncertainty achievable on a single local measurement. LQU is also the minimal
optimization of the LQC over all possible local observables, which can be written as
LQU(ρAB ) = min F (ρAB , KA ⊗ IB ) .
{KA }
(18)
The LQU has a simple expression for a 2 × d quantum system as [26]
UA (ρAB ) = 1 − λmax {WAB } ,
where λmax is the maximal eigenvalue of the 3 × 3 matrix WAB with the elements
√
√
(WAB )i j = Tr ρAB (σiA ⊗ IB) ρAB σ jA ⊗ IB ,
(19)
(20)
where σiA (i = x, y, z) are the Pauli matrixes of subsystem A.
Then we plot the LQC and LQU as functions of ξ (l0 ) /r0 and θ in Fig. 3(a) and Fig. 3(b). We
can see that the LQC and LQU decay fast with the increase of the ratio ξ (l0 ) /r0 when θ is close
to π /2 and then decrease slowly in a non-vanishing manner even when the ξ (l0 ) /r0 is large
enough. Therefore, we cannot see the phenomenon of sudden vanishing as for the entanglement
(ESD). The LQC and LQU as functions of ξ (l0 ) /r0 and γ are also displayed in Fig. 3(c) and
Fig. 3(d). When the purity γ 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). When γ 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 different θ are also shown in Fig. 4. We can clearly see the LQC
and LQU decay more and more slowly with the increase of the ratio ξ (l0 ) /r0 and the speed
become almost the same when θ is close to π /2. For certain initial state, for instance θ = π /3
shown in Fig. 4(b), the decay rate of LQU may be suddenly changed when the ξ (l0 ) /r0 is 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 quantum correlation for different values of phase correlation length which is determined by the
azimuthal quantum number l0 as shown in Eq. (16). The decays of LQC and LQC as functions
of ξ (l0 )/r0 for different values of l0 is shown in Fig. 5(a) and Fig. 5(b). It is shown that both
the LQC and LQU decay fastest at l0 = 1. When the azimuthal quantum number l0 increases,
the decays of LQC and LQU are slowed down and finally collapse onto two universal curves,
0.185
+ 0.075 and LQU(ρ ) ≈ g(x) = 0.92[exp(−3.50x1.90) + 0.08].
i.e., LQC(ρ ) ≈ f (x) = x3.4
+0.2
The LQC decays in an inverse proportional manner while the LQU decays in an exact exponential manner. Since the LQU is a kind of nonclassical correlation besides entanglement, then
it is natural to derive the universal exponential decay similar to that of entanglement reported
in Ref. [29]. From Eq. (18), one can verify that the LQU is equal to the minimal optimization
of the LQC, i.e., LQC ≥ LQU, such that LQC decays in a slower (polynomial) way instead of
the exponential. This result can be interpreted as that quantum coherence characterizes more
general quantumness while nonclassical correlations (such as quantum discord and entanglement) reveal only parts of quantumness, in which sense the LQC can be more robust against
atmospheric turbulence.
(a)
(b)
(c)
(d)
Fig. 3. (a,b) LQC and LQU as function of ξ (l0 )/r0 and θ and (c,d) LQC and LQU as
function of ξ (l0 )/r0 and γ . The parameters are chosen as p = 0, l0 = 1, ω0 = 1, (a,b) γ = 1
and (c,d) θ = π /2.
(a) 1.0
θ=π/6
(b)1.0
θ=π/4
0.8
θ=π/6
θ=π/4
0.8
0.6
θ=π/3
LQU
LQC
θ=π/3
0.6
0.4
0.4
0.2
0.2
0.0
0.0
0.2
0.4
0.6
0.8
ξ(l0 )/r0
1.0
1.2
1.4
0.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
ξ(l0 )/r0
Fig. 4. (a) LQC as function of ξ (l0 )/r0 and (b) LQU as function of ξ (l0 )/r0 . The parameters are chosen as p = 0, l0 = 1, ω0 = 1 and γ = 1.
4.
Conclusions
In conclusion, we have investigated the decay properties of quantum coherence and nonclassical
correlations (discord and entanglement) for photonic states carrying orbital angular momentum
(OAM) through Kolmogorov turbulent atmosphere via local quantum coherence (LQC), local
quantum uncertainty (LQU), and concurrence, respectively. By considering that the photonic
LQC
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ƺ
ͪ
ƶ
ƽƽ
ƺ
ƶ
ƽ
l
l
l
l
=1
=2
=5
=10
l =50
l =100
f(x)
0.2
0.4
0.6
0.8
ξ(l0 )/r0
1.0
1.2
1.4
0.8
LQU
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0.6
0.4
0.2
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ƽ l0 =1
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ƶ l0 =100
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ƽ
0.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
ξ(l0 )/r0
Fig. 5. (a) LQC as function of ξ (l0 )/r0 and (b) LQU as function of ξ (l0 )/r0 under different
l0 . The parameters are chosen as p = 0, ω0 = 1, θ = π /2 and γ = 1.
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 phenomena. 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 coherence 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
This work is supported by the National Natural Science Foundation of China (NSFC) (Grant
Nos. 11504140, 11504139 and 11364006), the Natural Science Foundation of Jiangsu Province
(Grant Nos. BK20140128 and BK20140167), the Fundamental Research Funds for the Central Universities (Grant No. JUSRP51517), the Natural Science and Technology Foundation of
Guizhou Province (Grant No.[2013]2231), and the Natural Science and Technology Foundation
of the Education Department of Guizhou Province (Grant No.[2014]242).
Quantum coherence and nonclassical
correlations of photonic qubits carrying
orbital angular momentum through
atmospheric turbulence
Mei-Song Wei1 , Jicheng Wang1 , Yixin Zhang1 , Qi-Liang He2 and
Zheng-Da Hu1,∗
1 Jiangsu
Provincial Research Center of Light Industrial Optoelectronic Engineering and
Technology, School of Science, Jiangnan University, Wuxi 214122, China
2 School
of Physics and Electronics, Guizhou Normal University, Guiyang 550001, China
∗
huyuanda1112@jiangnan.edu.cn
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 fact that photons can carry orbital angular momentum (OAM) to encode quantum states
makes them very useful for quantum information science (QIS) [1–8]. However, the decay of
quantumness will be unavoidable when the encoded photons with OAM transmit through turbulent atmosphere. Photons’ wave front will be distorted due to refractive index fluctuations of
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 dependent on the local observable K measured. 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, propagate 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 number l [8].
We assume that the input LG modes have a beam waist ω0 , a radial quantum number p0 = 0,
and azimuthal quantum numbers l0 and −l0 . The photon pair is initially prepared in an extended
Werner-like state defined as
1−γ
I + γ |Ψ0 i hΨ0 | ,
ρ (0) =
(1)
4
where 0 ≤ γ ≤ 1 denotes the purity of the initial state and |Ψ0 i is the Bell-like state given by
θ
θ
|Ψ0 i = cos
|l0 , −l0 i + eiφ sin
|−l0 , l0 i ,
(2)
2
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 purity γ = 1 and θ = 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 map Λ, in
terms of which the received state at the detectors reads as
ρ = (Λ1 ⊗ Λ2 ) ρ (0) ,
(3)
where Λ1 and Λ2 are 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 = ∑ ρ|i jihi′ j′ | |i ji i′ j′
ii′ , j j ′
=
′
′ ′
i j
′
(0)
′
′
(0)
ρii′ , j j′ |i ji
∑
′
′
ii , j j
=
∑ ′ Λll1ii′ Λmm
∑
2 j j ′ ρll ′ ,mm′ |i ji
′
′ ′
ii , j j ll ,mm
=
′ ′
i j
∑ ′ Λll1ii′ Λmm
∑
2 j j ′ ρ|lmihl ′ m′ | |i ji
′
′ ′
ii , j j ll ,mm
′ ′
i j .
(4)
Here, we let Λ1 = Λ2 = Λ due to the same effect of turbulent atmosphere on the photons. The
l ,l ′
0l ,0l ′
δl0 −l0′ ,l∓l Z
∞
0 0
elements Λl,l0 ′ 0 = ∑ p Λ pl,pl
′ of the linear map Λ is given by [29]
l0 ,l0′
Λl,±l
′
=
2π
0
drR pl0 (r) R∗pl0 (r) r
Z 2π
0
′
d ϑ e−iϑ [l±l−(l0 +l0 )]/2 e−0.5Dφ (2r|sin(ϑ /2)|) ,
(5)
where
p!
R pl0 (r) = 2
(|l0 + p|)
1/2
1
ω0
√ !|l0 |
2
2
r
r 2
|l | 2r
Ll 0
exp − 2
ω0
ω0
ω0
(6)
is the radial part of LG beam at z = 0 [10] with generalized Laguerre polynomials
|l |
Ll 0 (x) =
p
(|l0 | + p)!
∑ (−1)m (p − m)! (|l0 | + m)!m! xm .
(7)
m=0
Here, we consider Dφ = 6.88 (r/r0 )5/3 , which is the phase structure function of the Kolmogorov
model of turbulence and
−3/5
r0 = 0.423Cn2k2 L
(8)
is the Fried parameter, where Cn2 is the index-of-refraction structure constant, L is the propagation distance, and k is the optical wave number [33].
The density matrix of the input state (1) in the basis {|l0 , l0 i , |l0 , −l0 i , |−l0 , l0 i , |−l0 , −l0 i}
can be written as an X form
(0)
(0)
ρ11
ρ14
(0)
(0)
ρ22 ρ23
(9)
ρ (0) =
,
(0)
(0)
ρ32 ρ33
(0)
(0)
ρ41
ρ44
where
θ
1 − γ (0) 1 − γ
,
, ρ22 =
+ γ cos2
4
4
2
1−γ
1−γ
θ
(0)
(0)
ρ33 =
, ρ44 =
+ γ sin2
,
4
2
4
1 − γ γ −iφ
(0)
(0)
+ e sin θ ,
ρ14 =0, ρ23 =
4
2
(0)
(0)∗ (0)
(0)∗
ρ41 =ρ14 , ρ32 = ρ23 .
(0)
ρ11 =
(10)
According to Eq. (4), the density matrix of the output state (3) can also be expressed in the
X form as
ρ11
ρ14
ρ22 ρ23
,
ρ =
(11)
ρ32 ρ33
ρ41
ρ44
with
(0)
(0)
(0)
(0)
(0)
(0)
(0)
(0)
(0)
(0)
(0)
(0)
(0)
(0)
(0)
(0)
ρ11 =a2 ρ11 + abρ22 + abρ33 + b2 ρ44 ,
ρ22 =abρ11 + a2ρ22 + b2ρ33 + abρ44 ,
ρ33 =abρ11 + b2ρ22 + a2ρ33 + abρ44 ,
ρ44 =b2 ρ11 + abρ22 + abρ33 + a2 ρ44 ,
(0)
(0)
(0)
(0)
ρ14 =a2 ρ14 , ρ41 = a2 ρ41 ,
ρ23 =a2 ρ23 , ρ23 = a2 ρ32 ,
(12)
where
l ,l
−l ,−l
−l ,l
l ,−l
a =Λl00 ,l00 = Λ−l00 ,−l00 = Λ−l00 ,l00 = Λl00 ,−l00 ,
−l ,−l0
b =Λl0 ,l00
l ,l
0 0
.
= Λ−l
0 ,−l0
(13)
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
√
√
C (ρ ) = max {0, |ρ14 | − ρ22 ρ33 , |ρ23 | − ρ11 ρ44 } .
(14)
From Eqs. (12) and (13), we can derive the analytical form for the concurrence as
)
(
a2 sin θ − 2ab γ 1 − γ
.
C (ρ ) = max 0,
−
2
(a + b)2
(15)
For convenience, one can introduce the phase correlation length ξ (l0 ) which is defined as the
average distance between the points in the LG beam cross-section that have a phase difference
of π /2. The phase correlation length can be expressed as [29]
π
ω0 Γ (|l0 | + 3/2)
ξ (l0 ) = sin
,
(16)
2 |l0 | 2 Γ (|l0 | + 1)
with Γ (x) the Gamma function .
Then, we plot the concurrence C (ρ ) as a function of the ratio x = ξ (l0 ) /r0 and the initial
state parameter θ in 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 concurrence as a function of the ratio ξ (l0 ) /r0 and the purity parameter γ in Fig. 2(b). It is found that
the concurrence is zero when the initial state is unentangled for 0 ≤ γ ≤ 1/ (1 + 2 sin θ ), since
no entanglement can be created between the two photons under independent atmospheric turbulences 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 of ξ (l0 )/r0 and θ . (b) Concurrence as function of
ξ (l0 )/r0 and γ . 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 turbulence
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 system AB subject to a local measurement on the subsystem A, quantum uncertainty will yield if the subsystem
A has quantum coherence. Then, the LQC of an bipartite quantum state ρAB is given by [27]
1 √
LQC(ρAB ) = F (ρAB, KA ⊗ IB) = − Tr [ ρAB , KA ⊗ IB]2 ,
2
(17)
where KA is a local observable on system A and IB is the identity operator of subsystem B. Here,
we choose the local observable KA to be the third Pauli operator σ z .
Another measurement we use to express quantum correlation is LQU, which is the minimal quantum uncertainty achievable on a single local measurement. LQU is also the minimal
optimization of the LQC over all possible local observables, which can be written as
LQU(ρAB ) = min F (ρAB , KA ⊗ IB ) .
{KA }
(18)
The LQU has a simple expression for a 2 × d quantum system as [26]
UA (ρAB ) = 1 − λmax {WAB } ,
where λmax is the maximal eigenvalue of the 3 × 3 matrix WAB with the elements
√
√
(WAB )i j = Tr ρAB (σiA ⊗ IB) ρAB σ jA ⊗ IB ,
(19)
(20)
where σiA (i = x, y, z) are the Pauli matrixes of subsystem A.
Then we plot the LQC and LQU as functions of ξ (l0 ) /r0 and θ in Fig. 3(a) and Fig. 3(b). We
can see that the LQC and LQU decay fast with the increase of the ratio ξ (l0 ) /r0 when θ is close
to π /2 and then decrease slowly in a non-vanishing manner even when the ξ (l0 ) /r0 is large
enough. Therefore, we cannot see the phenomenon of sudden vanishing as for the entanglement
(ESD). The LQC and LQU as functions of ξ (l0 ) /r0 and γ are also displayed in Fig. 3(c) and
Fig. 3(d). When the purity γ 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). When γ 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 different θ are also shown in Fig. 4. We can clearly see the LQC
and LQU decay more and more slowly with the increase of the ratio ξ (l0 ) /r0 and the speed
become almost the same when θ is close to π /2. For certain initial state, for instance θ = π /3
shown in Fig. 4(b), the decay rate of LQU may be suddenly changed when the ξ (l0 ) /r0 is 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 quantum correlation for different values of phase correlation length which is determined by the
azimuthal quantum number l0 as shown in Eq. (16). The decays of LQC and LQC as functions
of ξ (l0 )/r0 for different values of l0 is shown in Fig. 5(a) and Fig. 5(b). It is shown that both
the LQC and LQU decay fastest at l0 = 1. When the azimuthal quantum number l0 increases,
the decays of LQC and LQU are slowed down and finally collapse onto two universal curves,
0.185
+ 0.075 and LQU(ρ ) ≈ g(x) = 0.92[exp(−3.50x1.90) + 0.08].
i.e., LQC(ρ ) ≈ f (x) = x3.4
+0.2
The LQC decays in an inverse proportional manner while the LQU decays in an exact exponential manner. Since the LQU is a kind of nonclassical correlation besides entanglement, then
it is natural to derive the universal exponential decay similar to that of entanglement reported
in Ref. [29]. From Eq. (18), one can verify that the LQU is equal to the minimal optimization
of the LQC, i.e., LQC ≥ LQU, such that LQC decays in a slower (polynomial) way instead of
the exponential. This result can be interpreted as that quantum coherence characterizes more
general quantumness while nonclassical correlations (such as quantum discord and entanglement) reveal only parts of quantumness, in which sense the LQC can be more robust against
atmospheric turbulence.
(a)
(b)
(c)
(d)
Fig. 3. (a,b) LQC and LQU as function of ξ (l0 )/r0 and θ and (c,d) LQC and LQU as
function of ξ (l0 )/r0 and γ . The parameters are chosen as p = 0, l0 = 1, ω0 = 1, (a,b) γ = 1
and (c,d) θ = π /2.
(a) 1.0
θ=π/6
(b)1.0
θ=π/4
0.8
θ=π/6
θ=π/4
0.8
0.6
θ=π/3
LQU
LQC
θ=π/3
0.6
0.4
0.4
0.2
0.2
0.0
0.0
0.2
0.4
0.6
0.8
ξ(l0 )/r0
1.0
1.2
1.4
0.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
ξ(l0 )/r0
Fig. 4. (a) LQC as function of ξ (l0 )/r0 and (b) LQU as function of ξ (l0 )/r0 . The parameters are chosen as p = 0, l0 = 1, ω0 = 1 and γ = 1.
4.
Conclusions
In conclusion, we have investigated the decay properties of quantum coherence and nonclassical
correlations (discord and entanglement) for photonic states carrying orbital angular momentum
(OAM) through Kolmogorov turbulent atmosphere via local quantum coherence (LQC), local
quantum uncertainty (LQU), and concurrence, respectively. By considering that the photonic
LQC
0.8
0.6
0.4
0.2
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0.0
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l
l
l
l
=1
=2
=5
=10
l =50
l =100
f(x)
0.2
0.4
0.6
0.8
ξ(l0 )/r0
1.0
1.2
1.4
0.8
LQU
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Ƹ l0 =5
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ͪ l0 =10
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l0 =50
Ƹ
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ͪ
ƶ l0 =100
ƽ ƻƸ
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ƶ
g(x)
ƺ
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ƶ
ƽ
0.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
ξ(l0 )/r0
Fig. 5. (a) LQC as function of ξ (l0 )/r0 and (b) LQU as function of ξ (l0 )/r0 under different
l0 . The parameters are chosen as p = 0, ω0 = 1, θ = π /2 and γ = 1.
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 phenomena. 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 coherence 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
This work is supported by the National Natural Science Foundation of China (NSFC) (Grant
Nos. 11504140, 11504139 and 11364006), the Natural Science Foundation of Jiangsu Province
(Grant Nos. BK20140128 and BK20140167), the Fundamental Research Funds for the Central Universities (Grant No. JUSRP51517), the Natural Science and Technology Foundation of
Guizhou Province (Grant No.[2013]2231), and the Natural Science and Technology Foundation
of the Education Department of Guizhou Province (Grant No.[2014]242).