Measurement of classical entanglement using interference fringes
MMeasurement of classical entanglement using interference fringes
Ziyang You, Zikang Tang, and Hou Ian ∗ Institute of Applied Physics and Materials Engineering, University of Macau, Macau S.A.R., China
Classical entanglement refers to non-separable correlations between the polarization direction andthe polarization amplitude of a light field. The degree of entanglement is quantified by the Schmidtnumber, taking the value of unity for a separable state and two for a maximally entangled state.We propose two detection methods to determine this number based on the distinguishable patternsof interference between four light sources derived from the unknown laser beam to be detected. Thesecond method being a modification of the first one has the interference fringes form discernableangles uniquely related to the entangled state. The maximally entangled state corresponds tofringes symmetric about the diagonal axis at either 45 ◦ or ◦ direction while the separable statecorresponds to fringes symmetric either about the X - or Y -axis or both simultaneously. Stateswith Schmidt number between unity and two have fringes of symmetric angles between these twoextremes. The detection methods would be beneficial to constructing transmission channels ofinformation contained in the classically entangled states. I. INTRODUCTION
Schr¨odinger introduced the concept of entanglement to describe non-separable correlations among differentquantum systems [1], in response to the Einstein, Podolsky, and Rosen (EPR) argument on the incompat-ibility between quantum mechanics and local realism [2]. Later, John Bell derived an inequality that canexperimentally confirm the nonlocality of quantum mechanics to settle the EPR argument [3]. Since then,various experiments have clearly shown that entangled quantum systems can violate the Bell inequality,where Clauser, Horne, Shimony, and Holt (CHSH) inequality is the most well-known example [4]. Apartfrom Bell-CHSH inequality, another measurement of entangled states relays on the Schmidt analysis, whichincludes calculation of the Schmidt number K to represent the degree of entanglement [5].In terms of nonlocality, entanglement has been viewed as a unique feature in quantum mechanics [6]. How-ever, non-separable correlations between different degrees of freedom in classical light fields, termed “classicalentanglement”, have been proved to exist [7–10]. Several pairs of degrees of freedom localized within a beamare correlated in a way analogous to quantum entanglement [11–13], such as that between polarization andspatial parity, between polarization and temporal amplitude, and between polarization and path. Moreover,recent experiments have supported the violation of Bell-CHSH inequality among the classical degrees offreedom measured under classical entanglement and their correlation levels reach as high as those obtainedby quantum entanglement [14–17]. Such a statistical and physical similarity has confirmed that classicalentanglement can provide some applications previously supported only by quantum entanglement [18]. Forexample, some works in the teleportation protocol for distributing information through classical entanglementhave shown potential applications in classical and quantum communication infrastructures [19, 20].Though the relevant studies, including those mentioned above, are abundant and substantial, quantifyingclassical entanglement in an experimentally accessible way is less explored. Here, we propose methods ofinterferometry to visualize the quantifiable degree of classical entanglement of a beam in its projected fringepattern. In particular, we use Schmidt analysis [5, 10, 15] to decide this degree between the polarizationdirection and the polarization amplitude of a laser beam [10] and design two experimental setups –“phasemethod” and “amplitude method”– to eventually reveal the Schmidt number in the specific symmetry axisof the fringe pattern. Being an improved version over the phase method on the optical path adopted, theamplitude method would show a more apparent fringe pattern. For instance, using the amplitude method, thefringes are symmetric exactly about the diagonal axis at 45 ◦ or 135 ◦ direction for the maximal entanglementcase. At the other extreme for separable states, the symmetry axes are either on the X -axis, the Y -axis, orboth simultaneously. The classical optical setup extends the entanglement analysis into the classical domainand would help the design of transmission channels for quantum information operated in the classical regime.In the following, we describe the entangled state and its relation to Schmidt number in Sec. II and then showthe fringe pattern simulations in Sec. III. Conclusions are given in Sec. IV. ∗ [email protected] a r X i v : . [ qu a n t - ph ] O c t II. CLASSICAL ENTANGLED STATE
A beam traveling in the Z -direction can be expressed by the electric field: E = E x ˆ e x + E y ˆ e y . (1)The E field in exhibits classical entanglement between the two degrees of freedom: one in the polarizationamplitude and one in the polarization direction [10]. The symbols ˆ e x and ˆ e y in Eq. (1) indicate the unitvectors of the polarization directions and E x and E y the wave amplitudes. With the intensity defined as I = (cid:104) E x E x + E y E y (cid:105) , the normalized electric field can be simplified as:ˆ e = E √ I = (cos θ Φ x ˆ e x + sin θ Φ y ˆ e y ) , (2)where we have Φ x = cos( kz − φ x ) , (3)Φ y = cos( kz − φ y ) , (4)being regarded as the unit vectors of a function space over the spatial coordinate z whose inner product isdefined as (Φ x , Φ y ) = 12 π π ˆ − π cos( kz − φ x ) cos( kz − φ y )d z = cos ∆ φ. (5)The phase difference ∆ φ = φ y − φ x between the two vectors generates a nonzero correlation. A pair oforthogonal functions can then be chosen as Φ = cos kz and Φ (cid:48) = sin kz , i.e. φ x = 0 while φ y = π/
2, toguarantee (Φ , Φ (cid:48) ) = 0. Then, the normalized field in Eq. (2) can be rewritten by the new orthogonal vectors:ˆ e = (cos φ x cos θ ˆ e x + cos φ y sin θ ˆ e y ) Φ + (sin φ x cos θ ˆ e x + sin φ y sin θ ˆ e y ) Φ (cid:48) . (6)From Eq. (6), the coefficient matrix can be derived, serving as background in the Schmidt analysis: C = (cid:18) cos φ x cos θ sin φ x cos θ cos φ y sin θ sin φ y sin θ (cid:19) . (7)The degree of entanglement decides how much separability the state is, which can be evaluated by aSchmidt analysis in both quantum and classical systems. According to Schmidt theorem [5], the Schmidtnumber K can be calculated to define the degree of entanglement precisely. In our case, the reduced densitymatrix of the lab frame can be obtained from Eq. (6) by tracing over the function frame: ρ lab = (cid:18) cos θ cos ∆ φ cos θ sin θ cos ∆ φ cos θ sin θ sin θ (cid:19) . (8)Then, the Schmidt number K can be obtained by summing the squared eigenvalues λ s of the reduced densitymatrix as weights, taking the form [5]: K = 1 (cid:80) s λ s = 11 − sin ∆ φ sin θ = 11 − C ) . (9) K is a function of the determinant of the coefficient matrix, and its value lies between 1 and 2 when twopolarized dimensions are involved.When K reaches the minimal value of unity, the electric field is in a separable state where the two polar-ization components are linearly polarized, letting the electric field become the productˆ e = (cos θ ˆ e x + sin θ ˆ e y ) Φ . (10)At the other extreme, when K reaches the maximal value of two, the electric field is in a maximally entangledstate that can be expressed as ˆ e = √
22 ˆ e x Φ + √
22 ˆ e y Φ (cid:48) , (11)which refers to the case of circular polarization. The intermediate value of K between these two extremesrepresents a partially entangled state called elliptic polarization. Generally speaking, the classical entan-glement between polarization amplitudes and polarization directions is intrinsically related to the polarizedstates. III. SIMULATION RESULTS AND DISCUSSION
In this part, the “phase method” and “amplitude method” are proposed to estimate the degree of entan-glement in a testing laser beam based on the fringe patterns of interference between four light sources.
A. Phase method
The patterns of interference between the four light sources are changed due to the constructive and de-structive effects at a different position. We employ the schematic setup in Fig.1. A testing laser beam firstlyimpinges on a polarizing beam splitter (PBS) to separate the horizontal and vertical polarization components,where a π/ π/ π phase shift. Both horizontal and vertical polarization components go through two 50:50 beam splitters (BS)to create identical copies separately. Then, a quarter-wave plate (QWP) is placed at the transmitted outputof the beam splitter to cause a π phase difference within components in the same polarization. Finally, thehorizontal and vertical polarization components are changed into a horizontal polarization by a 45 ◦ polarizercoupled to another 90 ◦ polarizer. Barrier ScreenUnknownlightSource MirrorPBS BS x-polarizedMirror BSHWP QWPy-polarized
Figure 1. Schematic diagram of the phase method. Firstly, the testing beam impinges on a polarizing beam splitter(PBS) to separate the horizontal and vertical polarization components. The horizontal and vertical polarizationcomponents then go through two 50:50 beam splitters (BS) to create identical copies separately. A quarter-wave plate(QWP) is placed at the transmitted output of the beam splitter to cause a π/ ◦ polarizer coupled to another 90 ◦ polarizer as inputs to the interference sources. The measurement setup decomposes the laser beam into the four inputs of the interference experiment,which can be expressed by the normalized input matrix S : S = (cid:18) S S S S (cid:19) = (cid:18) cos θ cos( kz − φ x ) cos θ sin( kz − φ x )sin θ cos( kz − φ y ) sin θ sin( kz − φ y ) (cid:19) = (cid:18) cos θ Φ x cos θ Φ (cid:48) x sin θ Φ y sin θ Φ (cid:48) y (cid:19) , (12) Figure 2. Interference patterns are the separable states for K = 1 in simulation. The fringes are horizontally symmetricalong the X -axis. The coefficients in the normalized input matrix are: (a) ∆ φ = 0, θ = π/
4; (b) ∆ φ = 0, θ = π/ φ = 0, θ = π/
10; (d) ∆ φ = π/ θ = π/ where the matrix elements ( S , S , S , S ) refer to the four input sources in Fig.1. The associated patternscan be considered as a straightforward response to different values of the phase difference ∆ φ and the polarizedangle θ , which can illustrate the degree of entanglement in Eq. (9) by a same determinant with the coefficientmatrix, det( C ) = det( S ). Since the information of ∆ φ is stored in the phase of input fields, this method isso called “phase method”.A model of interference between four light sources is constructed to simulate the interference patternsusing a Matlab program. The model setup includes: the laser wavelength of 600 nm, the point-point gap of d = 10 − m, the barrier-screen distance of L = 3 × − m, the screen area of 0 . × . . The simulationresults are shown in Fig.2 and Fig.3, where testing beams are in the separable states, maximally entangledstates, and intermediate scenarios. In Fig.2 (a)-(d), the separable states occur when K reaches the minimalvalue of one, where the polarized angle θ varies the fringe patterns. In this case, the interference fringesare horizontally symmetric along the X -axis due to a phase difference ∆ φ = 0. Meanwhile, the verticalsymmetry axis is shifted from the Y -axis due to a constant π phase difference between S and S , S and S , respectively.The maximally entangled states are obtained in Fig.3 (a),(b) when ∆ φ = π/ θ = π/ φ = − π/ θ = π/
4. Compared to the patterns of separable states, the horizontally symmetric axis is shifted fromthe X -axis due to a phase difference ∆ φ = ± π/
2. Simultaneously, the maximally entangled patterns havedestructive fringes equally distributed around the vertical axis caused by a polarized angle θ = π/
4. For theintermediate scenarios in Fig.3 (c),(d), the horizontally symmetric axis is shifted from the X -axis by differentvalues of ∆ φ . However, those changes in fringe patterns are not so obvious that they can distinguish themaximally entangled state.According to the simulation, the phase method is able to estimate the degree of entanglement by the fringepatterns and their horizontally symmetric axis, but with an insufficient distinction between different states.To improve detection accuracy, a modified strategy called “amplitude method” is introduced with the helpof a phase analyzer and rotatable polarizers. B. Amplitude method
In the phase method, the interference patterns are changed by the polarized angle and the phase differencesbetween the four inputs directly, which are similar for different states. To strengthen the distinction ofpatterns, the normalized input matrix S is changed to the same form as the coefficient matrix in Eq.7: Figure 3. Interference patterns in (a) (b) are the maximally entangled states for K = 2 in simulation. The patternsin (c) (d) are the intermediate scenarios for 1 < K <
2. The coefficients in the normalized input matrix are: (a)∆ φ = π/ θ = π/
4; (b) ∆ φ = − π/ θ = π/
4; (c) K = 1 .
6: ∆ φ = π/ θ = π/
3; (d) K = 1 .
3: ∆ φ = π/ θ = π/ S = (cid:18) S S S S (cid:19) = (cid:18) cos φ x cos θ sin φ x cos θ cos φ y sin θ sin φ y sin θ (cid:19) Φ x . (13)The new input matrix that requires to extract the phase information of φ x and φ y , converting into theamplitudes of the four sources, which is so called “amplitude method”. In actual experiment, the coefficientsof φ x and φ y refer to phase differences ∆ φ that are preserved within the beam propagation. Therefore, thephase coefficients are analyzed numerically with an indirect measurement by acquiring the phase difference ∆ φ with the help of a phase analyzer: sin φ y cos φ x − sin φ x cos φ y = cos ∆ φ, (14)where the values of φ x and φ y should be chosen as the positive coefficients of amplitudes in Eq.13.The schematic diagram of the experiment is shown in Fig.4. The testing beam is firstly sent to a 50:50beam splitter (BS) whose reflected part is directed to a phase analyzer. The transmitted beam is separatedinto the horizontal and vertical polarization components by passing through a polarizing beam splitter (PBS).A half-wave plate (HWP) and a phase compensator are then placed at the output of the polarizing beamsplitter (PBS) to compensate for a π phase shift introduced by the reflection in the mirror and PBS, andits original ∆ φ phase difference. After the phase compensation, both horizontal and vertical polarizationcomponents are decomposed by a 50:50 beam splitter (BS), where another half-wave plate (HWP) is placedat the transmitted output to cancel the reflected phase shift. Then, the four components are changedinto a horizontal polarization by the polarizer array that consists of rotatable polarizers and 90 ◦ polarizers.Meanwhile, the amplitude coefficients of the phases φ x and φ y are realized by the angle of rotation θ n in therotatable polarizers due to Malus’ law, for example:cos φ x = sin θ cos θ . (15)This method eventually decomposes the testing beam into four coherent fields as: S = cos φ x cos θ Φ x , S = sin φ x cos θ Φ x , S = cos φ y sin θ Φ x , and S = sin φ y sin θ Φ x .Another model is established with the same setup in the first method: the laser wavelength of 600nm,the point-point gap of d = 10 − m, the barrier-screen distance of L = 3 × − m, the screen area of 0 . × Phase compensator
Barrier ScreenUnknownlightSource BS BS MirrorPBS Rotatable polarizersPhase analyzer BS y-polarizedMirrorHWP HWPHWP Polarizersx-polarized
Figure 4. Schematic diagram of the amplitude method. The testing beam firstly goes through a 50:50 beam splitter(BS) whose reflected beam is directed to the phase analyzer. The transmitted beam is separated into the horizontaland vertical polarization components by passing through a polarizing beam splitter (PBS). Then, the horizontal andvertical polarization components are decomposed by 50:50 beam splitters and directed to the rotatable polarizer array.The amplitude coefficients of the phases φ x and φ y are realized by the angle of rotation θ n in the rotatable polarizers.Figure 5. Interference patterns are the separable states for K = 1 in simulation. The fringes are symmetric alongthe X - or Y - direction. The coefficients in the normalized field matrix are: (a) sin φ x = sin φ y = 0 and θ = π/
4; (b)sin φ x = sin φ y = cos φ x = cos φ y = √ / θ = π/
2; (c) sin φ x = sin φ y = cos φ x = cos φ y = √ / θ = π/ φ x = cos φ y = 1 and θ = π/ . . In the simulation, testing beams are still chosen as separable states, maximally entangled states, andintermediate scenarios.For the case of K = 1 shown in Fig. 5(a)-(d), the interference fringes of the separable states are symmetricalong about the X - or Y - axis or simultaneously when the determinant of the corresponding normalized inputmatrix is zero. The maximally entangled states are demonstrated in Fig. 6(a) and (b) for K = 2, where theinterference patterns are symmetric along the diagonal axis at 45 ◦ or 135 ◦ direction. The symmetric structureis due to the normalized input matrix’s determinant that reaches the maximum absolute value. Comparedto the patterns of the separable states, the maximally entangled fringes are asymmetric along the X - or Y -axis and in the form of smooth curves without crossing. For the intermediate scenarios with 1 < K < ◦ or 135 ◦ cases from the X -or Y - axis with K increasing, varying between those two extremes. Meanwhile, the fringes’ roughness also Figure 6. Interference patterns are the maximally entangled state for K = 2 in simulation. The fringes are symmetricalong the axis at 45 ◦ or 135 ◦ direction and in the form of smooth curves without crossing. The coefficients in thenormalized field matrix are: (a) sin φ y = cos φ x = 1 and θ = π/ φ x = cos φ y = 1 and θ = π/ < K <
2. The components in the coefficientmatrix are: (a) K = 1 .
3: cos φ x = 0 . , sin φ x = 0 . , cos φ y = 0 . , sin φ y = 0 . θ = π/
5; (b) K = 1 . φ x = 0 . , sin φ x = 0 . , cos φ y = 0 . , sin φ y = 0 .
995 and θ = π/ decreases with an increase in the value of K , which can easily distinct from the maximally entangled state.The amplitude method can measure the degree of entanglement by the interference fringes with discern-able angles uniquely related to the entangled state. Compared to the phase method, the fringe patterns’variations are much clearer to distinguish different states of entanglement but at the cost of complexity inthe experimental setup.The interference patterns in both the phase and amplitude methods expose the entanglement informationhidden in testing beams successfully. In our classical entanglement case, the entanglement degree can beascribed to the polarized state, commonly measured by the classical Stokes approach [21]. Relatively, ourmethodologies are accomplished by a statistical method, although they have a complex configuration todecompose the testing beam and reformulate the inputs of interference. However, their connection to Schmidttheorem can be categorized as a universal analysis method in both quantum and classical entanglement, wherethe measurement of entanglement degree can be achieved without any quantum contexts. Generally speaking,two detection methods based on the presented scheme’s interference patterns extend the physical concept ofentanglement analysis while agreeing with the statistical characterization of classical entanglement described. IV. CONCLUSIONS
We employed the concept of classical entanglement between the polarization amplitude and the polarizationdirection of an optical beam to study the relation between the degree of entanglement and the interferencefringe pattern hidden in this beam. Two methods (we named them the phase method and the amplitudemethod, respectively) based on optical manipulations to separate and interefere the polarization componentsof the beam are proposed to generate the fringe pattern. Both methods demonstrate differing patternscorresponding to distinct entangled states measured by Schmidt number. The amplitude method improvesover the phase method in terms of the easiness in differentiating the patterns, where the maximally entangledstates for instance correspond to fringes exactly symmetric about the diagonal axes. Along these lines, themethods proposed would be beneficial to constructing transmission channels of information in the classicallyentangled states.
ACKNOWLEDGMENTS
H.I. thanks the support by FDCT of Macau under Grant 0130/2019/A3, University of Macau underMYRG2018-00088-IAPME. [1] E. Schr¨odinger, " Discussion of probability relations between separated systems, " Math. Proc. Cambridge Philos.Soc. , 555-563 (1935).[2] A. Einstein, B. Podolsky, and N. Rosen, " Can Quantum-Mechanical Description of Physical Reality Be ConsideredComplete? " Phys. Rev. , 777-780 (1935).[3] J. S. Bell, " On the Einstein Podolsky Rosen paradox, " Physics , 195-200 (1964).[4] J. F. Clauser and A. Shimony, " Bell’s theorem. Experimental tests and implications, " Rep. Prog. Phys. ,1881-1927 (1978).[5] J. H. Eberly, " Schmidt Analysis of Pure-State Entanglement, " Laser Phys. , 921–926 (2006).[6] D. Paneru, E. Cohen, R. Fickler, R. W. Boyd, and E. Karimi, " Entanglement: Quantum or Classical? " Rep.Prog. Phys. , 064001 (2020).[7] K. F. Lee and J. E. Thomas, " Entanglement with classical fields, " Phys. Rev. A , 052311 (2004).[8] R. J. C. Spreeuw, " A Classical Analogy of Entanglement, " Found. Phys. , 361-374 (1998).[9] B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, " Nonquantum entan-glement resolves a basic issue in polarization optics, " Phys. Rev. Lett. , 023901 (2010).[10] X. F. Qian and J. H. Eberly, " Entanglement and classical polarization states, " Opt. Lett. , 4110-4112 (2011).[11] F. De Zela, " Relationship between the degree of polarization, indistinguishability, and entanglement, " Phys. Rev.A , 013845 (2014).[12] A. Aiello, F. T¨oppel, C. Marquardt, E. Giacobino, and G. Leuchs, " Quantum − like nonseparable structures inoptical beams, " New J. Phys. , 043024 (2015).[13] A. Forbes, A. Aiello, and B. Ndagano, " Classically Entangled Light, " Prog. Opt. , 99-153 (2019).[14] K. Kagalwala, G. Di Giuseppe, A. Abouraddy, and B. E. A. Saleh, " Bell’s measure in classical optical coherence, " Nat. Photonics , 72–78 (2013).[15] X. F. Qian, B. Little, J. C. Howell, and J. H. Eberly, " Shifting the quantum-classical boundary: theory andexperiment for statistically classical optical fields, " Optica , 611-615 (2015).[16] Y. Sun, X. Song, H. Qin, X. Zhang, Z. Yang, and X. Zhang, " Non-local classical optical correlation and imple-menting analogy of quantum teleportation, " Sci. Rep. , 9175 (2015).[17] J. Gonzales, P. S´anchez, D. Barberena, Y. Yugra, R. Caballero, and F. D. Zela, " Experimental Bell violationswith classical, non-entangled optical fields, " J. Phys. B: At., Mol. Opt. Phys. , 045401 (2018).[18] A. Holleczek, A. Aiello, C. Gabriel, C. Marquardt, and G. Leuchs, " Classical and quantum properties of cylin-drically polarized states of light, " Opt. Express , 9714-9736 (2011).[19] D. Guzman-Silva, R. Br¨uning, F. Zimmermann, C. Vetter, M. Gr¨afe, M. Heinrich, S. Nolte, M. Duparr´e, A.Aiello, M. Ornigotti, and A. Szameit, " Demonstration of local teleportation using classical entanglement, " LaserPhotonics Rev. , 317–321 (2016).[20] B. P. Silva, M. A. Leal, C. E. R. Souza, E. F. Galv˜ao, and A. Z. Khoury, " Spin–orbit laser mode transfer via aclassical analogue of quantum teleportation, " J. Phys. B: At., Mol. Opt. Phys. , 055501 (2016).[21] R. M. A. Azzam, " Arrangement of four photodetectors for measuring the state of polarization of light, " Opt.Lett.10