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Experimental controlled-NOT gate simulation with thermal light


We report a recent experimental simulation of a controlled-NOT gate operation based on polarization correlation measurements of thermal fields in photon-number fluctuations. The interference between pairs of correlated paths at the very heart of these experiments has the potential for the simulation of correlations between a larger number of qubits.


The discovery of the Hanbury Brown and Twiss (HBT) effect1,2 in 1956 triggered the development of the field of quantum optics. Indeed, this phenomenon motivated numerous studies of multiphoton entanglement and interference not only from a fundamental point of view3,4,5,6,7,8 but also toward applications in information processing9,10,11,12,13, metrology14,15 and imaging16,17,18.

Recent efforts have been made to simulate quantum entanglement using classical light8,19,20,21,22,23,24. These studies are important toward achieving a deeper understanding of the differences between classical and quantum systems. Moreover, although such schemes may suffer of an exponential scaling in the number of resources comparing with the quantum systems25,26,27, they make it possible to simulate small-scale quantum systems with simple interferometers without being affected by decoherence. We have recently developed a novel detection scheme that measures the photon-number fluctuation correlation(PNFC) of thermal light28. This scheme has been applied to the study of the multi-photon coherence of thermal states8,23,24,28,29,30, leading to effects similar to the nonlocal interference characterizing entangled states.

Motivated by these results, we experimentally demonstrate here how multiphoton interference of pairs of correlated optical paths emerges from the measurement of photon-number fluctuations of thermal fields. This phenomenon is not only interesting from a fundamental point of view but also opens the way to the simulation of quantum gate operations. In particular, by using only a pseudo-thermal source31,32 and a linear optical interferometer, a controlled-NOT (CNOT) gate operation33,34,35,36,37,38 is experimentally simulated. The experimental setup (Fig. 1) is a realization in the spatial domain24 of the theoretical proposal of Tamma and Seiler8. In particular, we demonstrate how correlation measurements in the fluctuations of the number of photons at the output of the interferometer not only simulate (Fig. 2) the truth-table of a CNOT-gate (Table 1) but also the Bell correlations (Fig. 3) typical of a CNOT-gate operation.

Table 1 Truth table for a CNOT gate operation.
Figure 1

Schematic setup for the CNOT gate experimental simulation.

The light emitted by a He-Ne laser is set to be left-circularly polarized. A rotating ground glass (GG) is then used to “thermalize” the coherent laser light into a large number of incoherent subfields. A beamsplitter (BS) splits the wavepackets into two beams. Polarizers Pi and half-wave plates HWPi (i = c,t) are used to prepare the “control” and “target” beams at polarization angles ϕc and ϕt, respectively, with respect to the horizontal direction. Each beam interacts with a mask with two pinholes Li and Ri separated beyond the spatial coherence length of the thermal field. Two polarizers oriented in horizontal () and vertical () directions, respectively, are placed in front of pinholes Lc and Rc. A half-wave plate , implementing a flip from H to V polarization and vice versa, is placed in front of the pinhole Rt. AcDc and AtDt are two independent polarizer-detectors performing single-photon detections at arbitrary polarization angles θi. A photon-number fluctuation correlation (PNFC) circuit is used to measure the photon-number fluctuation correlations between detectors Dc and Dt.

Figure 2

Experimental observation of the polarization correlation 〈Δncc, θcntt, θt)〉 in the photon-number fluctuations for the input polarizations (ϕc, ϕt) = (H, H), (H, V), (V, H), (V, V) and the output polarizations (θc, θt) = (H, H), (H, V), (V, H), (V, V). For each input polarization (ϕc, ϕt), the plotted data are normalized by .

Figure 3

Experimental observation of the polarization correlation 〈Δncc, θcntt, θt)〉 in the photon-number fluctuations for the input polarizations ϕc = π/4 and ϕt = 0. The black dots are experimental data normalized by 〈ncc, θc)〉〈ntt, θt)〉 and the continuous red sinusoidal curve is a theoretical fitting based on Eq. (2). In this measurement, θc was fixed at π/4 and the values of θt range from −π/4 to 7π/4.


Description of the experiments

We describe the experimental setup, depicted schematically in Fig. 1. The light source is a standard pseudo-thermal source consisting of a circularly polarized 633 nm CW laser beam and a rotating ground glass (GG). The diameter of the laser beam is ~2 mm. The size of the tiny diffusers on the GG is roughly a few micrometers. A large number of circularly polarized incoherent wavepackets, or subfields, are scattered from a large number of diffusers. The second-order coherence time of the source is measured to be ~90 ms. The randomly scattered wavepackets are then split by a non-polarizing beamsplitter into two beams, the “control beam” c and the “target beam” t. A polarizer Pi and a half-wave plate HWPi prepare each beam i = c, t at an arbitrary polarization direction corresponding to an angle ϕi with respect to the horizontal direction. The control beam goes through a mask with two polarizers in the horizontal () and vertical () directions placed in front of the two pinholes Lc and Rc, respectively. The target beam passes through two pinholes Lt and Rt. A half-wave plate , interchanging the H with the V polarization components, is placed in front of Rt. The double-pinhole at the control arm and the double-pinhole at the target arm of the interferometer are spatially “overlapped”, i.e., Lc (Rc) and Lt (Rt) have equal longitudinal-transverse positions with respect to the correspondent optical axis. However, at each arm, the two pinholes are separated beyond the coherence length of the thermal field. The two light beams are then detected at the single-photon level by the two detectors Dc and Dt after passing through the polarizers Ac and At, respectively. We consider a number N ~ 4 × 105 of consecutive detection time intervals with width Δt = 800 μs. The value of Δt is small compared with the coherence time of the source, but large enough to guarantee enough counts per window. The registration times and the number niji, θi) of photodetection events at each detector Di within the jth time window, with j = 1, …, N, are recorded for given output polarization angles θi by two independent but synchronized event timers. At each detector Di the mean photon number is obtained by averaging over all the values of photon number niji, θi) recorded in each of the N time windows j. The photon number fluctuation for each time window is calculated as28

Finally, for given input polarization angles ϕc and ϕt of the control and target beams, respectively, the correlation

in the photon-number fluctuations is measured at the output for arbitrary polarization angles θc and θt.

Interference between pairs of correlated paths and CNOT-gate simulation

We consider first the case of input and output polarizations either in the horizontal direction or in the vertical directions . In this case, the experimental outcomes in Fig. 2 for the correlation in the photon number fluctuations in Eq. (1) simulate the truth table (Table 1) of a CNOT-gate. The initial polarization direction of the control beam remains always unchanged at the output. In particular, if the control beam is H-polarized then it can pass only through the pinhole Lc and a non-zero correlation in Eq. (1) is measured only when the target beam passes through the pinhole Lt without changing its initial polarization. On the other hand, a V-polarized control beam can only propagate through the pinhole Rc and a nontrivial correlation at the output occurs only if the target beam, by taking the path Rt, flips its polarization direction from to or vice versa. These experimental results witness the emergence of two pairs of correlated paths corresponding to the propagation through either the pinhole pair (Lc, Lt) or the pair (Rc, Rt). Can these pairs of correlated paths actually interfere? One may think that this is not possible since the two pinhole pairs are placed with respect to each other beyond the source coherence length. Interestingly, we show here experimentally that interference not only occurs but allows also us to fully simulate the entanglement operation of a CNOT gate. For this purpose, we consider the case where the control beam is polarized at an angle ϕc = π/4 corresponding to the direction . In this case, by considering a target beam in the initial polarization direction , the correlation of the photon-number fluctuations in Fig. 3 measured at the interferometer output is given by

Indeed, the measurement simulates with ~100% visibility the polarization correlations typical of the Bell state produced at the output of a “genuine” CNOT gate with input state |ϕc〉|ϕt〉, where and |ϕt〉 = |V〉. Interestingly, entanglement correlations analogous to a CNOT operation are simulated here by using only a separable input state and taking advantage of the interference between two pairs (Lc, Lt) and (Rc, Rt) of correlated paths, as will become more evident in the theoretical description in the next section.

Theoretical description

Here we provide a theoretical analysis based on the Glauber-Scully theory39,40 of the experimental results described in the previous section. We start from modeling the state of the pseudo-thermal field. The ground glass contains a large number of tiny randomly shaped scattering diffusers, roughly a few micrometers in size. A large number of subfields or wave packets are scattered from the laser beam with random phases by these tiny diffusers. We consider each scattering diffuser as a sub-source. By considering, for simplicity, monochromatic light, the state of the pseudo-thermal field can be expressed in the coherent state representation as41

where k is the transverse wavevector. |αm(k)〉 is an eigenstate of the annihilation operator with an eigenvalue αm(k) which contains a real-positive amplitude am(k) and a random phase φm(k) arising from the scattering process associated with the mth diffuser.

We can then evaluate, for given input polarization angles ϕc and ϕt, the photon-number correlation

where 〈…〉Es denotes the ensemble average over all the possible values of αm(k). Here, the field operator can be expressed as the sum

with i = c, t, where is an effective spatial transfer function (to be defined later) which takes into account the polarization dependent evolution from the mth pointlike diffuser to the pointlike detector Di at position .

By introducing the “effective wavefunction”

Eq. (3) becomes

leading to the correlation between the photon-number fluctuations:

Here, the approximation in the second step of Eq. (5), given the large number of subfields, is used to simplify the notation.

We explicitly address the propagation through the two pinholes Li and Ri at positions and , respectively, at each interferometric arm in Fig. 1 by rewriting Eq. (4) as


where is the Green’s function associated with the spatial propagation from the mth subfield to the detector Di passing through the pinhole Pi (P = L, R) and “F” indicates the flip in the polarization components ( to and vice versa) of the polarization direction performed by the waveplate .

By substituting Eq. (6) in Eq. (5) we obtain

where, in the second step of Eq. (7), the value |αm(k)|2 was assumed to be the same for each subfield m.

Since the pinholes Li and Ri are placed with respect to each other beyond the transverse coherence length of the thermal field, Eq. (7) reduces to24


Interestingly, the measured correlation in the photon-number fluctuations emerges from the interference between only two multiphoton contributions and associated with the propagation through the two pairs of pinholes (Lc, Lt) and (Rc, Rt), respectively.

We recall now that in the experiment, the two detectors are placed along the optical axes in the control and target arms of the interferometer and the two pinholes in each arm are at the same distances from the axes. In these conditions Eq. (8) becomes24

We now compare this result with a genuine CNOT entangling operation on the input state |ϕc〉|ϕt〉, where and , leading to the output entangled state

where . Polarization correlation measurements over the state |ψc,t〉 occur with a probability

Comparing Eq. (10) with Eq. (9), it is clear that the measurement of correlations between the photon-number fluctuations at the two output ports leads to the simulation of a CNOT gate operation.


In summary, we have experimentally demonstrated for the first time thermal light interference between two pairs of correlated paths, where each path in a pair is spatially incoherent with the paths in the other pair. This counterintuitive effect is at the very heart of the experimental simulation of a CNOT gate operation described here.

In particular, the simulation of the entanglement correlations typical of a CNOT-gate operation emerges from the interference between the two pairs of paths (Lc, Lt) and (Rc, Rt) in Fig. 1 propagating through two corresponding pairs of pinholes when correlation measurements in the photon-number fluctuations are performed at the output. Interestingly, this interference phenomenon occurs even if the pinholes in one pair are separated by more than the source coherence length with respect to the pinholes in the other pair.

Furthermore, the correlation in the photon-number fluctuations between the polarizations measured by the two distant detectors resembles the typical nonlocal behavior of entangled states even if no entanglement process occurs in the interferometer. Indeed, by not relying on complex non classical interferometers, the interference operation demonstrated here is apparently insensitive to photon losses and decoherence.

Lastly, by taking advantage of the abundant source of input states characterizing a thermal source with respect to single photon sources, this phenomenon can be used, in principle, to simulate correlations between a larger number of qubits, with potential applications in novel optical algorithms8,42,43,44,45,46, imaging and metrology8,18,24.

Additional Information

How to cite this article: Peng, T. et al. Experimental controlled-NOT gate simulation with thermal light. Sci. Rep. 6, 30152; doi: 10.1038/srep30152 (2016).


  1. Brown, R. H. & Twiss, R. Correlation between photons in two coherent beams of light. Nature 177, 27–29 (1956).

    ADS  Article  Google Scholar 

  2. Brown, R. H. & Twiss, R. A test of a new type of stellar interferometer on sirius. Nature 178, 1046–1048 (1956).

    ADS  Article  Google Scholar 

  3. Alley, C. & Shih, Y. Foundations of quantum mechanics in the light of new technology, edited by m. namiki et all. Physical society of Japan, Tokyo 47 (1986).

  4. Shih, Y. & Alley, C. O. New type of einstein-podolsky-rosen-bohm experiment using pairs of light quanta produced by optical parametric down conversion. Physical Review Letters 61, 2921 (1988).

    CAS  ADS  Article  Google Scholar 

  5. Hong, C., Ou, Z. & Mandel, L. Measurement of subpicosecond time intervals between two photons by interference. Physical Review Letters 59, 2044 (1987).

    CAS  ADS  Article  Google Scholar 

  6. Tamma, V. & Laibacher, S. Multiboson correlation interferometry with multimode thermal sources. Physical Review A 90, 063836 (2014).

    ADS  Article  Google Scholar 

  7. Tamma, V. & Laibacher, S. Multiboson correlation interferometry with arbitrary single-photon pure states. Physical review letters 114, 243601 (2015).

    ADS  Article  Google Scholar 

  8. Tamma, V. & Seiler, J. Multipath correlation interference and controlled-not gate simulation with a thermal source. New Journal of Physics 18, 032002 (2016).

    ADS  MathSciNet  Article  Google Scholar 

  9. Nielsen, M. A. & Chuang, I. L. Quantum computation and quantum information (Cambridge university press, 2010).

  10. Aaronson, S. & Arkhipov, A. The computational complexity of linear optics. In Proceedings of the forty-third annual ACM symposium on Theory of computing, 333–342 (ACM, 2011).

  11. Laibacher, S. & Tamma, V. From the physics to the computational complexity of multiboson correlation interference. Physical review letters 115, 243605 (2015).

    ADS  Article  Google Scholar 

  12. Tamma, V. Sampling of bosonic qubits. International Journal of Quantum Information 12, 1560017 (2014).

    ADS  MathSciNet  Article  Google Scholar 

  13. Tamma, V. & Laibacher, S. Boson sampling with non-identical single photons. Journal of Modern Optics 63, 41–45 (2016).

    ADS  MathSciNet  Article  Google Scholar 

  14. Boto, A. N. et al. Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit. Physical Review Letters 85, 2733 (2000).

    CAS  ADS  Article  Google Scholar 

  15. D’Angelo, M., Garuccio, A. & Tamma, V. Toward real maximally path-entangled n-photon-state sources. Physical Review A 77, 063826 (2008).

    ADS  Article  Google Scholar 

  16. Pittman, T., Shih, Y., Strekalov, D. & Sergienko, A. Optical imaging by means of two-photon quantum entanglement. Physical Review A 52, R3429 (1995).

    CAS  ADS  Article  Google Scholar 

  17. Gatti, A., Brambilla, E., Bache, M. & Lugiato, L. Correlated imaging, quantum and classical. Physical Review A 70, 013802 (2004).

    ADS  Article  Google Scholar 

  18. Valencia, A., Scarcelli, G., D’Angelo, M. & Shih & Y. Two-photon imaging with thermal light. Physical review letters 94, 063601 (2005).

    ADS  Article  Google Scholar 

  19. Lee, K. & Thomas, J. Experimental simulation of two-particle quantum entanglement using classical fields. Physical review letters 88, 097902 (2002).

    CAS  ADS  Article  Google Scholar 

  20. Fu, J., Si, Z., Tang, S. & Deng, J. Classical simulation of quantum entanglement using optical transverse modes in multimode waveguides. Physical Review A 70, 042313 (2004).

    ADS  Article  Google Scholar 

  21. Kagalwala, K. H., Di Giuseppe, G., Abouraddy, A. F. & Saleh, B. E. Bell’s measure in classical optical coherence. Nature Photonics 7, 72–78 (2013).

    CAS  ADS  Article  Google Scholar 

  22. Qian, X.-F., Little, B., Howell, J. C. & Eberly, J. Shifting the quantum-classical boundary: theory and experiment for statistically classical optical fields. Optica 2, 611–615 (2015).

    ADS  Article  Google Scholar 

  23. Peng, T. & Shih, Y. Bell correlation of thermal fields in photon-number fluctuations. EPL (Europhysics Letters) 112, 60006 (2016).

    ADS  Article  Google Scholar 

  24. Cassano, M. et al. Spatial interference between pairs of optical paths with a chaotic source. arXiv:1601.05045 (2016).

  25. Knill, E., Laflamme, R. & Milburn, G. J. A scheme for efficient quantum computation with linear optics. Nature 409, 46 (2001).

    CAS  ADS  Article  Google Scholar 

  26. Duan, L. M. & Kimble, H. J. Scalable photonic quantum computation through cavity-assisted interactions. Physical review letters 92, 127902 (2004).

    ADS  Article  Google Scholar 

  27. Wei, H. R. & Deng, F. G. Scalable photonic quantum computing assisted by quantum-dot spin in double-sided optical microcavity. Optics express 21, 17671–17685 (2013).

    ADS  Article  Google Scholar 

  28. Chen, H., Peng, T. & Shih, Y. 100% correlation of chaotic thermal light. Physical Review A 88, 023808 (2013).

    ADS  Article  Google Scholar 

  29. Peng, T., Chen, H., Shih, Y. & Scully, M. O. Delayed-choice quantum eraser with thermal light. Physical review letters 112, 180401 (2014).

    ADS  Article  Google Scholar 

  30. Peng, T., Simon, J., Chen, H., French, R. & Shih, Y. Popper’s experiment with randomly paired photons in thermal state. EPL (Europhysics Letters) 109, 14003 (2015).

    ADS  Article  Google Scholar 

  31. Martienssen, W. & Spiller, E. Coherence and fluctuations in light beams. American Journal of Physics 32, 919–926 (1964).

    ADS  Article  Google Scholar 

  32. Arecchi, F., Gatti, E. & Sona, A. Time distribution of photons from coherent and gaussian sources. Physics Letters 20, 27–29 (1966).

    CAS  ADS  Article  Google Scholar 

  33. Pittman, T., Jacobs, B. C. & Franson, J. Probabilistic quantum logic operations using polarizing beam splitters. Physical Review A 64, 062311 (2001).

    ADS  Article  Google Scholar 

  34. Pittman, T., Fitch, M., Jacobs, B. & Franson, J. Experimental controlled-not logic gate for single photons in the coincidence basis. Physical Review A 68, 032316 (2003).

    ADS  Article  Google Scholar 

  35. O’Brien, J. L., Pryde, G. J., White, A. G., Ralph, T. C. & Branning, D. Demonstration of an all-optical quantum controlled-not gate. Nature 426, 264–267 (2003).

    ADS  Article  Google Scholar 

  36. Gasparoni, S., Pan, J.-W., Walther, P., Rudolph, T. & Zeilinger, A. Realization of a photonic controlled-not gate sufficient for quantum computation. Physical review letters 93, 020504 (2004).

    ADS  Article  Google Scholar 

  37. Okamoto, R., Hofmann, H. F., Takeuchi, S. & Sasaki, K. Demonstration of an optical quantum controlled-not gate without path interference. Physical review letters 95, 210506 (2005).

    ADS  Article  Google Scholar 

  38. Bao, X. H., Chen, T. Y., Zhang, Q., Yang, J., Zhang, H., Yang, T. & Pan, J. W. Optical nondestructive controlled-NOT gate without using entangled photons. Physical review letters 98, 170502 (2007).

    ADS  Article  Google Scholar 

  39. Glauber, R. J. The quantum theory of optical coherence. Physical Review 130, 2529 (1963).

    ADS  MathSciNet  Article  Google Scholar 

  40. Scully, M. O. & Zubairy, M. S. Quantum optics (Cambridge university press, 1997).

  41. Shih, Y. An introduction to quantum optics (CRC press, 2011).

  42. Tamma, V., Zhang, H., He, X., Garuccio, A. & Shih, Y. New factorization algorithm based on a continuous representation of truncated gauss sums. Journal of Modern Optics 56, 2125–2132 (2009).

    ADS  Article  Google Scholar 

  43. Tamma, V. et al. Factoring numbers with a single interferogram. Physical Review A 83, 020304 (2011).

    ADS  Article  Google Scholar 

  44. Tamma, V. Alley, C., Schleich, W. & Shih, Y. Prime number decomposition, the hyperbolic function and multi-path michelson interferometers. Foundations of Physics 42, 111–121 (2012).

    ADS  Article  Google Scholar 

  45. Tamma, V. et al. Analogue algorithm for parallel factorization of an exponential number of large integers: I. theoretical description. Quantum Information Processing 1–22 (2015).

  46. Tamma, V. et al. Analogue algorithm for parallel factorization of an exponential number of large integers: II. optical imple- mentation. Quantum Information Processing 1–15 (2015).

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The authors wish to thank M. Cassano, W.P. Schleich and J. Sprigg for helpful discussions. This work was supported by NSF. V.T. acknowledges the support of the German Space Agency DLR with funds provided by the Federal Ministry of Economics and Technology (BMWi) under grant no. DLR 50 WM 113.

Author information




T.P. constructed and performed the experiment. V.T. and Y.H.S. conceived the original idea. T.P. and Y.H.S. designed the experiment. T. P. analyzed the results. T. P., V.T. and Y.H.S. wrote the paper. All authors reviewed the manuscript.

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Peng, T., Tamma, V. & Shih, Y. Experimental controlled-NOT gate simulation with thermal light. Sci Rep 6, 30152 (2016).

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