Multipath Correlation Interference and Controlled-NOT Gate Simulation with a Thermal Source

We theoretically demonstrate a counter-intuitive phenomenon in optical interferometry with a thermal source: the emergence of second-order interference between two pairs of correlated optical paths even if the time delay imprinted by each path in one pair with respect to each path in the other pair is much larger than the source coherence time. This fundamental effect could be useful for experimental simulations of small-scale quantum circuits and of $100\%$-visibility correlations typical of entangled states of a large number of qubits, with possible applications in high-precision metrology and imaging. As an example, we demonstrate the polarization-encoded simulation of the operation of the quantum logic gate known as controlled-NOT gate.

Multiphoton interference and correlations [1][2][3][4] are fundamental phenomena at the heart of quantum mechanics. Their applications range from fundamental tests of quantum mechanics to quantum information processing [5,6].
In particular, the realization of a quantum computer has raised great interest in the scientific community because of its potential to outperform classical protocols in terms of efficiency [7]. However, toward the implementation of a scalable quantum computer several experimental challenges need to be addressed: 1) the realization of a large number of indistinguishable and scalable quantum bits (qubits), given by two-level quantum systems like single photons; 2) controlled interaction of such qubits in isolation from the environment in order to reduce decoherence; 3) efficient implementation of controlled-NOT(CNOT) gate operations [8][9][10][11] where entanglement of two independent qubits is obtained by nonlinear processes. In the case of photonic qubits, nonlinear interactions between single photons are extremely challenging to achieve, since the single-photon electric field is very low. The approach introduced by Knill, Laflamme and Milburn [7] circumvents this problem by taking advantage of the nonlinearity of the measurement process itself. However, such an approach relies on the realization of complex optical networks [8] with a large number of additional ancilla photons.
In this letter, we demonstrate how a multiphoton interferometer with only a thermal source is able to simulate controlled-NOT(CNOT) gate operations overcoming all the experimental challenges previously described. Indeed, a thermal source is one of the most natural sources and, differently from single photon sources, can be easily obtained in a laboratory by using laser light impinging on a fast rotating ground glass [12]. This avoids the need of producing indistinguishable single photons. Moreover, we will demonstrate CNOT-like correlations emerging from the statistics of polarization-correlated measurements in the fluctuation of the number of detected photons. Such correlations are intimately connected with entanglement correlations but have the advantage of not being affected by the decoherence typical of entanglement processes. Moreover, our scheme does not rely on any non linear process or complex network with additional ancilla. The calibration of the interferometric paths and of the detection times (see Eqs. (9), (10) and (11)) can be easily achieved in the case of a thermal source, where the coherence time can range from the order of ns to µs. All these advantages make our approach appealing from an experimental point of view, given the present difficulty in the implementation of quantum networks even for a reduced number of quantum gates.
We now describe in detail how the interferometer depicted in Fig. 1 operates as a "CNOT gate interferometer". Such interferometer consists of a first subinterferometer devoted to the preparation of the initial polarization state, a second sub-interferometer implementing a suitable polarization-dependent evolution and the final measurement process.
At the input port S of the first sub-interferometer a horizontally(H)-polarized thermal light source is described by the state [13,14] with the Glauber-Sudarshan probability distribution [15,16] Pρ where n ω is the average photon number at frequency ω. For simplicity, but without losing generality, we consider a Gaussian frequency distribution [13] n with the mean photon rate r, average frequency ω 0 and spectral width ∆ω. At the second input port S there is no source. The H-polarized thermal light impinges on the balanced beam splitter BS1 and, by using half-wave plates, is prepared at the "control" port C and "target" port T in two general polarization CNOT-gate interferometer: 1) The first subinterferometer transformation U1 prepares the initial polarization state; 2) The second sub-interferometer transformation U2 implements the CNOT polarization-dependent evolution; 3) At the interferometer output the polarization correlations in the photon number fluctuations are measured.
(cos φ C sin φ C ) T and φ T = (cos φ T sin φ T ) T , respectively. Here, the H and V polarization directions are indicated by the vectors (1 0) T and (0 1) T , respectively. Thereby, the interferometer transformation associated with the first sub-interferometer connecting the input ports S, S with the ports C, T is given by with the polarization rotations and the balanced beam-splitter transformation where 1 is the two-dimensional identity matrix. The second sub-interferometer consists of a "control" interferometer connecting the ports C and C and a "target" interferometer connected the ports T and T . Therefore, the global interferometric evolution is described by the two transformations U C,C and U T,T defining the diagonal matrix In particular, in the control interferometer, the light in the polarization modes H and V , at the output of the first polarizing beam splitter PBS1, acquire the time delays S C /c and L C /c, respectively, with c the speed of light, before being recombined at the output of the second polarizing beam splitter PBS2. This lead to the control transformation On the other hand, in the target interferometer the light is coherently split in two different paths and recombined by the balanced beam splitters BS2 and BS3, respectively, independently of the polarization. In the path of length S T the light polarization is unchanged. Instead, in the path of length L T the polarization modes H and V are flipped (H ↔ V ) by the NOT-gate operation implemented by a half-wave plate (HWP), whose axes are rotated by π/4 with respect to the H and V axes. Thus the overall target evolution is described by the transformation By using Eqs. (2) and (3), we can now derive the total interferometer matrix We finally address the detection process, consisting of polarization correlation measurements in the fluctuations of the number of photons ∆n θ C (t C ) and ∆n θ T (t T ) detected at the control and target ports d = C, T , respectively, with polarization θ d . . = (cos θ d sin θ d ) T at time t d , for detection integration-time intervals δt d 1/∆ω. The expectation value for the product of the photonnumber fluctuations at the two output ports is [13,17] with the polarization-dependent first-order correlation function The electric field operatorsÊ , for direction of propagation perpendicular to the H-V plane, are given, for simplicity in the narrow bandwidth approximation, by the operator with d = C, T and a constant K, and its respective Hermitian conjugate. Here, the annihilation operatorsâ where the 2 × 2 matrices U d,s are the elements of the total interferometer matrix (4). By direct substitution, we show in the appendix that, in the limits for the interferometric optical paths and for the detection times, the first-order correlation function (6) reads with the two contributions where K is a constant. By using Eq. (12), Eq. (5) becomes (13) Similarly to a Franson-type interferometer [18], the expectation value in Eq. (13) depends on the multiphoton interference associated with the two path configurations (S C , S T ) and (L C , L T ). By introducing the relative phase Eq. (13) reads We first point out that it is possible to probe by interference the relative phase ϕ L−S . Indeed, if we substitute φ C = φ T = θ C = θ T = π/4 in Eq. (14) the resulting expectation value oscillates with the phase ϕ L−S . More interestingly, we demonstrate now that, for ϕ L−S 1, the interferometer described so far can reproduce a CNOT gate operation. We consider a general A CNOT gate operation on the input state |φC C |φT T , defined by Eqs. (16) and (17), leads to the entangled output state |ψ C,T in Eq. (18). Entanglement correlations emerge from polarization correlation measurements, where θC and θT are the detected polarization angles at the ports C and T , respectively.
CNOT gate, as depicted in Fig. 2, with the two-qubit input state |φ C C |φ T T , where and are expressed as superpositions of the polarization states |H and |V , corresponding to single-photon occupations of the H and V modes, respectively. A CNOT-gate operation on this input state leads to the output entangled state where Polarization correlation measurements over the state |ψ C,T occur with a probability Comparing Eq. (19) with Eq. (14) in the limit ϕ L−S 1, we obtain demonstrating how our scheme is intimately connected with a CNOT gate operation.
As an example, if we fix the polarization angles φ C = π/4 and φ T = 0 in the setup in Fig. 1, the expectation value in Eq. (20) reads reproducing the 100%-visibility correlations typical of the Bell state |Φ + = 1 One can finally generalize the interferometer analyzed in this letter to a multiphoton network based on Norder correlation measurements, as depicted in Fig. 3. Here, we first prepare a polarization-dependent input state for an arbitrary number N of ports: from Hpolarized thermal light impinging on a symmetric 2Nport beam-splitter (generalization of the balanced beam splitter BS1 in Fig. 1), it is possible to prepare the light in the N output ports at arbitrary polarization angles φ i , with i = 1, 2, ..., N , by using N half-wave plates. The light then propagate through a 2N -port polarizationdependent interferometer. In analogy with a universal set of quantum gates in quantum information processing [19], given by CNOT gates and single qubit gates, this interferometer consists of polarization rotations and U 2 -type transformations, as the one characterizing the interferometer in Fig. 1. For a given sequence of U 2type transformations, the distance l i .
2 , needs to be twice as long as . This condition is not necessary for consecutive U 2 -type transformations along the same channels where no polarization rotation occurs between the output control port of the first U 2 -type interferometer and the input control port of the consecutive one. Finally, polarization correlations in the photon number fluctuations are measured at the N output ports at approximately equal detection times.
Multiphoton networks of the type in Fig. 3 can already be used, in principle, to simulate quantum circuits where a reduced number of CNOT-like operations is needed, as the ones introduced in [20,21]. In particular, our tech-nique can lead to the implementation of entanglementlike correlations for a given number N of channels [22], as demonstrated in Eq. (20) for N = 2, without depending on entanglement processes with the associated decoherence.
We also point out a fascinating phenomena emerging from the result in Eqs. (13) and (12). For the polarization angles φ C = 0, π and φ C = π/2, 3π/2 in Fig. 1 the expectation value of the product of the photon number fluctuations depends on the evolution of the light only through the path configurations (S C , S T ) and (L C , L T ), respectively. Differently, for any polarization angle φ C = π/2, π, 3π/2, with 0 < φ C < 2π, such a value depends on the interference associated with the two multi-photon path configurations (S C , S T ) and (L C , L T ). This novel interference effect emerge from the fundamental nature of Hanbury Brown and Twiss correlations with a thermal source. Interestingly, the paths S C and S T (L C and L T ) are independent on each other before the measurement. However, when correlation measurements in the fluctuation of the number of photons are performed at approximately equal detection times such paths become, on average, correlated. The interference associated with the corresponding path configurations (S C , S T ) and (L C , L T ) depends entirely on the initial polarization angles φ C and φ T . Such a phenomena is at the heart of both the Franson-like correlations (see Eq. (15)) and CNOT-like correlations (see Eq. (20)) demonstrated in this letter.
In conclusion, we have demonstrated how multiphoton interference with thermal light allows us to reproduce CNOT-like operations without the need of entanglement processes with associated decoherence, complex interferometers and indistinguishable single photons. Moreover, the entanglement-like correlations characterizing our scheme can be used, in principle, in high-precision metrology for biomedical applications [23]. On the other hand, the ability to share information between distant parties by simply performing correlated measurements can lead to entanglement-free schemes for secure communications [22]. Finally, our approach can be easily extended from the temporal to the spatial domain by taking advantage of the spatial distribution of a thermal light source. It is also feasible, in principle, with other types of bosonic sources, such as atomic sources where correlated measurements can be performed in the spin degree of freedom.  Here, we calculate the polarization-dependent firstorder correlation function in Eq. (6) for the interfero-metric setup in Fig. 1. By using Eqs. (7) and (8), and defining where in each second equality we used Eq. (4), we obtain the effective field operatorŝ with d = C, T . Indeed, given the presence of only one source, with polarization H at the port S, Eq. (6) can be rewritten as By direct substitution we obtain By using the property [2,13] tr ρ H dω dω e i(ωt1−ω t2)â (H) †

S
(ω)â Finally, by applying the conditions