Multi-core fiber integrated multi-port beam splitters for quantum information processing: supplementary material

Multi-core fiber integrated multi-port beam splitters for quantum information processing: supplementary material J. CARIÑE1,2, G. CAÑAS3, P. SKRZYPCZYK4, I. ŠUPI 5, N. GUERRERO1,2, T. GARCIA1,2, L. PEREIRA1,2, M. A. S.-PROSSER6, G. B. XAVIER7, A. DELGADO1,2, S. P. WALBORN1,2,8, D. CAVALCANTI5, AND G. LIMA1,2 1Departamento de Física, Universidad de Concepción, 160-C Concepción, Chile 2Millennium Institute for Research in Optics, Universidad de Concepción, 160-C Concepción, Chile 3Departamento de Física, Universidad del Bio-Bio, Avenida Collao 1202, Concepción, Chile 4H. H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol, BS8 1TL, United Kingdom 5ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels, Barcelona, Spain 6Departamento de Ciencias Físicas, Universidad de la Frontera, Temuco, Chile 7Institutionen för Systemteknik, Linköpings Universitet, 581 83 Linköping, Sweden 8Instituto de Física, Universidade Federal do Rio de Janeiro, Caixa Postal 68528, Rio de Janeiro, Rio de Janeiro 21941-972, Brazil


PROCESS TOMOGRAPHY
In this section we explain the process tomography method introduced in [1], which we adopted for characterizing the N × N MCF MBS. A unitary matrix is given by U = ∑ jk u jk e iφ jk |j k|. (1) where 0 ≤ u jk ≤ 1 and 0 ≤ φ jk < 2π. The protocol is based on measuring the probabilities I jk of detecting the photons at the port represented by |j when the core-mode state |k was sent through the MBS. In this way, we determinate the split ratios r(j|k) = I jk ∑ k I jk . ( The parameters u jk are the square root of the split ratios, On the other hand, the phases φ jk are determined by sending states of the form through the MBS. The probability distribution p(j|k) to detect photons at the output port |k is given by Hence, since we know the coefficient u jk form the split ratios, by recording the probabilities p(k|j) with respect to ϕ, one can obtain the relative phases φ kj − φ k1 . However, note that only N 2 − 2N + 1 of the phases φ jk are physically significant since 2N − 1 phases can be included into the basis vectors or externally controlled by phase modulators (PM) [1][2][3]. Therefore, without loss of generality, we can consider that φ 0k = φ j0 = 0, or equivalently, the matrix U has real border. The procedure allows us to determine uniquely the phases φ jk , and one can obtain an experimentally estimated matrixŨ of the MCB. Nonetheless, due to inherent experimental errors, this matrix is never unitary. In order to obtain the unitary matrix describing the N × N MCF MBS, one can optimize a cost function of the experimental data. For this purpose, we use the fidelity between two matrices [4,5], given by This function is equivalent to the fidelity between the quantum states corresponding to A and B by the Choi-Jamiolkowski map. The unitary estimate of the MBS is obtained by solving the following optimization problem with the restriction that V is a real-border unitary matrix. This can be converted into an unconstrained optimization problem using the fact that, for an arbitrary complex matrix Z, we have that Z(Z † Z) −1/2 is always a unitary matrix. Therefore, this optimization can be solved numerically by standard optimization methods.

4X4 MCF MBS
The 4x4 MCF MBS is illustrated in Fig. 1 b) of the main text. The use of the protocol described before to characterize the 4x4 MCF MBS gives the following experimental estimate: while the corresponding unitary one iŝ The fidelity between the experimental estimate and the unitary estimate is Note that the unitary estimate is almost a symmetric unitary matrix, or equivalently, the absolute value of each coefficient of the matrix is approximately 1/2. Comparing the unitary estimate with the symmetric unitary matrix with φ = 0, we have the fidelity

7X7 MCF MBS
A diagram of the geometry of the 7 × 7 MCF MBS is shown in Fig. 1. Using a procedure analogous to the 4 × 4 case, we obtain that experimental estimated matrixŨ 7 for the 7 × 7 MCF MBS: Following the procedure outlined above, the corresponding unitary matrixÛ 7 is: The fidelity between the experimental matrix and the unitary matrix is F(Ũ,Û) = 0.992 ± 0.008. (13) Note that this matrix is not symmetric, its coefficients have absolute value different than 1/ √ 7, and this is what one intuitively expects based on simple considerations concerning the geometry of the cores in the fiber. Nonetheless, recent results [6] have shown that even non-symmetric N-port beam splitter devices can serve as primitives for the construction of a universal quantum circuit that implements any N × N unitary.

MATRIX ERROR ANALYSIS
We perform Monte Carlo simulations to quantify the error of the estimated matrices. We employ the Gaussian distribution N(µ, σ) for this task, where µ is the mean and σ is the standard deviation. Considering the error as 3 times the standard deviation of the Gaussian distribution, approximately 99.7% of the realizations are inside of the interval µ ± 3σ. Experimentally, we measure the intensities I jk ± ∆I jk and the phases φ jk ± ∆φ jk , with ∆I jk and ∆φ jk being their respective experimental errors. Thereby, the simulated split ratios and phases are given byĪ respectively. We generate a sample of 10 5 MBS matricesŨ andÛ independently, and with them we calculate the average fidelity and their respectively errors, which were consider as 3 times the standard deviations.

QUANTUM GATES USING MBS DEVICES
As discussed in the main text, a universal circuit for the generation of N × N unitaries can be built using multi-arm interferometers composed of a mesh of conventional 2 × 2 50:50 beamsplitters [7,8], requiring N(N − 1) of them, in addition to phase shifters. In the configuration optimized for losses and efficiency, these N(N − 1) devices are arranged into a sequence of 2N layers, concatenated with layers of phase shifters [8]. Recently, the replacement of the 2 × 2 devices with MBS devices has been considered [6,9]. On those works it was shown, based on numerical evidence, that N + 1 layers of N × N MBSs, concatenated with N + 2 layers of phase shifters P N acting on the N modes, namely P N U N P N . . . U N P N , is sufficient to generate any N × N unitary up to high numerical precision. Moreover, Ref. [6] gave positive numerical results for randomly chosen N × N devices. Thus, we expect that both the 4 × 4 and 7 × 7 MCF MBS devices will serve as optical primitives for the construction of efficient photonic quantum information circuits.
Here we experimentally implemented the circuit U 4 P 4 U 4 , corresponding to the operation where A = e iφ 0 + e iφ 1 + e iφ 2 + e iφ 3 , B = e iφ 0 + e iφ 1 − e iφ 2 − e iφ 3 , C = e iφ 0 − e iφ 1 − e iφ 2 + e iφ 3 , D = e iφ 0 − e iφ 1 + e iφ 2 − e iφ 3 , and φ j is the total phase applied in mode j. To give very simple examples of some operations that can be realized, we can see that by choosing appropriate phases, one can set all but one of these elements (A, B, C, D) to zero, resulting in transformations of the form I 4 , X 10 X 23 , X 20 X 13 , X 03 X 12 , where I 4 is the 4 × 4 identity matrix and X jk is a 2 × 2 Pauli X operator acting on the subspace defined by basis states |j and |k . We note that X 03 X 12 is equivalent to the shift operator |j → |j + 1 mod 4 , if one exchanges modes 1 and 3 at the output. We also note that this type of optical circuit was used recently to implement a 4-path quantum switch, where in this case the matrix elements above are operators acting on the polarization degree of freedom of the photon [10].

MDI RNG PROTOCOL DETAILS
In a MDI RNG scenario, an end-user possesses a characterised preparation device P used to prepare a set of quantum states {ω x }, which are measured by the uncharacterized measuring device M, leading to a classical outcome a. It is assumed that an eavesdropper, Eve, can be quantum-correlated with M, by holding half of an entangled state ρ AE , the other half of which is inside the device. M performs a measurement (which can be known by Eve) on the input state ω x and a part of ρ AE , while Eve uses a positive operator valued measure (POVM) N E e to measure her part of ρ AE . After many uses of the device the user estimates the probabilities p(a|ω x ). In [11] it was shown that the maximal guessing probability of Eve, for a given input x * , compatible with p(a|ω x ), is bounded by the solution of the following semi-definite program (SDP) [12] where the maximisation is over the POVM N = {N ae } ae and probability distribution q = {q(e)} e , and the second constraint encodes no-signalling between the measuring device and Eve (see [11] for details). The amount of randomness that it certified is given by the min-entropy, assuming that Eve carries out individual attacks (i.e. does not share entanglement between rounds). In order to account for effects due to finite statistics, we use the Chernoff-Hoeffding tail inequality [13]. It asserts that with high probability where ξ(a|ω x ) are the frequencies observed in the experiment, and t x ( ) = log(1/ ) 2n x depends on a confidence parameter and the total number of measurement rounds n x in which the input was ω x . The confidence parameter corresponds to the probability that Eq. (19) is not satisfied. A typical choice is to take = 10 −9 . Using this, Eq. (17) can be strengthened, so that it depends only upon the observed frequencies, namely

EXPERIMENTAL STATES
In this section we give the states that are prepared in the experiment when two photons are emitted at the source. In the main text these states are labelled φ (2) x . In the main text, the notation used it that |x referred to one photon in mode x. Here, since we want to have multiple photons in a given mode, we will use the notation |1000 to denote one photon in mode zero, |0200 to denote 2 photons in mode 1, etc. With this notation, the first four states have two photons in each mode, namely while for the final state we have that

THEORETICAL BOUNDS ON RANDOMNESS GENERATION
Our MDI RNG is designed for an ideal n = 1 Fock state input. In this case there are 4 detection events, which gives an upper limit of log 4 = 2 bits of randomness/detected photon. In our experiment, we approximate this Fock state with a weak coherent state. An eavesdropper can take advantage of the multi-photon components, which decreases the amount of private randomness that can be produced. We partially overcome this by using the multi-photon detection events in addition to single-photon events to generate randomness. As discussed in the manuscript, we truncate at n=2 photon events, resulting in a state given by Eq. (7) of the paper, with n = 1, 2 photon components. Moreover, we consider the 4 single-photon detection events, as well as the 6 coincidence (two-photon) events at different detectors. If these 10 events were equiprobable, we could thus have log 10 = 3.3 bits of randomness/detection event. However, the two-photon detections are much less likely than the single photon events. Taking into account the actual probabilities p(1) and p(2) in our experiment, as well as the detection efficiency η = 0.1, we have an upper bound of H the min ≈ 2.03 for µ = 0.4 and H the min ≈ 2.02 for µ = 0.2, where we have taken Eve's guessing probability P g (x) to be equal to the largest recorded event probability.