Experimental quantum teleportation of propagating microwaves

Description

11 MHz by image rejection mixers. The signal is then digitized by a National Instruments NI-5782 transceiver module and processed with a National Instruments PXIe-7975R FPGA module in real time. The digital data processing consists of digital down-conversion (DDC), finite-impulse response (FIR) filtering with a full bandwidth of 400 kHz, and calculation as well as averaging of all quadrature moments I n 1 I m 2 Q k 1 Q l 2 with n + m + k + l ≤ 4 for n, m, k, l ∈ N.
During each measurement cycle, the moments of JPAs 1-4 are used to calculate the squeezing angles γ exp i for each JPA "on the fly" in order to obtain the angle correction δγ i = γ ext i − γ target i which is used to adjust the phase of the microwave pump tone by 2δγ i . Similarly, the phase of the microwave signal tone is adjusted by δθ = θ ext − θ target where θ is the displacement angle of the input coherent state for quantum teleportation. During data analysis on the PC, the recorded traces are separated into different parts according to the DTG timings as indicated in Figure S1. The data within a single averaging cycle consists of 7.7 × 10 8 raw data points per part of the trace and is used to perform a reference state reconstruction for each pulse in order to obtain the signal moments (â † ) nâm with n + m ≤ 4. Finally, this averaging cycle is repeated 10 times. The vector network analyzer, DTG, FPGA card and local oscillator are synchronized to a 10 MHz rubidium frequency standard. The pump microwave sources are daisy chained to the local oscillator with a 1 GHz reference signal.

Supplementary note 2: Theory Model
The quantum teleportation protocol is theoretically modelled in an iterative way, as illustrated in Figure S2. The scheme consists of 3 signal paths where path 1 and path 2 form the input for the TMS generation and the input signal is applied at path 3. The weak thermal environment in each path is modelled by the bosonic operatorf with the average noise photon number JPA 3 Figure S2: Scheme for the simulation of the quantum teleportation protocol. Each iteration step consists either of a unitary operation such as squeezing (Ŝ 12 ,Ŝ 34 ) and beam splitter operations (B 1 ,B 2 ,Ĉ), or of a non-unitary operation such as path losses (L 1 ,L 2 ,L 3 ,L 4 ,L 5 ) and added noise. The squeezing parameter of JPA 1 (JPA 2) is denoted by r 1 (r 2 ) and γ 1 (γ 2 ) is the squeezing angle in radians. The degenerate gain of JPA 3 (JPA 4) is described by G 3 (G 4 ) and γ 3 (γ 4 ) is the respective squeezing angle in radians. The input state is reconstructed at the position indicated by green dot. Red dot denotes the reconstruction point for the teleported output state.
Weak thermal states form inputs for squeezed state generation with JPA 1 and JPA 2 and subsequent path-entanglement at the outputs of the hybrid ring. As a result, the input noise photon numbers n 1 and n 2 consist of the environmental thermal noise and the noise added by the JPAs, which monotonically increases with increasing degenerate gain G i , i ∈ {1, 2}. Based on the existing experimental evidence and related phenomenological theory (25), we assume that this gain dependence obeys a power law The initial thermal coherent stateâ with displacement α in path 3 is modelled by applying the displacement operatorD(α) to the operatorf 3 which describes the initial bosonic noisê The number of noise photons n 3 in the coherent input is quantified by the purity µ = 1/(1 + 2n 3 ). As a result, we formally write the input state as |n 1 ; n 2 ; α, µ ≡ |n 1 ⊗ |n 2 ⊗ |α, µ . In each iteration step, we apply either a unitary squeezing/beam splitter operation or a non-unitary operation to model losses and noise. In the first step, a thermal squeezed state is created by JPA 1 and JPA 2. This operation is modelled with the squeezing operatorŜ 12 , acting on path 1 whereâ j is the signal operator in path j. In the next step, path 1 and path 2 are entangled with a hybrid ring, which we model as a 50:50 beam splitter bŷ The losses are modelled with a beam splitter model according tô with j ∈ {1, 2, 3, 4, 5}, cf. Figure S2. The bosonic bath modesv 3(j−1)+1 model the thermal environment. The power losses are denoted by ε 3(j−1)+3 . The 50:50 beam splitter which entangles path 2 and path 3 is modelled aŝ In the next step, JPAs 3 (JPA 4) acts as a degenerate parametric amplifier to realize the Bell measurement by amplifying orthogonal signal quadratures with respective degenerate gain G 3 (G 4 ). This amplification process is modelled with a squeezing operator, acting on path 2 and path 3, according tô where the squeezing parameters r 3 (r 4 ) are related to the degenerate gain via G 3 = e 2r 3 (G 4 = e 2r 4 ) and the squeezing angle of JPA 3 (JPA 4) is denoted by γ 3 (γ 4 ). In the following, we assume equal degenerate gain for both JPAs, G 3 = G 4 = G. The noise added by JPA 3 (JPA 4) is modelled with a random classical variable ζ 3 (ζ 4 ). We assume that JPA 3 and JPA 4 have equal noise properties, ζ 3 = ζ 4 = ζ and assume ζ to obey a centralized Gaussian distribution with ζζ * = n 34 (G) and Re(ζ 2 ) = Im(ζ 2 ) = n 34 (G)/2. Similar as in Eq. S2, we assume that the JPA noise depends on the gain G as where χ 1 and χ 2 are phenomenological constants characterizing noise properties of the JPAs.
The displacement operation for Bob's quantum state in path 1 is realized by a directional coupler, acting as an asymmetric beam splitter with a reflectivity β (S10) In the ideal case, as it will be shown later, β needs to fulfill the analog teleportation condition where ε models the path losses after JPA 3 and JPA 4. Finally, the teleportation protocol can be expressed by the operator and the final state |Ψ is given by The moments of the output signalb can then be calculated by We assume that all quantum states are Gaussian, which implies that only moments up to second order are required for full state tomography. As a result, it is sufficient to analyze the effect of the teleportation protocol on the respective displacement vector d and covariance matrix V . We realize this by rewriting Eq.S13 specifically for the first-and second-order quadrature moments.
For the initial displacement vector d 0 , we have where n d is the number of displacement photons and ϕ d describes the displacement angle. In the following, I 2 denotes the 2 × 2 identity matrix and 0 2 is the 2 × 2 zero matrix. For the initial covariance matrix, we write The squeeze operation for JPA 1 and JPA 2 is modelled by The beam splitter operations can be expressed as To model the phase-sensitive amplification of JPA 3 and JPA 4, we define the matrices  . (S19) The phase-sensitive amplification of JPA 3 and JPA 4 is then described by the matrices The noise added by JPA 3 and JPA 4 is described by To model the losses, we define the corresponding matrices and with j ∈ {1, 2, 3, 4, 5}. For the directional coupler, we write We define The displacement d after the directional coupler can then be calculated by For the covariance matrix V of the final state, we find the expression where the added matrix A can be analytically expressed as The displacement for the reference input state is calculated by We rewrite d and d i as The covariance matrix of the input state is given by We rewrite the resulting covariance matrices in block form where V j denotes the final covariance matrix in path j and V j denotes the respective input covariance matrix . In the next step, we calculate the Uhlmann-fidelity between the states (d 3 , V 3 ) and (d 1 , V 1 ), which can be expressed for single mode Gaussian states as where Λ = det(V 3 + V 1 ), ∆ = 16(detV 3 − 1/16)(detV 1 − 1/16) and β = d 3 − d 1 . To fit the experimental data, as shown in Fig. 3 in the main text, we use a least-square fit, where the JPA noise coefficients χ 1 , χ 2 , and the environmental temperature T are treated as fit parameters.
As a result, we minimize the squared distance between the experimentally determined fidelities {F (r i , G i )} for squeezing r i and degenerate gain G i and the theoretically predicted values where χ 1 , χ 2 , and T are treated as variables and {x i } is the parameter set including the JPA 1 and JPA 2 squeezing parameters r i , the degenerate gain G i of JPA 3 and JPA 4, the amplification angles γ i as well as the constant system parameters such as path losses, and coupling strength β. Thus, we fit the measurement data by minimizing the function The parameters used in the numerical model are summarized in Tab. S1. The parameters χ 1 , χ 2 , and T are extracted from the fit routine and the losses are estimated from the data sheets of the respective passive microwave components. The loss values ε 3 , ε 6 , ε 7 , ε 10 and ε 13 are set to zero since they have been artificially introduced to keep the block matrix structure which simplifies the numerical calculation. Table S1: Model parameters used for the QT protocol fit in the main article. The loss values ε i are estimated from individual losses of various components. During the fit we have varied three parameteres, χ 1 , χ 2 , and T .  Fig. S2) and the degenerate gain G of JPA 3 and JPA 4. In the following, we show that the quantity Gβ allows us to distinguish between coherent state quantum teleportation and vacuum teleportation. In the limit β → 0, G → ∞, Alice's state detection becomes ideally projective if Gβ = 4. Simultaneously, the regime of vacuum teleportation is characterized by G → 1. For Gβ 4, the measurement apparatus acts as an amplifier, leading to imperfect interference at the directional coupler. To investigate these regimes, we consider the ideal quantum teleportation protocol (i.e. without losses and noise, no thermal noise) with non-ideal state detection and finite two-mode squeezing. We write the ideal protocol in terms of block matrices. In the following, I 2 denotes the 2×2 identity matrix,σ z denotes the Pauli z-matrix, 0 2 is the 2 × 2 zero matrix and 0 = (0, 0) T . The covariance matrix V and the displacement d of the tripartite input state can then be expressed as V = 1 4   I 2 cosh 2rσ z sinh 2r 0 2 σ z sinh 2r I 2 cosh 2r 0 2 The ideal 50:50 beam splitters, acting on Alice's side, are described by To model the phase-sensitive amplification of JPA 3 and JPA 4, we define the matrices For the directional coupler, we write As a result, the ideal teleportation protocol can be expressed as The final covariance matrix according to Eq. S47 is calculated by Since only the cumulants up to second order are significantly different from zero, it is justified to treat the teleported states as Gaussian ones.

Supplementary note 5: Effect of Imperfections on Teleportation Fidelity
Detailed investigation of an error budget of microwave quantum teleportation is a very complicated task due to the highly nonlinear nature of teleportation fidelity as a function of losses, noise, and other imperfections. However, most important and general sources of infidelity can be identified by using the developed theory model. Figure S5 illustrates the impact of two most strongest sources of imperfections on the maximum fidelity of microwave quantum teleportation protocol. The maximum fidelity is defined as F max (S) = max(F (G, S = const)). As a baseline we use the default parameter set which has been obtained by fitting the experimental data with our theory model (see Fig. 3). Next, we consider three particular scenarios: (i) tenfold reduced losses ε i → ε i /10, (ii) ten-fold reduced noise parameters in JPAs χ i → χ i /10, and (iii) combination of both. The corresponding results are shown in Fig. S5. First of all, one can notice that a small improvement in terms of fidelity should be possible even with the current experimental parameters by slightly increasing the squeezing level S, since the black line maximum in Fig. S5 lies slightly outside of the experimentally studied range of the squeezing levels. Next, one can observe that the impact of the transmission losses reduction is greater than the impact from the noise reduction. Technologically, this is an important finding, since transmission losses are often more straightforward to improve.
One particular source of transmission losses is due to finite insertion losses in the current cryogenic hybrid rings. These losses are around 0.4 dB per hybrid ring and unavoidably reduce the amount of quantum correlations, couple extra thermal noise, and therefore, reduce the purity and squeezing of the propagating TMS signals. Based on the predictions of our theory model, one can roughly estimate that the individual hybrid ring insertion losses account for ap-proximately 2 − 4 % of the teleported state infidelities in our experiment (assuming the optimal measurement gain G and squeezing level S). In the end, an obvious improvement of the current experimental set-up would be to use superconducting hybrid rings with negligible insertion losses, which could potentially improve the current teleportation fidelities by an extra 6 − 10 % Optimization of JPA noise is a far more subtle and complicated task. Ultimately, the improvement of both losses and JPA noise properties is required in order to reach fidelites beyond 90 %, as illustrated by cyan line in Fig. S5.  Figure S5: Effect of imperfections in microwave teleportation. Here, we predict the maximum teleportation fidelity as a function of the initial squeezing level S for four different sets of theory model parameters. Default parameters correspond to the set of theory parameters used in Fig. 3 for fitting of the experimental data and serve as a reference. Reduced losses correspond to the ten-fold flat reduction of all transmission losses ε i → ε i /10. Reduced JPA noise corresponds to the ten-fold reduction in terms of noise parameters χ i → χ i /10. Grey area highlights the range of the squeezing levels studied in the current experiment.