Secure quantum remote state preparation of squeezed microwave states

Quantum communication protocols based on nonclassical correlations can be more efficient than known classical methods and offer intrinsic security over direct state transfer. In particular, remote state preparation aims at the creation of a desired and known quantum state at a remote location using classical communication and quantum entanglement. We present an experimental realization of deterministic continuous-variable remote state preparation in the microwave regime over a distance of 35 cm. By employing propagating two-mode squeezed microwave states and feedforward, we achieve the remote preparation of squeezed states with up to 1.6 dB of squeezing below the vacuum level. Finally, security of remote state preparation is investigated by using the concept of the one-time pad and measuring the von Neumann entropies. We find nearly identical values for the entropy of the remotely prepared state and the respective conditional entropy given the classically communicated information and, thus, demonstrate close-to-perfect security.

The Josephson parametric amplifiers (JPA) used in this work consist of a quarter-wavelength superconducting microwave resonator in a coplanar waveguide geometry which is short-circuited to the ground plane via a direct current superconducting quantum interference device (dc-SQUID) 1,2 . The JPAs were designed and fabricated at NEC Smart Energy Research Laboratories, Japan and RIKEN, Japan. The resonator and pump line are patterned into a 50 nm thick Nb film which has been deposited by magnetron sputtering onto 300 µm thick silicon substrates covered by a thermal oxide. The dc-SQUID is fabricated using an aluminum shadow evaporation technique. The flux-tunable resonant frequency f 0 of the JPA can be tuned by an external magnetic flux applied to the dc-SQUID loop via an external coil or via an on-chip antenna acting as the pump line. In order to squeeze incoming vacuum fluctuations or perform phase-sensitive amplification of the mode f 0 , we apply a strong coherent pump tone at frequency f p = 2f 0 to the pump line. The squeezing strength (degenerate gain) and squeezing angle (amplified quadrature) are controlled by the pump amplitude and pump phase, respectively, when the JPA is operated as squeezer (degenerate amplifier). For each JPA, a commercial cryogenic circulator is used to separate the incoming from the outgoing signal (see Supplementary Figure 1). In order to pre-characterize the JPAs and find a suitable working frequency f 0 for all JPAs, we perform spectroscopic measurements 2 . The extracted parameters are summarized in Supplementary Table 1.
Supplementary Table 1: JPA Parameters extracted by fitting the dependence of the resonant frequency f0 of the JPAs on the applied magnetic flux 2 . Here, Ic and EJ = IcΦ0/2π are the critical current and coupling energy of a single Josephson junction, respectively, L loop and βL = 2L loop Ic/Φ0 are the loop inductance and screening parameter of the dc-SQUID, respectively, and fr is the resonant frequency of the bare resonator. The Josephson junctions of the dc-SQUID are assumed to be identical. The external quality factors Qext and internal quality factors Qint are obtained from independent fits of the JPA spectral linewidths 2 . The parameters of JPA 3 are similar to the ones of JPA 1 and JPA 2.

Sample
Ic ( The experimental room temperature and cryogenic setup is shown in Supplementary Figure 1. The digitizer card and the microwave pump sources for each JPA are pulsed with a data timing generator (DTG). JPA 1 and JPA 2 are both temperature stabilized at 50 mK in order to ensure a stable JPA operation and produce squeezed states with orthogonal squeezing angles. The two squeezed states are superimposed by a cryogenic hybrid ring (50:50 beam splitter) in order to produce path-entangled two-mode squeezed (TMS) states at the outputs of the hybrid ring. By operating JPA 1 and JPA 2 at the same squeezing level, we are able to produce symmetric TMS states with local statistics of a thermal state. The entanglement strength of the TMS states is characterized by the two-mode squeezing level of S TMS = 7.1 dB and the negativity criterion of N = 2.2. Additionally, a lower bound on the entanglement of formation, E F = 1.2, can be calculated for our experimental TMS states 3 . One output path of the beam splitter is connected to JPA 3 which is operated as a phase-sensitive amplifier. The JPA 3 output signal is then either detected or sent to a directional coupler which couples to the other hybrid ring output. The first amplification stage of a highelectron-mobility transistor (HEMT) is followed by additional rf-amplifiers which are temperature stabilized with a Peltier cooler. We use a vector network analyzer for the characterization of the JPAs and a heterodyne detection setup for the tomographic measurements. The heterodyne detection setup and data processing is similar to those described in Supplementary References 4,5 where the signal is roughly filtered around the working frequency and down-converted to 11 MHz by image rejection mixers. The signal is then digitized with analog-to-digital (ADC) converters on an Acqiris DC440 card. After sending the digitized data to a computer, digital data processing is performed where digital down-conversion (DDC) and finite-impulse response (FIR) filtering with a full bandwidth of 430 kHz is applied. Finally, all correlation 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 are calculated and averaged. The data within a single averaging cycle consists of 4 × 10 8 averaged sample points per part of the pulse 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. During each measurement cycle, the moments of JPAs 1-3 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 . Finally, the described averaging cycle is repeated 10 times. The vector network analyzer, DTG,   Acqiris 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. The experimental states are reconstructed under the assumption that the states are Gaussian, and thus fully described by their signal moments up to the second order. In order to check for the Gaussianity of the states, we verify that the cumulants of third and fourth order are vanishingly small, as expected for Gaussian states 6 . The cumulants (â † ) nâm are calculated from the signal moments (â † ) nâm according to where ∂ n x is the n-th partial derivative with respect to x and ln is the natural logarithm 7,8 . Figure 2: Scheme for theoretical description of the RSP setup with used parameters. The experimentally varied feedforward gain G f and angle γ f are marked in blue. Due to a carefully designed symmetric implementation in the experiment, the losses 1 = 2 = in both paths before the beam splitter are assumed to be equal and include the insertion loss of the beam splitter. The angles γ1, γ2 and γ f are used in units of radians for all equations.
In order to theoretically describe the remote state preparation (RSP) setup, we use an input-output model for each component as shown in Supplementary Figure 2. JPA 1 and JPA 2 are modeled as squeezers with the same squeezing parameter r 1 = r 2 = r but different squeezing angles γ 1 and γ 2 in order to produce symmetric TMS states after the beam splitter. In the experiment, the pump of JPA 3 leaks through to JPA 1 and JPA 2 which results in a finite crosstalk between JPA 3 and the other two JPAs. Since experimentally the crosstalk to JPA 1 dominates, we approximate the effect of the crosstalk by a linear dependence of γ 1 on the gain G f and angle γ f of JPA 3 where γ 1 is the unperturbed squeezing angle of JPA 1. This approximation is consistent with independent measurements of the crosstalk. The squeezing operatorŜ 12 for JPA 1 and JPA 2 acting on the annihilation operatorsâ i of path 1 (Alice) and path 2 (Bob) is given by 9 The added noise of JPA 1 and JPA 2 is taken into account by an effective thermal state with a noise photon number n 1 = n 2 = n incident to the JPAs. In order to describe the action of JPA 3, we assume that classical noise is added to the JPA input signal followed by ideal phase-sensitive amplification where G f is related to r f as G f = e 2r f and θ f is the theoretically optimal JPA 3 amplification angle. The classical noise is described by the complex Gaussian random variable ζ with ζ = 0, ζζ * = n f and Re(ζ) 2 = Im(ζ) 2 = n f /2, where n f is the effective thermal noise photon number. In general, the JPA noise is gain dependent which we take into account by a linear dependence on G f for JPA 3. For that, we use n f = n f G f , where n f is a proportionality constant. Losses and η i of the microwave components are modeled with a beam splitter 10 wherev i is the operator describing the environment for path i. The environment can be safely approximated to be in the vacuum state due to the low temperature of the lossy components in the experiment. The hybrid ring is described by a 50:50 beam splitter 11 The displacement on Bob's side is implemented with a directional coupler and is described as an asymmetric beam where τ = 1 − 10 β/10 is the transmissivity and β is the coupling in decibel.
In order to describe the realistic setup, we need to take the electrical length of the different components into account. The total electrical lengths as well as different path lengths after the beam splitter are compensated with a rotationR by the angle θ rp of the final remotely prepared state on Bob's sidê With the operator definitions in Supplementary Equations (3)-(9), we can write the overall RSP protocol as where n is the noise photon number of JPA 1 and JPA 2, and |Ψ is the final state on both paths. The moment matrices for both paths of the final state are calculated as where (b † ) nbm 1 are the moments of the second directional coupler output signal and (b † ) nbm 2 are the moments of the remotely prepared state. With the definition of the quadraturesq = (b +b † )/2 andp = (b −b † )/2i, the moments (b † ) nbm 2 are used to calculate the squeezing angle γ rp , squeezed variance σ 2 s and antisqueezed variance σ 2 a of the remotely prepared state as where arg(·) is the argument of a complex number and the first order moments are taken to be zero. These quantities are then fitted simultaneously to the corresponding quantities of the experimental remotely prepared states (see Supplementary Figure 3 for antisqueezed variance). We are able to describe the RSP protocol presented in the main article with the parameters shown in Supplementary Table 2. We emphasize that the bare model only requires three fitting parameters (n, r, n f ) in order to obtain a good fit when estimating the remaining parameters from independent measurements. Including β, θ f , θ rp , and the crosstalk parameters (γ (0) 1 , κ, λ) as fitting parameters, only slightly improves the quantitative agreement between the experiment and the theory. In order to derive equation (1) in the main article, we choose γ 1 = γ f = 0 • , γ 2 = θ f = 90 • , same losses after the beam splitter (η 1 = η 2 ), neglect the effect of the electrical path lengths (θ rp = 0), and define the total loss χ 1 = χ 2 = χ, where Furthermore, we do not consider the experimental crosstalk (κ = λ = 0). The protocol works optimally for fixed resources if a state with the highest purity is   remotely prepared. In the limit of high JPA 1 and JPA 2 squeezing, r 1, we reach this optimal point for G f = τ /(1−τ ) and obtain for the optimally remotely prepared state by using Supplementary Equations (12)-(14) In general, the optimal JPA 3 gain depends on r in a nontrivial manner and converges to G f = τ /(1 − τ ) for r → ∞. However, the latter expression offers a good approximation to the optimal JPA 3 gain even for r ≈ 1 since the deviation ofγ rp ,σ 2 s , andσ 2 a between the optimal JPA 3 gain and G f = τ /(1−τ ) is below 1% for the parameters in Supplementary  Table 2.

SUPPLEMENTARY NOTE 4: PHASE SPACE OF PREPARABLE STATES
The model described in the previous section allows us to theoretically investigate the phase space of the preparable states of our RSP protocol in the phase space spanned by (S rp , γ rp ) and (S rp , A rp ). For this purpose, we use the parameters from Supplementary Table 2 and calculate the contour around the remotely prepared states for the experimental range of JPA 3 gain G f and amplification angle γ f (green contour in Supplementary Figure 4a,b). Alternatively, we use an iterative method to calculate the maximum error contour (blue contour in Supplementary  Figure 4a,b). Here, we randomly select a value from the 95% confidence intervals of each fitting parameter and calculate the resulting contour. If the current contour lies partly or fully outside the maximum error contour, the latter is expanded so that it includes the current contour. The process is repeated until the change of the area of the maximum error contour is negligible between iterations. We observe that the direct contour does not include all experimentally prepared states but shows a good qualitative agreement. The maximum error contour includes all measured remotely prepared states. We note that all remotely prepared states inside the contour can be continuously prepared. However, the position in the phase space does not uniformly depend on G f and γ f . Since we select a finite and uniform step size of G f and γ f in the experiment, the measured remotely prepared states do not uniformly occupy the phase space. In Supplementary Figure 4c,d, we show the expected contours of the accessible prepared states upon reducing the JPA noise photon numbers by one order of magnitude and the different losses ( , η 1 and η 2 ) to 0.1 dB each. The range of preparable A rp and γ rp is only slightly affected by the optimized parameters while the squeezing level S rp is significantly increased.

SUPPLEMENTARY NOTE 5: FEEDFORWARD SIGNAL
The feedfoward signal is characterized by toggling the cryogenic switch into position B (see Supplementary Figure 1) and measure both the signal from JPA 3 (Alice's side) as well as the signal on Bob's side while all JPAs are pumped. The squeezing level of JPA 1 and JPA 2 for these measurements is S = 7.3 dB which results in an entangled state with a negativity kernel 6,13 N k = 1.8 after the beam splitter. The JPA 3 amplification angle is fixed to the optimal angle γ f = 0 • . As shown in Supplementary Figure 5, we observe no entanglement (N k ≤ 0) between Alice and Bob after the local amplification for G f ≥ 11 dB. The feedforward signal is squeezed below the vacuum for low G f and becomes non-squeezed above G f 13 dB. Our theory model and experimental evidence show that the deamplified, and possibly squeezed, quadrature has a negligible effect on the prepared state. This can be understood by considering that the feedforward signal is only weakly coupled to Bob's part of the entangled state by the directional coupler. Therefore, only the strongly amplified quadrature in the feedforward signal will affect the prepared state on Bob's side. We consider the feedforward signal as classical if it has a positive Wigner function, is not squeezed below the vacuum and is not entangled with the signal on Bob's side. Therefore, all feedforward signals with G f ≥ 13 dB are classical. For G f < 13 dB, the information about the to-be-prepared state in the feedforward signal can be described classically as well since it is only encoded in the strongly amplified quadrature which, on its own, does not show any quantum signatures. The von Neumann entropy H(X) = − Tr(ρ x logρ x ) of a quantum state X with density matrixρ x is the quantum information analogue of the entropy used in thermodynamics (up to a factor of the Boltzmann constant k B ). For a Gaussian state, H(X) can be calculated from its covariance matrix V . The von Neumann entropy of a single-mode Gaussian state is given by 14 where f (x) = 2x + 1 2 log 2x + 1 2 − 2x − 1 2 log 2x − 1 2 . For a two-mode Gaussian state AB, the covariance matrix can be expressed in the form where A, B and C are 2 × 2 matrices describing the local state A, local state B and cross-correlations between both parties, respectively. From V , one can calculate the two symplectic eigenvalues of the bipartite Gaussian state where ∆ = det A + det B + 2det C. The joint entropy of the whole bipartite state is given by is called the conditional entropy. If both parties are correlated, knowledge about B will reveal information about A, and thus decrease its entropy, H(A|B) < H(A).