



Josephson Parametric Amplifiers
===============================

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)@Yamamoto2008 [@Pogorzalek2017]. 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 thick Nb film which has been deposited by magnetron sputtering onto $\SI{300}{\mu m}$ 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 $f_\mathrm{p}\,{=}\,2f_0$ 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 Fig. \[fig:ExpSetup\]). In order to pre-characterize the JPAs and find a suitable working frequency $f_0$ for all JPAs, we perform spectroscopic kevin00@example.net. The extracted parameters are summarized in Tab. \[tab:JPA\_params\].

\[h\]

<span>l\*<span>8</span><span>c</span>r</span> Sample & $I_\mathrm{c}$ ($\mu$A) & $\beta_\mathrm{L}$& $L_\mathrm{loop}$ (pH) & $f_\mathrm{r}/2\pi$ (GHz) & $E_\mathrm{J}/h$ ($\mathrm{THz}$) & $Q_\mathrm{ext}$ & $Q_\mathrm{int}$\
JPA1 & 2.45 & 0.09 & 35.8 & 5.808 & 1.22 & 300–360 & $>$30000\
JPA2 & 2.41 & 0.10 & 40.7 & 5.838 & 1.20 & 240–260 & $>$30000\

\[tab:JPA\_params\]

Experimental setup
==================

![Experimental scheme for the measurements. The JS3-25-8P rf-amplifiers are removed for the measurements concerning the quantum one-time pad. The RSP and quantum one-time pad measurements are performed with the cryogenic switch in position A. The intertwined lines between the outputs of the hybrid ring symbolize the entanglement. JPA3 and the directional coupler are separated by of superconducting cable.<span data-label="fig:ExpSetup"></span>](Supp_Fig_ExpSetup.pdf){width="88.00000%"}

The experimental room temperature and cryogenic setup is shown in Fig. \[fig:ExpSetup\]. The digitizer card and the microwave pump sources for each JPA are pulsed with a data timing generator (DTG). JPA1 and JPA2 are both temperature stabilized at 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 JPA1 and JPA2 at the same squeezing level, we are able to produce symmetric TMS states with local statistics of a thermal state. One output path of the beam splitter is connected to JPA3 which is operated as a phase-sensitive amplifier. The JPA3 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 high-electron-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 Refs.  where the signal is roughly filtered around the working frequency and down-converted to 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 is applied. Finally, all correlation quadrature moments $\langle I_1^n I_2^m Q_1^k Q_2^l\rangle$ with $n\,{+}\,m\,{+}\,k\,{+}\,l\,{\leq}\,4$ for $n,m,k,l\in \mathbb{N}$ are calculated and averaged. The data within a single averaging cycle consists of $~4\times10^8$ 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 $\langle(\hat{a}^\dagger)^n\hat{a}^m\rangle$ with $n\,{+}\,m\,{\leq}\,4$. During each measurement cycle, the moments of JPAs1-3 are used to calculate the squeezing angles $\gamma_i^\mathrm{exp}$ for each JPA "on the fly" in order to obtain the angle correction $\delta\gamma_i\,{=}\,\gamma_i^\mathrm{ext}-\gamma_i^\mathrm{target}$ which are used to adjust the phase of the microwave pump tone by $2\delta\gamma_i$. Finally, the described averaging cycle is repeated 10 times. The vector network analyzer, DTG, Acqiris card and local oscillator are synchronized to a rubidium frequency standard. The pump microwave sources are daisy chained to the local oscillator with a 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 kevin00@example.net. The cumulants $\langle\langle(\hat{a}^\dagger)^n\hat{a}^m\rangle\rangle$ are calculated from the signal moments $\langle(\hat{a}^\dagger)^n\hat{a}^m\rangle$ according to $$\langle\langle(\hat{a}^\dagger)^n\hat{a}^m\rangle\rangle = \partial^n_x \partial^m_y \mathrm{ln} \sum_{\alpha,\beta} \left. \frac{\langle (\hat{a}^\dagger)^\alpha\hat{a}^\beta \rangle x^\alpha y^\beta}{\alpha !\beta !}\right\rvert_{x=y=0}\,,$$ where $\partial^n_x$ is the n-th partial derivative with respect to $x$ and ln is the natural logarithm@Eichler2014 [@Xiang2018].

Theoretical modeling and fitting procedure {#sec:theory}
==========================================

![Scheme for theoretical description of the RSP setup with used parameters. The experimentally varied feedforward gain $G_\mathrm{f}$ and angle $\gamma_\mathrm{f}$ are marked in blue. Due to a carefully designed symmetric implementation in the experiment, the losses $\epsilon_1\,{=}\,\epsilon_2\,{=}\,\epsilon$ in both paths before the beam splitter are assumed to be equal and include the insertion loss of the beam splitter. The angles $\gamma_1$, $\gamma_2$ and $\gamma_\mathrm{f}$ are used in units of radians for all equations.<span data-label="fig:RSP_Scheme"></span>](Supp_Fig_ModelScheme.pdf)

In order to theoretically describe the remote state preparation (RSP) setup, we use an input-output model for each component as shown in Fig.\[fig:RSP\_Scheme\]. JPA1 and JPA2 are modeled as squeezers with the same squeezing parameter $r_1\,{=}\,r_2\,{=}\,r$ but different squeezing angles $\gamma_{1}$ and $\gamma_{2}$ in order to produce symmetric TMS states after the beam splitter. In the experiment, the pump of JPA3 leaks through to JPA1 and JPA2 which results in a finite crosstalk between JPA3 and the other two JPAs. Since experimentally the crosstalk to JPA1 dominates, we approximate the effect of the crosstalk by a linear dependence of $\gamma_{1}$ on the gain $G_\mathrm{f}$ and angle $\gamma_\mathrm{f}$ of JPA3 $$\gamma_{1}=\gamma_1^{(0)} + \kappa G_\mathrm{f} + \lambda \gamma_\mathrm{f} \,,
 \label{eqn:gamma1}$$ where $\gamma_1^{(0)}$ is the unperturbed squeezing angle of JPA1. This approximation is consistent with independent measurements of the crosstalk. The squeezing operator $\hat{S}_{12}$ for JPA1 and JPA2 acting on the annihilation operators $\hat{a}_i$ of path 1 (Alice) and path 2 (Bob) is given by@Scully1997 $$\hat{S}_{12}^\dagger \begin{pmatrix}  \hat{a}_1   \\ \hat{a}_2  \end{pmatrix} \hat{S}_{12} =  \begin{pmatrix} \hat{a}_1\mathrm{cosh}\,r - \hat{a}_1^\dagger e^{-2i\gamma_1} \mathrm{sinh}\,r  \\ \hat{a}_2\mathrm{cosh}\,r - \hat{a}_2^\dagger e^{-2i\gamma_2} \mathrm{sinh}\,r  \end{pmatrix} \, .
\label{eqn:S12}$$ The added noise of JPA1 and JPA2 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 JPA3, we assume that classical noise is added to the JPA input signal followed by ideal phase-sensitive amplification $$\hat{S}_{3}^\dagger \begin{pmatrix}  \hat{a}_1   \\ \hat{a}_2  \end{pmatrix} \hat{S}_{3} =  \begin{pmatrix} \left(\hat{a}_1+\zeta\right)\mathrm{cosh}\,r_\mathrm{f} - \left(\hat{a}_1^\dagger+\zeta^*\right) e^{-2i(\gamma_\mathrm{f}+\theta_\mathrm{f})} \mathrm{sinh}\,r_\mathrm{f}  \\ \hat{a}_2 \end{pmatrix} \, ,$$ where $G_\mathrm{f}$ is related to $r_\mathrm{f}$ as $G_\mathrm{f}\,{=}\,e^{2r_\mathrm{f}}$ and $\theta_\mathrm{f}$ is the theoretically optimal JPA3 amplification angle. The classical noise is described by the complex Gaussian random variable $\zeta$ with $\langle\zeta\rangle\,{=}\,0$, $\langle\zeta\zeta^*\rangle\,{=}\,n_\mathrm{f}$ and $\langle \mathrm{Re}(\zeta)^2\rangle\,{=}\,\langle \mathrm{Im}(\zeta)^2\rangle\,{=}\,n_\mathrm{f}/2$, where $n_\mathrm{f}$ 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 $G_\mathrm{f}$ for JPA3. For that, we use $n_\mathrm{f}\,{=}\,n'_\mathrm{f}G_\mathrm{f}$, where $n'_\mathrm{f}$ is a proportionality constant.

Losses $\epsilon$ and $\eta_i$ of the microwave components are modeled with a beam splitter@DiCandia2015 $$\hat{L}_1^\dagger \begin{pmatrix}  \hat{a}_1   \\ \hat{a}_2  \end{pmatrix} \hat{L}_1 =  \begin{pmatrix} \sqrt{1-\epsilon}\hat{a}_1 + \sqrt{\epsilon}\hat{v}_1  \\ \sqrt{1-\epsilon}\hat{a}_2 + \sqrt{\epsilon}\hat{v}_2  \end{pmatrix} \, ,$$ $$\hat{L}_2^\dagger \begin{pmatrix}  \hat{a}_1   \\ \hat{a}_2  \end{pmatrix} \hat{L}_2 =  \begin{pmatrix} \sqrt{1-\eta_1}\hat{a}_1 + \sqrt{\eta_1}\hat{v}_1  \\ \sqrt{1-\eta_2}\hat{a}_2 + \sqrt{\eta_2}\hat{v}_2  \end{pmatrix} \, ,$$ where $\hat{v}_i$ 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@Braunstein2005 $$\hat{B}^\dagger \begin{pmatrix}  \hat{a}_1   \\ \hat{a}_2  \end{pmatrix} \hat{B} =  \frac{1}{\sqrt{2}}\begin{pmatrix} \hat{a}_1 + \hat{a}_2 \\ -\hat{a}_1 + \hat{a}_2  \end{pmatrix} \, .$$

The displacement on Bob's side is implemented with a directional coupler and is described as an asymmetric beam splitter@Paris1996 $$\hat{C}^\dagger \begin{pmatrix}  \hat{a}_1   \\ \hat{a}_2  \end{pmatrix} \hat{C} =  \begin{pmatrix} \sqrt{\tau}\hat{a}_1 + \sqrt{1-\tau}\hat{a}_2 \\ -\sqrt{1-\tau}\hat{a}_1 + \sqrt{\tau}\hat{a}_2  \end{pmatrix} \, ,$$ where $\tau=1-10^{\beta/10}$ is the transmissivity and $\beta$ 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 rotation $\hat{R}$ by the angle $\theta_\mathrm{rp}$ of the final remotely prepared state on Bob's side $$\hat{R}^\dagger \begin{pmatrix}  \hat{a}_1   \\ \hat{a}_2  \end{pmatrix} \hat{R} =  \begin{pmatrix} \hat{a}_1 \\ \hat{a}_2 e^{-i\theta_\mathrm{rp}}  \end{pmatrix} \, .
\label{eqn:R}$$ With the operator definitions in equations (\[eqn:S12\])–(\[eqn:R\]), we can write the overall RSP protocol as $$|\Psi\rangle=\hat{R} \, \hat{C} \, \hat{S}_3 \, \hat{L}_2 \, \hat{B} \, \hat{L}_1 \, \hat{S}_{12}|n,n\rangle\, ,$$ where $n$ is the noise photon number of JPA1 and JPA2, and $|\Psi\rangle$ is the final state on both paths. The moment matrices for both paths of the final state are calculated as $$\label{eq:RSPstate}
\begin{pmatrix} \langle(\hat{b}^\dagger)^n \hat{b}^m \rangle_1 \\ \langle(\hat{b}^\dagger)^n \hat{b}^m \rangle_2 \end{pmatrix}=\langle\Psi|\begin{pmatrix}(\hat{a}^\dagger)_1^n \hat{a}_1^m \\ (\hat{a}^\dagger)_2^n \hat{a}_2^m \end{pmatrix} |\Psi\rangle\, ,$$ where $\langle(\hat{b}^\dagger)^n \hat{b}^m \rangle_1$ are the moments of the second directional coupler output signal and $\langle(\hat{b}^\dagger)^n \hat{b}^m \rangle_2$ are the moments of the remotely prepared state. With the definition of the quadratures $\hat{q}=(\hat{b}+\hat{b}^\dagger)/2$ and $\hat{p}=(\hat{b}-\hat{b}^\dagger)/2i$, the moments $\langle(\hat{b}^\dagger)^n \hat{b}^m \rangle_2$ are used to calculate the squeezing angle $\gamma_\mathrm{rp}$, squeezed variance $\sigma_\mathrm{s}^2$ and antisqueezed variance $\sigma_\mathrm{a}^2$ of the remotely prepared state as $$\begin{aligned}
\gamma_\mathrm{rp} &= -\frac{1}{2}\mathrm{arg}\left(-\langle\hat{b}^2 \rangle_2\right)\, , \label{eq:RSPgamma}\\
\sigma_\mathrm{s}^2 &= \frac{1}{4}\left( \langle\hat{b}^2 \rangle_2 e^{2i\gamma_\mathrm{rp}}+\langle(\hat{b}^\dagger)^2 \rangle_2 e^{-2i\gamma_\mathrm{rp}}+2\langle\hat{b}^\dagger\hat{b}\rangle_2+1 \right)\, , \label{eq:RSPsigma_s}\\
\sigma_\mathrm{a}^2 &= \frac{1}{4}\left( -\langle\hat{b}^2 \rangle_2 e^{2i\gamma_\mathrm{rp}}-\langle(\hat{b}^\dagger)^2 \rangle_2 e^{-2i\gamma_\mathrm{rp}}+2\langle\hat{b}^\dagger\hat{b}\rangle_2+1 \right)\, , \label{eq:RSPsigma_a}\end{aligned}$$ where $\mathrm{arg}(\cdot)$ 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 Fig. \[fig:AntiSq\] for antisqueezed variance). We are able to describe the RSP protocol presented in the main article with the parameters shown in Tab. \[tab:Fit\_Params\]. We emphasize that the bare model only requires three fitting parameters ($n$, $r$, $n_\mathrm{f}'$) in order to obtain a good fit when estimating the remaining parameters from independent measurements. Including $\beta$, $\theta_\mathrm{f}$, $\theta_\mathrm{rp}$, and the crosstalk parameters ($\gamma^{(0)}_1$, $\kappa$, $\lambda$) 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 $\gamma_1\,{=}\,\gamma_\mathrm{f}\,{=}\,\SI{0}{\degree}$, $\gamma_2\,{=}\,\theta_\mathrm{f}\,{=}\,\SI{90}{\degree}$, same losses after the beam splitter ($\eta_{1}\,{=}\,\eta_{2}\,{=}\,\eta$), and neglect losses before the beam splitter ($\epsilon\,{=}\,0$) as well as the effect of the electrical path lengths ($\theta_\mathrm{rp}\,{=}\,0$). Furthermore, we do not consider the experimental crosstalk ($\kappa\,{=}\,\lambda\,{=}\,0$). The protocol works optimally for fixed resources if a state with the highest purity is remotely prepared. In the limit of high JPA1 and JPA2 squeezing, $r\,{\gg}\,1$, we reach this optimal point for $G_\mathrm{f}\,{=}\,\tau/(1-\tau)$ and obtain for the optimally remotely prepared state by using equations (\[eq:RSPgamma\])-(\[eq:RSPsigma\_a\]) $$\begin{aligned}
\tilde{\gamma}_\mathrm{rp} &= \gamma_1 \,, \\
\tilde{\sigma}_\mathrm{s}^2 &= \frac{1}{4}\left[2(1+2n)e^{-2r}(1-\eta)\tau+2(\eta+n_\mathrm{f})\tau\right]\,, \label{eq:sigmaOptSqueezed} \\ 
\tilde{\sigma}_\mathrm{a}^2 &= \frac{(1-\eta)(1+2n)}{4 \tau\left(1+2n_\mathrm{f}\right)^2}\left\lbrace
\frac{\left(2\tau n_\mathrm{f}+1\right)^2}{2 e^{-2r}}\,{+}\,\frac{\left(2\tau n_\mathrm{f}+2\tau - 1 \right)^2}{2 e^{2 r}}\,{+}\,\frac{4 \eta  \tau ^2 n_\mathrm{f}^2+2n_\mathrm{f}\left[(1-\tau)^2+2\eta\tau^2 \right]+\eta (2\tau^2-2\tau+1)}{(1-\eta)(1+2n)}\right\rbrace\,. \label{eq:sigmaOptAnti}\end{aligned}$$ In general, the optimal JPA3 gain depends on $r$ in a nontrivial manner and converges to $G_\mathrm{f}\,{=}\,\tau/(1-\tau)$ for $r\,{\to}\,\infty$. However, the latter expression offers a good approximation to the optimal JPA3 gain even for $r\,{\approx}\,1$ since the deviation of $\tilde{\gamma}_\mathrm{rp}$, $\tilde{\sigma}_\mathrm{s}^2$, and $\tilde{\sigma}_\mathrm{a}^2$ between the optimal JPA3 gain and $G_\mathrm{f}\,{=}\,\tau/(1-\tau)$ is below $1\%$ for the parameters in Tab. \[tab:Fit\_Params\]

![**a** and **b**, Antisqueezed variance $A_\mathrm{rp}$ of the remotely prepared states as a function of the feedforward parameters for experiment and fit, respectively. The optimal point is marked by the blue star.<span data-label="fig:AntiSq"></span>](Supp_Fig_AntiSq.pdf){width="70.00000%"}

<span>l\*<span>13</span><span>c</span>r</span> $n$ & $r$ & $\gamma_{1}^{(0)}$ ($^\circ$) & $\gamma_{2}$ ($^\circ$) & $n'_\mathrm{f}$ & $\beta$ (dB) & $\epsilon$ (dB) & $\eta_{1}$ (dB) & $\eta_{2}$ (dB) & $\theta_\mathrm{f}$ ($^\circ$) & $\theta_\mathrm{rp}$ ($^\circ$) & $\kappa$ ($^\circ$) & $\lambda$\
$0.04$ & $1.20$ & $49.6$ & $135.0$ & $0.0059$ & $-14.6$ & $0.72$ & $0.35$ & $0.30$ & $136.5$ & $68.5$ & $-0.17$ & $0.02$\

\[tab:Fit\_Params\]

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. For this purpose, we use the parameters from Tab. \[tab:Fit\_Params\] and calculate the contour around the remotely prepared states for the experimental range of JPA3 gain $G_\mathrm{f}$ and amplification angle $\gamma_\mathrm{f}$ (green contour in Fig. \[fig:Contour\]). Alternatively, we use an iterative method to calculate the maximum error contour (blue contour in Fig. \[fig:Contour\]). 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.

![Phase space of experimental remotely prepared states spanned by $S_\mathrm{rp}$ and $\gamma_\mathrm{rp}$. The red line marks the threshold for squeezing below the vacuum limit. The green and blue shaded area indicate the direct contour from the fit and maximum error contour, respectively. The color code indicates the purity $\mu$ of the remotely prepared states. The optimal point is marked by the white star.<span data-label="fig:Contour"></span>](Supp_Fig_PhaseSpaceContour.pdf){width="65.00000%"}

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 $G_\mathrm{f}$ and $\gamma_\mathrm{f}$. Since we select a finite and uniform step size of $G_\mathrm{f}$ and $\gamma_\mathrm{f}$ in the experiment, the measured remotely prepared states do not uniformly occupy the phase space.

Feedforward signal
==================

The feedfoward signal is characterized by toggling the cryogenic switch into position B (see Fig. \[fig:ExpSetup\]) and measure both the signal from JPA3 (Alice's side) as well as the signal on Bob's side while all JPAs are pumped. The squeezing level of JPA1 and JPA2 for these measurements is $S\,{=}\,\SI{7.3}{\dB}$ which results in an entangled state with a negativity kernel@Menzel2012 [@Adesso2005] $N_\mathrm{k}\,{=}\,1.8$ after the beam splitter. The JPA3 amplification angle is fixed to the optimal angle $\gamma_\mathrm{f}\,{=}\,\SI{0}{\degree}$. As shown in Fig. \[fig:FF\], we observe no entanglement ($N_\mathrm{k}\,{\le}\,0$) between Alice and Bob after the local amplification for $G_\mathrm{f}\ge\SI{11}{dB}$. The feedforward signal is squeezed below the vacuum for low $G_\mathrm{f}$ and becomes non-squeezed above $G_\mathrm{f}\,{\simeq}\,\SI{13}{\dB}$. 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 $G_\mathrm{f}\,{\ge}\,\SI{13}{\dB}$ are classical. For $G_\mathrm{f}\,{<}\,\SI{13}{\dB}$, 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.

![Negativity kernel $N_\mathrm{k}$ (blue) and squeezing level (green) of the feedforward signal.<span data-label="fig:FF"></span>](Supp_Fig_Feedforward.pdf){width="65.00000%"}

Entropy of Gaussian states
==========================

The von Neumann entropy $H(X)\,{=-}\,\mathrm{Tr}(\hat{\rho}_x\mathrm{log}\,\hat{\rho}_x)$ of a quantum state $\hat{\rho}_x$ is the quantum information analogue of the entropy used in thermodynamics (up to a factor of the Boltzmann constant $k_\mathrm{B}$). For Gaussian states, $H(X)$ can be calculated from the covariance matrix $\mathbf{V}$. The von Neumann entropy of a single-mode Gaussian states is given by@Serafini2004 $$H(X) = f\left(\sqrt{\mathrm{det}\,\mathbf{V}}\right)\, ,$$ where $f(x)\,{=}\,\left(2x+\frac{1}{2}\right)\mathrm{log}\left(2x+\frac{1}{2}\right)-\left(2x-\frac{1}{2}\right)\mathrm{log}\left(2x-\frac{1}{2}\right)$.

For a two-mode Gaussian state $\hat{\rho}_{AB}$, the covariance matrix can be expressed in the form $$\mathbf{V}= \begin{pmatrix} \mathbf{A} & \mathbf{C}   \\ \mathbf{C}^\mathrm{T} & \mathbf{B} \end{pmatrix}\, ,$$ where $\mathbf{A}$, $\mathbf{B}$ and $\mathbf{C}$ are $2\times2$ matrices describing the local state A, local state B and cross-correlations between both parties, respectively. From $\mathbf{V}$, one can calculate the two symplectic eigenvalues of the bipartite Gaussian state $$\nu_{\pm} = \sqrt{\frac{\Delta\pm\sqrt{\Delta^2-4\mathrm{det}\,\mathbf{V}}}{2}}\, ,$$ where $\Delta\,{=}\,\mathrm{det}\,\mathbf{A}+\mathrm{det}\,\mathbf{B}+2\mathrm{det}\,\mathbf{C}$. The entropy of the whole bipartite state is given by $$H(A,B) = f\left(\nu_+\right)+f\left(\nu_-\right)\, ,$$ and the entropy of $A$ conditioned on knowing $B$ $$H(A|B) = H(A,B)-H(B)$$ is called the conditional entropy. If both parties are correlated, knowledge about $B$ will reveal information about $A$, and thus decrease its entropy, $H(A|B)\,{<}\,H(A)$.

{'source': 'arXiv_out/1902'}