Witnessing single-photon entanglement with local homodyne measurements: analytical bounds and robustness to losses

The general principle of the witness is shown in figure 1. The two distant entangled modes are detected by Alice and Bob via homodyne detection, which allows one to measure any quadrature component of the optical field (i.e., $X\;{\rm cos} (\phi )+P\;{\rm sin} (\phi )$) by varying the relative phase, ϕ, between the optical mode and the local oscillator [20]. Two phase settings are required on each side: Alice performs a measurement among two quadratures, $\{X,P\},$ while Bob makes a measurement in a basis rotated by ${{45}^{{}^\circ }}$ to access the quadratures $\{X+P,X-P\}$. The measurement outcomes, which are real numbers, are then sign-binned to obtain binary results $\pm 1$. The scenario is thus similar to the usual Bell test, where two parties can perform two possible measurements of two outcomes each. From the four possible combinations of quadratures, the witness parameter, S, is finally determined from the Clauser–Horne–Shimony–Holt (CHSH) polynomial [21],

$$\begin{eqnarray}&&S={{E}_{X,X+P}}+{{E}_{X,X-P}}+{{E}_{P,X+P}}-{{E}_{P,X-P}},\end{eqnarray} \tag{ 1 }$$

where the correlations are defined by ${{E}_{a,b}}=p(1,1)+p(-1,-1)-p(1,-1)\;-\;p(-1,1),$ and $p(i,j)$ are the conditional probabilities to obtain the outcomes i and j if the quadratures a and b are chosen.

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Additionally, the phase of the local oscillators can be averaged while keeping the relative phases between Alice and Bobʼs measurements fixed. This averaging can only lead to an underestimation of the entanglement, as it can be realized by local operations and classical communications. The S parameter reduces thus to two terms—one where the relative phase differs by $\frac{\pi }{4}$, and the other by $-\frac{\pi }{4}$:

$$\begin{eqnarray}&&S=2{{E}_{+\pi /4}}+2{{E}_{-\pi /4}}.\end{eqnarray} \tag{ 2 }$$

As shown in [11], this phase-averaging is actually crucial in the protocol, as it enables us to access the local photon-number probabilities with the same homodyne measurements. These probabilities are then used to further constrain the set of density matrices that we consider in our optimization of the separable bounds.

Sign-binning of homodyne measurement in the qubit subspace, $\{|0\rangle ,|1\rangle {{\}}^{\otimes 2}},$ is equivalent with a noisy spin measurement [22, 23]. For instance, the operator associated with a sign-binned X-measurement corresponds to $\sqrt{2/\pi }\;{{\hat{\sigma }}_{x}},$ where ${{\hat{\sigma }}_{x}}$ is the standard Pauli matrix. A maximally entangled state, $\left( |1\rangle |0\rangle +|0\rangle |1\rangle \right)/\sqrt{2}$, thus leads to ${{S}_{{\rm max} }}=2\sqrt{2}\;.2/\pi \simeq 1.8$, the maximal value that one can obtain using the aforementioned measurements. Note that since this value is lower than 2, a violation of the well-known local bound for the CHSH polynomial is not possible in this context. While this would have been sufficient to demonstrate entanglement, it is not necessary if the separable bound is lower.

The next question that arises is the value of the separable bound. It can be shown that the maximal value over the set of all the separable states is equal to ${{S}_{{\rm sep}}}=\sqrt{2}\;.2/\pi \simeq 0.9$ [24]. In the qubit space, an observed S parameter above 0.9 allows one to conclude that the two modes are entangled. Importantly, this separable bound can be optimized further if additional knowledge about the state is available, as this knowledge constrains the set of compatible separable states. In our case, the phase-averaged homodyne measurements provide the local photon number distributions. These local photon-number distributions, $p_{0}^{A},p_{1}^{A}$ (vacuum and single-photon component on the Alice side) and $p_{0}^{B},p_{1}^{B}$ (Bob side), allow us to optimize the bound, as we will now show.

First, thanks to the averaging of the local phases, many off-diagonal terms of the measured state do not contribute to the measurement results. Since our goal is to reveal entanglement, it is therefore sufficient to consider density matrices of the following form in the Fock basis [13]:

$$\begin{eqnarray}\hat{\rho }=\left( \begin{array}{ccccccccccccccc} {{p}_{00}} & 0 & 0 & 0 \\ 0 & {{p}_{01}} & {\boldsymbol{d}} & 0 \\ 0 & {{{\boldsymbol{d}} }^{*}} & {{p}_{10}} & 0 \\ 0 & 0 & 0 & {{p}_{11}} \\ \end{array} \right).\end{eqnarray} \tag{ 3 }$$

Then, for any state within the qubit subspace, it can be shown that the S parameter can be rewritten as [11]

$$\begin{eqnarray}&&S=\frac{16}{\pi \sqrt{2}}\Re \left[ \langle 01|\hat{\rho }|10\rangle \right]=\frac{16}{\pi \sqrt{2}}\Re \left[ {\boldsymbol{d}} \right].\end{eqnarray} \tag{ 4 }$$

When Alice and Bob measure the value of S, they can also extract the local probabilities, p₀^A and p₀^B, from the quadrature measurements. Hence, only a reduced set of states are compatible with these probabilities. It can be translated formally as:

• $p_{0}^{A}={{p}_{00}}+{{p}_{01}}$ and $p_{0}^{B}={{p}_{00}}+{{p}_{10}}$ (relationship between joint probabilities and local probabilities)
• ${\rm tr}[\hat{\rho }]=1$ (conservation of probabilities)
• $\hat{\rho }\geqslant 0$ (physical state, all eigenvalues are positive, i.e., ${{p}_{01}}{{p}_{10}}\geqslant |{\boldsymbol{d}} {{|}^{2}}$)
• $0\leqslant {{p}_{ij}}\leqslant 1$ (regular probabilities)

The maximization of $|{\boldsymbol{d}} |$ under all these constraints gives the upper bound, ${{S}^{{\rm max} }},$ for the witness parameter,

$$\begin{eqnarray}{{S}^{{\rm max} }}=\frac{16}{\pi \sqrt{2}}\left\{ \begin{array}{ccccccccccccccc} \sqrt{p_{0}^{A}p_{0}^{B}} & \ {\rm if}\;{\rm p}_{0}^{A}+{\rm p}_{0}^{B}\leqslant 1, \\ \sqrt{(1-p_{0}^{A})(1-p_{0}^{B})} & \ {\rm if}\;{\rm p}_{0}^{A}+{\rm p}_{0}^{B}\geqslant 1. \\ \end{array} \right.\end{eqnarray} \tag{ 5 }$$

We now derive the separable bound, $S_{{\rm sep}}^{{\rm max} }$. Separable states remain positive under partial transposition (PPT criterion) [25, 26]. This additional constraint leads to the condition $|{\boldsymbol{d}} {{|}^{2}}\leqslant {{p}_{00}}{{p}_{11}}$ for separable states. Hence, the maximization of $|{\boldsymbol{d}} |$ provides the maximal value of S but, this time, for the separable states only,

$$\begin{eqnarray}&&S_{{\rm sep}}^{{\rm max} }=\frac{16}{\pi \sqrt{2}}\sqrt{p_{0}^{A}p_{0}^{B}(1-p_{0}^{A})(1-p_{0}^{B})}.\end{eqnarray} \tag{ 6 }$$

We now study the use of the proposed witness in the case where the entangled state undergoes loss (e.g., propagates through lossy communication channels). What are the acceptable losses in this case? With the help of the analytical bounds derived previously, we detail how the proposed witness is affected.

The situation is sketched in figure 1. We consider the entanglement to be initially generated from an ideal single-photon state, and the channel transmissions are denoted by ${{\eta }_{A}}$ from the source to Alice, and ${{\eta }_{B}}$ from the source to Bob. One can write the full transmission between Alice and Bob as ${{\eta }_{AB}}={{\eta }_{A}}{{\eta }_{B}}$. After propagation, the resulting state shared by Alice and Bob can be written as

$$\begin{eqnarray}{{\hat{\rho }}_{AB}}=\frac{1}{2}\left( \begin{array}{ccccccccccccccc} 2-{{\eta }_{A}}-{{\eta }_{B}} & 0 & 0 & 0 \\ 0 & {{\eta }_{A}} & \sqrt{{{\eta }_{A}}{{\eta }_{B}}} & 0 \\ 0 & \sqrt{{{\eta }_{A}}{{\eta }_{B}}} & {{\eta }_{B}} & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} \right).\end{eqnarray} \tag{ 7 }$$

As given by equation (4), the CHSH polynomial value, S, can be written as

$$\begin{eqnarray}&&S({{\hat{\rho }}_{AB}})=\frac{16}{\pi \sqrt{2}}\frac{\sqrt{{{\eta }_{A}}{{\eta }_{B}}}}{2}\ .\end{eqnarray} \tag{ 8 }$$

Furthermore, the local probabilities are given by

$$\begin{eqnarray}&&p_{1}^{A}={{\eta }_{A}}/2\quad {\rm and}\quad p_{1}^{B}={{\eta }_{B}}/2.\end{eqnarray} \tag{ 9 }$$

The maximal value of equation (5) is saturated by the state given in equation (7), and the corresponding separable bound is

$$\begin{eqnarray}&&S_{{\rm sep}}^{{\rm max} }=\frac{8}{\pi \sqrt{2}}\sqrt{{{\eta }_{A}}{{\eta }_{B}}(1-{{\eta }_{A}}/2)(1-{{\eta }_{B}}/2)}\ .\end{eqnarray} \tag{ 10 }$$

With this simple model in hand, one can distinguish two different experimental scenarios. First, when the source is placed on Aliceʼs site, the losses are thus asymmetric, and ${{\eta }_{A}}=1$ and ${{\eta }_{B}}={{\eta }_{AB}}$. For this configuration, the separable bound is

$$\begin{eqnarray}&&S_{{\rm sep}}^{{\rm max} }(asym.)=\frac{4}{\pi }\sqrt{{{\eta }_{AB}}(1-{{\eta }_{AB}}/2)}\ .\end{eqnarray} \tag{ 11 }$$

The second scenario places the source at an equal distance from Alice and Bob, so that the state will propagate along the same distance on both arms. The two modes are thus affected by the same losses, ${{\eta }_{A}}={{\eta }_{B}}=\sqrt{{{\eta }_{AB}}}$, leading to the following separable bound:

$$\begin{eqnarray}&&S_{{\rm sep}}^{{\rm max} }(sym.)=\frac{8}{\pi \sqrt{2}}\sqrt{{{\eta }_{AB}}}(1-\sqrt{{{\eta }_{AB}}}/2)\ .\end{eqnarray} \tag{ 12 }$$

In order to compare both cases, we fix the full transmission, ${{\eta }_{AB}}={{\eta }_{A}}{{\eta }_{B}}$. In other words, the position of the source changes, but the total distance between Alice and Bob does not. Furthermore, we note that the symmetric situation can equivalently correspond to losses on the source itself. Indeed, it is formally equivalent to attribute these losses to the two transmission channels.

Figure 2 provides the CHSH polynomial as a function of the transmission, together with the two separable bounds. As shown before, the parameter S depends only on the total loss, while the separable bound additionally depends on whether the losses are symmetric or asymmetric. As we can see, the distance between the witness and the bound decreases with the losses but reaches zero only for infinite ones, meaning that in principle the witness can detect entanglement for any losses. Furthermore, we note that the distance of S from the separable bound is always larger for the asymmetric case than for the symmetric case. The witness is thus slightly more efficient in the latter case.

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When the number of photons per mode is not restricted to one, we use the local photon number distributions obtained via phase-averaged quantum state tomography to determine an upper bound on the joint probability, ${{p}_{{\rm joint}}}=p({{n}_{A}}\geqslant 2\cup {{n}_{B}}\geqslant 2),$ so that at least one of the modes is populated with more than one photon. Indeed, this probability can be bounded by the local probabilities of the zero and one-photon components on each side as

$$\begin{eqnarray}&&{{p}_{{\rm joint}}}\leqslant {{p}^{\star }},\end{eqnarray} \tag{ 13 }$$

where ${{p}^{\star }}=p_{\geqslant 2}^{A}+p_{\geqslant 2}^{B}$ and $p_{\geqslant 2}^{A(B)}=1-p_{0}^{A(B)}-p_{1}^{A(B)}$, denoting the probability that one party observes at least two photons.

In the following, we present some separable bounds in terms of ${{p}_{{\rm joint}}}.$ These can be re-expressed in terms of local photon distributions by substituting ${{p}^{\star }}$ for p_joint, thus slightly overestimating the bound.

Following a similar argument as presented in section 2.2, we provide here a separable bound for S that is valid outside of the qubit space.

In this larger Hilbert space, S can be bounded as follows (see [11]):

$$\begin{eqnarray}&&S\leqslant \frac{16}{\pi \sqrt{2}}d+\frac{8}{\pi }e+\frac{8}{\pi }f+2\sqrt{2}\ {{p}_{{\rm joint}}},\end{eqnarray} \tag{ 14 }$$

where $d=\Re \left[ \langle 01|\hat{\rho }|10\rangle \right]$, $e=\Re \left[ \langle 20|\hat{\rho }|11\rangle \right]$, and $f=\Re \left[ \langle 02|\hat{\rho }|11\rangle \right]$ denote different contributions to the witness. Due to the positivity of $\hat{\rho }$ and ${{\hat{\rho }}^{{{T}_{B}}(0,1)}}$, each of these contributions can be bounded as a function of a single density matrix variable p₀₀:

$$\begin{eqnarray}&&{{d}^{2}}\leqslant {{p}_{01}}{{p}_{10}}\leqslant (p_{0}^{A}-{{p}_{00}})(p_{0}^{B}-{{p}_{00}})\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}&&{{d}^{2}}\leqslant {{p}_{00}}{{p}_{11}}\leqslant {{p}_{00}}\left[ {{p}_{00}}+1-p_{0}^{A}-p_{0}^{B}+p_{\geqslant 2}^{A}+p_{\geqslant 2}^{B} \right]\end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray}&&{{e}^{2}}\leqslant {{p}_{02}}{{p}_{11}}\leqslant p_{\geqslant 2}^{B}\left[ {{p}_{00}}+1-p_{0}^{A}-p_{0}^{B}+p_{\geqslant 2}^{A}+p_{\geqslant 2}^{B} \right]\end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray}&&{{f}^{2}}\leqslant {{p}_{20}}{{p}_{11}}\leqslant p_{\geqslant 2}^{A}\left[ {{p}_{00}}+1-p_{0}^{A}-p_{0}^{B}+p_{\geqslant 2}^{A}+p_{\geqslant 2}^{B} \right].\end{eqnarray} \tag{ 18 }$$

The maximum value of S_sep can thus be obtained by optimizing equation (14) over the p₀₀ variable. Recall that here, we do not impose the state $\hat{\rho }$ to be fully PPT, but only PPT within the 0/1 subspace. This allows us to verify the presence of entanglement in the single-photon subspace [11].

For small $p_{\geqslant 2}^{A(B)}$, the choice $p_{00}^{c}=p_{0}^{A}p_{0}^{B}/z$ is optimal, where $z=1+p_{\geqslant 2}^{A}+p_{\geqslant 2}^{B}$. This gives the following separable bound:

$$\begin{eqnarray}\begin{array}{rcl} S_{{\rm sep}}^{{\rm max} } & = & \frac{16}{\pi \sqrt{2}}\sqrt{p_{0}^{A}p_{0}^{B}\left( 1-\frac{p_{0}^{B}}{z} \right)\left( 1-\frac{p_{0}^{A}}{z} \right)} \\ {} & {} & +\frac{8}{\pi }\left( \sqrt{p_{\geqslant 2}^{A}}+\sqrt{p_{\geqslant 2}^{B}} \right)\sqrt{z+\frac{p_{0}^{A}p_{0}^{B}}{z}-p_{0}^{A}-p_{0}^{B}} \\ {} & {} & +2\sqrt{2}\ {{p}_{{\rm joint}}}. \\ \end{array}\end{eqnarray} \tag{ 19 }$$

One can verify that this expression reduces to the qubit bound given by equation (6) in the case $p_{\geqslant 2}^{A(B)}=0$. Equation (19) provides an analytical estimation of the value of the witness needed to demonstrate single-photon entanglement as a function of the local observed probabilities. We emphasize that this bound is also valid in the presence of multiphoton components.

Here we derive a separable bound which only depends on the p_joint variable. Writing matrices M and N such that

$$\begin{eqnarray}\begin{array}{rcl} {\rm tr}(M\hat{\rho }) & = & \frac{16}{\pi \sqrt{2}}\Re \left[ \langle 01|\hat{\rho }|10\rangle \right]+\frac{8}{\pi }\left( \Re \left[ \langle 20|\hat{\rho }|11\rangle \right]+\Re \left[ \langle 02|\hat{\rho }|11\rangle \right] \right) \\ {\rm tr}(N\hat{\rho }) & = & \langle 00|\hat{\rho }|00\rangle +\langle 01|\hat{\rho }|01\rangle +\langle 10|\hat{\rho }|10\rangle +\langle 11|\hat{\rho }|11\rangle , \\ \end{array}\end{eqnarray} \tag{ 20 }$$

the maximum separable value of equation (14) given p_joint can be found by maximizing ${\rm tr}(M\hat{\rho })+2\sqrt{2}{{p}_{{\rm joint}}}$ under the constraints $\hat{\rho }\geqslant 0$, ${\rm tr}(\hat{\rho })\leqslant 1$, ${{\hat{\rho }}^{{{T}_{B}}(0,1)}}\geqslant 0,$ and ${\rm tr}(N\hat{\rho })=1-{{p}_{{\rm joint}}}.$

Any matrices A and B, and variables λ and μ that satisfy $A+{{B}^{{{T}_{B}}(0,1)}}-\mu N-\lambda I=-M$, $A\geqslant 0$, $B\geqslant 0$ can provide an upper bound on the result of this optimization. Indeed, these constraints guarantee that

$$\begin{eqnarray}\begin{array}{rcl} {\rm tr}(M\hat{\rho }) & = & {\rm tr}[(\lambda I+\mu N-A-{{B}^{{{T}_{B}}(0,1)}})\hat{\rho }] \\ {} & = & \lambda {\rm tr}(\hat{\rho })+\mu {\rm tr}(N\hat{\rho })-{\rm tr}(A\hat{\rho })-{\rm tr}(B{{{\hat{\rho }}}^{{{T}_{B}}(0,1)}}) \\ {} & \leqslant & \lambda +\mu (1-{{p}_{{\rm joint}}}). \\ \end{array}\end{eqnarray} \tag{ 21 }$$

In the appendix, we describe matrices A and B that satisfy these constraints for ${{p}_{{\rm joint}}}\leqslant 1/2$, $\lambda =\frac{2}{\pi }\sqrt{\frac{2}{{{p}_{{\rm joint}}}}}{{x}_{+}}$, $\mu =\left( \frac{2}{\pi }\sqrt{2}x_{+}^{2}-\lambda \right)/(1-{{p}_{{\rm joint}}}),$ and ${{x}_{\pm }}=\sqrt{1-{{p}_{{\rm joint}}}}\pm \sqrt{{{p}_{{\rm joint}}}}$. This gives the following maximum for the separable bound:

$$\begin{eqnarray}&&S_{{\rm sep}}^{{\rm max} }=2\sqrt{2}\left[ \frac{1}{\pi }{{(\sqrt{1-{{p}_{{\rm joint}}}}+\sqrt{{{p}_{{\rm joint}}}})}^{2}}+{{p}_{{\rm joint}}} \right].\end{eqnarray} \tag{ 22 }$$

One can check that this bound is achievable for all ${{p}_{{\rm joint}}}\leqslant 1/2$ by some quantum states $\hat{\rho },$ which are PPT in the single-photon subspace. This guarantees that the bound is tight as a function of p_joint. However, this bound does not take into account the local probabilities.

In [11], a semidefinite program (SDP) is presented to compute separable bounds on S as a function of the local photon number probabilities. Here, we provide a refined version of this program, including two improvements.

The first improvement is to express p_joint in (14) as a function of the density matrix elements, rather than bounding it according to equation (13). This allows us to perform the optimization of S across all terms together.

The second step is to take into account all information about the local probability distributions. This can be achieved by using the Frechet inequalities [36]. In the form of the disjunction⁶ , these inequalities can be expressed as

$$\begin{eqnarray}&&{\rm max} (0,p(A)+p(B)-1)\leqslant P(A\cap B)\leqslant {\rm min} (p(A),p(B)).\end{eqnarray} \tag{ 23 }$$

Here, A(B) refers to any set that includes at least one photon number on Aliceʼs (Bobʼs) side. For instance, in the case that probabilities up to one photon component are observed, the possible choices for A and B consist of any nonempty combination from $\{0\ {\rm photon},\ 1\ {\rm photon},\ {\rm more}\ {\rm than}\ 1\ {\rm photon}\}$. This gives us a set of $({{2}^{3}}-1)$ by $({{2}^{3}}-1)$ separate Frechet inequalities.

Adding the usual conditions to the two we just mentioned leads to the following formulation for the refined bound:

$$\begin{eqnarray}\begin{array}{rlr} {\rm max} & {} & S(p_{0}^{A},p_{1}^{A},p_{0}^{B},p_{1}^{B}) \\ {\rm s}{\rm .t}{\rm .} & {} & \hat{\rho }\geqslant 0 \\ {} & {} & {\rm tr}(\hat{\rho })\leqslant 1 \\ {} & {} & {{{\hat{\rho }}}^{{{T}_{b}}(0,1)}}\geqslant 0 \\ {} & {} & P(A\cap B)\geqslant {\rm max} [\ 0,p(A)+p(B)-1\ ],\ \forall A,B \\ {} & {} & P(A\cap B)\leqslant {\rm min} [\ p(A),p(B)\ ],\ \forall A,B. \\ \end{array}\end{eqnarray} \tag{ 24 }$$

The program described in [9] can be seen as a relaxation of this program.

As presented here, it should be clear that the program 24 can be extended to take into account additional local photon numbers. In this case, the expression (14) needs to be modified to fit the newly considered Hilbert space. Similarly, the definition of p_joint can be adapted. However, the program remains the same. This presents the possibility of enhancing the bounds by taking into account additional information. We come back to this possibility in the experimental part of this paper.

Finally, we note that uncertainties in the local probabilities can be taken into account in this method by following the same procedure presented in [11].

Until now, we have presented four separable bounds for the witness. Let us briefly highlight their differences and mention the context in which it would be appropriate to use each of them.

The first bound, given in equation (6), is valid only for qubit states, and is thus not applicable in practice. However, it takes advantage of the observed local photon number distributions. This is the bound we used in section 2.3 to first illustrate the effect of losses on the witness.

The second bound, given in equation (19), also takes advantage of the knowledge of the photon number distributions and applies outside of the qubit space. However, one can check that this bound is not always tight. This stems from the fact that only some of the Frechet inequalities were taken into account in its derivation. Moreover, this bound can be very sensitive to uncertainties in the local probabilities, making it hardly applicable in practice. Nevertheless, it can be useful to quickly estimate the value of the bound that can be derived from equation (24).

The third bound, given in equation (22), is tight as a function of p_joint alone. It behaves well in presence of uncertainties, but does not take advantage of the knowledge of the local photon number probabilities.

The fourth bound, given in equation (24), is expressed as a semidefinite program. It does not assume a qubit structure and computes the tightest separable bound compared to all other methods by taking all physical constraints into account. Moreover, since it includes an exact modelization of the underlying quantum state, it behaves well in the presence of uncertainties on the local probabilities. Therefore, this is the kind of bound that we use in the next section to analyze the experimental data.

The single-photon source is based on a type-II optical parametric oscillator (OPO) pumped far below threshold by a continuous-wave frequency-doubled Nd:YAG laser at 532 nm [27]. The frequency-degenerate signal and idler modes are orthogonally polarized and can be easily separated. The detection of a single photon on one mode then heralds the preparation of a single photon on the other mode [28–30]. Importantly, the photon is generated in a very well-defined spatiotemporal mode due to the OPO cavity. Experimental details, including the filterings required in the conditioning path and the definition of the temporal mode, have been presented elsewhere [31–34]. In the current experiment, the heralding efficiency (i.e., the single-photon component at the output of the OPO) is equal to 90 %, and the two-photon component is limited to a few percents. If one includes the total propagation and detection losses, the single-photon component reaches 68±2%. The initial effective transmission, ${{\eta }_{AB}},$ is thus $\sim 0.68.$

Entanglement is obtained by impinging the heralded single-photon state on a balanced polarizing beam splitter. To check the entanglement, the two modes are then directed to two homodyne detections, as shown in figure 1. By using the previous notations, without introducing additional communication channel losses, $p_{1}^{A}=p_{1}^{B}=0.34\pm 0.01$, in comparison to $p_{1}^{A}=p_{1}^{B}=0.5$ for entanglement generated from an ideal single-photon source. In the following, we include additional losses to decrease the transmission in both symmetric and asymmetric ways.

To perform the homodyne detections, a bright beam impinges on the balanced polarizing beam splitter mentioned above in order to distribute the two required local oscillators. Thus, the classical and quantum channels have orthogonal polarizations but the same spatial modes up to the detections. This configuration allows one to easily adjust the relative phase between the two detections by choosing an appropriate elliptical polarization for the bright beam before the splitting [9, 34]. By also sweeping its phase, both homodyne detections have a fixed relative phase but are locally phase-averaged, as required.

We now detail the full experimental procedure for implementing the proposed witness. The steps are as follows:

• Acquiring homodyne data. Phase-averaged homodyne tomography is performed on both modes. Four relative phase settings are required, but phase-averaging enables us to reduce them to two (i.e., $\pm \pi /4$). The recorded data are then used for the next steps.
• Extracting the local probabilities. The local photon-number distributions are extracted from the previous data. Importantly, no additional measurements are required. The estimation is obtained via pattern functions that relax any assumptions on the size of the Fock space [35].
• Determining the separable bound. The local probabilities are used to constrain the set of separable states and calculate the separable bound following the program given in equation (24).
• Calculating the S parameter. The homodyne data are sign-binned and the S parameter is then determined from equation (1). If S is above the separable bound, the bipartite state is entangled.

We now turn to the study of the effect of losses on the proposed witness. Losses have been simulated here by changing the temporal modes. Indeed, the experiment is based on continuous-wave homodyne detection (i.e., the quadrature measurement is a continuous signal, x(t)). In order to measure the mode in which our state lies, a temporal filtering is required, leading to ${{x}_{\psi }}=\int \psi (t)x(t)dt$. The optimal temporal mode, $\psi (t),$ contains the generated state, and all the other orthogonal modes contain a vacuum state [32]. We can thus generate controlled and tunable losses by mismatching the temporal mode and the optimal mode we chose. The overlap, $\int \psi (t+\tau )\psi (t)dt,$ provides the additional losses, $\eta (\tau ),$ on each channel. As completed with the same raw data, the original state is always the same. Only the losses are tuned by this procedure.

Experimental results are displayed in figure 3. The measured CHSH parameter is given for different values of losses, together with the corresponding separable bounds. In the asymmetric case, the losses have been increased on one of the homodyne detections, while for the symmetric case, the losses are generated equally on both of them. However, as we did in the model, we can compare both situations in a relevant fashion only if we consider the full losses; we estimate the local losses with the help of the experimental vacuum components, $p_{0}^{A}=1-{{\eta }_{A}}$ and $p_{0}^{B}=1-{{\eta }_{B}}$, and then obtain the corresponding overall transmission, ${{\eta }_{AB}}={{\eta }_{A}}{{\eta }_{B}}$.

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The obtained results are in very good agreement with the expected behavior (i.e., $S\propto \sqrt{{{\eta }_{AB}}}$), and show that the bound of the single-photon entanglement witness can be violated unless very significant losses have ocurred. The figure provides two separable bounds determined by using the program outlined in section 3.4. The first bound takes as constraint the local probabilities up to one photon, as considered in [11], while the second bound considers the two-photon component. Clearly, more losses can be tolerated thanks to this enhanced separable bound. In the asymmetric case, the limit is pushed experimentally from 90% to 95% (corresponding to around 65 km of fiber at telecom wavelength if one starts with an ideal single-photon), while in the symmetric case a rise from 95% to 97% (77 km of fiber) is obtained. This small difference between the symmetric and asymmetric cases can be explained by the higher photon number component and the sensitivity of the bound to this parameter. Indeed, in the asymmetric case, the mode that does not experience loss keeps a larger two-photon component, which allows for a separable state with a higher S parameter.