Virtual PDL of the apparatus

The schematics of our experiment are illustrated in Fig. 1a. Our source generates a nearly perfect two-photon \(\left| {{\mathrm{\Phi }}^ + {\rangle }} \right.\) state with fidelity of about 0.95. To characterize it, we connect both detector stations (DSs) directly back-to-back (B2B) to the source, perform state tomography measurements, and then calculate the concurrence of the resulting density matrix. The B2B concurrence varies slightly between different days and power off/on cycles. Overall, the B2B concurrence falls within the tight interval of 0.925 ± 0.008. The deviation from unity occurs mostly due to the presence of noise photons generated in the dispersion shifted fiber (DSF) by Raman scattering and pump leakage into the entangled photon band.32,33 Absolute values of a typical density matrix obtained by the B2B state tomography measurements are shown in Fig. 1b. The plot reveals a slight imbalance between the |HH〉 and |VV〉 modes, which is constantly present. The ratio of the |HH〉〈HH| and |VV〉〈VV| matrix elements is about 1.38 (or 1.4 dB). We attribute this asymmetry to the specification tolerances of the off-the-shelf components comprising the entangled photon source (EPS).

Fig. 1 a Experiment schematics. b Absolute value of back-to-back density matrix with PDLE/PDLC removed Full size image

The state produced by the EPS could be viewed as an ideal Bell state \(\left| {\Phi ^ + } \right\rangle = \frac{1}{{\sqrt 2 }}\left( {\left| {HH} \right\rangle + \left| {VV} \right\rangle } \right)\) that undergoes external mode damping by a virtual source PDL element whose axis of maximal transmission is aligned to the |H〉 polarization, or in other words, points in the σ 3 direction in Stokes space. For the special case of such alignment between the PDL element and the basis of the input Bell state, there is ambiguity in the exact placement of the source PDL element. Indeed, positioning the source PDL in either of the channels as indicated by \(\vec \gamma ^S\) in Fig. 2a, b would result in the experimentally observed imbalanced density matrix. In fact, any combination of PDL elements in channels A and B would produce the same state as long as the PDL elements are aligned to |H〉 and their PDL values add up to 1.4 dB. For the more general case of an arbitrarily oriented PDL element acting on a Bell state, the equivalence mapping of Fig. 2a, b becomes more involved.

Fig. 2 a, b The EPS of Fig. 1 is modeled by a combination of an ideal Bell state source and a virtual PDL \(

ormalsize {\vec{\boldsymbol \gamma }}^{\boldsymbol{S}}\), which can be positioned either in channel A (a) or channel B (b). \(

ormalsize {\vec{\boldsymbol \gamma }}^{\boldsymbol{E}}\) denotes the PDL vector produced by the emulator PDLE in channel A. c Shaded circles: concurrence C vs PDL of channel A (both computed from the measured tomography). Color denotes purity with lighter shades corresponding to higher purity values. Black line shows the theoretical dependence of Eq. 1. d Black circles: the same values of concurrence C as in (c) but presented as a function of the magnitude of PDL of the emulator PDLE and the value of κ (Eq. 7). The values of C and κ are computed from the measured tomography. The surface is calculated based on Eq. 6 Full size image

Externally applied PDL

We start with an experimental setup that has only one PDL module inserted into channel A, shown as the red PDL emulating (PDLE) box in Fig. 1a. We sequentially set the emulator to five different PDL magnitudes of 1.25, 2.55, 3.7, 5.1, and 6.3 dB. At each PDL value, the polarization controller PCE (see Fig. 1a) goes through 50 different arbitrary settings such that the resulting PDL Stokes vector \(\vec \gamma ^E\) covers the entire Poincaré sphere. Full tomography is performed at each setting. From each of the 250 experimentally obtained density matrices, we compute the state’s concurrence and purity. There are two ways to present the concurrence data depending on the placement of the virtual source PDL \(\vec \gamma ^S\) depicted in Fig. 2a, b, allowing us to verify our theoretical description (see Eq. 6 of the Methods section) in two different scenarios.

When the virtual source PDL \(\vec \gamma ^S\) and the emulator PDL \(\vec \gamma ^E\) are in the same channel as depicted in Fig. 2a, the PDL of channel B is zero. In this case, the concurrence of Eq. 6 reduces to a simple orientation-independent form given by:

$$C\left( {\rho^\prime } \right) = \frac{{C\left( \rho \right)}}{{\cosh \left( {\gamma _A} \right)}},$$ (1)

where γ A is the magnitude of the aggregate PDL \(\overrightarrow {{\gamma }} _{{A}}\) in channel A. The magnitude γ A can be extracted from a properly rotated experimental density matrix for each of our 250 tomography data points. Figure 2c plots the experimental concurrence C vs the extracted values of γ A (shaded circles) along with the theoretical dependence of Eq. 1 (black line). The figure shows nearly perfect agreement between experiment and theory. In general, the orientation \(\hat \gamma _A\) and the magnitude γ A of the aggregate PDL \(\overrightarrow {{\gamma }} _{{A}}\) are governed by the cumbersome concatenation rules of the two PDL vectors \(\vec \gamma ^S\) and \(\vec \gamma ^E\).34,35,36,37,38 These rules allow us to compute the angle ϑ between vectors \(\vec \gamma ^S\) and \(\vec \gamma ^E\) from the measured value γ A , which we use below. Experimentally, by controlling ϑ, we sample a wide range of γ A values. This also ensures that the orientation of the aggregate PDL \(\hat \gamma _A\) changes dramatically between the data points shown on the plot. This verifies that the concurrence is independent of the orientation \(\hat \gamma _A\) of the aggregate PDL. To address the vertical spread in the data, we shade the symbols by their corresponding purity values with lighter colors corresponding to higher purity. The source performance varies slightly from day-to-day and between power cycles. Those variations result in the spread, as data points with a slightly higher purity naturally correspond to somewhat higher concurrence values.

In the alternative scenario, the virtual source PDL \(\vec \gamma ^S\) is placed in channel B while the emulator PDL \(\vec \gamma ^E\) is in channel A, as depicted in Fig. 2b. Now channel B has a constant PDL, while the PDL of channel A is variable in magnitude and orientation. The latter can be characterized by \(\kappa = \left( {T\hat \gamma ^S} \right) \cdot \hat \gamma ^E\), where κ compactly describes the complex dependence of the concurrence of the final state on the relative orientation of the two PDL elements (see Eq. 7 of the Methods section). It follows from Eq. 5 of the Methods section that the correlation matrix element t 3 = 1, because our input state is |Φ+ 〉. Interestingly, as the vector \(\vec \gamma ^S\) points in the σ 3 direction, κ can be easily related to the measured angle ϑ described above as: \(\kappa = \left( {T\hat \gamma ^S} \right) \cdot \hat \gamma ^E = \hat \gamma ^S \cdot \hat \gamma ^E = \cos \vartheta\). Figure 2d shows the same experimental data seen in Fig. 2c, but now the concurrence is shown as a function of both the magnitude of applied PDL (γE) and the relative orientation of two vectors \(\vec \gamma ^S\) and \(\vec \gamma ^E\). The surface is calculated based on Eq. 6. Again, the data clearly support our theoretical results. For κ = −1, the PDL of both channels are counter aligned such that they partially cancel each other, and the concurrence is maximized (for γE = γS). On the other hand, κ = +1 corresponds to the case where the PDL of each channel is aligned in the same direction, thus increasing the effective PDL and minimizing concurrence.

Nonlocal PDL compensation

An arbitrarily oriented PDL element \(\overrightarrow {{\gamma }} _{{A}}\) in channel A can be mapped to a certain PDL element \(\overrightarrow {{\gamma }} _{{B}}\) in channel B. In other words, for a two-photon state resulting from application of \(\overrightarrow {{\gamma }} _{{A}}\) on a Bell state, an equivalent two-photon state can be obtained by applying a properly chosen \(\overrightarrow {{\gamma }} _{{B}}\) to the same initial Bell state. That is, the two density matrices post-selected by the coincidence measurement are equal. A general relationship for finding the equivalence mapping of an arbitrarily oriented PDL element from one qubit to the other can be found by solving the equation:

$$\left( {P_1 \otimes \sigma _0} \right)\rho \left( {P_1 \otimes \sigma _0} \right)^{\dagger} = \left( {\sigma _0 \otimes P_2} \right)\rho \left( {\sigma _0 \otimes P_2} \right){^\dagger} \cdot$$ (2)

We find that when the initial state is a Bell state, identical density matrices result from replacing PDL on one qubit of magnitude γ A and orientation \(\hat \gamma _A\) in Stokes space with an element of equal magnitude and different orientation given by \(\hat \gamma _B = T\hat \gamma _A\) in channel B (shown schematically in Fig. 3b). Here, T is the correlation matrix. As all elements of T satisfy |t j | = 1 for Bell states, application of T to a Stokes vector constitutes an inversion of some or all axes in Stokes space. Figure 3a illustrates these transformations for an arbitrary unit vector \(\hat \gamma _A\), which is shown on the sphere (red) together with four equivalently mapped vectors \(\hat \gamma _B\) (green). Each vector \(\hat \gamma _B = T\hat \gamma _A\) corresponds to a particular input Bell state and is labeled accordingly. When the input state is a singlet, the inversion occurs in all three axes; however, only one axis is inverted for each of the triplets.

Fig. 3 a An arbitrary real unit vector in channel A (red) together with four virtual vectors in channel B (green), each corresponding to a particular input Bell state, given by its label. b Equivalence mapping of a real PDL element \(

ormalsize \overrightarrow {\boldsymbol{\gamma }} _{\boldsymbol{A}}\) in channel A onto a virtual PDL element \(

ormalsize \overrightarrow {\boldsymbol{\gamma }} _{\boldsymbol{B}} = {\boldsymbol{T}}\overrightarrow {\boldsymbol{\gamma }} _{\boldsymbol{A}}\) in channel B for an ideal Bell state source Full size image

The existence of an equivalence mapping suggests that PDL can be compensated nonlocally. Locally, two concatenated PDL elements of equal magnitude and anti-parallel Stokes vectors in the same channel simply reduce to pure loss, which attenuates each mode equally. In a more general case, one of these elements could be considered as an image mapped from a real element in the other channel. Then, to compensate for a PDL element of magnitude γ A and orientation \(\hat \gamma _A\) in channel A, we use an additional element in channel B with magnitude γ B = γ A and orientation \(\hat \gamma _B\) such that \(T\hat \gamma _B \cdot \hat \gamma _A = - 1\). The latter indicates that in channel A, the real element \(\hat \gamma _A\) and the element mapped from channel B (\(T\hat \gamma _B\)) are anti-parallel. Indeed, it follows from Eq. 6 that when these conditions are satisfied, C(ρ′) = C(ρ) and complete PDL compensation is achieved nonlocally.

Now we modify our setup to demonstrate PDL compensation. We keep the same PDL module in channel A, called the “emulator” (red PDLE box in Fig. 1a), and add a functionally similar PDL module in channel B as the “compensator” (green PDL compensating (PDLC) box in Fig. 1a). The following procedure is employed to vary the “PDL of channel A.” With the magnitude γE of the emulator PDL \(\vec \gamma ^E\) being set to a fixed value of 5.1 dB, we change its orientation \(\hat \gamma ^E\) using the polarization controller PCE. As the “PDL of channel A” results from the concatenation of the virtual source PDL \(\vec \gamma ^S\) and the PDL of the emulator \(\vec \gamma ^E\) (Fig. 2a), we obtain a range of the magnitude of the “PDL of channel A” from 4.1 dB to 6.4 dB. Notice that this range indicates that we cover nearly all possible angles ϑ between \(\hat \gamma ^E\) and \(\hat \gamma ^S\) within our experimental accuracy. For each of the values of PDL in channel A, we set the emulator in channel B to zero, perform tomography, and calculate the concurrence, which is plotted as empty symbols in Fig. 4c. These uncompensated results are similar to those of Fig. 2c. In fact, the solid lines in both figures plot exactly the same dependence of Eq. 1. The experimental density matrices obtained by quantum tomography also allow us to establish the value of the PDL of channel A: γ A . We then set the magnitude of the PDL of the compensator PDLC to be equal to that of channel A: γ B = γ A . By searching through various orientations of PDLC \(\hat \gamma _B\), we find the one resulting in maximal concurrence, which is plotted with filled symbols in Fig. 4c. The horizontal dashed line marks a level of C = 0.925, corresponding to the measured B2B concurrence without any PDL in either channel. As the plot shows, we are able to recover the initial concurrence for all experimentally available values and orientations of PDL in channel A within our experimental accuracy.

Fig. 4 a Absolute value of a back-to-back density matrix with PDLE/PDLC removed. b Absolute value of a density matrix with PMD (τ ~ 6.6 ps) in channel A and no applied PDL. c Concurrence C vs PDL of channel A (both computed from the measured tomography). Open symbols denote the uncompensated case with PDLC removed, and closed symbols mark the maximum achieved concurrence. d Data similar to that shown in (c) but with additional PMD-induced decoherence in channel A as in (b) Full size image

As our final setup modification, we keep the compensating PDL module in channel B intact (green PDLC box in Fig. 1a) while replacing the emulator PDL module in channel A (red PDLE box in Fig. 1a) with a different unit that has a fixed amount of first-order PMD (differential group delay of τ = 6.6 ps) in addition to the variable PDL. Note that this emulator, as a result of how it is constructed, produces PDL and PMD vectors that are collinear with each other. First-order PMD partially reduces the input state coherence,23,24 thus transforming a nearly perfect |Φ+〉 Bell state into a rank two Bell diagonal state \(\rho = \frac{{1 + C}}{2}\left| {\Phi ^ + } \right\rangle \left\langle {\Phi ^ + } \right| + \frac{{1 - C}}{2}\left| {\Phi ^ - } \right\rangle \left\langle {\Phi ^ - } \right|\), where C is the resulting Bell diagonal state concurrence. To characterize this state experimentally, we perform quantum state tomography with the PDL magnitudes of both the emulator and compensator dialed to zero: γ A = γ B = 0. Figure 4b presents the resulting density matrix, whereas Fig. 4a shows a typical density matrix obtained from the B2B measurements without any PMD present. The reduced coherence due to PMD exhibits itself in the smaller off-diagonal elements of the matrix in the right panel and the correspondingly smaller value of computed concurrence given by C = 0.69. As described earlier, a slight imbalance between the |HH〉 and |VV〉 modes arises from the intrinsic source PDL \(\vec \gamma ^S\).