Single-qubit measurement of two-qubit entanglement in generalized Werner states

Traditional methods of characterizing entanglement in two-photon mixed states (e.g., the violation of Bell’s inequalities Freedman and Clauser (1972); Aspect et al. (1982); Ou et al. (1992); Giustina et al. (2015), quantum state tomography James et al. (2001), etc.) require detection of both photons (see, for example, Gühne and Tóth (2009); Friis et al. (2019) and references therein). These methods, therefore, involve coincidence measurement or postselection. Methods that do not require detection of both photons rely on the assumption that the quantum state is pure (see, for example, Walborn et al. (2006); Sahoo et al. (2020); Bhattacharjee et al. (2022); Di Lorenzo Pires et al. (2009); Just et al. (2013); Sharapova et al. (2015)).

Recently, it has been demonstrated theoretically Lahiri et al. (2021) and experimentally Lemos et al. (2023) that it is possible to measure entanglement of a special class of two-photon mixed states — obtained by generalizing Bell states — without detecting one of the photons and without employing coincidence measurement or postselection. Density matrices representing such states have two generally non-vanishing coherence terms (off-diagonal elements). The states are entangled if and only if these coherence terms are nonzero. Furthermore, these two coherence terms are complex conjugate of each other. Therefore, entanglement of the states considered in Refs. Lahiri et al. (2021); Lemos et al. (2023) is fully characterized by one coherence term of the corresponding density matrix. However, entanglement of most two-qubit states are not dependent on their density matrix elements in such a trivial manner, e.g., the Werner state that can be entangled without violating Bell’s inequalities Werner (1989). Whether the entanglement of any two-qubit photonic mixed state can be verified by detecting one qubit remains an open question of fundamental importance.

Here, we take an important step toward answering this question by extending the method to two-qubit mixed states that can be obtained by generalizing Werner states. We find that albeit the same principle introduced in Refs. Lahiri et al. (2021); Lemos et al. (2023) applies, the measurement procedure requires considerable modifications. Our results also suggest that the method to account for experimental losses would require significant adaptation (Supplementary Material).

The article is organized as follows: In Sec. II, we discuss the class of quantum state we address and their entanglement. In Sec. III, we provide an outline of the entanglement measurement scheme. In Sec. IV we present a detailed theoretical analysis and our main results. We also illustrate the results by numerical examples. In Sec. V we compare our results with existing ones. In Sec. VI, we discuss the effect of anticipated experimental imperfections. Finally, we summarize and conclude in Sec. VII.

We work with a two-photon polarization state that is a common test bed for two-qubit systems. Throughout this paper, we call the two photons forming a pair signal ($S$) and idler ($I$). We use $H,V,D,A,R$, and $L$ to represent horizontal, vertical, diagonal, antidiagonal, right-circular, and left-circular polarization, respectively. The ket, $|\mu_{I},\nu_{S}\rangle$, represents a photon pair where the idler photon has polarization $\mu$ and the signal photon has polarization $\nu$.

The quantum state considered here is obtained by generalizing the Werner state. The density matrix takes the following form in the computational basis $\{|H_{I}H_{S}\rangle,|H_{I}V_{S}\rangle,|V_{I}H_{S}\rangle,|V_{I}V_{S}\rangle\}$:

$\displaystyle\widehat{\rho}=\begin{pmatrix}\eta I_{H}+\frac{1-\eta}{4}&0&0&\eta\mathscr{I}\sqrt{I_{H}I_{V}}e^{-i\phi}\\ 0&\frac{1-\eta}{4}&0&0\\ 0&0&\frac{1-\eta}{4}&0\\ \eta\mathscr{I}\sqrt{I_{H}I_{V}}e^{i\phi}&0&0&\eta I_{V}+\frac{1-\eta}{4}\end{pmatrix},$

(1)

where $0\leq I_{H}\leq 1$, $I_{V}=1-I_{H}$, $0\leq\eta\leq 1$, $0\leq\mathscr{I}\leq 1$, and $\phi$ represents a phase. It can be immediately checked that this state becomes a Werner state when $I_{H}=I_{V}=1/2$ and $\mathscr{I}=1$. For $I_{H}=I_{V}=1/2$, $\mathscr{I}=1$ and $\eta=1$, it reduces to Bell states $|\Phi^{+}\rangle=(|H_{I}H_{S}\rangle+|V_{I}V_{S}\rangle)/\sqrt{2}$ and $|\Phi^{-}\rangle=(|H_{I}H_{S}\rangle-|V_{I}V_{S}\rangle)/\sqrt{2}$ when $\phi=0$ and $\phi=\pi$, respectively. Therefore, the state given by Eq. (1) can also be obtained by the convex combination of a fully mixed state with the generalization of Bell states considered in Refs. Lahiri et al. (2021); Lemos et al. (2023).

The entanglement of this state can be verified by testing the positive partial transpose (PPT) criterion Peres (1996). A partial transposition of the density matrix ($\widehat{\rho}$) is a transposition taken with respect to only one of the photons. The density matrix has a positive partial transpose if and only if its partial transposition does not have any negative eigenvalues. Since we have a $2\times 2$ system, according to the PPT criterion the state is entangled if and only if it does not have a positive partial transpose Horodecki (1997).

By determining the eigenvalues of a partial transpose of the density matrix given by Eq. (1), we find that three of them are always positive (see Appendix A). The only eigenvalue that can be negative is given by

$$\alpha_{1}=\frac{1-\eta-4\eta\mathscr{I}\sqrt{I_{H}I_{V}}}{4}.$$

(2)

Figure 1 illustrates entanglement of generalized Werner states ($\widehat{\rho}$) characterized by the PPT criterion for various choices of state parameters $\eta$, $I_{H}$, and $\mathscr{I}$.

The amount of entanglement present in the quantum state can be quantified by the concurrence Wootters (1998). We find that the concurrence of the state given by Eq. (1) is (see Appendix B)

$\displaystyle C(\widehat{\rho})=\text{max}\left\{\frac{\eta-1+4\eta\mathscr{I}\sqrt{I_{H}I_{V}}}{2},0\right\}.$

(3)

We show below how to determine the concurrence without detecting one of the photons.

The principle of our method is based on a unique quantum interference phenomenon that was first demonstrated by Zou, Wang, and Mandel Zou et al. (1991); Wang et al. (1991) and is sometimes called the interference by path identity Hochrainer et al. (2022). The method employs a nonlinear interferometer Chekhova and Ou (2016) that contains two identical twin-photon sources. Each source produces the same quantum state [Eq. (1)].

A conceptual arrangement of the entanglement measurement scheme is illustrated in Fig. 2. The two photon-pair sources are denoted by $Q_{1}$ and $Q_{2}$. A photon pair is in superposition of being created at the two sources — for sources made of nonlinear crystals, such a situation can be achieved by weakly pumping them with mutually coherent laser beams. The sources do not produce the states simultaneously, i.e., the probability of having more than one photon pair in the system between an emission and a detection is negligible. That is, in this situation the effect of stimulated (induced) emission is negligible Zou et al. (1991); Wang et al. (1991); Wiseman and Mølmer (2000); Liu et al. (2009); Lahiri et al. (2019).

Signal beams ($S_{1}$ and $S_{2}$) from the two sources are superposed by a balanced beamsplitter and the single-photon counting rate (intensity) is measured at one of the outputs of the beamsplitter. The idler beam from $Q_{1}$ is sent through $Q_{2}$ and is aligned with the idler beam generated by $Q_{2}$. (Such an alignment was originally suggested by Z. Y. Ou Zou et al. (1991).) This alignment makes it impossible to know from which source the signal photon originated. Consequently, a single-photon interference pattern appears at a detector placed at an output of the beamsplitter. Details of experimental conditions to obtain the interference pattern have been discussed in numerous publications (see, for example, Wang et al. (1991); Lemos et al. (2022a, b)).

We apply a unitary transformation on the field representing the idler photon between the two sources. Such transformations can readily be implemented in a laboratory using quarter- and half-wave plates. We choose a unitary transformation that is characterized by two controllable parameters ($\delta$ and $\theta$) and has the following form in the $\{|H\rangle,|V\rangle\}$ basis:

$\displaystyle U(\theta,\delta)=\begin{pmatrix}e^{-i\delta}\cos 2\theta&e^{-i\delta}\sin 2\theta\\ e^{i\delta}\sin 2\theta&-e^{i\delta}\cos 2\theta\end{pmatrix},$

(4)

where $0\leq\theta\leq\pi$ and $0\leq\delta/2\leq\pi$ can be understood as two half-wave plate angles.

The fact that we can fully control the choice of $\theta$ and $\delta$ plays crucial role in our measurement scheme. We show below that the effect of this unitary transformation appears in the interference patterns generated by the signal photon after it is projected onto appropriate polarization states. We also show how the entanglement of $\widehat{\rho}$ can be fully characterized from these single-photon interference patterns with the knowledge of the unitary transformation. For simplicity, we initially assume that there is no experimental loss. In the supplementary material, we provide a detailed description of how to treat dominant experimental losses and imperfections. We discuss the effects of experimental imperfections in Sec. VI.

We emphasize that the signal photon never interacts with the device performing the unitary transformation and the idler photon is never detected. These are two unique features of our entanglement measurement scheme in addition to the fact that no postselection or coincidence measurement is required.

In order to put our work into context with existing work, we now compare our results to those presented in Refs. Lahiri et al. (2021); Lemos et al. (2023). The entanglement of quantum states considered in Refs. Lahiri et al. (2021); Lemos et al. (2023) is fully characterized by one coherence term (off-diagonal element) of the density matrix. Consequently, such states are entangled if $\mathcal{V}_{D}|_{\theta=\frac{\pi}{4}}\neq 0$ or $\mathcal{V}_{R}|_{\theta=\frac{\pi}{4}}\neq 0$ and vice versa Lahiri et al. (2021); Lemos et al. (2023). This is no longer the case for generalized Werner states as illustrated by state $\widehat{\rho}_{2}$ in Table 1. State $\widehat{\rho}_{2}$ is characterized by state parameters $\eta=0.2$, $\mathscr{I}=1$, and $I_{H}=0.5$. We find that this state is separable (not entangled) even if $\mathcal{V}_{D}|_{\theta=\frac{\pi}{4}}\neq 0$ and $\mathcal{V}_{R}|_{\theta=\frac{\pi}{4}}\neq 0$. Furthermore, for the states considered in Refs. Lahiri et al. (2021); Lemos et al. (2023), one does not require to measure visibilities for $H$ and $V$ polarized signal photons to characterize entanglement in a loss-less scenario: measurements in the $H/V$ basis are required only to account for experimental losses. On the contrary, our results show that measurements in the $H/V$ basis are absolutely essential for characterizing entanglement of a generalized Werner state even in the absence of experimental losses.

In an experiment, numerous imperfections may appear. However, results of Ref. Lemos et al. (2023) show that most dominant ones are the misalignment of idler beams and loss of idler photons between the two sources. These imperfections need separate attention because they result in the loss of interference-visibility that cannot be compensated in any way.

In the Supplementary Material, we provide a detailed analysis showing how to treat such experimental imperfections. We represent the PPT criterion and the concurrence in terms of experimentally measurable quantities by taking these imperfections into account. Equations (G4) and (G5) of Appendix G display the corresponding formulas. Considering the five quantum states given by Table 1, we test the PPT criterion and determine their concurrence in the presence of high experimental loss (Supplementary Material, Table S1) and find that the results are identical to those presented in Table 1 and Fig. 3. Our analysis thus shows that the proposed method is experimentally implementable and resistant to experimental losses.

It is a common perception that one must detect both photons to measure entanglement of a two-photon mixed state. Only recently it has been demonstrated that by the application of “quantum indistinguishability by path identity” Hochrainer et al. (2022) it is possible to measure entanglement in a special class of two-qubit mixed states without detecting one of the qubits Lahiri et al. (2021); Lemos et al. (2023). We have generalized the method to cover a wider class of states. In particular, we have shown how to measure entanglement in a two-qubit generalized Werner state without detecting one of the qubits and without employing coincidence measurement or postselection. Our analysis shows that the generalization requires non-trivial adaptation of the previously introduced measurement procedure, especially when experimental imperfections are present. Our work also marks an important step toward the generalization of this unique method to an arbitrary two-qubit mixed state.

In a recent publication Zhan (2021), Zhan has proposed to extend the measurement procedure introduced in Refs. Lahiri et al. (2021); Lemos et al. (2023) to mixed high-dimensional Bell states. We expect that our work will inspire novel proposals to measure entanglement in high-dimensional generalized Werner states, which would not require detection of all entangled particles. We hope that such a generalization will contribute toward reducing the resource requirement for entanglement measurement of high-dimensional entangled states.

Finally, since our analysis is based on quantum field theory, it can in principle be applied to non-photonic quantum states. As the method relies on detection of only one of the two entangled particles, it can be practically advantageous when adequate detectors are not available for both entangled particles.

This material is based upon work supported by the Air Force Office of Scientific Research under award number FA9550-23-1-0216.

We obtain a partial transposition of the density matrix, $\widehat{\rho}$ [Eq. (1)], by taking the transposition with respect to only one of the photons. It can be readily found that the eigenvalues of the resulting matrix are

	$\displaystyle\alpha_{1}$	$\displaystyle=\frac{1-\eta-4\eta\mathscr{I}\sqrt{I_{H}I_{V}}}{4},$		(A1a)
	$\displaystyle\alpha_{2}$	$\displaystyle=\frac{1-\eta+4\eta\mathscr{I}\sqrt{I_{H}I_{V}}}{4},$		(A1b)
	$\displaystyle\alpha_{3}$	$\displaystyle=\frac{1+3\eta-4\eta I_{H}}{4},$		(A1c)
	$\displaystyle\alpha_{4}$	$\displaystyle=\frac{1-\eta+4\eta I_{H}}{4}.$		(A1d)

We show below that only $\alpha_{1}$ can take negative values and the remaining eigenvalues must be non-negative.

Since $0\leq I_{H}\leq 1$ and $I_{V}=1-I_{H}$, we must have $0\leq\sqrt{I_{H}I_{V}}\leq 1/2$. Furthermore, since $0\leq\mathscr{I}\leq 1$, we obtain the condition

$\displaystyle 0\leq\mathscr{I}\sqrt{I_{H}I_{V}}\leq 1/2.$

(A2)

Applying condition (A2) to Eq. (A1a), we immediately find that

$\displaystyle\frac{1-3\eta}{4}\leq\alpha_{1}\leq\frac{1-\eta}{4}.$

(A3)

The expression on the left of inequality (A3), is negative when $\eta>1/3$, that is, the lower bound of $\alpha_{1}$ can be negative. For example, if one sets $I_{H}=1/2$, $\mathscr{I}=1$, and $\eta=2/3$, one finds from Eq. (A1a) that $\alpha_{1}=-1/4$.

We now consider eigenvalue $\alpha_{2}$. We note that $1-\eta\geq 0$ and $4\eta\mathscr{I}\sqrt{I_{H}I_{V}}\geq 0$. It now becomes evident from Eq. (A1b) that $\alpha_{2}\geq 0$.

Using Eq. (A1c), we can express $\alpha_{3}$ in the following form

$\displaystyle\alpha_{3}=\frac{1-\eta I_{H}}{4}+\frac{3\eta(1-I_{H})}{4}.$

(A4)

Since $0\leq\eta\leq 1$ and $0\leq I_{H}\leq 1$, it immediately follows that $\alpha_{3}\geq 0$.

Finally, since $1-\eta\geq 0$ and $4\eta I_{H}\geq 0$, we readily obtain from Eq. (A1d) that $\alpha_{4}\geq 0$.

The concurrence is determined following the prescription given in Wootters (1998). First, the spin flipped density matrix $\widehat{\widetilde{\rho}}$ is determined using the formula

$\displaystyle\widehat{\widetilde{\rho}}=(\widehat{\sigma}_{y}\otimes\widehat{\sigma}_{y})\widehat{\rho}^{*}(\widehat{\sigma}_{y}\otimes\widehat{\sigma}_{y}),$

(B1)

where $\widehat{\sigma}_{y}$ is the second Pauli matrix, $\otimes$ represents the Kronecker product, the asterisk $(\ast)$ implies complex conjugation, and $\widehat{\rho}$ is given by Eq. (1). It is well known that $\widehat{\widetilde{\rho}}$ has only non-negative eigenvalues Wootters (1998). We denote these eigenvalues by $\lambda_{1}^{2}$, $\lambda_{2}^{2}$, $\lambda_{3}^{2}$, and $\lambda_{4}^{2}$. We find these eigenvalues to be given by

	$\displaystyle\lambda_{1}^{2}=\lambda_{2}^{2}=\frac{(1-\eta)^{2}}{16},$		(B2a)
	$\displaystyle\lambda_{3}^{2}=\frac{(1-\eta)^{2}}{16}+c_{1}-c_{2},$		(B2b)
	$\displaystyle\lambda_{4}^{2}=\frac{(1-\eta)^{2}}{16}+c_{1}+c_{2},$		(B2c)

where

	$\displaystyle c_{1}=\frac{\eta}{4}\left(1-\eta\right)+\eta^{2}I_{H}I_{V}(1+\mathscr{I}^{2}),$		(B3a)
	$\displaystyle c_{2}=\frac{\eta\mathscr{I}}{2}\sqrt{I_{H}I_{V}[1+2\eta-\eta^{2}(3-16I_{H}I_{V})]}.$		(B3b)

It is evident from Eqs. (B3a) and (B3b) that $c_{1}\geq 0$ and $c_{2}\geq 0$. Therefore, we have from Eqs. (B2a)–(B2c) that $\lambda_{4}\geq\lambda_{3}$ and $\lambda_{4}\geq\lambda_{1}=\lambda_{2}$. Consequently, the standard formula for the concurrence, $C(\widehat{\rho})=\text{max}\{\lambda_{4}-\lambda_{3}-\lambda_{2}-\lambda_{1},0\}$ Wootters (1998), becomes

$\displaystyle C(\widehat{\rho})=\text{max}\{\lambda_{4}-\lambda_{3}-2\lambda_{1},0\},$

(B4)

where $\lambda_{1},\dots,\lambda_{4}$ are positive square-roots of $\lambda_{1}^{2},\dots,\lambda_{4}^{2}$.

We now observe from Eqs. (A1a) and (B2) that the following relation holds:

$\displaystyle\left[\lambda_{4}^{2}+\lambda_{3}^{2}-4(\lambda_{1}-\alpha_{1})^{2}\right]^{2}=4\lambda_{4}^{2}\lambda_{3}^{2}.$

(B5)

It immediately follows from Eq. (B5) that

$\displaystyle\lambda_{4}-\lambda_{3}-2\lambda_{1}=-2\alpha_{1}.$

(B6)

Combining Eqs. (A1a), (B4), and (B6) we immediately obtain the concurrence-formula given by Eq. (3).

We briefly discuss the procedure of obtaining the two-photon density matrix generated by our system. We start with the generalized Werner state [Eq. (5)]:

$\displaystyle\widehat{\rho}=\sum_{\mu,\nu}^{H,V}\sum_{\mu^{\prime},\nu^{\prime}}^{H,V}\langle\mu_{I}\nu_{S}|\widehat{\rho}|\mu_{I}^{\prime}\nu_{S}^{\prime}\rangle|\mu_{I}\nu_{S}\rangle\langle\mu_{I}^{\prime}\nu_{S}^{\prime}|.$

(C1)

Without any loss of generality, an arbitrary element of this density matrix can be expressed in the form

$\displaystyle\langle\mu_{I}\nu_{S}|\widehat{\rho}|\mu_{I}^{\prime}\nu_{S}^{\prime}\rangle=C_{\mu\nu}C_{\mu^{\prime}\nu^{\prime}}J_{\mu\nu}^{\mu^{\prime}\nu^{\prime}}\exp\big{(}i\phi_{\mu\nu}^{\mu^{\prime}\nu^{\prime}}\big{)},$

(C2)

where each quantity on the right-hand side is defined as follows. The real and non-negative quantity, $C_{\mu\nu}=\sqrt{\langle\mu_{I}\nu_{S}|\widehat{\rho}|\mu_{I}\nu_{S}\rangle}$, represents the square-root of a diagonal element of $\widehat{\rho}$. The quantity $J_{\mu\nu}^{\mu^{\prime}\nu^{\prime}}$ is also non-negative and given by the properties: (i) $J_{\mu\nu}^{\mu^{\prime}\nu^{\prime}}=J_{\mu^{\prime}\nu^{\prime}}^{\mu\nu}$, (ii) $J_{\mu\nu}^{\mu^{\prime}\nu^{\prime}}=1$ for all diagonal elements, i.e., for $\mu=\mu^{\prime}$ and $\nu=\nu^{\prime}$, (iii) for $\mu\neq\mu^{\prime}$, $\nu\neq\nu^{\prime}$, $\mu=\nu$ and $\mu^{\prime}=\nu^{\prime}$, it takes the following form:

$\displaystyle J_{HH}^{VV}=J_{VV}^{HH}=\frac{4\mathscr{I}\eta\sqrt{I_{H}I_{V}}}{\sqrt{(1-\eta+4\eta I_{H})(1-\eta+4\eta I_{V})}},$

(C3)

and (iv) $J_{\mu\nu}^{\mu^{\prime}\nu^{\prime}}=0$ for the remaining cases. Phases $\phi_{\mu\nu}^{\mu^{\prime}\nu^{\prime}}$ in Eq. (C2) obey the following relations: $\phi_{\mu\nu}^{\mu^{\prime}\nu^{\prime}}=-\phi_{\mu^{\prime}\nu^{\prime}}^{\mu\nu}$, $\phi_{HH}^{VV}=-\phi$, and values of $\phi_{\mu\nu}^{\mu^{\prime}\nu^{\prime}}$ for other choices of $\mu,\nu,\mu^{\prime},\nu^{\prime}$ are irrelevant since the corresponding matrix elements are zero. The quantum state given by Eqs. (C1)-(C3) is generated by an individual source.

We first consider the case in which the idler beams are not aligned. In our system, the two sources are mutually coherent and they emit in such a way that there are never more than one photon pair simultaneously. A prescription to write down the quantum state in such a scenario is given in Ref. Lahiri et al. (2021). Following this prescription, we find that the state jointly generated by the two sources is represented by the $8\times 8$ density matrix

	$\displaystyle\widehat{\rho}^{\prime\prime}=\sum_{j,k}^{1,2}b_{j}^{\ast}b_{k}\sum_{\mu,\nu}^{H,V}\sum_{\mu^{\prime},\nu^{\prime}}^{H,V}$	$\displaystyle C_{\mu\nu}C^{*}_{\mu^{\prime}\nu^{\prime}}J_{\mu\nu}^{\mu^{\prime}\nu^{\prime}}\exp\big{(}i\phi_{\mu_{k}\nu_{k}}^{\mu^{\prime}_{j}\nu^{\prime}_{j}}\big{)}$
		$\displaystyle\times|\mu_{I_{k}}\nu_{S_{k}}\rangle\langle\mu^{\prime}_{I_{j}}\nu^{\prime}_{S_{j}}|,$		(C4)

where $j=1,2$ and $k=1,2$ label the two sources, $|b_{j}|^{2}$ is the probability of emission from source $Q_{j}$ (i.e., $|b_{1}|^{2}+|b_{2}|^{2}=1$), quantities $C_{\mu\nu}$ and $J_{\mu\nu}^{\mu^{\prime}\nu^{\prime}}$ are defined below Eq. (C2), $\phi_{\mu_{k}\nu_{k}}^{\mu^{\prime}_{j}\nu^{\prime}_{j}}=-\phi_{\mu^{\prime}_{j}\nu^{\prime}_{j}}^{\mu_{k}\nu_{k}}$, and $\phi_{\mu_{j}\nu_{j}}^{\mu^{\prime}_{j}\nu^{\prime}_{j}}=\phi_{\mu\nu}^{\mu^{\prime}\nu^{\prime}}$.

When the idler beams are aligned and the unitary transformation [Eq. (4)] is performed on the state of the idler photon between $Q_{1}$ and $Q_{2}$, the transformation of kets are given by Eqs. (6a) and (6b). We rewrite these equations in the following form:

$\displaystyle|\mu_{I_{2}}\rangle=e^{-i\phi_{I}}\sum_{\lambda}^{H,V}U_{\mu\lambda}^{\ast}|\lambda_{I_{1}}\rangle,$

(C5)

where $U_{\mu\lambda}$ represents matrix elements of the unitary transformation given by Eq. (4).

The two-photon quantum state ($\widehat{\rho}^{(f)}$) generated by our system is obtained by substituting from Eq. (C5) into Eq. (Appendix C: The two-photon density matrix) and it is given by

		$\displaystyle\widehat{\rho}^{(f)}=|b_{1}|^{2}\sum_{\mu,\nu}^{H,V}\Big{[}\eta I_{\mu}|\mu_{I_{1}},\mu_{S_{1}}\rangle\langle\mu_{I_{1}},\mu_{S_{1}}|+\Big{(}\mathscr{I}\eta\sqrt{I_{H}I_{V}}e^{-i\phi}|H_{I_{1}},H_{S_{1}}\rangle\langle V_{I_{1}},V_{S_{1}}|+\text{H.C.}\Big{)}+\frac{1-\eta}{4}|\mu_{I_{1}},\nu_{S_{1}}\rangle\langle\mu_{I_{1}},\nu_{S_{1}}|\Big{]}$
		$\displaystyle+|b_{2}|^{2}\sum_{\mu,\nu}^{H,V}\Big{[}\eta I_{\mu}\sum_{\lambda,\lambda^{\prime}}^{H,V}U^{*}_{\mu\lambda}U_{\mu\lambda^{\prime}}|\lambda_{I_{1}},\mu_{S_{2}}\rangle\langle\lambda^{\prime}_{I1},\mu_{S_{2}}|+\Big{(}\sum_{\lambda,\lambda^{\prime}}^{H,V}\mathscr{I}\eta\sqrt{I_{H}I_{V}}e^{-i\phi}U^{*}_{H\lambda}U_{V\lambda^{\prime}}|\lambda_{I_{1}},H_{S_{2}}\rangle\langle\lambda^{\prime}_{I1},V_{S_{2}}|+\text{H.C.}\Big{)}$
		$\displaystyle+\frac{1-\eta}{4}\sum_{\lambda,\lambda^{\prime}}^{H,V}U^{*}_{\mu\lambda}U_{\mu\lambda^{\prime}}|\lambda_{I_{1}},\nu_{S_{2}}\rangle\langle\lambda^{\prime}_{I1},\nu_{S_{2}}|\Big{]}+\Big{\{}e^{i\phi_{I}}b_{1}b_{2}^{*}\Big{[}\sum_{\mu,\nu}^{H,V}\Big{(}\sum_{\lambda}^{H,V}\eta I_{\mu}e^{i\phi_{\mu_{1}\mu_{1}}^{\mu_{2}\mu_{2}}}U_{\mu\lambda}|\mu_{I_{1}},\mu_{S_{1}}\rangle\langle\lambda_{I_{1}},\mu_{S_{2}}|$
		$\displaystyle+\frac{1-\eta}{4}\sum_{\lambda}^{H,V}e^{i\phi_{\mu_{1}\nu_{1}}^{\mu_{2}\nu_{2}}}U_{\mu\lambda}|\mu_{I_{1}},\nu_{S_{1}}\rangle\langle\lambda_{I_{1}},\nu_{S_{2}}|\Big{)}+\mathscr{I}\eta\sqrt{I_{H}I_{V}}\sum_{\mu\neq\nu}^{H,V}\sum_{\lambda}^{H,V}e^{i\phi_{\mu_{1}\mu_{1}}^{\nu_{2}\nu_{2}}}U_{\nu\lambda}|\mu_{I_{1}},\mu_{S_{1}}\rangle\langle\lambda_{I_{1}},\nu_{S_{2}}|\Big{]}+\text{H.C.}\Big{\}}$		(C6)

We obtain the detection probability, $P_{\mu}$, of a signal photon with polarization $\mu$ (where $\mu=H,V,D,A,R,L$) at an output of $BS$ (Fig. 2) using Eqs. (IV.1)-(10). When the signal photon is projected onto $|H_{S}\rangle$ polarization state, we have from these equations

	$\displaystyle P_{H}$	$\displaystyle=\frac{1-\eta}{4}+\frac{\eta I_{H}}{2}+|b_{1}||b_{2}|\Big{\{}\sin(\phi_{\text{in}}+\phi_{H_{1}H_{1}}^{H_{2}H_{2}}-\delta)$
		$\displaystyle\times\Big{(}\eta I_{H}+\frac{1-\eta}{4}\Big{)}-\Big{(}\frac{1-\eta}{4}\Big{)}\sin(\phi_{\text{in}}+\phi_{V_{1}H_{1}}^{V_{2}H_{2}}+\delta)\Big{\}}$
		$\displaystyle\times\cos(2\theta),$		(D1)

where $\phi_{\text{in}}=\arg\{b_{1}\}-\arg\{b_{2}\}+\phi_{I}-\phi_{S}$. We now note the following trigonometric identity:

		$\displaystyle u\sin x+v\sin(x+\alpha)$
		$\displaystyle=\{u^{2}+v^{2}+2uv\cos\alpha\}^{\frac{1}{2}}\sin(x+\beta),$		(D2)

where $\tan\beta=u\sin\alpha/(u+v\cos\alpha)$. Now using Eqs. (Appendix D: General expressions for signal photon detection probabilities (photon counting rates)) and (Appendix D: General expressions for signal photon detection probabilities (photon counting rates)), we find that

		$\displaystyle P_{H}=\frac{1-\eta}{4}+\frac{\eta I_{H}}{2}+|b_{1}||b_{2}|\cos(2\theta)\Big{\{}\Big{(}\eta I_{H}+\frac{1-\eta}{4}\Big{)}^{2}$
		$\displaystyle+\Big{(}\frac{1-\eta}{4}\Big{)}^{2}-2\Big{(}\eta I_{H}+\frac{1-\eta}{4}\Big{)}\Big{(}\frac{1-\eta}{4}\Big{)}\cos(\chi+2\delta)\Big{\}}^{\frac{1}{2}}$
		$\displaystyle\times\sin(\phi_{\text{in}}+\phi_{0}),$		(D3)

where $\chi=\phi^{V_{2}H_{2}}_{V_{1}H_{1}}-\phi^{H_{2}H_{2}}_{H_{1}H_{1}}$, $\phi_{0}=\phi_{H_{1}H_{1}}^{H_{2}H_{2}}-\delta+\epsilon_{1}$ and $\epsilon_{1}$ is analogous to $\beta$ in Eq. (Appendix D: General expressions for signal photon detection probabilities (photon counting rates)). It can readily checked that Eq. (Appendix D: General expressions for signal photon detection probabilities (photon counting rates)) reduces to Eq. (IV.2) when $\theta=0$.