Proposal for a loophole-free Bell test based on spin–photon interactions in cavities

The correlations between separated observers performing local measurements on an entangled state cannot be explained by a classical mechanism. A signal is impossible, as it would have to travel faster than light. Pre-established correlations are ruled out via the violation of a Bell inequality [1]. This phenomenon termed quantum non-locality is at the very heart of quantum mechanics and arguably one of the most surprising and counterintuitive aspects of the theory (see [2] for a recent review). In recent years it has received growing attention, partly due to the development of device-independent (DI) quantum information processing [3–9], which is based on quantum non-locality. The idea is that the correct implementation of a protocol can be guaranteed without resorting to assumptions about the devices used in the protocol. For instance, in DI quantum key distribution [4], two distant parties can establish a secret key and guarantee its privacy, via a Bell inequality violation, without placing assumptions about the detailed functioning of their devices. Notably there is hope that this approach will be able to address some of the shortcomings of standard quantum key distribution (QKD) schemes [10]. Its practical implementation is however still challenging [11, 12].

Quantum non-locality has been demonstrated in numerous experiments. However, technical imperfections in these experiments open loopholes, which makes it still possible, in principle, to explain the data through a classical mechanism. Given the fundamental significance of quantum non-locality, it is highly desirable to perform a Bell experiment free of any loopholes. Realizing such a test is challenging as it requires (i) a space-like separation between the parties and (ii) a high enough detection efficiency. Failure to address (i) opens the locality loophole: the correlations can be explained by a sub-luminal signal. Failure to address (ii) opens the detection loophole [13]: a classical model exploiting the detector inefficiency can fake Bell inequality violations. Experiments carried out on photons could achieve (i) while atomic experiments could achieve (ii). However no experiment was yet able to close both loopholes simultaneously. In particular, typical photon detection efficiencies are still too low to close the detection loophole. Significant progress was nevertheless achieved in the last years, both from the experimental [14–17] and theoretical [18–23] point of view.

These issues must also be addressed for the implementation of DI protocols. Here loopholes (in particular the detection loophole) could in principle be exploited by the eavesdropper, and hence compromise the DI security of the protocol [5]. Therefore the perspective of a loophole-free Bell experiment, and more specifically of achieving detection-loophole-free Bell violations on long distances, is essential for implementing protocols such as DI-QKD.

Here we present a scheme for a loophole-free Bell test based on Faraday-type spin–photon interactions in cavities [26–28]. Our setup is hybrid, using both photons and 'atomic' systems, hence combining the best of both worlds. Using spin–photon interactions, the entanglement between two photons can be swapped to two atomic systems in cavities. Importantly this can be realized in a heralded way via the detection of the photons after interaction with the cavities, even in the weak coupling regime. Once the creation of an entangled pair is heralded, the measurement settings for the Bell test are generated. In this way the setup, being 'event-ready', is basically insensitive to the losses in the optical channel between Alice and Bob. Moreover, since spin measurements have efficiency close to unity, the setup is immune to the detection loophole. These features make our scheme particularly well adapted to the implementation of DI-QKD. In the following we start by presenting a simple theoretical model of our setup, estimating the effect of technical imperfections such as the amplitude of the spin–photon interaction and the decay rate of the atomic system in the cavity. Then, we discuss the implementation of our scheme in various platforms, including negatively charged nitrogen vacancy centres (NV⁻) in diamond, quantum dot spin systems and atoms. Finally, we discuss the implementation of DI-QKD based on our setup.

The setup is sketched in figure 1. A pair of polarization entangled photons is emitted by a central source and sent to two remote observers, Alice and Bob. The photons are in the singlet state

$$\begin{eqnarray} &&|\psi_-\rangle_{\rm AB} = \frac{1}{\sqrt{2}} (|R,L\rangle-|L,R\rangle), \end{eqnarray} \tag{ 1 }$$

where we have used the notation |R,L〉 = |R〉_A⊗|L〉_B, and |L〉 (|R〉) denote a left (right) circular polarization. At each observers laboratory, the photon is then sent through an optical cavity containing a spin 1/2 particle. The photon and spin couple via a Faraday type effect, whereby the polarization of the photon is rotated depending on the state of the spin. Here this interaction is represented by a unitary operation of the form

$$\begin{eqnarray} &&U(\alpha) ={\rm e}^{{\rm i} \alpha ( |L\rangle\langle L| \otimes |\kern-2pt\uparrow\rangle\langle\uparrow| + |R\rangle\langle R| \otimes |\kern-2pt\downarrow\rangle\langle\downarrow|)}, \end{eqnarray} \tag{ 2 }$$

where |↑〉 and |↓〉 represent the two orthogonal spin states. The parameter α represents the strength of the interaction. Experimentally, the achievable values of α depend on the physical system that is considered, as we will discuss below.

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Initially, both spins are prepared in the superposition state $(|\kern-2pt\uparrow \rangle +|\kern-2pt\downarrow \rangle )/\sqrt {2}$. Hence the initial state of the global system, photons and spins, is given by

$$\begin{eqnarray} &&|\Psi\rangle = |\psi_-\rangle_{\rm AB} \otimes \frac{|\kern-2pt\uparrow\rangle_{\rm A}+|\kern-2pt\downarrow\rangle_{\rm A}}{\sqrt{2}} \otimes \frac{|\kern-2pt\uparrow\rangle_{\rm B}+|\kern-2pt\downarrow\rangle_{\rm B}}{\sqrt{2}} \end{eqnarray} \tag{ 3 }$$

After each photon interacted with the spin in the cavity, the state becomes

$$\begin{eqnarray} \fl U(\alpha_{\rm A}) U(\alpha_{\rm B}) |\Psi\rangle &=& |R,L\rangle ({\rm e}^{{\rm i}\alpha_{\rm B}} |\kern-2pt\uparrow\uparrow\rangle + |\kern-2pt\uparrow\downarrow\rangle + {\rm e}^{{\rm i}(\alpha_{\rm A}+\alpha_{\rm B})} |\kern-2pt\downarrow\uparrow\rangle + {\rm e}^{{\rm i} \alpha_{\rm A}} |\kern-2pt\downarrow\downarrow\rangle ) \nonumber \\ && - |L,R\rangle ({\rm e}^{{\rm i}\alpha_{\rm A}} |\kern-2pt\uparrow\uparrow\rangle + {\rm e}^{{\rm i}(\alpha_{\rm A}+\alpha_{\rm B})} |\kern-2pt\uparrow\downarrow\rangle + |\kern-2pt\downarrow\uparrow\rangle + {\rm e}^{{\rm i} \alpha_{\rm B}} |\kern-2pt\downarrow\downarrow\rangle),\qquad \end{eqnarray} \tag{ 4 }$$

where we have omitted subscripts and normalization.

After exiting the cavity each photon is measured in the horizontal–vertical polarization basis. For simplicity we will focus here on the case in which both photons are detected in the horizontal mode, i.e. projected onto the state $|H\rangle = \frac {1}{\sqrt {2}}(|L\rangle +|R\rangle )$. The final state of the two spins (here unnormalized) is then found to be

$$\begin{eqnarray} |\Psi_{\mathrm{f}}\rangle &=& ({\rm e}^{{\rm i}\alpha_{\rm A}}-{\rm e}^{{\rm i}\alpha_{\rm B}}) |\phi_-\rangle + ({\rm e}^{{\rm i}(\alpha_{\rm A}+\alpha_{\rm B})}-1) |\psi_-\rangle, \end{eqnarray} \tag{ 5 }$$

where

$$\begin{eqnarray} &&|\phi_-\rangle=\frac{1}{\sqrt{2}} ( |\kern-2pt\uparrow \uparrow\rangle- |\kern-2pt\downarrow\downarrow\rangle), \quad |\psi_-\rangle= \frac{1}{\sqrt{2}}(|\kern-2pt\uparrow \downarrow\rangle- |\kern-2pt\downarrow\uparrow\rangle) \end{eqnarray} \tag{ 6 }$$

are the usual two-qubit Bell states. The state can now be conveniently rewritten as

$$\begin{eqnarray} &&|\Psi_{\mathrm{f}}\rangle = \frac{1}{\sqrt{N}} \left( \sin(\Delta \alpha) |\phi_-\rangle + \sin(\bar{\alpha} \right) |\psi_-\rangle), \end{eqnarray} \tag{ 7 }$$

where Δα = (α_A − α_B)/2 and $\bar { \alpha } = (\alpha _{\mathrm { A}}+\alpha _{\mathrm { B}})/2$ are the difference and average values of the spin–photon interactions (of Alice and Bob), respectively, and $N= \sin ^2(\Delta \alpha )+\sin ^2(\bar {\alpha })$ is a normalization factor. The state (7) is entangled, unless $\Delta \alpha = \bar {\alpha }$. Note that the probability of entangling the two spins, i.e. to detect both photons in the horizontal polarization mode, is given by N/4. Upon heralding of the spin entanglement, the measurement settings for the Bell test are generated. This operation takes an amount of time t during which the spins decohere. Here we model this effect via an exponential decay. After a time t, the state of the spins becomes

$$\begin{eqnarray} &&\rho_{\mathrm{f}} = {\rm e}^{-t/\tau} |\Psi\rangle\langle\Psi| + (1-{\rm e}^{-t/\tau}) \frac{\mathbb{I}}{4},\end{eqnarray} \tag{ 8 }$$

where $\mathbb {I}/4$ is the fully mixed state of two qubits.

Now that we have characterized the state of the two spins, we will investigate its ability to violate a Bell inequality. Indeed our goal is to define the range of parameters that will lead to non-local correlations. Here we will focus on the Clauser–Horne–Shimony–Holt (CHSH) Bell inequality [24]. In this case, both Alice and Bob can perform two possible binary measurements. The choice of measurement settings of Alice (Bob) is denoted x = 0,1 (y = 0,1), and the outcome a = ± 1 (b = ± 1). For a pair of measurement settings, x, y, the correlation between the measurement outcomes of Alice and Bob is given by

$$\begin{eqnarray} &&E_{xy} = p(a=b|x,y) - p(a \neq b|x,y). \end{eqnarray} \tag{ 9 }$$

The CHSH inequality is then given by

$$\begin{eqnarray} &&{\rm CHSH} = E_{00} + E_{01} + E_{10} - E_{11} \leqslant 2, \end{eqnarray} \tag{ 10 }$$

which holds for any local model. Violation of this inequality hence witnesses non-locality.

For our final state (8), we now determine what set of parameters (i.e. strength of spin–photon interaction and spin decoherence) in order to obtain a violation of the CHSH inequality. Note first that in the (unrealistic) case in which both spin–photon interaction can be set to be perfectly equal, i.e. α_A = α_B > 0, the state (8) takes the particularly simple form

$$\begin{eqnarray} &&\rho_{\mathrm{f}} = {\rm e}^{-t/\tau} |\psi_-\rangle\langle\psi_-| + (1-{\rm e}^{-t/\tau}) \frac{\mathbb{I}}{4},\end{eqnarray} \tag{ 11 }$$

that is, a two-qubit Bell state mixed with white noise. Thus, Alice and Bob should use the optimal CHSH measurements, tailored for the singlet state |ψ₋〉. In this case the CHSH inequality is violated whenever ${\mathrm { e}}^{-t/\tau }>1/\sqrt {2}$, i.e. $t/\tau < \ln {\sqrt {2}} \,\simeq\, 0.3466$. Importantly, CHSH violation occurs here regardless of the strength of the spin–photon interaction, i.e. for any α_A = α_B > 0.

In the more realistic case in which Δα is non-zero, the state (8) takes a more complicated form. Luckily, the maximal CHSH violation of any two-qubit state can be evaluated using the criterion of Horodecki et al [25]. The results are presented in figure 2. Notably we observe now a trade-off between mean interaction strength $\bar {\alpha }$ and the spin decoherence t/τ. In particular, and as expected intuitively, for weak interaction quick spin measurements are demanded, whereas for stronger interactions slower spin measurements can be tolerated. In the following we will evaluate the feasibility of our setup, how stringent the requirement of figure 2 are, for various physical platforms.

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The realization of the unitary in equation (2) involves a degenerate polarization-dependent lambda transition strongly coupled to a cavity. The polarization dependent transitions arise in atomic transitions with spin half ground states [29], certain transitions in NV⁻ centres [30] and charged quantum dots [32]. Here we specialize to a reflection geometry because it can be shown [26] that reflected probe light will experience a polarization dependent phase shift α (and consequent Faraday rotation) which becomes large when cavity mode volume is small and the cavity storage time measured by its quality factor Q is high. The reflected photon is thus correlated with the ground state spin and coincident detection of two horizontal polarized photons can be used to herald entanglement of two remote spins.

We estimate the achievable $\bar {\alpha }$ assuming a (lossy) single sided cavity containing a single two level system (atom) in the weak excitation regime [26, 27]. The reflection coefficient and phase shift α at frequency ω is given by

$$\begin{eqnarray} &&r(\omega)=|r(\omega)|{\rm e}^{{\rm i}\alpha}=1-\frac{\kappa({\rm i}(\omega_{\mathrm{d}}-\omega)+\frac{\gamma}{2})}{({\rm i}(\omega_{\mathrm{d}}-\omega)+\frac{\gamma}{2})({\rm i}(\omega_{\mathrm{c}}-\omega)+\frac{\kappa}{2}+\frac{\kappa_{s}}{2})+g^{2}},\nonumber \end{eqnarray}$$

where g is coupling rate between the 'atom' and the cavity, γ is 'atom' decay rate, ω_c, ω_d, are the cavity and 'atom' resonances, respectively, and κ is the decay rate from the cavity κ_s the decay rate due to losses with κ + κ_s = ω_c/Q.

For demonstration of loophole-free Bell inequality violation we require that the spin is stored long enough for the herald photon events to be registered as a coincidence detection which sets a minimum value of measurement delay t ∼ D/c + T_l where D is the separation of observers and T_l is a minimum detection and electronic logic delay (of order 100 ns). Once a coincidence event is recorded, the observers generate locally their measurement settings. We also require the measurement time uncertainty, the time taken to manipulate and read out the spin state Δt, to be small compared to D/c to avoid the locality loophole. Thus we can summarize the requirements for a loophole free test as

$$\begin{equation} \Delta t<D/c \ll\tau. \end{equation} \tag{ 12 }$$

The read out of a spin cavity system that is strongly coupled involves the measurement of the Faraday rotation induced by the spin on an incident linear polarized beam. Using diagonally polarized probe light and measuring reflected light in a polarization beamsplitter oriented to differentiate horizontal (H) and vertical (V) polarizations the spin state can be estimated from the difference in photon counts detected in the H and V detectors N_H − N_V = ± N_tot sin α/2 with total detected photocounts N_H + N_V = N_tot and with positive (negative) outcome signalling spin up (down). The read out time is limited by the constraint of having < 0.1 photon on average in the cavity at any one time and having a reasonable certainty in the spin measurement which is guaranteed when $\sin \bar {\alpha } \gg 1/\sqrt { N_{\mathrm { tot}}}$. This constrains measurement time to be

$$\begin{equation} \Delta t \gg 10\tau_{\rm c}/\sin^2 \bar{\alpha}, \end{equation} \tag{ 13 }$$

where τ_c = Q/ω_c is the cavity lifetime. By suitably enforcing inequality (13), we can ensure the fidelity of the readout is near unity, this implies near unit efficiency (hence ensuring that the detection loophole is closed).

For the case of NV⁻ and atomic transitions the ground states are often separated by an energy such that only one state is in resonance with the cavity at a time. Equation (4) is simplified to U(α) = e^{iα(|L〉〈L|⊗|↑〉〈↑|)} but the underlying maths does not change if we halve $\bar {\alpha }$ and take this into account in the following calculations.

In the case of NV⁻ manufacturing high Q low loss cavities is also difficult. We could exploit recently fabricated low Q-factor microcavities [33] in order to perform a loophole free Bell test. Here using very small cavity volumes the interaction (g) between the NV⁻ and the light can be greatly enhanced while the cavity Q-factors remain low (κ ≪ g) and strong coupling is not reached. Instead we operate in the one-dimensional atom regime [35] where by suitably adjusting the cavity input–output coupling efficiency and looking close to resonance with the NV⁻ zero phonon line we can achieve a large phase shift approaching π albeit with lowered reflectivity (see the appendix). However when we move from strong coupling into the weak coupling regime the minimum readout time is now determined by the lifetime of the state τ_s rather than that of the cavity τ_c in equation (13). In such small cavities this can be reduced by an order of magnitude from bulk lifetimes due to Purcell enhancement. The readout time is also extended by the reduced cavity reflectivity in the weak coupling regime.

We summarize state of the art for the four systems in table 1 estimating parameters from the published work on strongly coupled atom [29], NV⁻ [31] and dot [32] cavity systems and the low-Q photonic crystal cavities [33]. For simplicity we have estimated t from twice the minimum readout time (assuming equality in (12)). The results show that the NV⁻ and atom cavity systems satisfy the requirement that the delay between storage and readout should be very short compared to decoherence time τ. However the long cavity storage times needed to achieve strong coupling in these systems mean that the minimum separation of measurement stations is of order 150 m for the atoms and 100 m for NV⁻. At first sight the measurement time on the dot cavity system is also short enough to mean that decoherence is small, however we have to take into account the inevitable electronic delays which mean the minimum t ∼ 100 ns and thus we expect t/τ ∼ 0.1. The low-Q cavity results are very encouraging as cavities with these specifications are already being fabricated [33]. We estimate that the reflectivity of these systems will still be > 30% and thus counting losses will not be severe. However we note that for all realizations the linewidth of the entangled pair photon source must be smaller than the cavity linewidth (or transition linewidth in the case of low-Q cavities) which will require sophisticated filtering if standard crystal sources are to be used and pair emission rates will be limited to R ≃ 0.1/τ_c (τ_s for low-Q case) which is hundreds of thousands per second for atoms and NV⁻ and many million per second for dot systems. We have plotted the nominal positions of the various realizations in figure 2 and can clearly see that loophole free violations can be expected when Δα < π/10. Hence the experiment could in principle be carried out in the weak coupling regime, although one should indeed ensure that the probability of a successful heralding is large compared to photo-detector dark counts.

Table 1. Estimated parameters for the implementation of our setup in various physical platforms.

System	Atoms	N–V centres	Dots	Low-Q
g/(2π) (MHz)	5	100	5000	3300
κ/(2π) (MHz)	3	13	3000	440 000
κs/(2π) (MHz)	0.5	39	7000	220 000
γ/(2π) (MHz)	3	0.6	1000	6
$\bar \alpha$ (Rad)	∼0.4π	∼0.1π	∼0.1π	∼0.4π
τ (μs)	10 000	1000	1	1000
Δt (ns)	500	300	1.5	1000
Min D (m)	150	100	<1	300
Min t (μs)	∼1	∼0.3	∼0.1	∼2

Finally, we give an estimate of the total time required to perform the experiment. As an example, we consider the case of a low-Q cavity with $\bar {\alpha } = 0.4\,\pi$ and Δα = π/10. The expected Bell violation is then CHSH ≃ 2.32. Considering a separation of 300 m between Alice and Bob and a transmission loss of 3 dB km⁻¹ (considering photons at wavelength 700 nm) we obtain a channel transmission of η_t ∼ 80%. The probability of a successful heralding is then $p_{\mathrm { herald}} = \eta _{\mathrm {t}} \eta _{\mathrm { c}}^2 \eta _{\mathrm { d}}^2 \frac {N}{4} \sim 7\times 10^{-4}$ per photon pair emitted from the source, where η_c is the reflectivity of the cavity and η_d is the efficiency of the photo-detectors (here we take η_c = η_d = 30%). To match the cavity, a narrow-band (cavity enhanced) source of entangled photons is required [34], which can produce ∼ 10⁶ pairs per second. We thus expect ∼ 10³ heralding events per second. Hence the main limitation comes from the measuring time, here estimated to be of order ms. Good statistics, say 10⁵ runs, would then necessitate a few minutes of data acquisition.

The setup discussed here could also be adapted for the implementation of DI protocols, such as DI-QKD. Two possibilities can be considered. First the scheme of figure 1, featuring a spin–photon interface on both sides, and a source of photons placed in the middle. Note however that the requirements for implementing DI-QKD are slightly different compared to the case of a loophole-free Bell test (see discussion in [5]). In particular, it is important to ensure that no information can leak out of Alice and Bob's labs, for instance by an adequate shielding. Thus, the setup will be immune against the locality loophole, and the timing constraints discussed in the previous section can be relaxed. Here the main requirement will be that Δt < τ. Another possibility consists in using the spin–photon interface only on Bob's side, and placing the source close to Alice's lab. Then the spin–photon interface is used to herald the presence of a successful transmission of a signal to Bob's lab, similarly to the 'qubit amplifier' of [11].

Finally, we give an estimate of the key rate following the first approach mentioned above (i.e. the scheme of figure 1). As we are considering much larger distances here compared to the case of a loophole-free Bell test, it will be important to choose a setup in which the wavelength of photons is in the telecom window to limit losses. Here, as an example, we consider an implementation based on quantum dots working with photons at λ = 1.3 μm, with Faraday interactions given by $\bar {\alpha } = 0.1 \pi$ and Δα = π/50. The expected Bell violation is then CHSH ≃ 2.45. We consider a source working at 10 GHz [37], with a probability of pair creation of 10⁻³, and expect transmission losses of 0.3 dB km⁻¹. Following Pironio et al [5], which considers security against collective attacks, the key rate is given by

$$\begin{eqnarray} &&R = 1- h(q) -h\left(\frac{1+ \sqrt{({\rm CHSH}/2)^2-1}}{2}\right), \end{eqnarray} \tag{ 14 }$$

where h(x) = −x log₂ x − (1 − x)log₂(1 − x) is the binary entropy. Here h(q) represents the fraction of the raw key that Alice and Bob must sacrifice in order to correct for errors; note that here the error is given by $q= \sin ^2{\Delta \alpha }/(\sin ^2{ \bar {\alpha }} + \sin ^2{\Delta \alpha }) \simeq 4 \%$. In figure 3, we plot the key rate as a function of the distance. We see that for distances up to 100 km we get decent key rates, considering reasonable parameters for the reflectivity of the cavity and the photo-detector efficiency. We believe that this opens promising perspectives for implementing DI-QKD using the present scheme. A detailed study of the performance of the spin–photon interface (and its implementation in different physical platforms) would be of great interest⁵.

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We discussed a hybrid scheme where entanglement between photons is transferred to solid state (spin) qubits mediated by cavity QED. The successful entanglement of the spins can be heralded by the detection of the photons. The spins can be measured efficiently within times that allow both the locality and the detection loopholes to be closed. The required apparatus separations are measured in hundreds of metres when we consider atom and NV⁻ centres ground states as our matter qubits in realizable cavities while this could be reduced to metres if the electron spin of singly charged quantum dots is used as the qubit. In this latter case the remaining challenge is to achieve electron spin decoherence times longer than 1 μs possibly through sophisticated spin echo schemes [36]. Finally, we discussed potential applications of these systems for the implementation of DI protocols.

We thank A Acin and C Simon for discussions. This work is supported by the UK EPSRC, the EU projects DIQIP and Q-ESSENCE, the Swiss National Science Foundation (grant no. PP00P2_138917) and ERC Advanced grant no. (QUOWSS 247462).

Here we show that a low q-factor photonic crystal microcavities fabricated in single crystal diamond around a single NV⁻ centre [21] could be suitable for performing a loophole free Bell test. The ratios of g:κ_T:γ are estimated from [21], and can be found in table 1 in the main document. We then calculate the reflected amplitude and phase when photons are reflected from the NV⁻ cavity system using equation (12) defining κ_T = κ + κ_s as the total decay rate from the cavity setting the Q-factor.

The calculated reflectivity and phase can now be seen in figure A.1. where we have kept κ_T constant and varied the ratio of κ:κ_s. Ideally for optimum efficiency we would like to work in the loss free regime where κ ≫ κ_s however this is not a realistic goal in present day microcavity structures. A more realistic target is to work in the regime where κ ≈ 4κ_s, this has the added benefit as at this point the empty cavity reflectivity (r_cold) is approximately equal to cavity with resonant NV⁻ centre (r_hot) so we do not need to modify the equations in the main document. At this point the maximum Faraday rotation is α = π/2 (sin(α) = 1) and r_hot = r_cold = 0.65 which leads to a high fidelity protocol with a decrease in efficiency.

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It is worth pointing out that whilst for simplicity here we have considered the case where r_hot = r_cold, the fidelity of the preparation of the |ψ⁻〉 state we are interested in is unaffected by non-equal reflectivity's between hot and cold cavities, and differences between the reflectivity of Alice and Bob's respective spin–photon-interfaces.