**4. Experimental tests of Bell's inequalities**

Let us now consider the following experiment wherein there are two experimental stations as discussed in the earlier Gedankenexperiment but with a small twist: Here, S is a SPDC source emitting a pair of polarization entangled photons in one of the Bell states and let *a* and *b* be randomly obtained from certified QRNGs located and securely isolated at the stations **a** and **b** respectively. The two inputs of are then used to choose between the two mutually unbiased {*V,H*} and {*D,A*} where, *V/H* refers to the vertical/horizontal basis and *D/A* refers to diagonal/anti-diagonal basis sets respectively (**Figure 2**).

The state emitted from the source is one of the Bell states and let us assume without loss of generality, that the state is *ψ*þ⟩ Spontaneous parametric downconversion is a probabilistic process with a very low probability of emitting an entangled photon pair. Hence, for optimal pump laser powers, the probability of multiple pairs being emitted is extremely low. After considering the travel time and setting a coincidence window, when both detectors detect photons, it most likely that the pair of photons were emitted simultaneously and are in an entangled state. The source may be suitably characterized through Quantum State Tomography

#### **Figure 2.**

*S is a entangled photon source emitting photon in a Bell state with one photon of each pair reaching stations A and B located in space-like separated regions. Inputs x, y to these boxes take value {0, 1}. The basis choice at either station is made based on the random inputs x, y. The output of these boxes are a, b assuming values {*�*1, +1} as earlier.*

#### **Figure 3.**

*Basis choices for measurements may be made independently at either station such as H/V, H\_α/V\_α for arbitrary α.*

(QST) [3]. Usually, local corrections of polarization may be corrected through suitable polarization controllers located at *A* and *B*. Once the steps are done and the source is well-characterized, projective measurements are carried out at each of the stations. To carry out measurements, the basis choice at each station is carried out rotating the polarizers by some angle. We may however choose to measure in rotated basis as illustrated (**Figure 3**).

The rotated basis vectors may be expressed as:

$$|H\_a\rangle = \cos a|H\rangle - \sin a|V\rangle, |V\_a\rangle = \sin a|H\rangle + \cos a|V\rangle \tag{12}$$

If the polarizers at *A* and *B* are rotated by angles *α* and *β* respectively and an ensemble of measurements are carried out on identically prepared states, quantum mechanics predicts the probability of obtaining coincidence counts when the vertical polarization is measured to be:

*Device Independence and the Quest towards Physical Limits of Privacy DOI: http://dx.doi.org/10.5772/intechopen.100364*

$$P\_{VV} = \left| \left< V\_a V\_\beta | \psi^+ \right> \right|^2 = \frac{1}{2} \cos^2(\beta - a) \tag{13}$$

and likewise,

$$P\_{HH} = \left| \left\langle H\_a H\_\beta | \psi^+ \right\rangle \right|^2 = \frac{1}{2} \cos^2(\beta - a) \tag{14}$$

$$P\_{VH} = \left| \left\langle V\_a H\_\beta | \psi^+ \right\rangle \right|^2 = \frac{1}{2} \sin^2(\beta - a) \tag{15}$$

$$P\_{HV} = \left| \left< H\_a V\_\beta | \psi^+ \right> \right|^2 = \frac{1}{2} \sin^2(\beta - a) \tag{16}$$

Defining:

$$E(\alpha, \beta) = P\_{HH} + P\_{VV} - P\_{VH} - P\_{HV} \tag{17}$$

and

$$\mathcal{S} = \left| E(a, b) - E(a, b') \right| + \left| E(a', b) - E(a', b') \right| \tag{18}$$

For certain angles of the polarizers, this parameter *S* can acquire values greater than 2. For instance, for *<sup>a</sup>* <sup>¼</sup> *<sup>π</sup>* 4, *<sup>a</sup>*<sup>0</sup> <sup>¼</sup> 0, *<sup>b</sup>* <sup>¼</sup> �*<sup>π</sup>* <sup>8</sup> , *<sup>b</sup>*<sup>0</sup> <sup>¼</sup> *<sup>π</sup>* <sup>8</sup>, the value of *<sup>S</sup>* <sup>¼</sup> <sup>2</sup> ffiffi 2 <sup>p</sup> . Carefully performed measurements on any of the Bell states are in good agreement with the quantum mechanical predictions This number can be easily shown to be ≤2 for any arbitrary local realistic theory [29]. This inequality is called the CHSH inequality. Thus, a value of *S* exceeding 2 is indicative of the presence of non-local correlations.

### **5. Loop-hole-free Bell tests**

The actual measurement of quantum states to check for violation of CHSH inequalities involves the use of devices that involves losses and detectors that have an efficiency *μ* much less than 1. In such a case, the CHSH inequality is obtained by evaluating the expectation values conditional to coincident counts in both the detectors. This is necessary because of finite losses in the communication channels and *μ* being less than 1 [].

$$\mathcal{S} = |E(a,b)|coin| - E(a,b')|coin| + |E(a',b)|coin| + E(a',b')|coin| \leq \frac{4}{\mu} - 1 \tag{19}$$

Therefore, S > 2 if and only if *μ*>0.828 and hence, Bell's inequality violation would be seen only if the detector efficiency is better than this value. Superconducting nanowire single photon detectors are nowadays commercially available. When detectors of this efficiency are not available, substantial number of coincidences indicating the presence of entangled pairs of photons go undetected. Under these conditions, the sub-ensemble of coincidence detected are assumed to truly representative of the statistics of the entangled photon pairs emitted by the source. This assumption is called the fair sampling assumption. If this assumption holds, then *S*≤ 2 for all local-realistic theories. There is in fact yet another assumption that pertains to detectors which would result in false positive entangled pair detections. Because of experimental expediency, a coincidence event is defined as pairs detected within a coincidence pre-assigned time window. Even uncorrelated pairs of events could result in a seeming coincident event when one or the other

photon is delayed by a suitable time interval. Such an occurrence could result in the Bell's inequality violation for classical source.

In an actual quantum key distribution protocol, it is assumed that the choice of the measurement basis is made randomly and independently. The actual detection set-up looks like that indicated in the **Figure 4** below when either of the following measurement basis choices f g *H=V* or f g *D=A* is made for polarization entangled photon at the source. With such an arrangement, the non-polarizing 50:50 beam splitter sends the incoming photon to either of the polarizing beam splitters with equal probability and hence a random basis choice f g *V*, *H* or f g *D*, *A* is made. This choice is not pre-determined and is perfectly random in nature.

Other than the loopholes mentioned earlier, there are very many other possible loopholes that could vitiate experimental demonstration of Bell's inequality violation. For example, the memory loophole wherein it may be posited that somehow the experimental apparatus retains the details of the previous measurements, thereby rendering the conclusions questionable. Another significant loophole is called the locality loophole under a presumed superluminal communication between the two stations. There are many other possible loopholes and remedial measures that could be taken to close them. We shall desist from going into each of them. The interested reader may refer to [1] and some references contained therein. Suffice to say that is experimentally demanding to demonstrate that all the loopholes have been closed in a single experimental. However, closing one or more loophole but not all have been demonstrated in numerous experiments. There are a couple of experiments which claim to have succeeded in closing all the loopholes. The possibility of someone coming up with an ingenious loophole proposal, however improbable, cannot be ruled out (**Figure 4**).

The watertight loop-hole-free experimental demonstration of information security requires a throughgoing analysis of the complete experimental conditions as well as characterization of components used in the experimental apparatus for deviation from the idealized system and a careful characterization of their imperfection. Alternately, the experimental apparatus is accepted as being unreliable. In the latter case, one is left with the option of having rely on a careful statistical

#### **Figure 4.**

*The experimental arrangement for choosing randomly between two possible polarization choices for polarization is indicated.*

analysis of observed nonlocal correlations of the data coupled with an experimental apparatus that comes close to being loophole free to the extent that is possible.
