Fig. 1. Hilbert space representation of Order E ects

Suppose that while judging Document d, the user has the order T opicality → Reliability in mind. Then the final probability of relevance is the projection from

(Fig. 1b).

Order E ects in decision making have been successfully modeled and predicted using the Quantum framework [7, 16].

3 Deriving a Bell Inequality for Documents

3.1CHSH Inequality

In Sect. 2, we showed how we can calculate the relevance probabilities of a document for di erent dimensions. We constructed a Hilbert space for each document, consisting of seven di erent basis, representing each dimension of relevance. Two or more such documents can be considered as a composite system by taking a tensor product of the document Hilbert spaces. If |d1 and |d2 are the state vectors of two documents, we can represent the tensor product as |d1 |d2 . Figure 2 shows the geometrical representation of two such Hilbert spaces. Here

irrelevance in the Habit basis.

In the CHSH inequality, we have observables A1 and A2 for a system taking values in ±1. For a document d1, we have observables corresponding to the di erent relevance dimensions. Taking the case of two relevance dimensions, Habit and Novelty, we have observables Rhab and Rnov which take values in

±1. Where Rhab = +1 corresponds to a projection on the basis vector |R hab,

− |

Rhab = 1 corresponds to the projection on its orthogonal basis vector R hab.

182 S. Uprety et al.

Fig. 2. Tensor product of two Hilbert spaces

Taking two documents as a composite system, we can write the CHSH inequality in the following way:

| Rhab1Rhab2 + Rhab1Rnov2 + Rnov1Rhab2 − Rnov1Rnov2 | ≤ 2 (8)

where the subscripts 1 and 2 denote that the observables belong to document 1 and document 2 respectively. Using the fact that AB = 1 P (AB = 1) + (−1) P (AB = −1) and P (AB = 1) + P (AB = −1) = 1, we can convert the above inequality into its probability form as:

We don’t have the joint probabilities P (AB) in our dataset, hence we assuming P (AB) = P (A)P (B) (this where the assumption of realism is incorrectly made, which will not lead to the CHSH inequality violation), we get:

1≤ P (Rhab1 = 1)P (Rhab2 = 1) + P (Rhab1 = −1)P (Rhab2 = −1)+ P (Rhab1 = 1)P (Rnov2 = 1) + P (Rhab1 = −1)P (Rnov2 = −1)+

P (Rnov1 = 1)P (Rhab2 = 1) + P (Rnov1 = −1)P (Rhab2 = −1)+

P (Rnov1 = 1)P (Rnov2 = −1) + P (Rnov1 = −1)P (Rnov2 = 1) ≤ 3 (10)

As we mentioned above, Rhab = +1 corresponds to the basis vector |Rhab and therefore P (Rhab1 = 1) corresponds to the probability that document d1 is relevant with respect to the H abit dimension of relevance. Therefore we can calculate these probabilities as projections in the Hilbert space:

and similarly for document d2.

3.2CHSH Inequality for Documents Using the Trace Method

Another way to define the CHSH inequality for documents is by directly calculating the expectation values using the trace rule. According to this rule, expectation value of an observable A in a state |d is given by

The document representations in another basis are as follows:

H and N are basically relevance with respect to two relevance dimensions, say Habit and Novelty. We can write the N basis in terms of the H basis (see Appendix A) as: