Материал: искусственный интеллект

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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:

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obtained in terms of the amplitudes a, b, c and d. Now the CHSH inequality for the observables H and N acting on the two documents can be written as:

Here H 1H 2 denotes that we measure the observable H on both the documents. In the language of tensor products,

in the standard (Habit) basis.

3.3N-Settings Bell Inequality

The CHSH inequality refers to two two-dimensional systems where each system has two measurement settings (or measurement basis). However this can be generalized for systems with multiple settings or basis [11]

where [x] denotes the largest integer smaller or equal to x.

For seven relevance dimensions, n = 7 and the bound is 25. We can convert Eq. (22) into its probability form as done in Sect. 3.1, or use the trace rule to directly calculate the expectation values as done in Sect. 3.2

4 Experiment and Results

Having obtained an equivalent representation of Bell inequalities in Sect. 3, we proceed to substitute the values in the inequalities and test for violation using relevance scores as calculated in Sect. 2. For each query, a user judges several documents to be relevant or non-relevant according to his or her information need. We investigate the correlations between these documents, with each document having multiple decision perspectives, using the Bell Inequalities. We consider the following types of document pairs to test for Quantum Correlations:

(I) We consider those queries where only two documents are SAT clicked (Satisfied Click - Those documents which are clicked and browsed for at least 30 s). Out of 55617 queries in our dataset, 1702 queries had exactly two SAT clicked documents. We consider a composite system of these two documents and measure (judge the relevance) along di erent basis (relevance dimensions) corresponding to each of the Bell inequalities described in Sects. 3.1, 3.2 and 3.3.

(II) We consider those queries for which we have at least one SAT clicked document. Out of 55617 queries in our dataset, we find 52936 queries with at least one SAT clicked document. We then consider a composite system of this SAT clicked document with all the unclicked documents for the query (one by one) and measure (judge the relevance) along di erent basis (relevance dimensions) corresponding to each of the Bell inequalities described in Sects. 3.1, 3.2, 3.3.

In both cases, we do not find the violation of the Bell inequalities for any query. While case (I) corresponds to correlated documents and case (II) corresponds to anti-correlated documents, it is to be noted that we are taking a composite system by taking a tensor product of two document states. This, in turn is separable back into the two document states. The reason why Quantum Mechanics violates Bell Inequalities is due to the existence of non-separable states like the Bell States. To get something similar to an entangled state, we consider another type of document pairs:

(III) Consider a pair of documents which are listed together for many queries, but are always judged in a correlated manner. That is, if one document of the pair is SAT clicked, the other one is also SAT clicked for that query. And similarly both might be unclicked for another query in which they appear together. Also, we find those documents which are SAT clicked together in half of the queries they occur in, and unclicked in the other half. This corresponds to the following Bell State:

We take such pairs of documents to test the Bell inequalities on them. Out of 774 pairs of documents, no pair show the violation of the inequalities discussed above.

The composite state of the two documents described in Eq. (23) appears to be like an entangled state of the documents - knowing that one document is SAT clicked or not can tell us about the other document. However, one fundamental property of the Bell states is their rotational invariance. Representing a Bell State in any basis, one gets the same probabilities of the two possible outcomes. For example,

where H, N and T are relevance with respect to the Habit, Topicality and Novelty basis. One can always hypothetically construct document Hilbert spaces in such

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a manner that the composite state is rotationally invariant, but that is not the case in the query log data, which is the target of our investigation.

As a formal test of non-separable states, we perform Schmidt decomposition [14] of the composite system of document pairs. We do not find any evidence of non-separable states for any type of document pairs, as described in cases (I), (II) and (III).

5 Conclusion and Future Work

We tested Bell inequalities for violation using data from Bing Query logs. Despite the presence of incompatible measurements, Bell inequalities are not violated. However, the incompatibility in measurement applies to the user’s cognitive state with respect to a single document. Hence there might exist a joint probability distribution governing user’s cognitive state for a pair of documents. The experiments in which the violation of Bell inequality has been reported for cognitive systems, the users are asked to report their judgments on composite states. Hence the joint probabilities can be directly estimated from the judgments. This may result in a “Conjunction Fallacy” [17] due to incompatible decision perspectives, thus violating the monotonicity law of probability by overestimating the joint probability, and therefore violating the Bell inequality. In our dataset, we don’t have judgments over the document pairs. That is, the user does not judge a pair of document to be relevant with respect to some dimensions. Instead we have got the probabilities of relevance of a single document with respect to di erent dimensions. When we use the relevance probability of individual documents to compute the joint probabilities for a pair of documents, we are forced to assume the existence of a joint probability distribution. Thus there might be a possibility of Bell inequality violation if we can obtain data for a pair of documents. For example, users can be asked to rate a document to be relevant with respect to Novelty and another document relevant with respect to Topicality. This would correspond to the E(Rnov , Rtop) term in the CHSH inequality. In this case, user’s judgment of a document may a ect judgment of the other document in the pair.

Another test of the quantum nature of relevance judgments can be to test the non-contextual inequalities like the KCBS inequality [12]. Bell inequalities are designed for a composite system with the assumption of locality and realism. The non-contextual inequalities are designed for a single system with multiple measurement perspectives, some of which are incompatible with each other. However, contextuality only exhibits in systems of more than two dimensions. Hence we need to modify our two-dimensional (two decision outcomes - relevant or not relevant) approach to test inequalities like the KCBS inequality. One can also test for violation of the Contextuality-by-Default inequality [4, 9]. This forms part of our future work.

Acknowledgment. This work is funded by the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 721321. We would like to thank Jingfei Li for his help in providing the processed dataset.