Modeling Meaning Associated with Documental Entities: Introducing the Brussels Quantum Approach

Abstract

We show that the Brussels operational-realistic approach to quantum physics and quantum cognition offers a fundamental strategy for modeling the meaning associated with collections of documental entities. To do so, we take the World Wide Web as a paradigmatic example and emphasize the importance of distinguishing the Web, made of printed documents, from a more abstract meaning entity, which we call the Quantum Web, or QWeb, where the former is considered to be the collection of traces that can be left by the latter, in specific measurements, similarly to how a non-spatial quantum entity, like an electron, can leave localized traces of impact on a detection screen. The double-slit experiment is extensively used to illustrate the rationale of the modeling, which is guided by how physicists constructed quantum theory to describe the behavior of the microscopic entities. We also emphasize that the superposition principle and the associated interference effects are not sufficient to model all experimental probabilistic data, like those obtained by counting the relative number of documents containing certain words and co-occurrences of words. For this, additional effects, like context effects, must also be taken into consideration.

Appendices

Appendix 1: Interference Plus Context Effects

In this appendix, we show that using the “interference plus context effects” formula (24), all data can in principle be modeled, by suitably choosing the different parameters. For simplicity, we start by assuming that $M^w_XN=NM^w_X$, i.e., that N and $M^w_X$ are compatible, so that the projection $N^\dagger M_X^w N$ can be simply written as $NM_X^w$, as is clear that $N^\dagger M_X^w N=N^\dagger N M_X^w=N^2M_X^w= NM_X^w$. In other words, we have (NM ^w)^† = NM ^w and (NM ^w)² = NM ^w. This means that we can define the following three orthogonal projectors:

$$\displaystyle \begin{aligned} P_1=M_X^wN,\quad \quad P_2=(\mathbb{I} -M_X^w)N,\quad \quad P_3=\mathbb{I}-N, \end{aligned} $$

(26)

which are orthogonal to each other:

$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle P_1P_2 = M_X^wN(\mathbb{I} -M_X^w)N = M_X^wN^2 - (M_X^wN)^2 = 0,\\ &\displaystyle &\displaystyle P_1P_3 = M_X^wN(\mathbb{I}-N)=M_X^wN-M_X^wN^2=0,\\ &\displaystyle &\displaystyle P_2P_3 =(\mathbb{I} -M_X^w)N(\mathbb{I}-N)=(\mathbb{I} -M_X^w)(N-N^2)=0. \end{array} \end{aligned} $$

(27)

Consequently, we can write the Hilbert space as the direct sum: ${\mathcal {H}}={\mathcal {H}}_1 \oplus {\mathcal {H}}_2\oplus {\mathcal {H}}_3$, where ${\mathcal {H}}_1= P_1 {\mathcal {H}}$, ${\mathcal {H}}_2= P_2 {\mathcal {H}}$, and ${\mathcal {H}}_3= P_3 {\mathcal {H}}$ are three orthogonal subspaces, and we can write |ψ _A〉 and |ψ _B〉 as linear combinations of vectors belonging to them:

$$\displaystyle \begin{aligned} \begin{array}{rcl} |\psi_A\rangle&\displaystyle =&\displaystyle ae^{i\alpha}|e\rangle+a'e^{i\alpha'}|e'\rangle+ a'' e^{i\alpha''} |e''\rangle, \\ |\psi_B\rangle&\displaystyle =&\displaystyle be^{i\beta}|f\rangle+b'e^{i\beta'}|f'\rangle+b'' e^{i\beta''} |f''\rangle, {} \end{array} \end{aligned} $$

(28)

where |e〉, |f〉 are unit vectors in ${\mathcal {H}}_1$, |e′〉, |f′〉 are unit vectors in ${\mathcal {H}}_2$, and |e″〉, |f″〉 are unit vectors in ${\mathcal {H}}_3$. Considering that the vectors in the expansions (28) are mutually orthogonal, it follows that:

$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle p_A\mu_A=\langle\psi_A | NM_X^w N |\psi_A\rangle = \langle\psi_A | P_1 |\psi_A\rangle =\langle\psi_A | ( ae^{i\alpha}|e\rangle ) =a^2,\\ &\displaystyle &\displaystyle p_B\mu_B=\langle\psi_B | NM_X^w N |\psi_B\rangle = \langle\psi_B | P_1 |\psi_B\rangle =\langle\psi_B | ( be^{i\beta}|f\rangle ) =b^2. \end{array} \end{aligned} $$

(29)

We also have:

$$\displaystyle \begin{aligned} \begin{array}{rcl} \mathfrak{R} \langle\psi_A | NM^wN |\psi_B\rangle &\displaystyle =&\displaystyle \mathfrak{R} \langle\psi_A | P_1 |\psi_B\rangle = \mathfrak{R} (\langle\psi_A | P_1)(P_1 |\psi_B\rangle) = \mathfrak{R} (\langle e|ae^{{-}i\alpha})(be^{i\beta}|f\rangle)\\ &\displaystyle =&\displaystyle ab\, \mathfrak{R}\, e^{i(\beta-\alpha)} \langle e|f\rangle = abc\,\mathfrak{R}\, e^{i(\gamma+\beta-\alpha)}= abc \cos\phi, \end{array} \end{aligned} $$

(30)

where for the second equality we have used $P_1=P_1^2$, and for the fifth equality we have defined the positive number c and the phase γ such that c e ^iγ = 〈e|f〉, whereas for the last equality we have defined ϕ = γ + β − α. In a similar way, we set $c'e^{i\gamma '} = \langle e'|f'\rangle$ and ϕ′ = γ′ + β′− α, and considering that $N=\mathbb {I} N=[M_X^w + (\mathbb {I}- M_X^w)]N= P_1 +P_2$, we have:

$$\displaystyle \begin{aligned} \mathfrak{R}\, \langle\psi_A | N |\psi_B\rangle = \mathfrak{R}\, \langle\psi_A | P_1 |\psi_B\rangle +\mathfrak{R}\, \langle\psi_A |P_2 |\psi_B\rangle = abc \cos\phi+a'b'c'\cos\phi'. \end{aligned} $$

(31)

In a similar way, we have:

$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle p_A = \langle\psi_A | N |\psi_A\rangle = \langle\psi_A | P_1 |\psi_A\rangle + \langle\psi_A |P_2 |\psi_A\rangle = a^2 + {a'}^2\\ &\displaystyle &\displaystyle p_B = \langle\psi_B | N |\psi_B\rangle = \langle\psi_B | P_1 |\psi_B\rangle + \langle\psi_B |P_2 |\psi_B\rangle = b^2 + {b'}^2, \end{array} \end{aligned} $$

(32)

from which it follows that:

$$\displaystyle \begin{aligned} {a'}^2 = p_A-a^2 = p_A(1-\mu_A)=p_A\bar{\mu}_A,\quad \quad {b'}^2 = p_B-b^2 = p_B(1-\mu_B) = p_B\bar{\mu}_B, \end{aligned} $$

(33)

where we have defined $\bar {\mu }_A =1-\mu _A$ and $\bar {\mu }_B =1-\mu _B$. We can thus rewrite (24) as:

$$\displaystyle \begin{aligned} \mu_{AB} ={p_A\,\mu_A +p_B\,\mu_B + 2\sqrt{p_Ap_B}\sqrt{\mu_A\mu_B} \, c \cos\phi \over p_A+p_B+ 2\sqrt{p_Ap_B}\, (\sqrt{\mu_A\mu_B} \, c \cos\phi+\sqrt{\bar{\mu}_A\bar{\mu}_B} \, c' \cos\phi')}. {} \end{aligned} $$

(34)

To relate (34) to the webpages’ counts, we consider the situation where states are uniform superpositions of states associated with manifest stories (characteristic function states). Different from the “only interference effects situation” of Sect. 4, we however now assume that the vectors represented by characteristic functions are those that are obtained following the action of the context N. Clearly, this should only be considered as a rough approximation meant to illustrate that the present approach can handle the probabilities calculated by performing webpages’ counts. So, we assume that $|\psi ^{\prime }_A\rangle = |\chi _A\rangle$ and $|\psi ^{\prime }_B\rangle = |\chi _B\rangle$, so that according to (13), (34) can be written as:

$$\displaystyle \begin{aligned} \mu_{AB} ={p_A\,{n_{A,X}\over n_A} +p_B\,{n_{B,X}\over n_B} + 2\sqrt{p_Ap_B}\sqrt{{n_{A,X}n_{B,X}\over n_An_B}} \,c \cos\phi \over p_A+p_B+ 2\sqrt{p_Ap_B}\left(\sqrt{{n_{A,X}n_{B,X}\over n_An_B}} \,c \cos\phi+\sqrt{{n_{A,X'}n_{B,X'}\over n_An_B}} \,c' \cos\phi'\right)}, {} \end{aligned} $$

(35)

where we have defined $n_{A,X'}= n_A - n_{A,X}$ and $n_{B,X'}=n_B - n_{B,X}$, which are the number of webpages containing the term “A” but not the term “X” and the term “B” but not the term “X,” respectively. The consistency of the model is therefore about finding values for p _A, p _B, c, c′∈ [0, 1] and ϕ, ϕ′∈ [0, 2π], such that (35) can be equal to ${n_{AB,X}\over n_{AB}}$. This will always be the case since (34) can in fact deliver all values between 0 and 1, as we are now going to show.

Consider first the limit case where (34) is equal to 0. Then its numerator has to vanish. If, say, we choose c = 1 and ϕ = π, this means that we must have $(\sqrt {p_A\,\mu _A}-\sqrt {p_B\,\mu _B})^2=0$, which is satisfied if ${p_A\over p_B}={\mu _B\over \mu _A}$. For the other limit case where (34) is equal to 1, if we choose c′ = 1 and ϕ′ = π, we have the condition: $(\sqrt {p_A\,\bar {\mu }_A}-\sqrt {p_B\,\bar {\mu }_B})^2=0$, which is clearly satisfied if ${p_A\over p_B}={\bar {\mu }_B\over \bar {\mu }_A}$. For the intermediate values between 0 and 1, if we set $\phi =\phi '={\pi \over 2}$ (no-interference condition), (34) becomes:

$$\displaystyle \begin{aligned} \mu_{AB} ={p_A\over p_A+p_B}\,\mu_A +{p_B\over p_A+p_B}\,\mu_B, {} \end{aligned} $$

(36)

which is a convex combination of μ _A and μ _B. Therefore, by varying p _A and p _B, by just considering context effects all values contained in the interval $[\min (\mu _A,\mu _B), \max (\mu _A,\mu _B)]$ can be obtained.

To be able to extend further the interval, the relative phases ϕ or ϕ′ have to be allowed to take values different from ${\pi \over 2}$. In this way, also the intervals $[0,\min (\mu (A),\mu (B))]$ and $[\max (\mu (A),\mu (B)),1]$ can be reached. To see this, we have to study the behavior of μ _AB = μ _AB(x, x′) as a function of the two variables $(x,x')=(\cos \phi , \cos \phi ')$. We know that μ(AB;0, 0) is given by (36), so we just have to show that, for suitable choices of p _A and p _B, by varying x and x′ we can reach the 0 value. For a given x, μ _AB(x, x′) monotonically decreases as x′ increases. Thus, we only have to consider μ _AB(x, 1), and by studying the sign of ∂ _x μ _AB(x, 1) one can easily check that (we leave this as an exercise) μ _AB(x, 1) monotonically increases with x. Thus, the minimum corresponds to μ _AB(−1, 1), which is 0 if c = 1 and ${p_A\over p_B}={\mu _B\over \mu _A}$. Similarly, we can consider μ _AB(x, −1) and check that μ _AB(x, −1) also monotonically increases with x. Thus, its maximum corresponds to μ _AB(1, −1), which is 1 if c′ = 1 and ${p_A\over p_B}={\bar {\mu }_B\over \bar {\mu }_A}$. In other words, for arbitrary μ _A, μ _B, and μ _AB, a quantum representation that can faithfully model the experimental data exist, if both interference and context effects are considered.

Appendix 2: Meaning Bond

In this appendix, we offer a more specific interpretation for the normalized weights a _j characterizing the linear combination in (9), in terms of a notion of meaning bond of a concept with respect to another concept, when the QWeb is in a given state |ψ〉. For this, let M _A and M _B be the projection operators onto the set of QWeb states that are “states of A” and “states of B,” respectively. We can then define the ψ-meaning bond M _ψ(B|A) of B towards A by the ratio:

$$\displaystyle \begin{aligned} M_{\psi}(B|A)={p_{\psi}(B|A)\over p_{\psi}(B)}, {} \end{aligned} $$

(37)

where p _ψ(B) = 〈ψ|M _B|ψ〉 is the probability for the QWeb’s state |ψ〉 to be successfully tested as being also a “state of B,” and

$$\displaystyle \begin{aligned} p_{\psi}(B|A)={\langle \psi|M_AM_BM_A |\psi \rangle\over \langle \psi|M_A|\psi \rangle} {} \end{aligned} $$

(38)

is the conditional probability of having the QWeb’s state being successfully tested as being a “state of B,” when it has been successfully tested to be a “state of A.” Indeed, if the QWeb state |ψ〉 was successfully tested to be a “state of A,” according to the projection postulate the state immediately following the test is $|\psi _A\rangle ={M_A |\psi \rangle \over \| M_A|\psi \rangle \|}$, which is now a “state of A.” And we have p _ψ(B|A) = 〈ψ _A|M _B|ψ _A〉, hence (38) possesses a sound interpretation as a conditional probability.

The ψ-meaning bond M _ψ(A|B) of A towards B can be similarly obtained by interchanging in (37) the roles of A and B, and since in general [M _A, M _B] ≠ 0, M _ψ(A|B) ≠ M _ψ(B|A), which means that the meaning bond of A towards B will not in general coincide with the meaning bond of B towards A. So, if $p_{\psi _A}(B)$ and p _ψ(B) are interpreted as measuring how much of the meaning of B is present in the QWeb, when the latter is in state |ψ _A〉 and |ψ〉, respectively, it is clear that the meaning bond $M_{\psi }(B|A)={p_{\psi _A}(B)\over p_{\psi }(B)}$, being their ratio, it measures the relative increase or decrease of the meaning presence of B when the QWeb state |ψ〉 is further contextualized by a concept A. In that respect, we can also say that if B is more (resp., less) meaning present in the QWeb, when its state is further contextualized by a concept A, then for such state there is an attractive (resp., repulsive) meaning bond of B towards A, whereas if $p_{\psi _A}(B)=p_{\psi }(B)$ the meaning bond can be said to be neutral. Also, since we have $p_{\psi _B}(B)=1$, the meaning bond of B towards itself is $M_{\psi }(B|B)=p^{-1}_{\psi }(B)$, so that there will be self-neutrality when p _ψ(B) = 1, and self-attraction if p _ψ(B) < 1 (but there cannot be self-repulsion).

We now observe that: $p_{\psi }(W_j)M_{\psi }(W_j|A)=p_{\psi }(W_j|A)=\langle \psi _A| P_j|\psi _A\rangle =a_j^2$, where P _j = |e _j〉〈e _j| is the projection operator onto the one-dimensional subspace generated by the ‘ground state of W _j,” i.e., of the story-concept indicated by the specific combination of words contained in the webpage W_j. Thus, we have that the coefficients a _j in the expansion of the state $|\psi _A\rangle ={M_A|\psi \rangle \over \|M_A|\psi \rangle \|} = \sum _{j=1}^n a_je^{i\alpha _j}|e_j\rangle$, which is a “state of A,” can be written as:

$$\displaystyle \begin{aligned} a_j=\sqrt{p_{\psi}(W_j)M_{\psi}(W_j|A)} {} \end{aligned} $$

(39)

and therefore are given by (the square root of) the “ψ-meaning bond of W _j towards A,” normalized by the probability p _ψ(W _j), and in that sense we can say that they express a meaning connection between A and the W _j. Note also that in the case where |ψ〉 corresponds to the uniform state $|\chi \rangle ={1\over \sqrt {n}} \sum _{j=1}^n e^{i\rho _j}|e_j\rangle$, (37) reduces to the ratio

$$\displaystyle \begin{aligned} M_{\chi}(B|A)={n\, n_{AB} \over n_An_B}, {} \end{aligned} $$

(40)

which corresponds to the more specific notion of meaning bond introduced in [3] (see also [5]).