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Looking at Vector Space and Language Models for IR Using Density Matrices

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 8369)


In this work, we conduct a joint analysis of both Vector Space and Language Models for IR using the mathematical framework of Quantum Theory. We shed light on how both models allocate the space of density matrices. A density matrix is shown to be a general representational tool capable of leveraging capabilities of both VSM and LM representations thus paving the way for a new generation of retrieval models. We analyze the possible implications suggested by our findings.


  • Matter Density
  • Diagonal Density Matrices
  • Classical Sample Space
  • Quantum Probability Measure
  • Conventional Probability Distribution

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  1. 1.

    The Dirac notation establishes that \(|u\rangle \) denotes a unit norm vector in \(\mathbb {H}^n\) and \(\langle u| \) its conjugate transpose.

  2. 2.

    In a more general formulation of the theory, a quantum probability measure reduces to a classical probability measure for any set \(\mathcal {M} = \{M_i\}\) of positive operators \(M_i\) such that \(\sum _i M_i = I_n\). The set \(\mathcal {M}\) is called Positive-Operator Valued Measure (POVM) [12]. Therefore, the properties reported in this paper which apply to a complete set of mutually orthogonal projectors equally hold for a general POVM.

  3. 3.

    In general, the dyads in the mixture don’t need to be orthogonal. However, in this case, the coefficients \(\upsilon _i\) cannot be easily interpreted as the probabilities assigned by the density matrix to each dyad.

  4. 4.

    In quantum physics, the meaning of i.i.d. can be associated to the physical notion of measurement. If a density matrix \(\rho \) represents the state of a system, an i.i.d. set of \(m\) quantum events is obtained by performing a measurement on \(m\) different copies of \(\rho \) and by recording the outcomes.

  5. 5.

    In this paper, we do not explicitly take into account situations in which the vectors could contain negative entries. For example, this could easily happen after the application of Rocchio’s algorithm [16] in feedback situations or by reducing the dimensionality of the vector space by LSI [3]. Besides the historically encountered difficulties in the interpretation of such negative entries [6], in these particular cases, the rank equivalence situations discussed here could not hold. However, we argue that ignoring these situations causes no harm to the generality of our conclusions on the need of an enlarged representation space.

  6. 6.

    This is indeed the practice of Query Expansion (QE), see for example [2].

  7. 7.

    In [8], each basis of a vector space is considered as describing a contextual property and the vectors in the basis as contextual factors. We prefer not to adopt such interpretation for two reasons: (1) in this paper, classical sample spaces are exclusively associated to orthonormal basis and (2) we believe that referring to concepts leads to a more general formulation, better tailored to our needs.


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Correspondence to Alessandro Sordoni or Jian-Yun Nie .

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Sordoni, A., Nie, JY. (2014). Looking at Vector Space and Language Models for IR Using Density Matrices. In: Atmanspacher, H., Haven, E., Kitto, K., Raine, D. (eds) Quantum Interaction. QI 2013. Lecture Notes in Computer Science(), vol 8369. Springer, Berlin, Heidelberg.

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