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Entanglement and Quantum Logical Gates. Part I.

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Abstract

Is it possible to give a logical characterization of entanglement and of entanglement-measures in terms of the probabilistic behavior of some gates? This question admits different (positive or negative) answers in the case of different systems of gates and in the case of different classes of density operators. In the first part of this article we investigate possible relations between entanglement-measures and the probabilistic behavior of quantum computational conjunctions.

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Notes

  1. The reduced state of \(\rho \in \mathfrak {D}(\mathcal {H}^{(m+n)})\) with respect to the subspace \(\mathcal {H}^{(m)}\) is defined as follows: \(Red^{(1)}_{[m,n]}(\rho ) = Tr_{\mathcal {H}^{(n)}}(\rho )\), where \(Tr_{\mathcal {H}^{(n)}}\) is the partial trace (i.e. the unique linear operator \(Tr_{\mathcal {H}^{(n)}}:\mathcal {H}^{(m+n)}\to \mathcal {H}^{(m)}\) such that \(Tr_{\mathcal {H}^{(n)}}(A\otimes B) = A\; Tr(B)\), where A and B are operators on \(\mathcal {H}^{(m)}\) and \(\mathcal {H}^{(n)}\), respectively). The reduced state \(Red^{(2)}_{[m,n]}(\rho )\) is defined in a similar way.

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Authors and Affiliations

  1. Dipartimento di Pedagogia, Psicologia, Filosofia, Università di Cagliari, Via Is Mirrionis 1, I-09123, Cagliari, Italy

    H. Freytes, R. Giuntini & G. Sergioli

  2. Dipartimento di Ingegneria Gestionale, dell’Informazione e della Produzione, Università di Bergamo, Viale Marconi 5, I-24044, Dalmine (BG), Italy

    R. Leporini

Authors
  1. H. Freytes
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  2. R. Giuntini
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  3. R. Leporini
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  4. G. Sergioli
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Corresponding author

Correspondence to R. Leporini.

Additional information

This article is dedicated to the memory of Peter Mittelstaedt. The authors will always remember the great scientist who has represented a noble example for all scholars of the IQSA-community.

Sergioli’s work has been supported by the Italian Ministry of Scientific Research within the FIRB project “Structures and dynamics of knowledge and cognition”, Cagliari unit F21J12000140001 and by RAS, within the project “Modeling the uncertainty: quantum theory and imaging processing”; Leporini’s work has been supported by the Italian Ministry of Scientific Research within the PRIN project “Automata and Formal Languages: Mathematical Aspects and Applications”. Hector Freytes is also affiliates to UNR-CONICET, Argentina.

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Freytes, H., Giuntini, R., Leporini, R. et al. Entanglement and Quantum Logical Gates. Part I.. Int J Theor Phys 54, 4518–4529 (2015). https://doi.org/10.1007/s10773-015-2668-1

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  • DOI: https://doi.org/10.1007/s10773-015-2668-1

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