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A family of generalized quantum entropies: definition and properties

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Abstract

We present a quantum version of the generalized \((h,\phi )\)-entropies, introduced by Salicrú et al. for the study of classical probability distributions. We establish their basic properties and show that already known quantum entropies such as von Neumann, and quantum versions of Rényi, Tsallis, and unified entropies, constitute particular classes of the present general quantum Salicrú form. We exhibit that majorization plays a key role in explaining most of their common features. We give a characterization of the quantum \((h,\phi )\)-entropies under the action of quantum operations and study their properties for composite systems. We apply these generalized entropies to the problem of detection of quantum entanglement and introduce a discussion on possible generalized conditional entropies as well.

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Notes

  1. It is assumed that \(M {\ge } N\), otherwise p is completed with zeros; when \(M > N\), the remaining \(N-M\) terms that do not appear in Eq. (14) are added in order to fulfill the unitary of U and \(\lambda \) is to be understood as completed with zeros (for more details, see the proof of the Schrödinger mixture theorem [25, pp. 222–223]).

  2. Recall that a POVM is a set \(\{E_k\}\) of positive definite operators satisfying the resolution of the identity

  3. By definition, the partial trace operation over B, \({\text {Tr}}_B: {\mathcal {H}}_A^{N_A} \otimes {\mathcal {H}}_B^{N_B} \rightarrow {\mathcal {H}}_A^{N_A}\), is the unique linear operator such that \({\text {Tr}}_B X_A \otimes X_B = ( {\text {Tr}}_B X_B)X_A\) for all \(X_A\) and \(X_B\) acting on \({\mathcal {H}}_A^{N_A}\) and \({\mathcal {H}}_B^{N_B}\), respectively. For instance, let us consider the bases \(\{|e_i^A\rangle \}_{i=1}^{N_A}\) and \(\{|e_j^B\rangle \}_{j=1}^{N_B}\) of \({\mathcal {H}}_A^{N_A}\) and \({\mathcal {H}}_B^{N_B}\) respectively, and the product basis \(\{|e_i^A\rangle \otimes |e_j^B\rangle \}\) of \({\mathcal {H}}_A^{N_A} \otimes {\mathcal {H}}_B^{N_B}\). Let us denote by \(\rho ^{AB}_{i j,i' j'}\) the components in the product basis of an operator \(\rho ^{AB}\) acting on \({\mathcal {H}}_A^{N_A} \otimes {\mathcal {H}}_B^{N_B}\). Thus, the partial trace over B of \(\rho ^{AB}\) gives the density operator of the subsystem A, \(\rho ^A = {\text {Tr}}_B \rho ^{AB}\), whose components are \(\rho ^A_{i,i'} = \sum _j \rho ^{AB}_{i j,i' j}\) in the basis \(\{|e_i^A\rangle \}\).

  4. Notice that the Cauchy equations \(g(x+y) = g(x) + g(y)\), \(g(xy) = g(x)+g(y)\) and \(g(xy) = g(x) g(y)\) are not necessarily linear, logarithmic or power type, respectively, without additional assumptions on the domain where they are satisfied and on the class of admissible functions (see e.g. [43, 64]). But, recall that the entropic functionals h and \(\phi \) are continuous and either increasing and concave, or decreasing and convex.

  5. For \(\alpha = 0\) this subadditivity is also satisfied, but note that in this special case, \(\phi \) is not continuous and moreover does not fulfill the conditions of the proposition.

  6. Equivalently, the pure states \(|\psi _m^A \rangle \langle \psi _m^A|\) and \(|\psi _m^B \rangle \langle \psi _m^B|\) can be replaced by mixed states defined on \({\mathcal {H}}^A\) and \({\mathcal {H}}^B\), respectively [70].

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Acknowledgments

GMB, FH, MP and PWL acknowledge CONICET and UNLP (Argentina), and MP and PWL also acknowledge SECyT-UNC (Argentina) for financial support. SZ is grateful to the University of Grenoble-Alpes (France) for the AGIR financial support.

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  1. Instituto de Física La Plata, UNLP, CONICET, Facultad de Ciencias Exactas, Calle 115 y 49, 1900, La Plata, Argentina

    G. M. Bosyk, F. Holik & M. Portesi

  2. Laboratoire Grenoblois d’Image, Parole, Signal et Automatique (GIPSA-Lab, CNRS), 11 rue des Mathématiques, 38402, Saint Martin d’Hères, France

    S. Zozor

  3. Facultad de Matemática, Astronomía y Física, UNC, CONICET, Avenida Medina Allende S/N, Ciudad Universitaria, X5000HUA, Córdoba, Argentina

    P. W. Lamberti

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Bosyk, G.M., Zozor, S., Holik, F. et al. A family of generalized quantum entropies: definition and properties. Quantum Inf Process 15, 3393–3420 (2016). https://doi.org/10.1007/s11128-016-1329-5

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