Top

Quantum Information Processing

Published in:

01-08-2016

A family of generalized quantum entropies: definition and properties

Authors: G. M. Bosyk, S. Zozor, F. Holik, M. Portesi, P. W. Lamberti

Published in: Quantum Information Processing | Issue 8/2016

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

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.

previous article Quantum teleportation of a generic two-photon state with weak cross-Kerr nonlinearities

next article Influence of environmental noise on the weak value amplification

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

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]).

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

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 \(\{|e_i^A\rangle \}\).

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.

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.

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].

Jozsa, R., Schumacher, B.: A new proof of the quantum noiseless coding theorem. J. Mod. Opt. 41(12), 2343 (1994). doi:10.1080/09500349414552191 ADSMathSciNetCrossRefMATH

Schumacher, B.: Quantum coding. Phys. Rev. A 51(4), 2738 (1995). doi:10.1103/PhysRevA.51.2738 ADSMathSciNetCrossRef

Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information, 10th edn. Cambridge University Press, Cambridge (2010)CrossRefMATH

Renes, J.M.: The physics of quantum information: complementary uncertainty, and entanglement. Int. J. Quantum Inf. 11, 1330002 (2013). doi:10.1142/S0219749913300027 MathSciNetCrossRefMATH

Ogawa, T., Hayashi, M.: On error exponents in quantum hypothesis testing. IEEE Trans. Inf. Theory 50(6), 1368 (2004). doi:10.1109/TIT.2004.828155 MathSciNetCrossRefMATH

Holevo, A.: Probabilistic and Statistical Aspects of Quantum Theory. Quaderni Monographs, vol. 1, 2nd edn. Edizioni Della Normale, Pisa (2011)CrossRef

Gill, R.D., Guţă, M.I.: On asymptotic quantum statistical inference. In: Banerjee, M., Bunea, F., Huang, J., Koltchinskii, V., Maathuis, M.H. (eds.) From Probability to Statistics and Back: High-Dimensional Models and Processes—A Festschrift in Honor of Jon A. Wellner, vol. 9, pp. 105–127. Institute of Mathematical Statistics collections, Beachwood, Ohio, USA (2013). doi:10.1214/12-IMSCOLL909

Yu, N., Duang, R., Ying, M.: Distinguishability of quantum states by positive operator-valued measures with positive partial transpose. IEEE Trans. Inf. Theory 60(4), 2069 (2004). doi:10.1109/TIT.2014.2307575 MathSciNetCrossRef

von Neumann, J.: Thermodynamik quantenmechanischer Gesamtheiten. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, pp. 273–291 (1927)

10.

Rényi, A.: On measures of entropy and information. In: Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability, vol. 1, p. 547 (1961)

11.

Tsallis, C.: Possible generalization of Boltzmann–Gibbs statistics. J. Stat. Phys. 52(1–2), 479 (1988). doi:10.1007/BF01016429 ADSMathSciNetCrossRefMATH

12.

Canosa, N., Rossignoli, R.: Generalized nonadditive entropies and quantum entanglement. Phys. Rev. Lett. 88(17), 170401 (2002). doi:10.1103/PhysRevLett.88.170401 ADSMathSciNetCrossRefMATH

13.

Hu, X., Ye, Z.: Generalized quantum entropy. J. Math. Phys. 47(2), 023502 (2006). doi:10.1063/1.2165794 ADSMathSciNetCrossRefMATH

14.

Kaniadakis, G.: Statistical mechanics in the context of special relativity. Phys. Rev. E 66(5), 056125 (2002). doi:10.1103/PhysRevE.66.056125 ADSMathSciNetCrossRef

15.

Maassen, H., Uffink, J.B.M.: Generalized entropic uncertainty relations. Phys. Rev. Lett. 60(12), 1103 (1988). doi:10.1103/PhysRevLett.60.1103 ADSMathSciNetCrossRef

16.

Uffink, J.B.M.: Measures of uncertainty and the uncertainty principle. Ph.D. thesis, University of Utrecht, Utrecht, The Netherlands (1990). See also references therein

17.

Wehner, S., Winter, A.: Entropic uncertainty relations—a survey. New J. Phys. 12, 025009 (2010). doi:10.1088/1367-2630/12/2/025009 ADSMathSciNetCrossRef

18.

Zozor, S., Bosyk, G.M., Portesi, M.: On a generalized entropic uncertainty relation in the case of the qubit. J. Phys. A 46(46), 465301 (2013). doi:10.1088/1751-8113/46/46/465301 ADSMathSciNetCrossRefMATH

19.

Zozor, S., Bosyk, G.M., Portesi, M.: General entropy-like uncertainty relations in finite dimensions. J. Phys. A 47(49), 495302 (2014). doi:10.1088/1751-8113/47/49/495302 MathSciNetCrossRefMATH

20.

Zhang, J., Zhang, Y., Yu, C.S.: Rényi entropy uncertainty relation for successive projective measurements. Quantum Inf. Process. 14(6), 2239 (2015). doi:10.1007/s11128-015-0950-z ADSMathSciNetCrossRefMATH

21.

Horodecki, R., Horodecki, P.: Quantum redundancies and local realism. Phys. Lett. A 194(3), 147 (1994). doi:10.1016/0375-9601(94)91275-0 ADSMathSciNetCrossRefMATH

22.

Abe, S., Rajagopal, A.K.: Nonadditive conditional entropy and its significance for local realism. Phys. A 289(1–2), 157 (2001). doi:10.1016/S0378-4371(00)00476-3 MathSciNetCrossRefMATH

23.

Tsallis, C., Lloyd, S., Baranger, M.: Peres criterion for separability through nonextensive entropy. Phys. Rev. A 63(4), 042104 (2001). doi:10.1103/PhysRevA.63.042104 ADSCrossRef

24.

Rossignoli, R., Canosa, N.: Violation of majorization relations in entangled states and its detection by means of generalized entropic forms. Phys. Rev. A 67(4), 042302 (2003). doi:10.1103/PhysRevA.67.042302 ADSCrossRef

25.

Bengtsson, I., Życzkowski, K.: Geometry of Quantum States: An Introduction to Quantum Entanglement. Cambridge University Press, Cambridge (2006)CrossRefMATH

26.

Huang, Y.: Entanglement detection: complexity and Shannon entropic criteria. IEEE Trans. Inf. Theory 59(10), 6774 (2013). doi:10.1109/TIT.2013.2257936 ADSMathSciNetCrossRef

27.

Ourabah, K., Hamici-Bendimerad, A., Tribeche, M.: Quantum entanglement and Kaniadakis entropy. Phys. Scr. 90(4), 045101 (2015). doi:10.1088/0031-8949/90/4/045101 ADSCrossRef

28.

Yeung, R.W.: A framework for linear information inequalities. IEEE Trans. Inf. Theory 43(6), 1924 (1997). doi:10.1109/18.641556 MathSciNetCrossRefMATH

29.

Zhang, Z., Yeung, R.W.: On characterization of entropy function via information inequalities. IEEE Trans. Inf. Theory 44(4), 1440 (1998). doi:10.1109/18.681320 MathSciNetCrossRefMATH

30.

Cardy, J.: Some results on the mutual information of disjoint regions in higher dimensions. J. Phys. A 28, 285402 (2013). doi:10.1088/1751-8113/46/28/285402 MathSciNetCrossRefMATH

31.

Gross, D., Walter, M.: Stabilizer information inequalities from phase space distributions. J. Math. Phys. 54(8), 082201 (2013). doi:10.1063/1.4818950 ADSMathSciNetCrossRefMATH

32.

Wilde, M.M., Datta, N., Hsieh, M., Winter, A.: Quantum rate-distortion coding with auxiliary resources. IEEE Trans. Inf. Theory 59(10), 6755 (2013). doi:10.1109/TIT.2013.2271772 MathSciNetCrossRef

33.

Datta, N., Renes, J.M., Renner, R., Wilde, M.M.: One-shot lossy quantum data compression. IEEE Trans. Inf. Theory 59(12), 8057 (2013). doi:10.1109/TIT.2013.2283723 MathSciNetCrossRef

34.

Ahlswede, R., Löber, P.: Quantum data processing. IEEE Trans. Inf. Theory 47(1), 474 (2001). doi:10.1109/18.904565 MathSciNetCrossRefMATH

35.

Salicrú, M., Menéndez, M.L., Morales, D., Pardo, L.: Asymptotic distribution of \((h,\phi )\)-entropies. Commun. Stat. Theory Methods 22(7), 2015 (1993). doi:10.1080/03610929308831131 MathSciNetCrossRefMATH

36.

Shannon, C.E.: A mathematical theory of communication. Bell Syst. Tech. J. 27, 623 (1948)MathSciNetCrossRefMATH

37.

Havrda, J., Charvát, F.: Quantification method of classification processes: concept of structural \(\alpha \)-entropy. Kybernetika 3(1), 30 (1967)MathSciNetMATH

38.

Daróczy, Z.: Generalized information functions. Inf. Control 16(1), 36 (1970)MathSciNetCrossRefMATH

39.

Rathie, P.N.: Unified \((r, s)\)-entropy and its bivariate measures. Inf. Sci. 54(1–2), 23 (1991). doi:10.1016/0020-0255(91)90043-T MathSciNetCrossRefMATH

40.

Burbea, J., Rao, C.R.: On the convexity of some divergence measures based on entropy functions. IEEE Trans. Inf. Theory 28(3), 489 (1982). doi:10.1109/TIT.1982.1056497 MathSciNetCrossRefMATH

41.

Li, Y., Busch, P.: Von Neumann entropy and majorization. J. Math. Anal. Appl. 408(1), 384 (2013). doi:10.1016/j.jmaa.2013.06.019 MathSciNetCrossRefMATH

42.

Csiszàr, I.: Information-type measures of difference of probability distributions and indirect observations. Studia Scientiarum Mathematicarum Hungarica 2, 299 (1967)MathSciNetMATH

43.

Kuczma, M.: An Introduction to the Theory of Functional Equations and Inequalities: Cauchy’s Equation and Jensen’s Inequality, 2nd edn. Birkhäuser, Basel (2009)CrossRefMATH

44.

Marshall, A.W., Olkin, I., Arnold, B.C.: Inequalities: Theory of Majorization and Its Applications, 2nd edn. Springer, New-York (2011). doi:10.1007/978-0-387-68276-1 CrossRefMATH

45.

Karamata, J.: Sur une inegalité relative aux fonctions convexes. Publications Mathématiques de l’Université de Belgrade 1, 145 (1932)MATH

46.

Bhatia, R.: Matrix Analysis. Springer, New-York (1997)CrossRefMATH

47.

Khinchin, A.I.: Mathematical Foundations of Information Theory. Dover Publications, New-York (1957)MATH

48.

Tempesta, P.: Beyond the Shannon–Khinchin formulation: the composability axiom and the universal-group entropy. Ann. Phys. 365, 180 (2016). doi:10.1016/j.aop.2015.08.013 ADSMathSciNetCrossRef

49.

Fadeev, D.K.: On the concept of entropy of a finite probabilistic scheme (Russian). Uspekhi Matematicheskikh Nauk 11(1(67)), 227 (1956)MathSciNet

50.

Tsallis, C.: Introduction to Nonextensive Statistical Mechanics—Approaching a Complex World. Springer, New-York (2009). doi:10.1007/978-0-387-85359-8 MATH

51.

Tempesta, P.: Formal groups and Z-entropies. arXiv preprint arXiv:1507.07436 (2016)

52.

Rastegin, A.E.: Rényi and Tsallis formulations of noise-disturbance trade-off relations. Quantum Inf. Comput. 16(3&4), 0313 (2016)

53.

Rastegin, A.E.: Some general properties of unified entropies. J. Stat. Phys. 143(6), 1120 (2011). doi:10.1007/s10955-011-0231-x ADSMathSciNetCrossRefMATH

54.

Fan, Y.J., Cao, H.X.: Monotonicity of the unified quantum \((r, s)\)-entropy and \((r, s)\)-mutual information. Quantum Inf. Process. 14(12), 4537 (2015). doi:10.1007/s11128-015-1126-6 ADSMathSciNetCrossRefMATH

55.

Sharma, N.: Equality conditions for the quantum \(f\)-relative entropy and generalized data processing inequalities. Quantum Inf. Process. 11(1), 137 (2012). doi:10.1007/s11128-011-0238-x MathSciNetCrossRefMATH

56.

Lieb, E.H.: Some convexity and subadditvity properties of entropy. Bull. Am. Math. Soc. 81(1), 1 (1975)MathSciNetCrossRefMATH

57.

Wehrl, A.: General properties of entropies. Rev. Mod. Phys. 50(2), 221 (1978). doi:10.1103/RevModPhys.50.221 ADSMathSciNetCrossRef

58.

Ohya, M., Petz, D.: Quantum Entropy and Its Use. Springer, Berlin (1993)CrossRefMATH

59.

Lamberti, P.W., Portesi, M., Sparacino, J.: Natural metric for quantum information theory. Int. J. Quantum Inf. 7(5), 1009 (2009). doi:10.1142/S0219749909005584 CrossRefMATH

60.

Bosyk, G.M., Bellomo, G., Zozor, S., Portesi, M., Lamberti, P.W.: Unified entropic measures of quantum correlations induced by local measurements. arXiv preprint arXiv:1604.00329 (2016)

61.

Nielsen, M.A.: Probability distributions consistent with a mixed state. Phys. Rev. A 62, 052308 (2000). doi:10.1103/PhysRevA.62.052308 ADSCrossRef

62.

Müller-Lennert, M., Dupuis, F., Szehr, O., Fehr, S., Tomamichel, M.: On quantum Rényi entropies: a new generalization and some properties. J. Math. Phys. 54(12), 122203 (2013). doi:10.1063/1.4838856 ADSMathSciNetCrossRefMATH

63.

Halmos, P.R.: A Hilbert Space Problem Book, 2nd edn. Springer, New-York (1982)CrossRefMATH

64.

Cauchy, A.L.: Cours d’analyse de l’école royale polytechnique, vol. 1: analyse algébrique (Imprimerie royale (digital version, Cambrige, 2009), Paris, 1821)

65.

Raggio, G.A.: Properties of \(q\)-entropies. J. Math. Phys. 36(9), 4785 (1995). doi:10.1063/1.530920 ADSMathSciNetCrossRefMATH

66.

Audenaert, K.M.R.: Subadditivity of \(q\)-entropies for \(q>1\). J. Math. Phys. 48(8), 083507 (2007). doi:10.1063/1.2771542 ADSMathSciNetCrossRefMATH

67.

Daróczy, Z., Járai, A.: On the measurable solution of a functional equation arising in information theory. Acta Mathematica Academiae Scientiarum Hungaricae 34(1–2), 105 (1979). doi:10.1007/bf01902599 MathSciNetCrossRefMATH

68.

Rényi, A.: Probability Theory. North-Holland Publishing Company, Amsterdand (1970)MATH

69.

Werner, R.F.: Quantum states with Einstein–Podolsky–Rosen correlations admitting a hidden-variable model. Phys. Rev. A 40(8), 4277 (1989). doi:10.1103/PhysRevA.40.4277 ADSCrossRef

70.

Nielsen, M.A., Kempe, J.: Separable states are more disordered globally than locally. Phys. Rev. Lett. 86(22), 5184 (2001). doi:10.1103/PhysRevLett.86.5184 ADSCrossRef

71.

Jaynes, E.T.: Information theory and statistical mechanics. Phys. Rev. 106(4), 620 (1957). doi:10.1103/PhysRev.106.620 ADSMathSciNetCrossRefMATH

72.

Horodecki, R., Horodecki, M., Horodecki, P.: Entanglement processing and statistical inference: the Jaynes principle can produce fake entanglement. Phys. Rev. A 59(3), 1799 (1999). doi:10.1103/PhysRevA.59.1799 ADSMathSciNetCrossRefMATH

73.

Svetlichny, G.: Distinguishing three-body from two-body nonseparability by a Bell-type inequality. Phys. Rev. D 35(10), 3066 (1987). doi:10.1103/PhysRevD.35.3066 ADSMathSciNetCrossRef

74.

Mermin, N.D.: Simple unified form for the major no-hidden-variables theorems. Phys. Rev. Lett. 65(27), 3373 (1990). doi:10.1103/PhysRevLett.65.3373 ADSMathSciNetCrossRefMATH

75.

Seevinck, M., Svetlichny, G.: Bell-type inequalities for partial separability in \(N\)-particle systems and quantum mechanical violations. Phys. Rev. Lett. 89(6), 060401 (2002). doi:10.1103/PhysRevLett.89.060401 ADSMathSciNetCrossRefMATH

76.

Horodecki, R., Horodecki, P., Horodecki, M., Horodecki, K.: Quantum entanglement. Rev. Mod. Phys. 81, 865 (2009). doi:10.1103/RevModPhys.81.865

77.

Furuichi, S.: Information theoretical properties of Tsallis entropies. J. Math. Phys. 47(2), 023302 (2006). doi:10.1063/1.2165744 ADSMathSciNetCrossRefMATH

78.

Rastegin, A.E.: Convexity inequalities for estimating generalized conditional entropies from below. Kybernetika 48(2), 242 (2012). http://eudml.org/doc/246939

79.

Teixeira, A., Matos, A., Antunes, L.: Conditional Rényi entropies. IEEE Trans. Inf. Theory 58(7), 4272 (2012). doi:10.1109/TIT.2012.2192713 MathSciNetCrossRef

80.

Fehr, S., Berens, S.: On the conditional Rényi entropy. IEEE Trans. Inf. Theory 60(11), 6801 (2014). doi:10.1109/TIT.2014.2357799 MathSciNetCrossRef

81.

Tomamichel, M., Berta, M., Hayashi, M.: Relating different quantum generalizations of the conditional Rényi entropy. J. Math. Phys. 55(8), 082206 (2014). doi:10.1063/1.4892761 ADSMathSciNetCrossRefMATH

82.

Gigena, N., Rossignoli, R.: Generalized conditional entropy in bipartite quantum systems. J. Phys. A 47(1), 015302 (2014). doi:10.1088/1751-8113/47/1/015302 ADSMathSciNetCrossRefMATH

83.

Ollivier, H., Zurek, W.H.: Quantum discord: a measure of the quantumness of correlations. Phys. Rev. Lett. 88(1), 017901 (2001). doi:10.1103/PhysRevLett.88.017901 ADSCrossRefMATH

84.

Jurkowski, J.: Quantum discord derived from Tsallis entropy. Int. J. Quantum Inf. 11(01), 1350013 (2013). doi:10.1142/S0219749913500135 MathSciNetCrossRefMATH

85.

Bellomo, G., Plastino, A., Majtey, A.P., Plastino, A.R.: Comment on “Quantum discord through the generalized entropy in bipartite quantum states”. Eur. Phys. J. D 68(337), 1 (2014)

86.

Berta, M., Seshadreesan, K.P., Wilde, M.M.: Rényi generalizations of quantum information measures. Phys. Rev. A 91(2), 022333 (2015). doi:10.1103/PhysRevA.91.022333 ADSMathSciNetCrossRefMATH

Title: A family of generalized quantum entropies: definition and properties
Authors: G. M. Bosyk
S. Zozor
F. Holik
M. Portesi
P. W. Lamberti
Publication date: 01-08-2016
Publisher: Springer US
Published in: Quantum Information Processing / Issue 8/2016
Print ISSN: 1570-0755
Electronic ISSN: 1573-1332
DOI: https://doi.org/10.1007/s11128-016-1329-5

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Springer Professional "Wirtschaft"

Other articles of this Issue 8/2016

Quantum walking in curved spacetime

Bit-oriented quantum public-key encryption based on quantum perfect encryption

Shortcuts to adiabatic passage for generation of W states of distant atoms

Influence of environmental noise on the weak value amplification

Enhancement of genuine multipartite entanglement and purity of three qubits under decoherence via bang–bang pulses with finite period

A quantum walk on the half line with a particular initial state