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Letters in Mathematical Physics

, Volume 84, Issue 1, pp 1–14 | Cite as

Quantum Random Walks and Vanishing of the Second Hochschild Cohomology

  • Debashish Goswami
  • Lingaraj Sahu
Article

Abstract

Given a conditionally completely positive map \({\mathcal{L}}\) on a unital *-algebra \({\mathcal{A}}\) , we find an interesting connection between the second Hochschild cohomology of \({\mathcal{A}}\) with coefficients in the bimodule \({E_{\mathcal L} = {\mathcal B}^a (\mathcal{A} \oplus M)}\) of adjointable maps, where M is the GNS bimodule of \({\mathcal{L}}\) , and the possibility of constructing a quantum random walk [in the sense of (Attal et al. in Ann Henri Poincar 7(1):59–104, 2006; Lindsay and Parthasarathy in Sankhya Ser A 50(2):151–170, 1988; Sahu in Quantum stochastic Dilation of a class of Quantum dynamical Semigroups and Quantum random walks. Indian Statistical Institute, 2005; Sinha in Banach Center Publ 73:377–390, 2006)] corresponding to \({\mathcal{L}}\) .

Mathematics Subject Classification (2000)

16E40 81S25 

Keywords

Noncommutative probability quantum random walk Hochschild cohomology 

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Copyright information

© Springer 2008

Authors and Affiliations

  1. 1.Stat-Math UnitIndian Statistical Institute, KolkataKolkataIndia
  2. 2.Stat-Math UnitIndian Statistical Institute, Bangalore CentreBangaloreIndia

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