Rights statement: This is a pre-copy-editing, author-produced PDF of an article accepted for publication in International Mathematics Research Notices following peer review. The definitive publisher-authenticated version J Martin Lindsay; Quantum Stochastic Lie–Trotter Product Formula II, International Mathematics Research Notices, , rnx306, https://doi.org/10.1093/imrn/rnx306 is available online at: https://academic.oup.com/imrn/article/2019/12/3901/4812375
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Research output: Contribution to Journal/Magazine › Journal article › peer-review
Research output: Contribution to Journal/Magazine › Journal article › peer-review
}
TY - JOUR
T1 - Quantum stochastic Lie-Trotter product formula II
AU - Lindsay, J. Martin
N1 - This is a pre-copy-editing, author-produced PDF of an article accepted for publication in International Mathematics Research Notices following peer review. The definitive publisher-authenticated version J Martin Lindsay; Quantum Stochastic Lie–Trotter Product Formula II, International Mathematics Research Notices, , rnx306, https://doi.org/10.1093/imrn/rnx306 is available online at: https://academic.oup.com/imrn/article/2019/12/3901/4812375
PY - 2019/6/1
Y1 - 2019/6/1
N2 - A natural counterpart to the Lie-Trotter product formula for norm-continuous one-parameter semigroups is proved, for the class of quasicontractive quantum stochastic operator cocycles whose expectation semigroup is norm continuous.Compared to previous such results, the assumption of a strong form of independence of the constituent cocycles is overcome. The analysis is facilitated by the development of some quantum Ito algebra. It is also shown how the maximal Gaussian component of a quantum stochastic generator may be extracted - leading to a canonical decomposition of such generators, and the connection to perturbation theory is described. Finally, the quantum Ito algebra is extended to quadratic form generators, and a conjecture is formulated for the extension of the product formula to holomorphic quantum stochastic cocycles.
AB - A natural counterpart to the Lie-Trotter product formula for norm-continuous one-parameter semigroups is proved, for the class of quasicontractive quantum stochastic operator cocycles whose expectation semigroup is norm continuous.Compared to previous such results, the assumption of a strong form of independence of the constituent cocycles is overcome. The analysis is facilitated by the development of some quantum Ito algebra. It is also shown how the maximal Gaussian component of a quantum stochastic generator may be extracted - leading to a canonical decomposition of such generators, and the connection to perturbation theory is described. Finally, the quantum Ito algebra is extended to quadratic form generators, and a conjecture is formulated for the extension of the product formula to holomorphic quantum stochastic cocycles.
KW - Lie-Trotter product formula
KW - quantum stochastic cocycle
KW - one-parameter semigroup
KW - series product
KW - concatenation product
KW - quantum Ito algebra
KW - quantum stochastic analysis
U2 - 10.1093/imrn/rnx306
DO - 10.1093/imrn/rnx306
M3 - Journal article
VL - 2019
SP - 3901
EP - 3939
JO - International Mathematics Research Notices
JF - International Mathematics Research Notices
SN - 1073-7928
IS - 12
ER -