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We present an Ito's formula for the one-dimensional discrete-time quantum walk and give some examples including a Tanaka's formula by using the formula.
Hudson, R. L., Parthasarathy, K. R.: Quantum Itô’s formula and stochastic calculus. Comm. Math. Phys
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Fujita, T.: Stochastic Calculus for Finance (in Japanese). Kodansha (2002)
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Kempe, J.: Quantum random walks - an introductory overview. Contemporary Physics
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Kendon, V.: Decoherence in quantum walks - a review. Math. Struct. in Comp. Sci
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Fujita, T.: Random Walks and Stochastic Calculus (in Japanese). Nippon-Hyoron-sha (2008)
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Fujita, T., Kawanishi, Y.: A proof of Itô’s formula using a discrete Itô’s formula. Stud. Sci. Math. Hungr
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Konno, N.: Quantum Walks. In: Quantum Potential Theory, Franz, U., and Schürmann, M., Eds., Lecture Notes in Mathematics: Vol. 1954, pp. 309–452, Springer-Verlag, Heidelberg (2008)
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Venegas-Andraca, S. E.: Quantum Walks for Computer Scientists. Morgan and Claypool (2008)
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Biane, P.: Itô’s stochastic calculus and Heisenberg commutation relations. Stochastic Process. Appl
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Cited in the paper.
Kunita, H.: Itô’s stochastic calculus: Its surprising power for applications. Stochastic Process. Appl
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Later among the works it cites.
Gudder, S.: Quantum measures and integrals. arXiv:1105.3781 (2011)
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Gudder, S.: Discrete quantum processes. arXiv:1106.0019 (2011)
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Gudder, S., Sorkin, R. D.: Two-site quantum random walk. Gen. Rel. Grav
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