By Eleanor Rieffel and Wolfgang Polak
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Extra info for An Introduction to Quantum Computing for Non-Physicists
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The Development of the Number Field Sieve, Vol. 1554 of Lecture Notes in Mathematics. Springer Verlag. LIBOFF, R. L. 1997. ). Addison-Wesley, Reading, Mass. -K. AND CHAU, H. F. 1999. Unconditional security of quantum key distribution over arbitrarily long distances. Science 283, 2050–2056. MAYERS, D. 1998. Unconditional Security in Quantum Cryptography. lanl. gov/abs/quant-ph/9802025. NIELSEN, M. , AND LAFLAMME, R. 1998. Complete Quantum Teleportation using Nuclear Magnetic Resonance. lanl. gov/abs/quant-ph/9811020.
Similarly, if A and B are transformations on n-dimensional and k-dimensional vectors respectively, then A ⊗ B9 is a transformation on nkdimensional vectors. The exact mathematical details of tensor products are beyond the scope of this paper (see Hungerford  for a comprehensive treatment). For our purposes the following algebraic rules are sufficient to calculate with tensor products. For matrices A, B, C, D, U , vectors u, x, y, and scalars a, b, the following hold: (A ⊗ B)(C ⊗ D) (A ⊗ B)(x ⊗ y) (x + y) ⊗ u u ⊗ (x + y) ax ⊗ by = = = = = U ∗ U = A∗1 ⊗ A∗2 ⊗ · · · ⊗ A∗n × (A1 ⊗ A2 ⊗ · · · ⊗ An ) ⊗U = A⊗U B⊗U C⊗U D⊗U ⊗U = aU bU cU dU .
An Introduction to Quantum Computing for Non-Physicists by Eleanor Rieffel and Wolfgang Polak