Nielsen and Chuang (Quantum Computation and Quantum Information, 2010) define the tensor product of linear operators acting on vector spaces on page 73. Therefore the tensor product used in the definition of the reduced density matrix on page 105 via $|a_1\rangle \langle a_2|\otimes|b_1\rangle \langle b_2|$ is defined, because both outer products are linear operators according to earlier statements. Although it seems to me that the tensor product used on page 102 via $\rho_1\otimes\rho_2 \otimes\dots\otimes\rho_n$ isn't defined, because in the book I can't find any hint that might link the density operator to linear operators. I'm sure the tensor product that is used on page 102 is defined properly, so what is it that I don't get?
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6Density operator is a linear operator. – Meng Cheng Mar 20 '22 at 22:16
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Are all operators used in "entry level" quantum mechanics (density operator, unitary operars for rotations in the bloch sphere, hermitian operaors for measurements) linear? – manuel459 Mar 20 '22 at 22:19
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5@manuel459 The answer is an unequivocal yes. – Maximal Ideal Mar 20 '22 at 22:23
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3The only exception is anti-unitary operators, which are anti-linear. – Meng Cheng Mar 20 '22 at 22:27
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Not Strictly Related :Total spin of two spin- 1/2 particles, SECOND & THIRD_ANSWER. – Frobenius Mar 20 '22 at 22:40