- #1

Danny Boy

- 49

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**Definition 1**

*The von Neumann entropy*of a density matrix is given by $$S(\rho) := - Tr[\rho ln \rho] = H[\lambda (\rho)] $$ where ##H[\lambda (\rho)]## is the

*Shannon entropy*of the set of probabilities ##\lambda (\rho)## (which are eigenvalues of the density operator ##\rho##).

**Definition 2**If a system is prepared in the ensemble ##\{ p_j, \rho_j \}## then we define the Holevo $\chi$ quantity for the ensemble by $$\chi := S(\rho) - \sum_j p_j S(\rho_j) $$

**Short Question**: Let ##A## and ##B## be two quantum systems in a state of the form $$\rho^{(AB)} = \sum_k q_i |a_{i}^{(A)} \rangle \langle a_{i}^{(A)} | \otimes \rho_{i}^{(B)} $$

where the states ##|a_{i}^{(A)} \rangle## are orthogonal. What property of the von Neumann entropy implies that the von Neumann entropy of the joint state is $$S(\rho^{(AB)}) = H(\vec{q}) + \sum_i q_i S(\rho_{i}^{(B)})? $$

Thanks for any assistance.