Exercise 2

from Example Sheet 4

Exercise 2.

Let \(|\psi\rangle_{ABE}\) be a pure state of a tripartite system \(ABE\). define the coherent information from \(A\) to \(B\) of \(\psi\) to be \[ I_c^{A>B}(\psi) = - S(A|B)_\psi. \] Here \(S(A|B)_\psi\) denotes the conditional entropy of the subsystem \(A\) with respect to subsystem \(B\), given that the composite system \(ABE\) is in the pure state \(|\psi\rangle_{ABE}\). Henceforth we shall omit \(\psi\).

Prove the following identities:

  1. \(\frac{1}{2} I(A:B) + \frac{1}{2} I(A:E)= S(A)\)

  2. \(\frac{1}{2} I(A:B) - \frac{1}{2} I(A:E)= I_c^{A>B}\)

  1. By definition of the quantum mutual information, \[\begin{aligned} \frac{1}{2} I(A:B) + \frac{1}{2} I(A:E) &= \frac{1}{2} [ S(A) + S(B) - S(AB) + S(A) + S(E) - S(AE)] \\ &= S(A) + \frac{1}{2} [ S(B) - S(AB) + S(E) - S(AE)]. \end{aligned}\] Since \(| \psi \rangle_{ABE}\) is pure, \(S(B) = S(AE)\) and \(S(E) = S(AB)\) (since for any bipartition of the systems ABE, the entropies of both reduced density matrices are equal). Thus, the term in brackets vanishes and we obtain \(S(A)\) as desired.
  2. Likewise, \[\begin{aligned} \frac{1}{2} I(A:B) - \frac{1}{2} I(A:E) &= \frac{1}{2} [ S(A) + S(B) - S(AB) - S(A) - S(E) + S(AE)] \\ &= \frac{1}{2} [ S(B) - S(AB) - S(E) + S(AE)] \\ &= \frac{1}{2} [ S(B) - S(AB) - S(AB) + S(B)] \\ &= S(B) - S(AB) = - S(A|B) = I_c^{A>B}. \end{aligned}\]