Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 50 / 2

Let $D=\dim {\mathcal{H}}_A$ and introduce an isomorphic reference system $R$. Choose a basis $\{|j⟩{\}}_{j=1}^D$ and the maximally entangled stateComplete positivity makesa positive density operator. This is the normalized Choi matrix, with the reference placed first. Its spectral decomposition can be writtenThe number $r$ is at most $D^2$.

Here an index state is a chosen basis vector $|j{⟩}_R$ in the reference. The corresponding relative state of $|v_{\ell }⟩$ is the generally unnormalized vectorThus $|v_{\ell }⟩={\sum }_j|j{⟩}_R|{\eta }_{\ell j}{⟩}_A$. Define a linear map $A_{\ell }$ by its columns:Then $(I_R\otimes A_{\ell })|\Omega ⟩=\sqrt{{\lambda }_{\ell }}|v_{\ell }⟩$, soExpand both sides in reference matrix units. The block with reference indices $i,j$ on the left is $D^{-1}\Phi (|i⟩⟨j|)$; on the right it is $D^{-1}{\sum }_{\ell }{A}_{\ell }|i⟩⟨j|A_{\ell }^{†}$. Since matrix units span all operators and $\Phi$ is linear,This constructs the Kraus representation explicitly from the relative states, rather than assuming it.

Finally, trace preservation implies for every density operator $\rho$ thatTaking every rank-one orthogonal projection as $\rho$ forces the Hermitian operator in brackets to have the same quadratic form as the identity. ThereforeThe argument is finite-dimensional, as appropriate to the quantum-information setting; a channel with a different output dimension has the same construction with rectangular Kraus operators.

Fix a basis for the transposition map $T$. If $M\ge 0$, write $M=B{B}^{†}$. ThenThus transposition is a positive linear map and also preserves the trace.

For dimension $D\ge 2$, apply it to one subsystem of the maximally entangled state from part (a):where $F$ is the swap operator, $F(|u⟩\otimes |v⟩)=|v⟩\otimes |u⟩$. It satisfies $F^{†}=F$ and $F^2=I$, so its eigenvalues are $+1$ on symmetric vectors and $-1$ on antisymmetric vectors. In particular,A positive input has therefore acquired a negative expectation value after adjoining an identity channel. Transposition is positive but not completely positive in dimension at least two. The one-dimensional case is the identity map and is completely positive; the dimension qualification is necessary.