Choi-Jamiołkowski matrix and the link product

When discussing networks of channels it is useful to introduce the notion of the Choi-Jamiołkowski matrix (CJ). For a linear map

(24)\[\mathrm{P} : \mathcal{L}\left( \mathcal{A} \right) \mapsto \mathcal{L}\left( \mathcal{B} \right),\]

the corresponding CJ matrix is given by [34, 36]:

(25)\[P = \left( \mathrm{P} \otimes \mathbb{I}_{\mathcal{A}} \right) \left( \ket{\Phi}\bra{\Phi} \right) \in \mathcal{L}\left( \mathcal{B} \otimes \mathcal{A} \right),\]

where \(\ket{\Phi}=\sum_i \ket{i}\otimes \ket{i}\) is a non-normalized maximally entangled state on \(\mathcal{A} \otimes \mathcal{A}\). In what follows, CJ matrices of maps (e.g. map \(\mathrm{P}\)) will be denoted with the same symbol but in italics (e.g. \(P\)). CJ matrix \(P\) has many useful properties [36]:

CJ operator properties

For a map \(\mathrm{P}\) from (24) and its CJ matrix defined by formula (25) the following properties are satisfied:

1. if \(\mathrm{P}\) represents a state preparation procedure then \(P\) is the density matrix of the state it produces,

2. if \(\mathrm{P} = \mathrm{Tr}_{\mathcal{A}}\) represents the partial trace with respect to \(\mathcal{A}\) then \(P\) is the identity matrix on \(\mathcal{A}\),

3. \(\mathrm{P}\) is completely positive if and only if \(P\) is positive semidefinite,

4. \(\mathrm{P}\) is trace-preserving if and only if \(\mathrm{Tr}_{\mathcal{B}} P = \mathbb{1}_{\mathcal{A}}\),

5. \(\mathrm{P}\) is unital, that is \(\mathrm{P}(\mathbb{1}_{\mathcal{A}}) = \mathbb{1}_{\mathcal{B}}\), if and only if \(\mathrm{Tr}_{\mathcal{A}} P = \mathbb{1}_{\mathcal{B}}\).

Additionally, for two maps:

(26)\[\begin{split}\begin{aligned} \mathrm{P} &: \mathcal{L}\left( \mathcal{H}_0 \right) \mapsto \mathcal{L}\left( \mathcal{H}_1 \otimes \mathcal{H}_2 \right), \\ \mathrm{Q} &: \mathcal{L}\left( \mathcal{H}_2 \otimes \mathcal{H}_3 \right) \mapsto \mathcal{L}\left( \mathcal{H}_4 \right), \end{aligned}\end{split}\]

the CJ matrix of their composition \(\mathrm{R}\):

Fig. 15 Composition of channels \(\mathrm{P}\) and \(\mathrm{Q}.\)

is the link product of \(P\) and \(Q\):

(27)\[R = P * Q = \mathrm{Tr}_{2}\left[ \left( \mathbb{1}_{34} \otimes P^{T_2} \right) \left( Q \otimes \mathbb{1}_{01} \right) \right],\]

where \(\mathbb{1}_{ij}\), \(\mathrm{Tr}_{ij}\) and \(T_{i}\) denote identity, partial trace and partial transposition on \(\mathcal{H}_i \otimes \mathcal{H}_j\) and \(\mathcal{H}_i\). In case of CJ matrices acting on more complicated Hilbert spaces for example:

\[M \in \mathcal{L}\left( \bigotimes_{i\in X} \mathcal{H}_i \right),\quad N \in \mathcal{L}\left( \bigotimes_{i\in Y} \mathcal{H}_i \right),\]

where \(X\) and \(Y\) are some sets of indices, the link product is

(28)\[M * N = \mathrm{Tr}_{X \cap Y}\left[ \left( \mathbb{1}_{Y \setminus X} \otimes M^{T_{X \cap Y}} \right) \left( N \otimes \mathbb{1}_{X \setminus Y} \right) \right],\]

where \(\mathrm{Tr}_Z\), \(\mathbb{1}_Z\) and \(T_Z\) are trace, identity matrix and transposition for \(\bigotimes_{i\in Z}\mathcal{H}_i\) and it is assumed that there are SWAP operators inserted inside the trace such that terms in round brackets act on the tensor product of spaces \(\mathcal{H}_i\) in the same order. Additionally, since the relation between channels and CJ matrices is isomorphic [34, 36] link product can be naturally extended to channels.