Choi-Jamiolkowski isomorphism

Mainly about Choi-Jamiolkowski isomorphism. It means that any quantum channel can found a correspoding matrix with it. Their connection is 1 to 1 and the map is CP iff the correspoding matrix, also called choi matrix or choi operator, is positive. And mind that when we say a matrix is positive, it can also be zero in the meaning of \(\langle\psi|A|\psi\rangle\). This connection also sometimes called Channel-State Duality.

The one to one correspondence can stated as follows^[1][2]:

\[\varepsilon\otimes I(|\Omega\rangle\langle\Omega|)=C\tag{1} \]

\[\varepsilon(\rho) = Tr_2((I\otimes\rho^T)C) \tag{2} \]

where \(|\Omega\rangle\equiv\sum_n|n\rangle|n\rangle\)(Mind that the first \(n\) might be different from the second \(n\), for example \(|0+\rangle+|1-\rangle\) is also alright!). The first equaition is the definition of the choi operator, while the second can be deducted from the first equation. For more examples refer to ^[3].

The 1-1 correspondence showing below. From eq.(1) to eq.(2) is easy. From eq.(2) to eq.(1):

\[ \sum_p|p\rangle\langle p|\otimes\sum_{mn}\rho_{mn}|n\rangle\langle m| \cdot\sum_{lk}\varepsilon(|l\rangle\langle k|)\otimes |l\rangle\langle k| \tag{3} \]

and by using \(Tr_2\), we can easy get the result of eq.(3) is \(\varepsilon(\sum_{mn}\rho_{mn})\).

Lemma Any positive matrix can be written as \(\sum_k|K\rangle\rangle\langle\langle K|\), where \(|K\rangle\rangle\equiv I\otimes K|\Omega\rangle\) or \(|K\rangle\rangle\equiv K^\prime\otimes I|\Omega\rangle\).

We will show this lemma with an example and without rigorously prove it, because the example will showing the way how we contruct so.

For 2 qubits case. for any state \((a,b,c,d)^T\), \(I\otimes K|\Omega\rangle=I\otimes K|00\rangle+I\otimes K|11\rangle\). So we can let \(K=\begin{pmatrix}a&c\\b&d\end{pmatrix}\). So easy to see that \(I\otimes K|\Omega\rangle = (a,b,c,d)^T\). So for every vector, we can write it as \(I\otimes K|\Omega\rangle\). And any positive matrix can be written as \(\sum_k|K\rangle\rangle\langle\langle K|\) using spectral decomposition, with \(|K\rangle\rangle\) the unnormalized vector. QED.

Theorem The map is CP iff the correspoding choi operator is positive.

If \(C\) is positive, very easy to prove that \(\varepsilon\) is CP. Another way, if \(C\) is positive, it has spectrum decomposition as \(\sum_j|K_j\rangle\rangle\langle\langle K_j|\), where \(|K_j\rangle\rangle\) stands for \(I\otimes K_j^\dagger|\Omega\rangle\). To prove that any positive operator can be decomposed like this, we only need to prove that \(C=\sum_i\sum_{mn}p_{mn}^i|mn\rangle\sum_{kl}p_{kl}^i\langle kl|\) is equal to \(\sum_j I\otimes K_j|\Omega\rangle\langle\Omega|I\otimes K_j^\dagger\), which is easy to verify. Using this connection and Eq.(2), we can get that:

\[\varepsilon(\rho) = \sum_iK_i\rho K_i^\dagger. \]

And since any map which can be stated with kraus operator is CP, we finish the proof. QED.

Relation between choi matrix and matrix form of kraus operators

\[J_T=T\otimes I\left( |\omega \rangle \langle \omega | \right) =\frac{1}{d}\sum_{ijk}{E_k|i\rangle \langle j|E_{k}^{\dagger}\otimes |i\rangle \langle j|} \\ \hat{T}=\sum_k{E_k\otimes \bar{E}_k} \\ \langle mn|J_T|xy\rangle =\frac{1}{d}\sum_k{\langle m|E_k|n\rangle \langle y|E_{k}^{\dagger}|x\rangle} \\ \langle mn|\hat{T}|xy\rangle =\sum_k{\langle m|E_k|x\rangle \langle y|E_{k}^{\dagger}|n\rangle} \\ \therefore d\langle mn|J_T|xy\rangle =\langle mx|\hat{T}|ny\rangle\]

From this we can actually see one truth:

\[J_T=\sum_{ij}{\frac{1}{d}}T\left( |i\rangle \langle j| \right) \otimes |i\rangle \langle j| \\ \hat{T}=\sum_{ij}{T\left( |i\rangle \langle j| \right) \otimes |j\rangle \langle i|}\]

And we can know that \(\hat{T}\) is basis independent while \(J_T\) may depend on basis.

As for how to transfer choi matrix quickly into kraus operators, we may refer to this link an answer of John Watrous.

Furthre Reading

link1
link2 If link failed, the pdf can be found in the zotera notes corriculum.

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1. G. Chiribella, G. M. D’Ariano, and P. Perinotti, Quantum Circuit Architecture, Phys. Rev. Lett. 101, 060401 (2008). ↩︎

2. John Preskill's lecture, Chap 3.3. ↩︎

3. A. R. U. Devi, A. K. Rajagopal, S. Shenoy, and R. W. Rendell, Interplay of Quantum Stochastic and Dynamical Maps to Discern Markovian and Non-Markovian Transitions, JQIS 02, 47 (2012). ↩︎