1. Introduction
2. Background
3. Channel entanglement detection
3.1. Bipartite channels
A bipartite channel ${{ \mathcal N }}_{{AB}}$ is entangled if the Choi state ${{\rm{\Phi }}}_{{AA}^{\prime} B^{\prime} B}^{{ \mathcal N }}$ is entangled.
• | Consider the eigenvector $| \phi \rangle $ with a negative eigenvalue of ${{\rm{\Phi }}}_{{AA}^{\prime} B^{\prime} B}^{{ \mathcal N }}$, the witness operators is $W=| \phi \rangle \langle \phi {| }^{{{\rm{T}}}_{{AA}^{\prime} }}$. |
• | Suppose ${{\rm{\Phi }}}_{{AA}^{\prime} B^{\prime} B}^{{ \mathcal N }}={\sum }_{k}{\lambda }_{k}{V}_{k}^{{AA}^{\prime} }\otimes {V}_{k}^{B^{\prime} B}$, the witness operator is $W={\mathbb{I}}-{\sum }_{k}{V}_{k}^{{AA}^{\prime} }\otimes {V}_{k}^{B^{\prime} B}$. |
• | For any observable ${O}_{{AA}^{\prime} B^{\prime} B}$, we can construct a witness as $W=\alpha {\mathbb{I}}-{O}_{{AA}^{\prime} B^{\prime} B}$ with $\alpha ={\max }_{{{\rm{\Phi }}}_{{AA}^{\prime} B^{\prime} B}^{{ \mathcal N }},{ \mathcal N }\ {\rm{is\; separable}}}\ \mathrm{Tr}[{{\rm{\Phi }}}_{{AA}^{\prime} B^{\prime} B}^{{ \mathcal N }}{O}_{{AA}^{\prime} B^{\prime} B}]$. |
3.2. Multipartite channel entanglement
A multipartite channel ${{\rm{\Theta }}}_{n}$ is (genuinely) entangled if the Choi state ${{\rm{\Phi }}}_{11^{\prime} ,22^{\prime} ,\ldots ,{nn}^{\prime} }^{{\rm{\Theta }}}$ (genuinely) entangled.
3.3. Entanglement detection via quantum games
4. Example
Given a multipartite channel ${{\rm{\Theta }}}_{n}$ with local unitary ${U}_{1},{U}_{2},\ldots ,{U}_{n}$ and ${V}_{1},{V}_{2},\ldots ,{V}_{n}$, ${{\rm{\Theta }}}_{n}$ is entangled if $({U}_{1}\otimes {U}_{2}\otimes ...\otimes {U}_{n})\circ {{\rm{\Theta }}}_{n}\circ ({V}_{1}\otimes {V}_{2}\otimes ...\otimes {V}_{n})$ is entangled.
4.1. Bipartite channels
Figure 1. The CNOT or SWAP gate with local depolarizing noise. Given maximally entangled input states ${{\rm{\Phi }}}_{{AA}}=| {\rm{\Phi }}\rangle {\left\langle {\rm{\Phi }}\right|}_{{AA}}$ and ${{\rm{\Phi }}}_{{BB}}=| {\rm{\Phi }}\rangle {\left\langle {\rm{\Phi }}\right|}_{{BB}}$, the output state corresponds to the Choi state of the noise channel. The task is to detect the entanglement of the output state between the bipartition between ${AA}^{\prime} $ and $B^{\prime} B$. |
Table 1. Quantum game with ${W}_{\mathrm{CNOT},1}$ for the noisy CNOT gate. |
α | ${\rho }_{A}^{{\rm{T}}}$ | ${\rho }_{B}^{{\rm{T}}}$ | ${O}_{A^{\prime} }$ | ${O}_{B^{\prime} }$ | α | ${\rho }_{A}^{{\rm{T}}}$ | ${\rho }_{B}^{{\rm{T}}}$ | ${O}_{A^{\prime} }$ | ${O}_{B^{\prime} }$ |
---|---|---|---|---|---|---|---|---|---|
1 | ${\mathbb{I}}$ | ${\mathbb{I}}$ | ${\mathbb{I}}$ | ${\mathbb{I}}$ | 1 | ${\mathbb{I}}$ | ${\sigma }_{x}$ | ${\mathbb{I}}$ | ${\sigma }_{x}$ |
1 | ${\mathbb{I}}$ | ${\sigma }_{y}$ | ${\sigma }_{z}$ | ${\sigma }_{y}$ | −1 | ${\mathbb{I}}$ | ${\sigma }_{z}$ | ${\sigma }_{z}$ | ${\sigma }_{z}$ |
−1 | ${\sigma }_{x}$ | ${\sigma }_{x}$ | ${\sigma }_{x}$ | ${\mathbb{I}}$ | −1 | ${\sigma }_{x}$ | ${\mathbb{I}}$ | ${\sigma }_{x}$ | ${\sigma }_{x}$ |
1 | ${\sigma }_{x}$ | ${\sigma }_{z}$ | ${\sigma }_{y}$ | ${\sigma }_{y}$ | 1 | ${\sigma }_{x}$ | ${\sigma }_{y}$ | ${\sigma }_{y}$ | ${\sigma }_{z}$ |
1 | ${\sigma }_{y}$ | ${\sigma }_{z}$ | ${\sigma }_{x}$ | ${\sigma }_{y}$ | 1 | ${\sigma }_{y}$ | ${\sigma }_{y}$ | ${\sigma }_{x}$ | ${\sigma }_{z}$ |
1 | ${\sigma }_{y}$ | ${\sigma }_{x}$ | ${\sigma }_{y}$ | ${\mathbb{I}}$ | 1 | ${\sigma }_{y}$ | ${\mathbb{I}}$ | ${\sigma }_{y}$ | ${\sigma }_{x}$ |
1 | ${\sigma }_{z}$ | ${\sigma }_{y}$ | ${\mathbb{I}}$ | ${\sigma }_{y}$ | −1 | ${\sigma }_{z}$ | ${\sigma }_{z}$ | ${\mathbb{I}}$ | ${\sigma }_{z}$ |
1 | ${\sigma }_{z}$ | ${\mathbb{I}}$ | ${\sigma }_{z}$ | ${\mathbb{I}}$ | 1 | ${\sigma }_{z}$ | ${\sigma }_{x}$ | ${\sigma }_{z}$ | ${\sigma }_{x}$ |
Table 2. Quantum game with ${W}_{\mathrm{CNOT},2}$ for the noisy CNOT gate. |
α | ${\rho }_{A}^{{\rm{T}}}$ | ${\rho }_{B}^{{\rm{T}}}$ | ${O}_{A^{\prime} }$ | ${O}_{B^{\prime} }$ | α | ${\rho }_{A}^{{\rm{T}}}$ | ${\rho }_{B}^{{\rm{T}}}$ | ${O}_{A^{\prime} }$ | ${O}_{B^{\prime} }$ |
---|---|---|---|---|---|---|---|---|---|
14 | ${\mathbb{I}}$ | ${\mathbb{I}}$ | ${\mathbb{I}}$ | ${\mathbb{I}}$ | −2 | ${\mathbb{I}}$ | ${\sigma }_{x}$ | ${\mathbb{I}}$ | ${\sigma }_{x}$ |
2 | ${\mathbb{I}}$ | ${\sigma }_{y}$ | ${\sigma }_{z}$ | ${\sigma }_{y}$ | −2 | ${\mathbb{I}}$ | ${\sigma }_{z}$ | ${\sigma }_{z}$ | ${\sigma }_{z}$ |
−2 | ${\sigma }_{x}$ | ${\sigma }_{x}$ | ${\sigma }_{x}$ | ${\mathbb{I}}$ | −2 | ${\sigma }_{x}$ | ${\mathbb{I}}$ | ${\sigma }_{x}$ | ${\sigma }_{x}$ |
2 | ${\sigma }_{x}$ | ${\sigma }_{z}$ | ${\sigma }_{y}$ | ${\sigma }_{y}$ | 2 | ${\sigma }_{x}$ | ${\sigma }_{y}$ | ${\sigma }_{y}$ | ${\sigma }_{z}$ |
2 | ${\sigma }_{y}$ | ${\sigma }_{z}$ | ${\sigma }_{x}$ | ${\sigma }_{y}$ | 2 | ${\sigma }_{y}$ | ${\sigma }_{y}$ | ${\sigma }_{x}$ | ${\sigma }_{z}$ |
2 | ${\sigma }_{y}$ | ${\sigma }_{x}$ | ${\sigma }_{y}$ | ${\mathbb{I}}$ | 2 | ${\sigma }_{y}$ | ${\mathbb{I}}$ | ${\sigma }_{y}$ | ${\sigma }_{x}$ |
2 | ${\sigma }_{z}$ | ${\sigma }_{y}$ | ${\mathbb{I}}$ | ${\sigma }_{y}$ | −2 | ${\sigma }_{z}$ | ${\sigma }_{z}$ | ${\mathbb{I}}$ | ${\sigma }_{z}$ |
−2 | ${\sigma }_{z}$ | ${\mathbb{I}}$ | ${\sigma }_{z}$ | ${\mathbb{I}}$ | −2 | ${\sigma }_{z}$ | ${\sigma }_{x}$ | ${\sigma }_{z}$ | ${\sigma }_{x}$ |
Table 3. Quantum game with ${W}_{\mathrm{SWAP},1}$ for the noisy SWAP gate. |
α | ${\rho }_{A}^{{\rm{T}}}$ | ${\rho }_{B}^{{\rm{T}}}$ | ${O}_{A^{\prime} }$ | ${O}_{B^{\prime} }$ | α | ${\rho }_{A}^{{\rm{T}}}$ | ${\rho }_{B}^{{\rm{T}}}$ | ${O}_{A^{\prime} }$ | ${O}_{B^{\prime} }$ |
---|---|---|---|---|---|---|---|---|---|
1 | ${\mathbb{I}}$ | ${\mathbb{I}}$ | ${\mathbb{I}}$ | ${\mathbb{I}}$ | 1 | ${\mathbb{I}}$ | ${\sigma }_{z}$ | ${\mathbb{I}}$ | ${\mathbb{I}}$ |
1 | ${\mathbb{I}}$ | ${\mathbb{I}}$ | ${\sigma }_{z}$ | ${\mathbb{I}}$ | 1 | ${\mathbb{I}}$ | ${\sigma }_{z}$ | ${\sigma }_{z}$ | ${\mathbb{I}}$ |
−1 | ${\sigma }_{x}$ | ${\mathbb{I}}$ | ${\mathbb{I}}$ | ${\sigma }_{x}$ | −1 | ${\sigma }_{x}$ | ${\sigma }_{z}$ | ${\mathbb{I}}$ | ${\sigma }_{x}$ |
−1 | ${\sigma }_{x}$ | ${\mathbb{I}}$ | ${\sigma }_{z}$ | ${\sigma }_{x}$ | −1 | ${\sigma }_{x}$ | ${\sigma }_{z}$ | ${\sigma }_{z}$ | ${\sigma }_{x}$ |
1 | ${\sigma }_{y}$ | ${\mathbb{I}}$ | ${\mathbb{I}}$ | ${\sigma }_{y}$ | 1 | ${\sigma }_{y}$ | ${\sigma }_{z}$ | ${\mathbb{I}}$ | ${\sigma }_{y}$ |
1 | ${\sigma }_{y}$ | ${\mathbb{I}}$ | ${\sigma }_{z}$ | ${\sigma }_{y}$ | 1 | ${\sigma }_{y}$ | ${\sigma }_{z}$ | ${\sigma }_{z}$ | ${\sigma }_{y}$ |
−1 | ${\sigma }_{z}$ | ${\mathbb{I}}$ | ${\mathbb{I}}$ | ${\sigma }_{z}$ | −1 | ${\sigma }_{z}$ | ${\sigma }_{z}$ | ${\mathbb{I}}$ | ${\sigma }_{z}$ |
−1 | ${\sigma }_{z}$ | ${\mathbb{I}}$ | ${\sigma }_{z}$ | ${\sigma }_{z}$ | −1 | ${\sigma }_{z}$ | ${\sigma }_{z}$ | ${\sigma }_{z}$ | ${\sigma }_{z}$ |
Table 4. Quantum game with ${W}_{\mathrm{SWAP},2}$ for the noisy SWAP gate. |
α | ${\rho }_{A}^{{\rm{T}}}$ | ${\rho }_{B}^{{\rm{T}}}$ | ${O}_{A^{\prime} }$ | ${O}_{B^{\prime} }$ | α | ${\rho }_{A}^{{\rm{T}}}$ | ${\rho }_{B}^{{\rm{T}}}$ | ${O}_{A^{\prime} }$ | ${O}_{B^{\prime} }$ |
---|---|---|---|---|---|---|---|---|---|
12 | ${\mathbb{I}}$ | ${\mathbb{I}}$ | ${\mathbb{I}}$ | ${\mathbb{I}}$ | −4 | ${\mathbb{I}}$ | ${\sigma }_{x}$ | ${\sigma }_{x}$ | ${\mathbb{I}}$ |
4 | ${\mathbb{I}}$ | ${\sigma }_{y}$ | ${\sigma }_{y}$ | ${\mathbb{I}}$ | −4 | ${\mathbb{I}}$ | ${\sigma }_{z}$ | ${\sigma }_{z}$ | ${\mathbb{I}}$ |
−4 | ${\sigma }_{x}$ | ${\mathbb{I}}$ | ${\mathbb{I}}$ | ${\sigma }_{x}$ | −4 | ${\sigma }_{x}$ | ${\sigma }_{x}$ | ${\sigma }_{x}$ | ${\sigma }_{x}$ |
4 | ${\sigma }_{x}$ | ${\sigma }_{y}$ | ${\sigma }_{y}$ | ${\sigma }_{x}$ | −4 | ${\sigma }_{x}$ | ${\sigma }_{z}$ | ${\sigma }_{z}$ | ${\sigma }_{x}$ |
4 | ${\sigma }_{y}$ | ${\mathbb{I}}$ | ${\mathbb{I}}$ | ${\sigma }_{y}$ | 4 | ${\sigma }_{y}$ | ${\sigma }_{x}$ | ${\sigma }_{x}$ | ${\sigma }_{y}$ |
−4 | ${\sigma }_{y}$ | ${\sigma }_{y}$ | ${\sigma }_{y}$ | ${\sigma }_{y}$ | 4 | ${\sigma }_{y}$ | ${\sigma }_{z}$ | ${\sigma }_{z}$ | ${\sigma }_{y}$ |
−4 | ${\sigma }_{z}$ | ${\mathbb{I}}$ | ${\mathbb{I}}$ | ${\sigma }_{z}$ | −4 | ${\sigma }_{z}$ | ${\sigma }_{x}$ | ${\sigma }_{x}$ | ${\sigma }_{z}$ |
4 | ${\sigma }_{z}$ | ${\sigma }_{y}$ | ${\sigma }_{y}$ | ${\sigma }_{z}$ | −4 | ${\sigma }_{z}$ | ${\sigma }_{z}$ | ${\sigma }_{z}$ | ${\sigma }_{z}$ |
Figure 2. Entanglement witness values for the noisy CNOT and SWAP gates as a function of the noise parameter p. Since the witnesses have different normalization, we only plot different witness values under normalization. In particular, for the noisy CNOT gate, the second and third witnesses have the same value after proper normalization. For the noisy SWAP gate, they all have the same value after normalization. |
4.2. Multipartite channels
Figure 3. (a) An example quantum circuit consisting of two CZ gates with local depolarizing noise. The Choi state of (a) is equivalent to (b) the graph state $\left|{G}_{\mathrm{CZ}}\right\rangle $ up to local unitary rotation. |
Figure 4. Stabilizer witness value $\left\langle {W}_{\left|{G}_{\mathrm{CZ}}\right\rangle }\right\rangle $ as a function about the noise parameter p. |
5. Conclusion
Appendix A. Proof of lemma 1
For a bipartite channel ${ \mathcal N }\in \mathrm{CPTP}\left({A}_{0}{B}_{0}\to {A}_{1}{B}_{1}\right)$, its Choi state is separable if ${ \mathcal N }$ is a separable channel.
If a bipartite channel ${ \mathcal N }\in \mathrm{CPTP}\left({A}_{0}{B}_{0}\to {A}_{1}{B}_{1}\right)$ is separable. Then, we have
For a given bipartite channel ${ \mathcal N }\in \mathrm{CPTP}\left({A}_{0}{B}_{0}\to {A}_{1}{B}_{1}\right)$. ${ \mathcal N }$ is separable if its Choi state is separable.
We first assume that the input state is a product state, i.e. ${\rho }_{{AB}}={\rho }_{A}\otimes {\rho }_{B}$. According to the Choi–Jamiolkowski isomorphism, we have