Timeline for What is the Kraus representation of the quantum channel with Choi $\lambda |\phi^+\rangle \langle\phi^+| + (1-\lambda )|\phi^-\rangle \langle\phi^-|$?
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| Mar 9, 2021 at 17:04 | comment | added | Adam Zalcman | Yes, that's right. If you define the Choi as $(T\otimes I)|m\rangle\langle m|$ then you can still pick out the output of $T$ on the standard basis matrices, but the process is less visually appealing because the elements become interspersed. However, in the case of $(1)$ both yield the same result because of all the zeros in the matrix. IOW, $(T\otimes I)|m\rangle\langle m|$ and $(I\otimes T)|m\rangle\langle m|$ have the same four corner elements. | |
| Mar 9, 2021 at 12:18 | comment | added | glS♦ | (2) is an interesting observation, but you need to define the Choi as $c=(I\otimes T)|m\rangle\!\langle m|$ for it to work (with the channel on the right), correct? This way you get $(\langle i|\otimes I)c(|k\rangle\otimes I)=T(E_{ik})$ which should correspond to the block structure you mention | |
| Mar 8, 2021 at 22:54 | comment | added | Adam Zalcman | (2) doesn't follow from (1). It's a property of every Choi–Jamiołkowski matrix that can be verified directly from definition. (3) doesn't follow from (2). It's a definition. (4) follows from (3) by substituting $\rho$ and carrying out the calculation. | |
| Mar 8, 2021 at 21:08 | comment | added | mikanim | My statement still holds. I still don't understand how your solution finds the Kraus operators how you get from (1) to (2), (2) to (3) and then (3) to (4). | |
| Mar 8, 2021 at 20:28 | comment | added | Adam Zalcman | FWIW, phase damping channel is one of the simplest quantum channels. Most textbooks describe it early on in discussion of quantum channels (Nielsen & Chuang, Kitaev & Shen & Vyalyi). OTOH, many textbooks either never get to Choi–Jamiołkowski at all or describe it after introducing a few example channels which generally include simple channels like phase damping. | |
| Mar 8, 2021 at 20:02 | comment | added | mikanim | Right but it's like using a result from Analysis III to solve an Analysis I exercise. Obviously it's easy to do but when it's not given (nor taught where I am getting this exercise from) then it makes the solution useless. | |
| Mar 8, 2021 at 19:58 | history | edited | Adam Zalcman | CC BY-SA 4.0 |
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| Mar 8, 2021 at 19:56 | comment | added | Adam Zalcman | As usual, there are many paths to solution. IMHO, recognizing and exploiting well-known objects when they appear in a chain of reasoning leads to simpler, faster and more interesting solutions than a run-of-the-mill calculation :-) | |
| Mar 8, 2021 at 19:49 | comment | added | mikanim | Yea but I think the solution shouldn't involve knowing what phase damping is. Just finding simply the Kraus matrices is what is being asked. | |
| Mar 8, 2021 at 19:43 | comment | added | Adam Zalcman | It isn't obvious. However, when you calculate the elements of $c$, i.e. $(1)$ then the elements $2\lambda - 1$ are very suggestive since they look like the $2p-1$ factors by which the phase damping channel multiplies the off-diagonal elements of the input density matrix, see $(4)$. | |
| Mar 8, 2021 at 19:39 | comment | added | mikanim | But in my question it isn't obvious that the C-J matrix is for the phase damping channel. I'm not seeing how we should use the phase damping channel. It's just "the quantum channel" in the question. | |
| Mar 8, 2021 at 19:11 | history | edited | Adam Zalcman | CC BY-SA 4.0 |
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| Mar 8, 2021 at 19:09 | comment | added | Adam Zalcman | As the answer shows, you are asking about the Choi-Jamiołkowski matrix of the phase damping channel. This realization gives us the Kraus representation. | |
| Mar 8, 2021 at 19:00 | comment | added | mikanim | Thanks for the answer. Why do we need the phase damping channel? | |
| Mar 8, 2021 at 18:51 | history | answered | Adam Zalcman | CC BY-SA 4.0 |