Deriving a controlled Kraus operator from an uncontrolled Kraus operator

I have a Kraus operator $$M$$. $$M$$ is composed of a list of matrices $$M_k$$ satisfying

$$\sum_{k} M_k^\dagger M_k = I$$

I would like to control the application of $$M$$ using a control qubit. This controlled operation will have a Kraus operator $$C(M)$$. Given $$M$$ as a list of matrices, how do I compute a list of matrices describing the Kraus operator $$C(M)$$?

For example, what are Kraus operators for the controlled amplitude damping channel?

Note that a perfectly valid answer to this question is "this concept of controlling a Kraus operator is ambiguous, here's why".

Let me clarify what I mean when I say "controlled Kraus operator". Any quantum operation can be translated into a unitary circuit acting on the system of interest as well as an external environment that will be traced out at the end. If you derive that circuit for the original operation, introduce a new system qubit and control every operation in the circuit using that new system qubit, then the circuit now implements the controlled Kraus operation.

My initial idea was to treat each $$M_k$$ as if it was a unitary operation and created a derived $$C(M_k) = \begin{bmatrix} I & 0 \\ 0 & M_k \end{bmatrix}$$, but this produces a list of matrices whose upper left corner violates the $$\sum_{k} C(M)_k^\dagger C(M)_k = I$$ requirement.

• Why don't you just arbitrary prefactors to the identities so they sum up to $1$? May 28 '20 at 17:42
• @NorbertSchuch I also considered that rule, and a few others. For each of them I found an operation where they gave the wrong controlled variant. For example, I wanted the controlled Reset's matrices to be [diag(1, 1, 1, 0), |10><11|]. Spreading out the matrix would have resulted in a different operation. May 28 '20 at 19:33

The concept of controlling a Kraus operator is not well defined. It produces ambiguous results.

For example, consider the dephasing operation. This operation can be represented as a circuit where the qubit-to-dephase is CNOT'd into the environment which is then traced out:

But another completely valid circuit representation uses the opposite kind of control:

If you produce the "controlled dephasing operation" by controlling the first circuit, you get a circuit that dephases the 11 subspace from the 00,01,10 subspace:

Whereas if you choose the other starting circuit, you get a circuit that dephases the 10 subspace from the 00,01,11 subspace:

These controlled operations are not equivalent. They do different things. But both were derived using the definition from the question. Therefore the definition is ambiguous and the problem cannot be solved.

In more detail, the problem comes down to the fact that, after the Kraus operator, you can apply any unitary operation $$U$$ to the environment. The uncontrolled operator is unaffected by $$U$$'s presence, but the controlled operator is affected. There would need to be some convention around fixing this $$U$$ in order to derive a specific controlled operation, similar to how there is a convention around how unobservable global phase becomes observable relative phase when controlling unitary operations.

• A little bit off-topic, but I was wondering about your statement 'similar to how there is a convention around how unobservable ... when controlling unitary operations.' If U is any unitary, is the convention then to take the representative of U in SU?
– JSdJ
May 27 '20 at 8:19
• @JSdJ The convention is to include an unnecessary detail in the definition of the original gate (the global phase) and then use that to decide the relative phase of the controlled operation. It's such a natural thing to do that it's almost not worth even calling it a convention. May 27 '20 at 11:56
• Just because sth. is ambigous, it does not mean it cannot be solved! For instance, the problem: What is the purification of $\rho$? is highly ambiguous. Yet, we can provide solutions, and even assess the nature of the ambiguity, and this is certainly a relevant and interesting and answerable problem! -- Basically, what you care for is a channel s.th. if the control register is 0, it is the identity, if the control register is 1, it does a certain channel. For any other control register you don't care. That seems well-defined and I would bet that there is always a solution. May 28 '20 at 17:37
• @NorbertSchuch The problem is that there's always many solutions, and they disagree about the behavior. You can define "controlling a channel" to be some particular mathematical operation, but it won't satisfy the criteria I laid out in the question (must be equivalent to controlling any unitary circuit implementation over system + environment). May 28 '20 at 19:36
• If you only look at linearity, it is evident that there is an ambiguity, since you only define the action of the CP map on the ancilla states |0><0| and |1><1|, but not on |0><1| (which is linearly independent). Of course, positivity can/will add in non-trivial constraints, and this is where the fun might start -- I would assume that the degree of ambiguity will relate to the original CP map, and it might be an interesting and non-trivial question to characterize that. --- As an example, if your CP map is the identity, the controlled-identity CP map could either also act ... May 31 '20 at 10:35

Unlike the concept of a controlled unitary, a controlled CP map is not uniquely defined.

As an example, consider the Identity map $$I$$, seen as a CP map. Then, the controlled-Identity map (as a CP map) can be defined in different ways, e.g. $$\mathcal E(\rho) = \rho$$ or $$\mathcal E(\rho) = \tfrac12I_A\otimes\mathrm{tr}_A(\rho)\ ,$$ where the $$A$$ (first) system is the control qubit. Specifically, you can think of the second map as the first map, where afterwards, a dephasing channel is applied to the control qubit.

Indeed, this is a degree of freedom which you always have: Dephasing the control qubit after the application of the channel. However, it is not clear whether this always gives a different channel (i.e. whether this is a actual degree of freedom.)

One approach would be that you want that the control qubit is affected as little as possible. For instance, you could demand that if the $$B$$ system is in a fixed point $$\sigma_B$$ of the CP map (such a $$\sigma_B$$ always exists), then the control qubit should remain unchanged. This e.g. fixes the controlled-identity channel uniquely. Whether this always uniquely fixed the channel, I am not sure.

I think there is an unambiguous (in some sense) definition of a controlled quantum channel (also see the update).

Any quantum channel from $$H$$ to $$H$$ (actually, from $$\mathcal{L}(H)$$ to $$\mathcal{L}(H)$$) has a representation $$\Phi(\rho) = \text{Tr}_2 (U \rho \otimes \rho_2 U^\dagger),$$ where $$U$$ is a unitary on $$H \otimes H_2$$, $$H_2$$ is the ancilla space, $$\rho_2$$ is a density matrix on $$H_2$$ and $$\rho$$ is on $$H$$.

Clearly, this formula doesn't depend on the phase of $$U$$. But we have similar situation in the simple case of controlled unitaries. Moreover, as I've learned from physicists, phases of physical unitaries matter $$-$$ we can distinguish them (e.g. if we are given some physical blackbox unitaries).

So, the controlled version of it is the channel $$C(\Phi)(\rho') = \text{Tr}_2\big(C(U) \cdot \rho' \otimes \rho_2 \cdot C(U)^\dagger\big),$$ where we introduce control qubit space $$H_0$$, so $$\rho'$$ is a density matrix on $$H_0 \otimes H$$ and $$C(U)$$ is the controlled unitary on $$H_0 \otimes H \otimes H_2$$.

It's hard to say what this means from the point of view of Kraus decomposition (which ignores the phase).

Update
Actually, not only the phase of $$U$$ matters. From the Craig Gidney's answer we see that if $$\rho_2 = |0\rangle\langle0|$$ on the 1-qubit ancilla, $$U_1 = CNOT$$, $$U_2 = X\otimes I \cdot CNOT \cdot X\otimes I$$ then $$\Phi_1(\rho) = \Phi_2(\rho),$$ but $$C(\Phi_1)(\rho') \neq C(\Phi_2)(\rho').$$

But I still think that this straightforward idea of fixing $$U$$ related to a quantum channel representation is the way to go.

• In my answer, I show that there are multiple difference choices you can make for $U$ (differing in more than just the global phase) to achieve $H$, and that these choices result in different controlled operations. Could you explain why this answer doesn't fall victim to the example I give? Jun 3 '20 at 3:52
• @CraigGidney yes, you're right. I've updated the answer. Jun 3 '20 at 6:03