Deriving a controlled Kraus operator from an uncontrolled Kraus operator

Question

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.

Answer 1

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.

Answer 2

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.

Answer 3

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.