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In the quantum process formalism, we assume that we begin with a product state $$\rho \otimes |e_0\rangle \langle e_0|$$ where $\rho$ is a mixed state in a system of interest and where $|e_0\rangle$ is a pure state of the environment. This seems like a generous assumption however since the system could be entangled with the environment.

Nielsen and Chuang mention in section 8.2.2 that the quantum process formalism works even in the case where system and environment are entangled, but I haven't been able to find a good justification for this.

Why is it safe to make this assumption?

Here's my best guess:

Suppose we have a state $\sigma_{s, env}$ where the system $s$ is entangled with the environment $env$. A quantum channel over $\rho:=\mathrm{Tr}_{env}(\sigma_{s, env})$ can be understood as a unitary evolution of $\sigma_{s, env}$:

$$U \sigma_{s, env} U^\dagger$$

I think (but I don't know) we can always cook up a unitary $V$ such that

$$V\sigma_{s,env}V^\dagger = \rho\otimes|e_0\rangle\langle e_0|$$

Where $|e_0\rangle$ is a pure state of the environment. If this is the case, we then have:

$$ UV \Big(\rho \otimes|e_0\rangle\langle e_0| \Big)V^\dagger U^\dagger$$

Which suggests to me that we can always "move" any initial entanglement out into the unitary transformation and call it a job done.

Is this basically correct or is there more to it?

Visipi
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1 Answers1

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"Why is it safe to make this assumption?"

tl;dr: In general it isn't, but often it is, which is why this formalism is so widely used.

Nielsen and Chuang pick up the point you mentioned again in Section 8.5: there they construct an example where a qubit is prepared in some unknown state $\rho$ while already correlated with environmental degrees of freedom, and the usual quantum operation formalism fails to describe the evolution as a single CPTP map. What is really happening in this formalism is that you're trying to describe the evolution of a joint system based on the marginals (which is practically motivated by the fact that this is what an experimenter has access to), i.e., you're trying to model the change ${\rm tr}_E(\rho_{SE})\mapsto {\rm tr}_E(U \rho_{SE} U^\dagger)$ via some map.

  • In the uncorrelated scenario $\rho_{SE}=\rho_S\otimes\rho_E$—meaning ${\rm tr}_E(\rho_{SE})=\rho_S$—this is exactly a map of environment form $\rho_S\mapsto {\rm tr}_E(U (\rho_S\otimes\rho_E) U^\dagger)$ which is equivalent to the map being linear, completely positive, and trace-preserving.
  • If $\rho_{SE}$ is correlated, however, then ${\rm tr}_E(\rho_{SE})\mapsto {\rm tr}_E(U \rho_{SE} U^\dagger)$ can fail in every way imaginable: it can fail to be completely positive, it can fail to be linear, and it can even fail to be a map in the sense that two states with the same marginal can result in different outputs, cf. this paper of Schmid et al. (arXiv). This paper goes on to argue that this is not a failure of the CPTP formalism per se, but rather that it was never meant to be applied blindly to initially correlated systems (and they make a bigger point that the "inference map"—which is the assignment we are considering—differs from the causal evolution map which is in fact still CPTP; but this is already beyond what is relevant for your question I think)

So really the product-state assumption is "safe" not because it always holds—which also shows that your point about universally disentangling system and environment for all inputs via a single global unitary $V$ cannot work—but because it is often accurate (or accurate enough) when ensuring during state-preparation that the environment is reset, isolated, or otherwise uncorrelated.

Frederik vom Ende
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