Degree of entanglement as a physically ill-posed problem: The case of entanglement with vacuum

We analyze an example of a photon in a superposition of different modes, and ask what is the degree of their entanglement with a vacuum. The problem turns out to be ill-posed since we do not know which representation of the algebra of canonical commutation relations (CCR) to choose for the field quantization. Once we make a choice, we can solve the question of entanglement unambiguously. So the difficulty is not with mathematics, but with physics of the problem. In order to make the discussion explicit we analyze from this perspective a popular argument based on a photon leaving a beam splitter and interacting with two two-level atoms. We first solve the problem algebraically in the Heisenberg picture, without any assumption about the form of representation of CCR. Then we take the $\infty$-representation and show in two ways that in two-mode states the modes are maximally entangled with the vacuum, but single-mode states are not entangled. Next we repeat the analysis in terms of the representation of CCR taken from Berezin’s book and show that two-mode states do not involve the mode-vacuum entanglement. Finally, we switch to a family of reducible representations of CCR recently investigated in the context of field quantization, and show that the entanglement with the vacuum is present even for single-mode states. Still, the degree of entanglement here is difficult to estimate, mainly because there are $N+2$ subsystems, with $N$ unspecified and large.

Figure 1

Qualitative properties of the the vacuum probability distribution corresponding to the vacuum wave function in the $N$-representation. For wave vectors in the “quantum optics regime” we expect $Z_k∕Z\approx 1$, where $Z$ is the maximal value of $Z_k$.