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

Abstract

We analyze an example of a photon in superposition of different modes, and ask what is the degree of their entanglement with 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 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 Heisenberg picture, without any assumption about the form of representation of CCR. Then we take the ∞-representation and show in two ways that in two-mode states the modes are maximally entangled with 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 vacuum is present even for single-mode states. Still, the degree of entanglement is here difficult to estimate, mainly because there are N + 2 subsystems, with N unspecified and large.

Tomographic test of Bell’s inequality for a time-delocalized single photon

“…Most important, our experiment sheds some light on the hotly debated concepts of single-photon two-mode entanglement and nonlocality [7,8,9,10,11,12,13,15]. On the one hand, singlephoton two-mode states of the kind described by Eq.…”

“…Intensive studies and debates have been dedicated to the meaningfulness of extending both the concepts of entanglement and nonlocality to TWC-type single-photon two-mode systems (see, e.g., [8,9,10,11,12,13,14,15]). Quite recently, Babichev, et al [11] and Zavatta, et al [14] have experimentally characterized a source of two-mode spatially and temporally delocalized single photons, respectively, by employing time-domain homodyne tomography [16] and reconstructing both density matrix and Wigner function of the measured system.…”

Deterministic teleportation using single-photon entanglement as a resource

“…We believe, however, that our theoretical sketch will help bring some closure to the debate that still surrounds the notion of single-particle nonlocality. The root of this debate can be traced to the perception of the vacuum |0 as a singular-if not pathological [6]-state whose similarity to the other Fock states has little physical meaning beyond mathematical isomorphism. By building on earlier discussions about mode entanglement and the nonlocality of this state [2,9,31,33], and by showing the full power of single-photon teleportation with no additional quantum resources, we hope to have proven the contrary.…”

“…Now that the qubits have been completely transferred from the photonic to the atomic modes, we can call upon the Hadamard transformation worked out in (6). This rotation gives…”

“…Ever since Tan et al [1] articulated the notion of singleparticle nonlocality [2,3], controversy has surrounded the ability of a single particle to exhibit entanglement [4][5][6][7][8]. Proposals [9][10][11][12][13][14][15][16][17][18][19][20] and experiments [21][22][23] demonstrating single-photon entanglement, nonlocality, and entanglement purification have been performed, but the prospect of teleportation using single-photon entanglement as the quantum resource has still not been regarded on equal footing with teleportation schemes involving "carrier" particles in each of the entangled modes.…”