From Wavefunctional Entanglement to Entangled Wavefunctional Degrees of Freedom

Abstract

The question of whether entanglement between photons is equivalent to entanglement between their characteristic field modes, specifically, the single-particle wavefunctions that are composed and superposed to describe particles in such modes, is a key, open problem concerning multi-partite optical degrees of freedom, and has profound implications for topics ranging from quantum foundations to quantum computation. Here, I offer a fresh, deeper, physical insight into this subtle, albeit enduring, issue by describing a situation in which the entangling interactions between optical modes, namely, the wavefunctions, can be distilled into genuine entanglement between the physical, observable properties of the photons, which are the wavefunctional degrees of freedom. This theoretical observation also highlights the salience of the measurement context, especially, of clearly disambiguating between the choice of the quantum subsystem and the decision to measure an observable along a particular axis of measurement, while quantifying and transforming quantum optical entanglement. This theoretical observation might be applied to formulate a new class of protocols for performing quantum information tasks, using entangled photons within inseparable field modes.

Figures (5)

• Figure 1: The problem of transforming entanglement between modes to useful entanglement between particles. (a) This paper raises and addresses the question of how entanglement between optical field modes (shown in red) can be—either, possibly, unitarily, or by making a measurement on an ancillary photon—distilled into entanglement between the photons (depicted in black) that are described by such modes. (b) A flow chart sequentially deconstructing this problem from its simplest—entangling modes—to its deepest—creating genuinely entangled states of interacting photons (see bottom left), which are useful for quantum computation, as opposed to states of non-interacting photons that are entangled merely due to symmetrization (see bottom right)—levels. (c) An interferometer—such as a beam splitter—should be able to provision the modes for constructing the states described in (b). Notice that this highly simplified situation is shown merely as an illustrative example; in practice, however, the two input and the two output modes are rarely useful for creating entangled states of two photons and four modes.
• Figure 2: Comparing and contrasting entanglement between indistinguishable photons, with entanglement between extrinsically distinguishable, albeit intrinsically identical, photons. (a) The phenomenon of entanglement due to symmetrization between two photons that arises as a consequence of the two photons' being identical and conforming to exchange symmetry. The spatial, optical modes of the two photons have been mapped onto the bound energy eigenmodes of a simple harmonic potential, so as to illustrate this situation. (b) Conversely, this figure describes genuine entanglement between two photons that are assumed to distinguishable, due to an extrinsic degree of freedom, such as a suitable component of the orbital angular momentum, which takes the values, $l$ and $l^{\prime}$ for photons 1 and 2, respectively. In such a situation, constructing genuinely entangled states of two photons, as indicated in this figure, requires accessing four spatial, optical modes—of which, $\left|a\right>$ and $\left|b\right>$ are mapped onto the bound energy eigenmodes of an harmonic potential, and $\left|a^{\prime}\right>$ and $\left|b^{\prime}\right>$ are mapped onto the bound energy eigenmodes of an anharmonic potential. See, for example, Sec. \ref{['sec:Equivalence']} for a discussion of a scheme that produces such entangled states.
• Figure 3: A simple Gedankenexperiment illustrating how entanglement between optical field modes—namely, wavefunctional entanglement—can be distilled into entanglement between physical properties of photons that are described by such modes—namely, the wavefunctional degrees of freedom. (a) A version of the two-slit diffraction experiment in which the superposition of an ancillary photon (labeled as 3)—of having traveled through the top and bottom slits—is amplified and transformed into a superposition of the photo-detecting apparatus—of having detected and not detected the photon—which, in turn, controls the anharmonicity of an initially harmonic potential, $x^2/2$. The bottom half of the detector screen has been discarded, so as to highlight that, unlike in the corresponding classical case, the location of detection of the particles—for example, the upper half of the screen—is not correlated with the slit of entry, for example, the top slit. (b) The encoding of $\widetilde{\left|\Psi\right>_{i}}$—that describes a pair of modes in a photon number entangled relationship with each other—onto the harmonic oscillator energy eigenmodes (shown in blue and red). (c) Due to the probabilistic nature of the photo-detection of the ancillary photon, $\widetilde{\left|\Psi\right>_{i}}$ transforms into $\widetilde{\left|\Psi\right>_{f}}$, a superposition of a state encoded onto the harmonic oscillator eigenmodes, as well as onto the anharmonic oscillator eigenmodes (shown in cyan and purple). This state is now a genuinely entangled state of two interacting photons (depicted in black, and labeled as 1 and 2), as emphasized by the depiction of an analogous spinor representation in the inset. The vertical and the horizontal lines—alongside the anharmonic potential, $\lambda x^4/4$—represent the increase in the eigenenergies and the spatial narrowing of the eigenfunctions—due to the anharmonicity—respectively.
• Figure 4: A form of the Gedankenexperiment shown in Fig. \ref{['fig:f3']} that has been further simplified, so as to clarify and highlight the importance of the notion of quantum pre-measurement in analyzing the entangling scheme described in Sec. \ref{['sec:Equivalence']}. (a) A version of the well-known Young's double-slit diffraction experiment, in which the superposition of an ancillary photon (labeled as 3)—of having traveled through the top and the bottom slits—is transformed into a superposition of product states of the macroscopically-delocalized, two-level atom, A and the single photon, 3. Notice that this atom acts as a quantum meter, namely, a microscopic, measuring apparatus, which is treated quantum mechanically, and the overall excitation number of this atom is the relevant pointer observable. The ground and the excited states of this two-level atom are supposed to correspond to an unionized and an ionized atomic state, respectively. Subsequent to the formation of such a superposition of an excited atom—with an accompanying ejected photo-electron—and a non-excited atom—without any ejected photo-electron—a quantum mechanical photocurrent is produced that probabilistically triggers the anharmonic potential controller. The parameter, $\lambda$ indicates the degree of triggering. The curved black curve that connects the detector atom and the anharmonic potential controller, along with the grounding symbol on the latter, is supposed to represent a quantum mechanical photo-current, specifically, the flow of a quantum mechanical state of photo-electrons through a closed circuit. (b) and (c) Same as Figs. \ref{['fig:f3']}(b) and (c), respectively. These sub-figures have been reproduced here, so that the reader can visualize the entire entangling scheme in one figure.
• Figure 5: Effects of relative phases in real and Hilbert spaces on inter-particle and inter-modal entanglement. (a) An optical re-interpretation of the E.P.R.B. Gedankenexperiment, which was originally devised and realized by Aspect et al., and enables the direct measurements of the single-particle, and the joint, two-particle detection probabilities, so as to verify the Bell nonlocality of the two-particle entangled state, $\left|\Psi\left(\nu_{\textrm{A}}, \nu_{\textrm{B}}\right)\right>$. The subsystems are the photons and the entangled property is the angle of linear polarization. The angles, $\theta_{\textrm{A}}$ and $\theta_{\textrm{B}}$ are in the $x-y$ plane. (b) The algebraic sum of the correlation coefficients, $S$ as a function of the relative orientation of the two linear analyzers in real space, $\theta = \theta_{\textrm{A}}-\theta_{\textrm{B}}$ (solid red curve). The solid green line and the hollow blue circles indicate the von Neumann entropies of entanglement of $\left|\Psi\left(\nu_{\textrm{A}}, \nu_{\textrm{B}}\right)\right>$, and its unitarily transformed version, $\left|\widetilde{\Psi}\left(\nu_{\textrm{A}}, \nu_{\textrm{B}}\right)\right>$, respectively. All local, realist, hidden-variables theories, formulated according to the Einsteinian worldview, predict $-2\leq S \leq2$. The violation of the Bell's inequalities on the left and right of the vertical, dashed lines is the telltale signature of quantum nonlocality. (c) The von Neumann entropy of entanglement of a two-particle, two-mode entangled state—where the quantum subsystems are the modes, and the entangled property is the photon occupation number—as a function of the angle of rotation in the Hilbert space, $\varphi$. Notice that a change in the relative angular orientation in the Hilbert space—as opposed to in the real space—can modulate the entanglement entropy. (d). Same as (b), but for a two-particle, four-mode entangled state, where the quantum subsystems are the modes, and the entangled property is the linear momentum of the particle. The inset shows a space-time arrangement that allows the output modes to be interfered two-by-two—with individually controlled phases, $\vartheta_{\textrm{A}}$ and $\vartheta_{\textrm{B}}$ at two distinct spatial locations—so as to verify this state's Bell nonlocality.