Quantum information theory of the Bell-state quantum eraser

Quantum systems can display particle- or wavelike properties, depending on the type of measurement that is performed on them. The Bell-state quantum eraser is an experiment that brings the duality to the forefront, as a single measurement can retroactively be made to measure particlelike or wavelike properties (or anything in between). Here we develop a unitary information-theoretic description of this and several related quantum measurement situations that sheds light on the trade-off between the quantum and classical features of the measurement. In particular, we show that both the coherence of the quantum state and the classical information obtained from it can be described using only quantum-information-theoretic tools and that those two measures satisfy an equality on account of the chain rule for entropies. The coherence information and the which-path information have simple interpretations in terms of state preparation and state determination and suggest ways to account for the relationship between the classical and the quantum world.

Figure 1

Schematic of the double-slit Bell-state quantum eraser experiment [17]. After production of the Bell state (1) by type-II spontaneous parametric down-conversion (SPDC) in the $\beta$-barium borate (BBO) crystal, photon $B$ travels along the upper branch to a polarizing beam splitter (PBS), with the optical axis oriented at angle $\theta$ relative to the $|h\rangle ,|v\rangle$ basis. Its polarization is subsequently measured in the rotated basis by the photodetectors $D_B^{\left(0\right)},D_B^{\left(1\right)}$. Photon $A$ travels down to the quarter-wave plates (QWPs) and double slit and then to a CCD camera, denoted $D_X$, which plays the role of an interference screen.

Figure 2

Entropic Venn diagrams showing the effect of the tagging operation. (a) Before tagging, the polarization of photons $A$ (denoted by variable $P$) and $B$ are entangled in a Bell state. (b) After tagging, the spatial degree of freedom of photon $A$ (denoted by variable $Q$) becomes entangled with the polarizations of $A$ and $B$ according to Eq. (9).

Figure 3

Entropic Venn diagrams for (a) state preparation with detector $D_B$ [see Eq. (14)] and (b) state determination with detector $D_A$ [see Eq. (23)]. Here, S is the von Neumann entropy of the density matrix in Eq. (21).

Figure 4

Top: Relationship between coherence and path information in Eq. (38) for the Bell-state quantum eraser as a function of the erasure angle $\theta$. Shown are the information-theoretic quantities for the path information $S(Q:D_A|D_B)=S$ (solid) and coherence $S(Q:D_B)=1-S$ (dashed). Bottom: The square of the distinguishability $D$ (solid) and of the fringe visibility $V$ (dashed), defined in text, as a function of the erasure angle $\theta$. When $\theta =0$ ($\theta =\pi /4$), there is full (no) path information and no (full) coherence.

Figure 5

Geometry of the double-slit apparatus. The slit width is $a$, the distance from the origin to the center of slit $j$ is $x_j$, and the distance between slits is $d=|x_2-x_1|$. The distance from the slits to the screen is $L$, while the angle from the center of slit $j$ to point $x$ on the screen is $tan{\varphi }_j=(x-x_j)/L$.

Figure 6

Conditional probability distributions $p_k\left(x\right)$ plotted as a function of the position on the screen $x$ (in meters) and normalized so that the maximum is at 1. The solid gray (black dotted) oscillations describe the interference pattern $p_0\left(x\right) \left[p_1\left(x\right)\right]$ of photon $A$ conditional on outcome $k=0$ ($k=1$) of detector $D_B$. The solid black line is the total distribution $p\left(x\right)$ and is the single-slit diffraction result. Parameters for this specific case are $a=10\mu \mathrm{m}, d=20\mu \mathrm{m}, L=1$ m, and $\lambda =702$ nm. Top to bottom: probability distributions for three erasure angles, $\theta =0, \theta =\pi /16, \theta =\pi /4$.