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Delayed-Choice Quantum Erasers and the Einstein-Podolsky-Rosen Paradox

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Abstract

Considering the delayed-choice quantum eraser using a Mach-Zehnder interferometer with a nonsymmetric beam splitter, we explicitly demonstrate that it shares exactly the same formal structure with the Einstein-Podolsky-Rosen-Bohm (EPR-Bohm) experiment. Therefore, the effect of quantum erasure can be understood in terms of the standard EPR correlation. Nevertheless, the quantum eraser still raises a conceptual issue beyond the standard EPR paradox, if counterfactual reasoning is taken into account. Furthermore, the quantum eraser experiments can be classified into two major categories: the entanglement quantum eraser and the Scully-Drühl-type quantum eraser. These two types are formally equivalent to each other, but conceptually the latter presents a “mystery” more prominent than the former. In the Scully-Drühl-type quantum eraser, the statement that the which-way information can be influenced by the delayed-choice measurement is not purely a consequence of counterfactual reasoning but bears some factual significance. Accordingly, it makes good sense to say that the “record” of the which-way information is “erased” if the potentiality to yield a conclusive outcome that discriminates the record is eliminated by the delayed-choice measurement. We also reconsider the quantum eraser in the many-worlds interpretation (MWI), making clear the conceptual merits and demerits of the MWI.

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Notes

  1. Denoting the two detectors as \(D_+\) and \(D_-\), instead of \(D_1\) and \(D_2\), underscores the analogy to the Stern-Gerlach apparatus, as will be seen shortly.

  2. In the double-slit experiment, we can place two polarizers with horizontal and vertical polarizations in front of the two slits to emulate the function of \(\textrm{PBS}\). A polarization rotator is then placed behind the second slit to make the paths from the two slits interfere with each other. The different positions on the screen correspond to different values of \(\phi \). We can also place two optical attenuators with different attenuation rates behind the two slits to emulate different values of \(\theta \).

  3. In the same spirit, the modified Mach-Zehnder interferometer parameterized by \(\theta \) and \(\phi \) is also analogous to a single polarizing beam splitter that splits the incident beam into two beams of orthonormal polarizations \(\vert {\hat{n}_{\theta ,\phi },+}\rangle \) and \(\vert {\hat{n}_{\theta ,\phi },-}\rangle \). We focus on the Stern-Gerlach apparatus oriented in \(\hat{n}_{\theta ,\phi }\) as the representative example for the analogy, as it is easier to adjust both \(\theta \) and \(\phi \) for the Stern-Gerlach apparatus than for a polarizing beam splitter.

  4. However, because of the unitary freedom for density matrices, \(\rho ^{(s)}\) also admits infinitely many different interpretations. Therefore, it is only an interpretation, not an objective reality, to assert that \(\gamma _s\) travels either \({\textrm{Path}_{1}}\) or \({\textrm{Path}_{2}}\) with equal probability. We will come back to this point shortly.

  5. We assume maximum entanglement between \(\gamma _s\) and \(\gamma _i\) in this paper, whereas the analysis in [5] also considers the extension that the degree of entanglement is adjustable.

  6. In the literature of quantum erasure, the extension with a variable \(\phi _2\) on Bob’s side is seldom considered. Thus, unfortunately, the fact that Alice and Bob are on equal footing is obscured and not often noted.

  7. In [33], it is claimed that “the delayed-choice quantum eraser leaves no choice” in a particular situation that is equivalent to the configuration with \(\theta _1=\pi /2\), \(\phi _1=\pi /2\) considered in this paper. In this situation, because each individual \(\gamma _s\) registered in \(D_\pm \) corresponds to the initial state \(\vert {\hat{n}_{\theta _1=\pi /2,\phi _1=\pi /2}}\rangle \equiv \frac{1}{\sqrt{2}}\left( \vert {\leftrightarrow }\rangle \pm i\vert {\updownarrow }\rangle \right) \), which is sensibly said to have traveled both paths equally, it is argued in [33] that Bob hence “no longer has the choice to seek either which-path information or quantum eraser”. This sensibly interpreted history of \(\gamma _s\), however, is still compatible with alternative interpretations as far as the probabilistic outcomes obtained by Alice and Bob are concerned. In fact, Bob always have the choice to yield a different history of \(\gamma _s\) unless Alice sets \(\theta _1=0\) or \(\theta _1=\pi \), only for which the history of \(\gamma _s\) becomes factual.

  8. In the Scully-Drühl-type quantum eraser, we do not consider the polarization degree of freedom for photons, and all beam splitters are non-polarizing ones, unlike \(\textrm{PBS}\) used in Fig. 1.

  9. In the literature of quantum mechanics and especially quantum information, the purity of a matrix density \(\rho \) is usually defined as \(\gamma :={{\,\textrm{Tr}\,}}(\rho ^2)\), which satisfies \(1/d\le \gamma \le 1\), where d is the dimension of the Hilbert space. Here, we adopt a different definition used in [34], which is for the two-dimensional case and satisfies \(0\le \mu _s\le 1\).

  10. In most cases of real experiments, the remaining degrees and the degrees of external parties altogether should not be described by a pure state, but by a density matrix (most likely a thermal density matrix), because repeated runs of an experiment, although considered to be in the same setting, are in fact carried out in different environmental states due to thermal or other uncontrollable fluctuations. Therefore, the observed interference patten is the probabilistic average of (4.26) averaged over the density matrix. Averaging the phase \(\delta \) appearing in (4.26) over the probabilities given by the density matrix will smear the modulation in response to \(\phi _1\). As a result, the interference visibility is further diminished much more than merely by the factor of \(\mu _s\).

  11. The work of [54] analyzed the complementarity duality in detail, but the distinguishability considered therein is neither \(\mathcal {D}_u\) nor \(\mathcal {D}_m\) but that as defined in [34], which in the literature is also referred to as the predictability and bears a different meaning as noted in [42].

  12. By contrast, in (2.14) for the Mach-Zehnder interferometer in Fig. 1, asserting that the state \(\vert {\leftrightarrow }\rangle \) is projected into \(\vert {\hat{n}_{\theta ,\phi },+}\rangle :=\cos (\theta /2)\vert {\leftrightarrow }\rangle + e^{i\phi } \sin (\theta /2)\vert {\updownarrow }\rangle \) or \(\vert {\hat{n}_{\theta ,\phi },-}\rangle :=\sin (\theta /2)\vert {\leftrightarrow }\rangle - e^{i\phi } \cos (\theta /2)\vert {\updownarrow }\rangle \) is perhaps counterintuitive, but not self-contradictory. Recall the discussion after (2.14).

  13. This does not violate causality, because the overriding resets a counterfactual history into a factual one, not a factual history into a different factual one. In the case that Alice sets \(\theta _1=0,\pi \) and Bob sets \(\theta _2=0,\pi \), the histories deduced by Alice and Bob both are factual, and they just agree with each other — no overriding upon each other.

  14. In a real experiment, the interference pattern recovered by the quantum erasure is further diminished much more than () because of the reason addressed in Footnote .

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Acknowledgements

The author would like to express his gratitude to the following people who have helped this work. Tabish Qureshi and Tai Hyun Yoon brought their related works to the author’s attention. Bo-Hung Chen and Hsiu-Chuan Hsu had various discussions with the author. Additionally, an anonymous reviewer gave valuable suggestions that have significantly improved the manuscript. This work was supported in part by National Science and Technology Council, Taiwan under the Grant MOST 111-2112-M-110-013.

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  1. Department of Physics, National Sun Yat-sen University, 80424, Kaohsiung, Taiwan

    Dah-Wei Chiou

  2. Center for Condensed Matter Sciences, National Taiwan University, 10617, Taipei, Taiwan

    Dah-Wei Chiou

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Chiou, DW. Delayed-Choice Quantum Erasers and the Einstein-Podolsky-Rosen Paradox. Int J Theor Phys 62, 120 (2023). https://doi.org/10.1007/s10773-023-05370-4

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