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Null-Result Detection and Einstein-Podolsky-Rosen Correlations

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Abstract

It follows from Bell’s theorem and quantum mechanics that the detection of a particle of an entangled pair can (somehow) “force” the other distant particle of the pair into a well-defined state (which is equivalent to a reduction of the state vector): no property previously shared by the particles can explain the predicted quantum correlations. This result has been corroborated by experiment, although some loopholes still remain. However, it has not been experimentally proved—and it is far from obvious—that the absence of detection, as in null-result (NR) experiments could have the very same effect. In this paper a way to try to bridge this gap is suggested.

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Notes

  1. The terms “null” measurement, “null-result” measurement, and “negative-result” measurement are also used.

  2. Home and Whitaker [2] discuss different kinds of negative-result experiments, and argue that they can be explained without the need to invoke the collapse of the wave function.

  3. See Ryff [4], in which the term “indirect detection” is used. But, in general, indirect detection involves the inference of the existence of something (e.g., elementary particles, dark matter, via Weakly Interacting Massive Particles (WIMPs), et cetera) from the detection of some other thing.

  4. See Ryff [5], in which the term “interaction-free measurement” is used. However, this term has become linked to the kind of experiment discussed in [6], which, as will become evident, is actually quite different in spirit from the kind of experiment which is being discussed here. In the Elitzur-Vaidman proposal, whenever a photon is detected in the “forbidden” output port, it is possible to infer the presence of a light absorbing object in one of the arms of a Mach-Zehnder interferometer. Here, whenever a detector does not click, it is possible to infer the path followed by a photon.

  5. In an interview in [8], Bohm explicitly entertains the idea of FTL signaling.

  6. It may sound strange to consider the lack of detection as an event; however, taking into account the context, there seems to be no reason for any confusion.

  7. Actually, the transmitted and reflected probability amplitudes can be recombined by means of a Mach-Zehnder interferometer, and the detectors can be placed at the interferometer output ports, so that interference can be observed. As a result, we don’t, strictly speaking, have a classical situation. However, as has been stressed, it admits, in principle, a simple interpretation based on the de Broglie-Bohm pilot-wave approach [10], in which a wave with a photon (“full wave”) follows one arm of the interferometer and a wave without a photon (“empty wave”) follows the other [11]. The possibility of introducing photon trajectories has also been investigated, for example, in [12], and references there in. As a consequence, for this kind of experiment, the consideration of time-like events (by introducing a detour on the photon path leading to the distant detector, for example) would not change it into an indisputable NR-detection experiment. Similarly, by simply changing the position of the detectors, placing the first distant from and the second close to the beam splitter, this does not improve the situation.

  8. Here, “objective” change of probability is to be understood as a change in probability as a consequence of a change in the physical properties of the system. A “subjective” change of probability is associated to a Bayesian point of view, in which the probability is automatically updated when our knowledge changes. According to Polkinghorne, “One must acknowledge that a true case of action at a distance is involved, and not merely some gain in additional knowledge. Putting in a learned language, the EPR effect is ontological and not simply epistemological.” [13] (a very lucid and concise explanation of the foundations of quantum mechanics). Maudlin [14] is also a good read.

  9. Renninger has been considered to be the first to draw attention to null measurements (as in [16, p.59]); however, in an interesting article [17], Whitaker discusses the gedanken experiment suggested in a precursory paper by Epstein [18].

  10. In which—as far as I know—the term “interaction-free” measurement has been introduced.

  11. Since some loopholes still remain, it follows that, strictly speaking, we cannot affirm that we know from the empirical violations of Bell’s inequalities that the successful detection of a particle can collapse the wave function and thus force the other distant particle of the pair into a well-defined state. In my opinion, from a Popperian standpoint, the only thing we can say is that the quantum mechanical predictions have been corroborated by experiment. Exactly when and where this “forcing” (which, in principle, can be accepted even for those who reject the idea of collapse) can be considered accomplished is a question still open to debate. It can be conjectured, based on the pilot-wave interpretation, for instance, that it takes place whenever the first photon is split at the two-channel polarizer into a wave with a photon and a wave without a photon. However, according to Bohm and Hiley [10], in the case of boson fields we should give up the notion of particle–and, naturally, also of pilot wave–and consider the field variables themselves as fundamental ontological entities. But it does not seem clear in this approach how to deal with NR-detections. As acknowledged by Holland [20], a detailed treatment of the optical experiments testing Bell’s inequalities has not yet been given in the deBroglie-Bohm causal interpretation. On the other hand, in the theories based on the idea of spontaneous collapse [21] it seems that the detector must be, during an exceedingly short time, in a superposition of fired and not fired states. There are also other possible interpretations, based on the decoherence approach [22], on many-worlds [23], and so forth. I think that to discuss all these alternatives would lead me astray from the main point, namely, that independently of the explanation we prefer, NR detections have not been satisfactorily tested.

  12. To avoid any conceptual confusion, it is important to distinguish between the terms collapsing and forcing in the present context. Although ν 2 is not forced into a well-defined polarization state (and, therefore, no “ordinary” reduction or collapse of the state vector occurs), ν 1 can no longer be detected at D 1′, as a consequence, Eq. (1) is not valid. More specifically, ν 1 can no longer be detected in state |a ⊥〉, otherwise, we would have the strange situation in which, by introducing a second polarizer after the first one with the same orientation, ν 1 could be transmitted at the first polarizer and reflected at the second. Whether this should be considered a sort of collapse, I prefer, by the moment, to leave it as an open question. It is also important to notice that, when writing the probabilities p(b|a) and p′(b|a), we are assuming that whenever ν 1 is not detected at D 1′ it will necessarily be detected at D 1. Otherwise, we would have more detections at D 1′ than at D 1.

  13. Scarani and Gisin [25] have shown that the assumption of finite speed superluminal interaction leads, assuming that the correlations are purely nonlocal, and for more than two entangled particles, to FTL signalling. The same subject has been approached, from a different standpoint in [26]. An interesting discussion of this theme is in [27]. For more recent and general results, see [28].

  14. Actually, to have FTL communication we have to consider space-like events (no detours in Fig. 1) and all ν 2 that are detected, for instance, independently of ν 1 also being detected or not (no classical communication channel to inform on that). We also have to implicitly assume that there is a preferred frame in which one of the photons (ν 1, in this example) is really detected first. Arguments in favor of a preferred frame have been presented by Bell in [29], in Bell [3], and in an interview in [30]. Bohm and Hiley also advocate a preferred frame, in [10]. The possibility of introducing and identifying a preferred frame is examined in [31], in [32], and more recently in [33].

  15. The use of the no signaling condition as an axiom to build the fundamental structure of quantum mechanics has also been discussed in [35].

  16. Although this paper can be considered a contribution to the important “psi-ontic vs. psi-epistemic” issue debate ([36], and references therein), it is not intended to settle it. My personal reasons to prefer an ontic point of view are briefly expounded in [37]. Retrocausal interpretations are also interesting ([38], and references therein), but, to me, it is not yet clear how to avoid a trend of strong determinism in this case, since, to the extent that the future interferes with the present, our capability of changing the future becomes limited (Maudlin’s arguments on this subject (see Ref. [14]) still sound pertinent). As already mentioned, I think that to discuss all alternative interpretations of quantum mechanics would lead us to deviate of our main point; furthermore, most probably I would not be able to present the arguments of their proponents in a correct and unbiased way. Please note that it is not my intention to put forward a new interpretation of the observed quantum mechanical phenomena and to compare it with other alternative interpretations. The purpose of this paper is to investigate quantum mechanics in a new experimental context. For divergent opinions on foundational questions on quantum mechanics see [39], and also [40].

  17. For an interesting analysis of the interconnection between quantum and classical theories, including historic aspects.

  18. My own experience has shown me that this is a very controversial subject. The interested reader can consult [44], where it is argued (in the appendix) that it is possible to avoid a causal paradox and the abandonment of Lorentz transformations by introducing a preferred frame, as suggested by Bell [29], and assuming that the equivalence between passive and active Lorentz transformations is violated when superluminal signaling is considered (we have a similar situation in the case of the violation of parity, in which some mirror-reflected phenomena have no counterpart in the real world). The importance of Poincaré’s ideas [45] are emphasized.

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Acknowledgements

I thank Paulo Henrique Souto Ribeiro for helpful conversations.

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    Luiz Carlos Ryff

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Ryff, L.C. Null-Result Detection and Einstein-Podolsky-Rosen Correlations. Found Phys 44, 58–70 (2014). https://doi.org/10.1007/s10701-013-9762-0

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