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# Quantum violation of the pigeonhole principle and the nature of quantum correlations

Contributed by Yakir Aharonov, November 12, 2015 (sent for review April 3, 2015; reviewed by Charles H. Bennett and Lucien Hardy)

### This article has a letter. Please see:

## Significance

We show that quantum mechanics violates one of the fundamental principles of nature: If you put three particles in two boxes, necessarily two particles will end up in the same box. We find instances when three quantum particles are put in two boxes, yet no two particles are in the same box, a seemingly impossible and absurd effect. This is only one of a host of related quantum effects which we discovered and which point to a very interesting structure of quantum mechanics that was hitherto unnoticed and has major implications for our understanding of nature. It requires us to revisit some of the most basic notions of quantum physics––the notions of separability, of correlations, and of interactions.

## Abstract

The pigeonhole principle: “If you put three pigeons in two pigeonholes, at least two of the pigeons end up in the same hole,” is an obvious yet fundamental principle of nature as it captures the very essence of counting. Here however we show that in quantum mechanics this is not true! We find instances when three quantum particles are put in two boxes, yet no two particles are in the same box. Furthermore, we show that the above “quantum pigeonhole principle” is only one of a host of related quantum effects, and points to a very interesting structure of quantum mechanics that was hitherto unnoticed. Our results shed new light on the very notions of separability and correlations in quantum mechanics and on the nature of interactions. It also presents a new role for entanglement, complementary to the usual one. Finally, interferometric experiments that illustrate our effects are proposed.

- weak value and weak measurement
- entanglement and quantum nonlocality
- correlations
- two-state vector formalism
- foundations of quantum mechanics

## Quantum Pigeonhole Principle

Arguably the most important lesson of quantum mechanics is that we need to critically revisit our most basic assumptions about nature. It all started with challenging the idea that particles can have, at the same time, both a well-defined position and a well-defined momentum, and went on and on to similar paradoxical facts. But, the pigeonhole principle that is the subject of our paper seems far less likely to be challenged. Indeed, although on one hand it relates to physical properties of objects––it deals, say, with actual pigeons and pigeonholes––it also encapsulates abstract mathematical notions that go to the core of what numbers and counting are so it underlies, implicitly or explicitly, virtually the whole of mathematics. [In its explicit form the principle was first stated by Dirichlet in 1834 (1) and even in its simplest form its uses in mathematics are numerous and highly nontrivial (2).] It seems therefore to be an abstract and immutable truth, beyond any doubt. Yet, as we show here, for quantum particles the principle does not hold.

Consider three particles and two boxes, denoted

Let us now check whether two of the particles are in the same box. Because the state is symmetric, we could focus on particles 1 and 2 without any loss of generality; any result obtained for this pair applies to every other pair.

Particles 1 and 2 being in the same box means the state being in the subspace spanned by

Crucially, as noted before, the state is symmetric under permutation, hence what is true for particles 1 and 2 is true for all pairs. In other words, given the above pre- and postselection, we have three particles in two boxes, yet no two particles can be found in the same box––our quantum pigeonhole principle.

## Related Effect for Every Final Outcome

In the previous section we focused on what happens when at the final measurement all particles are found in the state *Supporting Information*). Similar patterns occur in all other cases.

## Generalizing the Quantum Pigeonhole Principle

The above effect is but one of a multitude of similar effects. For example, in the case of *Supporting Information*).

## Nature of Quantum Correlations

Before analyzing our paradox in more detail, we would like to comment more on the nature of quantum correlations.

The first thing to notice is that neither the preselected state nor the postselected state is correlated (they are both direct products and each particle is prepared and postselected individually), yet the particles are correlated.

The second thing to notice is that if we measure the location of each particle individually, they appear to be completely uncorrelated. Indeed, suppose we measure separately the location of particles 1 and 2. There are four possible outcomes of this measurement:

The above shows a fundamental difference in the way in which the probabilities work in the standard, “preselected only” experiment and in a “pre- and postselected” one (i.e., when we only consider the cases in which a final measurement gave a particular answer).

Indeed, consider first the standard situation, that is, consider that the particles are prepared in the state

On the other hand, suppose we compare the above two measurement methods but in the case of a pre- and postselected ensemble. In full generality, when one measures an arbitrary operator

Using this result, in the case of separate measurements on each particle we find

The third thing to notice is that the global measurement which only asks about correlations but no other detailed information is, in some sense, better than the detailed measurement as it delivers the information about correlations while minimizing the disturbance that it produces to the state. Indeed, suppose two particles are in an arbitrary superposition

Finally, and most importantly, we note that the global measurement is a measurement of an operator with entangled eigenstates and it requires either to put the particles in interaction or consume some entanglement resources to perform it. The quantum pigeonhole effect is thus an example of a new aspect of entanglement: Entanglement in the measurement is needed to reveal correlations existing in a direct product state.

## Nature of Quantum Interactions: A First Experiment

The quantum pigeonhole effect has major implications for the understanding of the very nature of quantum interactions. Consider again three particles and two boxes. Let the particles interact with each other by bipartite short-range interactions, i.e., any two particles interact when they are in the same box and do not interact otherwise. Then, as there are three particles and only two boxes we expect that always at least two of the particles should interact. But, due to our pigeonhole effect, this is not so, as shown in the following experiments.

Consider a Mach–Zender interferometer for electrons, as depicted in Fig. 1. It consists of two beam-splitters BS_{1} and BS_{2}, a phase shifter (PS) that introduces a phase shift of

Suppose now that we inject simultaneously three electrons in the interferometer from the left side, such that they travel in parallel beams. The beams are arranged in an equilateral triangle configuration, as can be seen by the zoomed-in cut-away section of the beam in Fig. 1.

When two electrons go through the same arm of the interferometer they repel each other and their trajectories are deflected. Indeed, the force that one electron exerts on the other produces a change in momentum and this in turn leads to the deflection of the beams by an amount depending on the original separation of the beams, the length of the interferometer, and the speed of the electrons. When the electrons go through different arms they effectively do not interact (because the arms are separated by a large distance). Because we have three electrons and only two arms, we expect to always have interactions, regardless of which detectors the electrons end up at after traversing the interferometer.

We are interested in what happens in the cases when all three electrons end up at **2**, we can find particles 1 and 2 in the same box. In this case the state collapses to **3**, so if we were to make the postselection now this even would not be selected; this was our main proof of the quantum pigeonhole effect. However, suppose that instead of making the postselection at this point we now make a measurement of the pair 1, 3. We can also find 1 and 3 in the same box. The state now collapses to **3**, hence these events are not eliminated from the postselected ensemble. To be able to see the effect we need to limit the mutual disturbance of the measurements.

In our interferometer case, the interaction between the electrons leads to the deflection of the beams whenever two electrons are in the same arm. The deflection of the beams is therefore effectively like a measurement of whether or not two electrons are in the same arm. To limit the disturbance produced by the simultaneous existence of the interaction between all three pairs, we just need to reduce the strength of the effect of the interaction. Technically, we want to ensure that the change in the momentum of an electron due to the force produced by the other electrons is smaller than the spread in its momentum. Due to this, the deflection of a beam––if it occurs––is smaller than its spatial spread, hence, by seeing where one electron landed on

Naively, we would expect to see the three beams deflected outward and deformed; each electron should move radially outward when all three are present in the same arm, and sideways when only two electrons are present. We expect the deviation to be by less than the cross-section but nevertheless by a noticeable amount. Instead (see the *Supporting Information*) what we find is that the beams are completely undeflected and undisturbed (up to second-order perturbations), indicating that indeed there was no interaction whatsoever between the electrons.

## Second Experiment

A second experiment uses a similar interferometer as described in the first experiment, but instead of electrons we now inject atoms. Let all of the atoms start in an excited state and arrange the setting in such way that there is a very significant probability for the atoms to spontaneously emit photons while traversing the interferometer. We surround the interferometer with photon detectors that could detect the emitted photons. Importantly, we chose the energy separation between the ground and excited state such that the wavelength of the emitted photons is much larger than the separation between the arms, so that by detecting a photon we cannot tell whether the atom that emitted it went through the left or right arm. Again, we inject all three atoms from the left and are interested only in the cases in which all three end up at detector

When two atoms are close to each other (being in the same arm) they interact with each other (for example by dipole–dipole interactions) and the energy levels are shifted. Observing the wavelength of the emitted photons we can then tell if the atoms were in the same arm or not. Following the same reasoning as used in the previous experiment, we also arrange for the frequency shift to be smaller than the spread of the spectral line, so that the three pairwise interactions should not disturb each other. Again, due to this, one single photon observation cannot tell us whether the frequency was shifted or not, but by accumulating the statistics we can detect the shift.

Because there are three atoms but only two arms, similarly to the previous experiment, we expect that in each run of the experiment at least two atoms will be in the same arm, so the photons they emit will be frequency shifted, no matter which detector the atoms end up in. However, according to our quantum pigeonhole effect, when we look at the cases in which all three atoms end up at

## Conclusions

In conclusion, we presented a new quantum effect that requires us to revisit some of the most basic notions of quantum physics––the notions of separability, of correlations, and of interactions. It is still very early to say what the implications of this revision are, but we feel one should expect them to be major because we are dealing with such fundamental concepts.

## Generalizing the Pigeonhole Effect

Consider *N* particles in *N* particles are in

## Interferometric Experiment

To describe our interferometric experiment it is useful to separate the degrees of freedom into a “which-arm” degree of freedom and the relative positions *V* is the interaction potential. The projector *T* is the total time of the interaction (the time taken for the electrons to go through the interferometer) and where for simplicity, we set

## Acknowledgments

Y.A. acknowledges support (in part) by the Israel Science Foundation Grant 1311/14, the Israeli Centers of Research Excellence Center “Circle of Light” of Deutsch-Israelische Projektkooperation, the German–Israeli Project cooperation, and the European Research Council (ERC) Advanced Grant Nonlocality in Space and Time (NSLT). Y.A., D.C.S., and J.T. acknowledge support (in part) by the Fetzer Franklin Fund of the John E. Fetzer Memorial Trust. S.P. acknowledges the ERC Advanced Grant NSLT.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: aharonov{at}chapman.edu.

Author contributions: Y.A., F.C., S.P., I.S., D.C.S., and J.T. performed research; and S.P. and J.T. wrote the paper.

Reviewers: C.H.B., IBM Thomas J. Watson Research Center; and L.H., Perimeter Institute.

The authors declare no conflict of interest.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1522411112/-/DCSupplemental.

Freely available online through the PNAS open access option.

## References

- ↵.
- Dirichlet PGL,
- Dedekind R

*Vorlesungen über Zahlentheorie*(Vieweg, Braunschweig, Germany); trans (1999) [*Lectures on Number Theory*] (American Mathematical Society, Providence, RI). German. - ↵.
- Allenby RBJT,
- Slomson AB

- ↵.
- Nielsen M,
- Chuang I

- ↵

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- Abstract
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- Generalizing the Quantum Pigeonhole Principle
- Nature of Quantum Correlations
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- Second Experiment
- Conclusions
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- Interferometric Experiment
- Acknowledgments
- Footnotes
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