Experimental demonstration of quantum pigeonhole paradox

Significance

We have demonstrated the quantum pigeonhole paradox with three single photons. The effect of variable-strength quantum measurement is experimentally analyzed order by order and a transition of violation of the pigeonhole principle is observed. We find that the different kinds of measurement-induced entanglement are responsible for the photons’ abnormal collective behavior in the paradox. The experimental violation of pigeonhole principle presents a challenge to the fundamental counting principle of nature.

Abstract

We experimentally demonstrate that when three single photons transmit through two polarization channels, in a well-defined pre- and postselected ensemble, there are no two photons in the same polarization channel by weak-strength measurement, a counterintuitive quantum counting effect called the quantum pigeonhole paradox. We further show that this effect breaks down in second-order measurement. These results indicate the existence of the quantum pigeonhole paradox and its operating regime.

Quantum paradoxes (1) are at the frontier in the understanding of the fundamental quantum mechanics and have uncovered many strong conceptual conflicts transitioning from quantum to classical physics. Prominent examples include the Einstein–Podolsky–Rosen (EPR) paradox that has introduced new physical concepts such as nonlocality (2, 3), and the quantum teleportation paradox that has brought numerous applications in quantum information sciences (4–6).

The pigeonhole principle (7), also known as the drawer principle or shelf principle, states that if there are more pigeons than boxes, at least one box must contain more than one pigeon. Recently, Aharonov et al. (8) put forward a quantum pigeonhole paradox where the classical pigeonhole counting principle in some case may break down. It was shown that three two-state quantum particles pre- and postselected in a particular subensemble could result in no two particles being in the same quantum states.

To observe the quantum pigeonhole effect, an elaborated quantum measurement is required to probe the single-quantum particles. Quantum measurement (9–11) that bridges the quantum and classical world has played a key role in many counterintuitive quantum protocols, ranging from quantum teleportation (4, 12, 13) to interaction-free measurement (14, 15). While projection measurements are routinely used to estimate the expectation of physical observables, they inevitably disturb or destroy the studied physical system. To interrogate the states of quantum particles but with minimal measurement-induced disturbance, the weak-strength measurement invented by Aharonov et al. (16) ∼30 years ago provides a feasible way.

Here we adopt the weak-measurement concept to probe the underlying physics of the quantum pigeonhole paradox. In our experiment, we use three quantum pigeons, which are three single photons, and two “boxes,” which are denoted by the two orthogonal polarization states horizontal (H) and vertical (V) (Fig. 1A). The photons are initially prepared in the state $|\psi \rangle ={|+\rangle }_1{|+\rangle }_2{|+\rangle }_3$ and finally measured and postselected in circular polarization state $|\varphi \rangle ={|R\rangle }_1{|R\rangle }_2{|R\rangle }_3$, where $|+\rangle =\left(|H\rangle +|V\rangle \right)/\sqrt{2}$, $|R\rangle =\left(|H\rangle +i|V\rangle \right)/\sqrt{2}$, and the subscripts labels the three photons. The success probability to observe the postselected state is ${\left|\langle \psi |\varphi \rangle \right|}^2=1/8$. To probe whether two quantum pigeons are in the same box, we introduce parity check observables $S_{ij}=|H_i{H}_j\rangle \langle H_i{H}_j|+|V_i{V}_j\rangle \langle V_i{V}_j|$ to query whether the ith and jth photons are in the same polarization state but without measuring which polarization, as shown in Fig. 1B. Similarly, we use a three-body parity check observable $S_{123}=|H_1{H}_2{H}_3\rangle \langle H_1{H}_2{H}_3|+|V_1{V}_2{V}_3\rangle \langle V_1{V}_2{V}_3|$ to query whether the three photons are all in the same polarization. It is obvious that nonzero observable $S_{123}$ implies nonzero observables $S_{ij}$, but the converse is not true.

Fig. 1.

Quantum pigeonhole paradox. (A) Transmitting three photons through two polarization channels. In the classical world, there must be at least two photons through the same channel. In the quantum world, this will be no longer always true. (B) The proposed experiment. Three photons are prepared and detected in certain polarization bases, and noninvasive quantum measurements are used to probe the photons’ polarization state.

To observe the pre- and postselected quantum systems without disturbance, weak-strength measurement of the observables is required (8). We introduce a quantum meter system with initial state $|\phi \rangle$, which will be changed by nonzero observables $S_{ij}$ according to the system-meter interacting Hamiltonian:

$$H=S_{12}\otimes F_m+S_{23}\otimes F_m+S_{31}\otimes F_m.$$ [1]

Here the meter state $|\phi \rangle$ is a narrow wave packet which can be moved by the displacement operator $F$, as shown in Fig. 2A. With an interaction strength $\lambda$, the meter is moved to a quantum state

$$|{\phi }^{\prime }\rangle =\langle \varphi |e^{-i\lambda H}|\psi \rangle |\phi \rangle .$$ [2]

We expand the exponential operator according to the interaction strength $\lambda$ up to the second order:

$$\begin{array}{l}|{\phi }^{\prime }\rangle \approx \langle \varphi |\psi \rangle |\phi \rangle -i\lambda (\langle \varphi |S_{12}+S_{23}+S_{31}|\psi \rangle )\otimes F_m|\phi \rangle \\ -\frac{{\lambda }^2}{2}(\langle \varphi |S_{12}+S_{23}+S_{31}+6S_{123}|\psi \rangle )\otimes F_m^2|\phi \rangle +O\left({\lambda }^3\right)\cdots .\end{array}$$ [3]

This formula shows that the shift of meter is proportional to $\langle \varphi |S_{12}+S_{23}+S_{31}|\psi \rangle$ and $\langle \varphi |S_{12}+S_{23}+S_{31}+6S_{123}|\psi \rangle$ in the first and second order of $\lambda$, respectively.

Fig. 2.

The experimental scheme. (A) A strength-variable measure system. A virtual meter is used to probe the polarization parity of photons. (B) Experimental setups. UV femtosecond laser pulses (394 nm, 80 MHz) pass through two beam-like type-II compound BBO crystals to produce four single photons by SPDC process. The photons are prepared in diagonal polarization product states with PBSs and HWPs. The parity observations are implemented by overlapping two single photons on PBSs. Finally, the photons are detected in circular polarization bases with QWPs, HWPs, and PBSs, and the fourfold photon events are recorded.

For small $\lambda$, the system is almost not disturbed, and we expect to observe first-order meter shifts when two photons appear in the same polarization. If no first-order shift is observed, it implies that arbitrary two photons are not in the same polarization and proves the emergence of the quantum pigeonhole paradox. In this first-order effect, the coefficients $\langle \varphi |S_{ij}|\psi \rangle$ completely determine the meter shift, and coefficient $\langle \varphi |S_{123}|\psi \rangle$ plays a nonnegligible role in high-order effect.

In the past few years, weak measurement on a single photon has been implemented in numerous experiments (17–24). However, directly performing a weak measurement on three photons simultaneously is still out of current photonic quantum technology. Here we circumvented this challenge by directly analyzing the expected first- and second-order meter shift. Below we describe our designed photonic experiments to probe the parity observables of the pre- and postselected ensembles.

The experimental setup is shown in Fig. 2B. Three polarization-encoded single photons are prepared from two pairs of entangled photons produced by spontaneous parametric down-conversion (SPDC) process (25). The experiment constitutes three stages: (i) preparing the photons in the polarization state $|\psi \rangle$; (ii) probing the photons polarization parities; and (iii) measuring the photons in circular polarization bases.

In the first stage, two beam-like type-II beta-barium borate (BBO) crystals are pumped by UV laser pulses with a central wavelength of 394 nm (26). The emitted four photons are filtered by polarizing beam splitters (PBSs) and prepared by half-wave plates (HWPs) in the diagonal polarization states. The fidelity of the prepared polarization state of three photons is 0.992(4), as shown in Fig. 3A. One extra photon is traced out and the other three photons are passed to the second stage.

Fig. 3.

The experimental calibrations. (A) The prepared diagonal polarization states. The fidelity is 0.992(4). (B) The HOM interference of SPDC photons. The visibility is 0.989(1) for same SPDC photons and is 0.90(2) for different SPDC photons.

In the second stage, PBSs are used to check the polarization parities. A PBS has two input and two output ports, and transmits a photon’s H component and reflects its V component. To ensure the two injected photons emit from two output ports, the two photons must both transmit or both reflect simultaneously. That is, when we register two photons in two output ports, we know that the photons have the same polarization components but still do not know which one (27). In this sense, a PBS performs the quantum operator $S_{ij}$ on the two coming photons. Adding a second PBS, the three-body operator $S_{123}=S_{23}{S}_{12}$ can also be implemented. We performed two-photon Hong–Ou–Mandel (HOM) interference (28, 29) to guarantee the photons arriving at the PBS are of maximal indistinguishability. The HOM interference visibilities of the same SPDC photons and different SPDC photons are 0.989(1) and 0.90 (2), respectively, as shown in Fig. 3B. The interference visibilities are stable during the experimental period and the reduction of interference visibility between different SPDC photons is due to the spectral correlation between the interference photon and its twins SPDC photon (26).

In the final stage, the three single photons are measured in circular polarization bases by quarter-wave plates (QWPs) followed by PBSs. We recorded the threefold coincident counts shown in Fig. 4. In Fig. 4A, a PBS is used to perform the quantum operation $S_{12}$ on photons 1 and 2. Totally, 1,105 events are recorded and there are only three events projected to the postselected state $|\varphi \rangle$. So, the expectation of parity observable ${\left|\langle \varphi |S_{12}|\psi \rangle \right|}^2/{\left|S_{12}|\psi \rangle \right|}^2$ is estimated to be 0.003(2), which is of significant zero probability. The small residual value above zero is mainly contributed by the imperfect multiphoton destructive interference, characterized by the finite HOM interference visibilities of two individual single photons. Furthermore, due to the apparent symmetry between the three photons in our test system, we can conclude that the parity of any pair of two photons is all zero. In Fig. 4B, We show the counts of threefold events using two PBSs to estimate the three-body parity observable. Here 1,152 events are recorded, where 269 events are projected to the postselected state $|\varphi \rangle$. We estimate that ${\left|\langle \varphi |S_{123}|\psi \rangle \right|}^2/{\left|S_{123}|\psi \rangle \right|}^2=0.23\left(1\right)$, which is significantly nonzero.

Fig. 4.

The experimental results. (A) Probing two-photon parity. The probability of projection onto postselected state $|RRR\rangle$ is significantly zero. (B) Probing three-photon parity. The probability of projection onto postselected state $|RRR\rangle$ is significantly nonzero.

Our experimental results show that in the defined pre- and postselected three photons subensemble, an imaginary noninvasive weak measurement will not observe a first-order meter shift due to the zero observables ${\left|\langle \varphi |S_{ij}|\psi \rangle \right|}^2$ and indicates that no two photons are in the same polarization channel, proving the quantum pigeonhole paradox. When the measurement to probe the parity becomes stronger, the nonzero second-order meter shift (proportional to ${\left|\langle \varphi |S_{123}|\psi \rangle \right|}^2$) is not negligible and we can observe that the probability of three photons in the same polarization is significantly nonzero, which naturally implies two photons could in the same box. Therefore, the phenomenon of quantum pigeonhole paradox can only be observed by noninvasive measurement and will disappear in strong measurement.

We note that the parity observations induce quantum correlations in the single photons. With small measurement strength, the measurement back-action of observables $S_{ij}$ produces an EPR-type entanglement (2), while at stronger measurement, high-order back-action $S_{ij}{S}_{jk}$ produces a Greenberger–Horne–Zeilinger-type entanglement (30). These induced quantum entanglements are responsible for the collective properties of independent quantum particles in the quantum pigeonhole paradox, controlling the occurrence or disappearance of quantum pigeonhole effects.

In summary, we have implemented the quantum pigeonhole paradox by transmitting three single photons through two polarization channels, which demonstrates the pigeonhole paradox definitely at the level of individual quantum particles (31). The experiments show that in addition to the condition of well-defined pre- and postselected subensemble, noninvasive weak measurement is also required to observe the counting paradox. We implement the desired measurement indirectly by analyzing the measurement effects order by order and reveal the paradox will not survive under high-order measurement.