Explaining Quanta with Optical Illusions

Quantum mechanics demonstrated that an elementary particle has qualities, called “quantum properties,” that can only take discrete values upon measurement. An electron, for example, possesses a property called “spin” that, when measured, can only be found in one of two “quantum states,” commonly called “up” and “down,” and formally indicated as |+〉 and |−〉, or |0〉 and |1〉 in the qubit notation that I adopt hereafter. Other particles can show more than two exclusive states after measurements of their properties. The set of states assumed by a particle for its various properties defines its overall “quantum state.”

A classical object too has a “state,” which is the collection of its qualities, like the color, the size, the weight, and so on. However, a classical object can only exist in one state at a time (e.g., it cannot be simultaneously yellow and blue), while experiments clearly show that a quantum object can exist in a “superposition state,” which means that it can be at the same time in more than one state. Specifically, the “superposition principle” of quantum mechanics says that one can express any quantum state as the superposition of other quantum states. However, when someone “observes” a particle in superposition, i.e., takes a measurement of a quantum property, (s)he sees only one of the possible discrete outcomes, that value being random and inherently unpredictable, though the probabilities of possible outcomes might be known.

The fact that classical objects cannot behave in this way makes it very difficult for people to realize what the “quantum object” actually is, because they are used to imagining something having exclusive qualities at any given time. To help with this, scientists have proposed cute explanation methods, which use diagrams, colors, games, simulations, or everyday objects.^1–5 Nonetheless, the difficulty remains, largely due to the lack of a representation that is inherently dual. This is why I propose here to think of a quantum object as a “bistable drawing.”

A bistable drawing is a picture that creates ambiguity between two distinct shapes in the interpretation by the human visual system. Looking at such images causes a spontaneous and stochastic alternation between two mutually exclusive perceptual states.⁶ Such pictures are also called “reversible optical illusions,” the most famous of which are the Necker cube,⁷ “My wife and my mother-in-law,”⁸ and the Rubin vase,⁹ but there are many others.

The first of them (Fig. 1) is the 2D parallel projection of a wireframe cube, which can be interpreted to have either the lower-left or the upper-right square as its front side. In “My wife and my mother-in-law” (Fig. 2), the profile of a young girl appears when the face of an old woman disappears. Finally, in the Rubin vase (Fig. 3), which belongs to the category of “figure–ground illusions,” one can perceive a white vase on a dark background or, alternatively, the dark profile of two faces looking at each other on a white background.

The peculiarity of such drawings is that, every time one looks at them, one perceives a shape or its complement in a seemingly purely random way. Therefore, a bistable drawing behaves like a quantum object, which is dual and, when observed, instantly “collapses” into one of its exclusive states in a completely random way.

Some psychological studies have suggested such similarity between human perception and quantum mechanics implies that there exists a valid quantum-like model for brain mental states.^10–12 In contrast, I propose here to adopt this similarity to explain quantum concepts easily.

In fact, this apparently simple analogy includes a proper correspondence for a number of physical aspects of the quantum theory, namely:

• quantum object → bistable drawing

• pure quantum states |0〉 and |1〉 → each one of the two shapes

• superposition → coexistence of the two shapes

• quantum collapse → sudden perception of a single shape

• measurement → the act of looking at the drawing

• random result of a measurement → randomness in which shape is perceived

• orthogonality of the pure states → impossibility of perceiving the two shapes simultaneously

• quantum decoherence → loss of stable perception of one shape

• a quantum is neither a wave or a particle, it’s just a different thing → a drawing is neither of its perceived shapes, it’s just a distribution of black and white regions,

so that bistable drawings can help viewers realize in a personal way these very concepts.

Note that this analogy does not allow cutting out a single shape from the drawing, because the isolated shape is no longer a bistable drawing and so cannot represent a quantum object.

The last item in the list deserves some more words. The viewer is led to think of these drawings as being in a state of “suspended reality” that becomes real in a different way only after observation. However, one cannot ask whether the drawing of the Rubin vase depicts “in reality” a vase or two faces, or what is “in reality” the front face of the Necker cube. In fact, a drawing is not what it represents, but only a distribution of black and white regions, and only the interaction between the drawing and the observer generates a meaning. This concept helps to understand the wave–particle duality, suggesting how mutually exclusive properties like wave and particle can be aspects of something inherently different, the quantum object, exhibiting either wave or particle behavior depending on what is observed.

Of course, my paradigm remains an analogy and, like all the analogies, it has some limitations.

The main limitation is that it does not support measurement. In fact, it cannot reproduce the stability of a quantum state after collapse. If one performs the same measurement on a quantum state soon after a previous measurement, the result is always the same; instead, the perception of a bistable drawing continues to swap between the two alternatives as long as one looks at it.

The teacher would need to make it clear that repeating measurements would be akin to seeing the old woman and then only seeing the old woman each time the same drawing was sequentially observed. Furthermore, instead of seeing this as a problem, (s)he could leverage this feature to represent the “quantum decoherence,” a very important aspect in the field of quantum computing; this is why I included it in the previous list.

Finally, a clever observer knows well that the described perception is just a response of his visual system and brain, and that nothing really changes in a bistable drawing when someone looks at it. Such knowledge could lead the viewer to think that also the quantum state of a real particle only resides in the observer and that the quantum collapse is a pure illusion. This is wrong in the common Copenhagen interpretation, which holds that quantum descriptions are objective; however, different interpretations of quantum mechanics exist, for which there is no objective truth independent of the observer. So again, the reader can use this inadequacy of the model to illustrate those different interpretations.

It is worth noting that these drawings are only able to represent effectively dual states that have (nearly) equal probabilities of being measured.

I further exploit the proposed correspondence to get a representation of the “complementarity principle,”¹³ which states that quantum objects have certain pairs of complementary properties that cannot be observed simultaneously. For example, measurements of spin components along perpendicular axes are complementary, as the observation of one leaves the others in superposition.

In the proposed paradigm, one can represent a pair of complementary properties with two bistable drawings arranged to be seen exclusively.

For example, one could place the Rubin vase on the front side of a card and “My wife and my mother-in-law” on the back side (Fig. 4). If in the front picture the viewer randomly recognized a vase, and is now looking at the drawing on the back side, when turning the card back (s)he will perceive a vase or two faces again randomly, independently of the image seen before.

In practice, in the moment that the viewer observes one drawing, (s)he loses the identity seen in the other, as happens in a quantum object, where the measurement of a property turns its complement to the undefined state of superposition.

The complementarity principle, by excluding the possibility of a simultaneous collapse of two complementary properties, explains why quanta have to be intrinsically dual, and why all particles must have at least one property undefined at a given time, with the consequence that randomness is inherent in the fundamental nature of the universe.

Therefore, I add the following correspondences to the previous list of analogies:

• pair of complementary properties → pair of bistable drawings on the front and back sides of a card

• complementarity principle → loss of perception of a drawing while looking at the other.

As before, it’s not allowed to force the analogy by, e.g., placing a mirror to simultaneously see the rear face of the card, or flipping a card 360° without looking at the back.

When two quanta interact with each other, a relation is established between their quantum states and holds until one of them is subjected to a new interaction.¹⁴ This relation is called “entanglement,” or “quantum correlation,” and it is at the heart of the disparity between classical and quantum physics.

Consider an experiment in which two particles enter a region where they interact, and then go out in different directions.

Let us consider the simple case of a binary property, like the spin of an electron, and suppose that each particle enters the experiment in a superposition state. Before the interaction, the two particles are independent, which means that measuring their spins, one would find |0〉 or |1〉 randomly and independently.

After the interaction, however, the two particles are no longer independent, as laboratory experiments demonstrate. In particular, the measured spin of one particle is always concordant, or always discordant, with the measured spin of the other. In other words, after the interaction, the two spins can only be correlated (i.e., |0〉, |0〉 or |1〉, |1〉), or anti-correlated (i.e., |0〉, |1〉 or |1〉, |0〉), depending only on how the experiment is conceived. Moreover, such correlation is present even if the time delay between measurements of the two spins is less than the time a light ray would need to go from one particle to the other. This excludes any possibility of communication between them and implies quantum nonlocality.¹⁵ 

This behavior raises two questions: how can two things be concordant or discordant if they are still undefined? And, how can the second spin be dependent on the first one if there has been no communication between them?

The presented analogy offers to the viewer a visual answer to both questions.

To visualize the entanglement, one places two identical bistable drawings side by side, e.g., two copies of the Rubin vase (Fig. 5). If looking at the first, the viewer randomly sees a vase, after moving their glance to the other, they will also see a vase in it. Similarly, if somebody sees two faces in the first drawing, (s)he will also see two faces in the second. In other words, in the second drawing one always perceives the same shape randomly perceived in the first, as happens in a positively correlated entanglement.

Instead, if one aims to illustrate an anticorrelated entanglement, one just has to replace the second drawing with its inverse, where the vase is black and the faces are white (Fig. 6).

This experiment helps the viewer understand how two quanta in superposition can be concordant or discordant even if the result of the first measurement is purely random. Moreover, the viewer clearly realizes that the first drawing does not need to communicate anything to the other, because the mutual pattern of black and white regions encodes the type of correlation from the beginning.

In this way, my analogy suggests the correct understanding that quantum correlation is a property encoded in the couple, which has to be considered as a whole. Therefore, one can represent even the strangest of all quantum concepts, i.e., entanglement, by leveraging the following correspondences:

• pair of entangled quanta → pair of identical bistable drawings placed side by side

• pair of quanta in anticorrelation → a bistable drawing placed aside its inverted copy

• absence of communication between the quanta → absence of communication between the drawings.

The limitation in this correspondence is that it cannot reproduce the fact that two entangled particles are still correlated if measured by different observers, because there is no correlation in the shapes perceived by two different people that look at identical bistable drawings. Hence, one should not try to force the analogy in such a direction. In the last section hereafter, I suggest how to overcome this limitation.

In quantum physics, teleporting a particle from one place to another means transferring its quantum state to the destination without physically moving the particle from its location. This is achieved by exploiting quantum entanglement.¹⁶ 

To begin, one must prepare a concordant entangled state of two quanta (e.g., two photons) and hold one of them, A, in the departure lab while the other, B, is physically carried to the destination. This separated couple can be called a “quantum channel.”

What one wants to do is to transfer the quantum state of a third photon, C (called the “teleported photon”), from the departure point to the destination, using the quantum channel as a carrier.

To initiate this transfer, one correlates the photon C with the photon A of the quantum channel. Let us call this action the “teleportation entanglement.”

Quantum theory indicates that, after this correlation, the teleported photon, C, is entangled with the photon B at the destination, even if they have never interacted directly. Hence, if C is found, e.g., in state |0〉, B will also be found in state |0〉 after measurement, which means that the state of a quantum at the departure point has been transferred into the state of a quantum at the destination. That is, it has been “teleported.”

This happens because quantum correlation has a sort of “transitive” property: if one has correlated A with B and then correlates C with A, then C and B become correlated (while the original correlation between A and B breaks down).

However, all that is true only if the teleportation entanglement between A and C was concordant. In fact, if it was discordant, photon B receives the opposite state, and people at the destination need to flip the state of B before measurement if they want to get the same state as C.

The sign of correlation is random, so the teleportation can only work if the departure lab informs the destination lab about the sign effectively taken by the correlation. This is classical information that can only be sent at the speed of light, e.g., via radio, which is the reason why quantum teleportation cannot be instantaneous and needs the aid of a classical communication channel.

I have not found a way to use three bistable drawings to replicate the teleportation protocol and perceive in the drawing at the destination the same shape perceived in the drawing to be transferred. This probably goes beyond the power of the proposed analogy. Nonetheless, bistable drawings turn out to be very helpful to show that quantum teleportation does not actually transport anything, because the identity of the teleported photon is already at the destination since we set the quantum channel. In fact, the photon B that we carried to the destination was in a state of superposition, so that we physically carried all these identities in one move. An observer who looks at Fig. 5 will immediately understand this fact, because (s)he quickly realizes that physically moving either drawing moves both shapes together.

Bistable images are not the only type of optical illusions that show convenient correspondences with quantum physics. There are some static optical illusions that are perceived to rotate, and can be very effective in explaining the concept of quantum spin.

Books and documentaries commonly describe the spin of a particle like the “rotation” of a marble around its axis, which can rotate clockwise or counterclockwise, but this is completely wrong. Elementary particles such as electrons are point-like objects and, unlike marbles, do not have a surface, nor a volume or any structure, so that the classical notion of rotation has no sense for them. Nonetheless, a particle with nonzero spin like an electron does have a mechanical angular momentum and, correspondingly, a magnetic moment.

Moreover, an electron within an atom also possesses an orbital angular momentum and the corresponding magnetic moment, even if quantum mechanics clearly showed that electrons do not move around the nucleus as a planet does around the Sun.

Quantum spin is in no way a classical angular momentum, but it is an angular momentum of a different form. Like kinetic energy and potential energy, which are very different forms of energy but can be exchanged and produce similar effects, a particle with spin can transfer angular momentum to a macroscopic object and cause it to rotate. Quantum spin thus exhibits the characteristics of a rotation despite the particles not rotating in any way.

I propose to illustrate this very counterintuitive concept by means of “spinning illusion” drawings, like the ones in Fig. 7. While looking at these figures, the viewer perceives a clockwise or counterclockwise rotation even if the drawings do not rotate, in close correspondence with the notion of quantum spin that I outlined.

The illusion of movement in such images is generated by a suitable pattern of light and dark stripes as a result of the different response speed of retinal neurons to high-contrast and low-contrast colors. Variations of this pattern can lead to different directions of the apparent rotation (the first panel in Fig. 7 appears to rotate clockwise, while the others seem to rotate counterclockwise), which can be used to represent spin up and spin down in a spin ½ particle.

To summarize:

• quantum spin → spinning illusion drawing

• spin up or down for a spin ½ particle → clockwise or counterclockwise direction of rotation illusion.

A limitation of these drawings is that they cannot support spin superposition, since the same rotation is perceived when looking at them again.

As mentioned, both the bistable and spinning illusions presented so far show some major limitations, namely, i) the lack of support for the measurement process; ii) inadequacy in representing entanglement when the two cards are distant from each other, or seen by different people; and iii) lack of representation of spin superposition in the spinning illusions.

However, the world of visual ambiguities is quite large, so the teacher can adopt different illusions, getting different pros and cons. As an example, it’s worthwhile to present an illusion able to support the measurement process, and another one representing the spin superposition.

Figure 8 depicts the behavior of the “Close/Far” illusion, which belongs to the so-called “hybrid drawings.” Looking at it, the observer sees either the word “CLOSE” or the word “FAR” depending on the viewing distance from the drawing.

This happens because the drawer predominantly used high spatial frequencies to compose the shape to be viewed nearby, and low spatial frequencies for the shape to be viewed from a distance. When the drawing is close, the attention is captured by the high-frequency image, while when it is far, the high frequencies vanish beyond the eye’s resolution and the low-frequency image prevails.

For each person, there is a narrow range of intermediate distances where the hybrid drawing appears as bistable. To represent the measurement of a quantum state in superposition, the observer can place the drawing at a bistable distance, and suddenly move it closer or further away as soon as (s)he reads either “CLOSE” or “FAR,” respectively. Once moved, the drawing is no longer in a bistable condition, and any further observation will confirm the previously “measured” state.

We could even use this scheme to represent entanglement by placing two identical hybrid drawings, one beside the other on a common support. As the viewer perceives “CLOSE” or “FAR” in one of the drawings, (s)he correspondingly moves the support closer or further away, carrying them out of the bistable region, so that the perceived image will be viewed in both drawings.

The same happens if we place the two drawings on a very wide support, so that they stay at a large separation from each other. This way, a different person who will observe the second drawing after we have “measured” the first one will see the same shape we randomly perceived on our side.

In comparison to a bistable drawing, this hybrid illusion has the advantage of dealing with quantum measurement, but the disadvantage of being less practical and intuitive. Therefore, they complement each other.

Figure 9 instead presents frames of the “Spinning ballerina” illusion, which is a bistable animation showing a dancer that can be seen spinning in one direction or in the other. This animation is able to represent the superposition of a one-half spin particle (if a video device is available), but it is not suitable to represent the “spin without spinning” concept. Therefore, this illusion complements those shown in Fig. 7.

I present a new visual paradigm, which aims at exploiting optical illusions as an aid to explaining the concepts of quantum mechanics.

In particular, I demonstrate the effectiveness of using a specific set of optical illusions, called “bistable drawings,” to represent a quantum object. Thanks to their inherent duality, these drawings lead the viewer to understand the quantum concepts through a direct sensorial experience, which avoids most misconceptions.

Such analogy has a definite correspondence to a number of quantum concepts, like superposition, collapse, complementarity, and entanglement, which I discuss extensively together with their limits of applicability. Thanks to these correspondences, the viewer learns the meaning of state superposition, understands complementarity of the states, and realizes that entangled particles do not communicate with each other and that quantum teleportation does not actually transport anything.

I also introduce the “spinning drawings” optical illusion, and demonstrate its ability to represent the mysterious concept of quantum spin, which avoids the common, but misleading, comparison to classical rotation, and teaches that nothing is actually spinning in quantum spin.

For each illusion, I analyze the limitations of the analogy, and finally I give two examples of how to overcome these limits by exploring the larger world of visual ambiguities.

This work aims to invite the reader to exploit in lessons, presentations, and books the advantages of using inherently dual visual representations for explaining quantum concepts. Here I examined bistable, spinning, and hybrid optical illusions, but the reader could try exploring more sensorial ambiguities to represent the many counterintuitive concepts of modern physics.

I thank the team of the Italian podcast of physics “Fisicast,” and in particular professor Giovanni Organtini of La Sapienza University of Rome, for helpful discussions and comments.

Gianluca Li Causi is a senior researcher of the Rome Astronomical Observatory of the INAF, where he works in the field of astronomical technologies. Has worked for various instrument of different telescopes and is currently responsible for the data simulation of the next-generation multi-object spectrographs MOONS@VLT and MOSAIC@ELT, and for the data processing of the high-contrast camera SHARK-VIS@LBT, which has recently produced the highest-resolution ground-based image of the Jovian moon Io. He is active in outreach and didactics and has been co-founder and director of the Italian podcast for physics “FISICAST,” for which he developed new ideas for explaining modern physics to the people. [email protected]