How to play Quantum Mine Sweeper
To play Quantum Mine Sweeper you only have to read the first two sections. The last section helps to understand the physical ideas behind the game.
Quantum Mine Sweeper (QMS) is a logical game. The aim of the game is the same as that of interaction-free measurement: You want to detect a device (for reasons of dramaturgy usually a bomb) which explodes if exposed to the slightest amount of light -- and of course you don't want it to explode. Usually one would consider this task as impossible. Detection is only possible if you shed light on the bomb, but this implies that it will explode.
The trick to circumvent this dilemma is to use the wave-particle duality of light. In the context of Quantum Mine Sweeper, wave-particle duality means that a single quantum of light (a photon) used to detect the bomb can behave as a particle (like a billiard ball) or a wave (like a wave on the sea) simultaneously. More precisely, if you do not look for the position of a photon it will behave like a wave and is split into two at each semi-transparent mirror, in much the same way as a wave is split into two separate waves (called wave packets) by a rock. On the other hand, if you look where the photon actually is, it will always be at a single position (like a billiard ball). That is, looking for the position of the photon destroys all wave packets but the one where you have detected the photon: one may say that detecting the position of the photon "collects" it at a single position. After the detection the photon's position it moves again like a wave and can be split by semi-transparent mirrors until it is detected again.
The previous paragraph describes a physical fact which you simply have to accept. Do not let confuse you by seemingly contradictive concepts. To play the game you don't need to understand the paragraph but just to remember that a photon is split by semi-transparent mirrors and "collected" by a position detection. These concepts will become more clear in the rest of this section, but if you feel that you want to learn a bit more about the corresponding physics you may read the last section.
Let's now turn to the game itself. Initially there are just four mirrors present. The lower left and the upper right mirror are semi-transparent, while the others do fully reflect the light. You can check this by pushing the "all wavepackets" button to observe how the light is split and recombined at the semi-transparent mirrors. The "single photon" button shows a random path which a single wave packet would take if you could observe it.
There are several ways to manipulate the action of a mirror on an incoming wave packet. Obviously the direction of the mirror determines the direction of the reflected wave packet. You can change the direction of a mirror by left-clicking on it. Doing this several times can also change the mirror to a cross and vice versa (a cross has no effect on photons at all, it simply marks a grid position).
Another important parameter of a mirror is its transmission
factor T. This factor is equal to 100% if the mirror allows
the complete wave packet to fly through it without changing its direction.
The mirror is then just transparent like a window pane.
On the other hand, if T is equal to 0% the mirror reflects
the full wave packet and thus comes closest to the everyday-notion of
a mirror. For all values of T between 0% and 100% the mirror
is not fully transparent and not fully reflective. The special case
of T = 50% is called semi-transparent.
To change the transmission factor first right-click on the mirror. A small Panel will open over the mirror and you can vary T by using the corresponding scroll bar. Since in interaction-free measurements it is often more appropriate to write T as the cosine of an angle I did so in Quantum Mine Sweeper, too. The relation is simple, however. A transparent (window) mirror corresponds to a value of 0 for the angle, a fully reflective mirror corresponds to a value of 50, and a semi-transparent mirror corresponds to a value of 25.
Another important but less common feature of a mirror is the phase shift imprinted on light. Its only purpose in Quantum Mine Sweeper is to combine two wave packets which arrive at the same time at one mirror in such a way that they leave the mirror as a single wavepacket. In the initial configuration of Quantum Mine Sweeper this happens at the upper right mirror. If you push the "all wave packets" button, two wave packets arrive at it, but only one is leaving in the upward direction. By changing the phase shift of this mirror you can achieve that two wave packets are leaving, or that a single wave packet is leaving to the right. You can change the phase shift of a mirror by a right-click on it. Then you can use the corresponding scroll bar. If a single wave packet is leaving the mirror in one direction you can change the direction of the leaving wave packet by adding or subtracting 8 to the phase shift.
If you are not interested in the physical background of the phase
shift you can skip this subsection. It is not necessary to play the game.
The phase shift is a wave phenomenon.
a sea wave is split into two waves by a rock. One of the two
wave packets is
traveling down a shorter channel while the second is traveling
through a longer channel. Immediately after the splitting the two
wave packets have the same phase. This is the physicists language
for saying that the valleys of the first wave packet are at the same
position as those of the second packet. The amplitude of the combined
wave is just the sum of the two packet amplitudes and
is then maximal.
After they have travelled through their respective channels the two wave packets meet each other and are recombined, but now they have a different phase: the valleys do not meet anymore. In the special case that each valley of one packet meets a peak of the second packet the combined amplitude is zero so that the wave is erased. This is exactly what happens if at a mirror two wave packets are combined to a single one leaving the mirror: the absence of a second outgoing packet arises because of the cancellation of the amplitudes of the two ingoing wave packets.
Physically, a mirror produces a phase shift because one of the incoming wave packets effectively has to travel a longer way than the other one: the silver layer sits on one side of a glass pane with a certain thickness. One packet is immediately reflected, while the other one has to travel through the glass pane.
The bomb is essentially a photon absorber. If a wave packet hits the bomb the wave packet is simply removed. You can see this by first setting the bomb at a random position by pushing the "set bomb" button and then pushing the "all wave packets" button. If you observe the propagating wave packets you will notice that one of them is removed at the position of the bomb. To remove the bomb push the corresponding button. If you hit the "single photon" button a single wave packet takes a random path. If the bomb is in the path of the photon the latter is absorbed and the bomb explodes. In this case your intention to detect the bomb without an explosion is not achieved. The next section describes how this problem can be circumvented.
We have seen that the phase shift of a mirror can be adjusted in such a way that two incoming wave packets are combined to form just one outgoing wave packet. Interaction-free measurements use this fact in a tricky way to detect a bomb without an explosion.
Consider again the initial situation of Quantum Mine Sweeper with two semi-transparent mirrors (upper left and lower right) and two fully reflective mirrors (lower left and upper right). The phase shifts of the mirrors are adjusted in such a way that -- in absence of a bomb -- the two wave packets are combined at the upper right mirror in such a way that only one wave packets leaves the mirror in upward direction. In other words, if we follow the path of a single wave packet it will never leave the upper right mirror rightwards.
Now consider the situation that the bomb is placed in path of one of the two wave packets. As a consequence, the corresponding wave packet is removed so that only a single wave packet will arrive at the upper right mirror. Since this mirror is semi-transparent it will split the incoming wave packet into two so that there will be a wave packet which leaves the mirror rightwards.
If we now launch a single wave packet with the "single photon" button there will be a chance that it will leave the upper right mirror rightwards. Since this is impossible if the bomb is not there such a wave packet indicates the presence of the bomb -- even though the wave packet never met the bomb. It is the absence of the second wave packet which makes such a detection without explosion possible. If you want to learn more about the underlying mechanism I recommend to read the following section.
The task for you is now quickly stated: find an arrangement of mirrors which allows to detect the presence of a bomb on any path between the 9 mirror/cross positions displayed in Quantum Mine Sweepers. Good Luck, and if you solve the puzzle I'd be happy to hear about it. I have a solution, but you may have found a different one.
If you want to learn more about this topic you may visit the homepages of the scientists listed below. I myself am not working on this field, but I think that it provides a beautiful example of one of the deepest mysteries in science: wave-particle duality. The list is incomplete as I wasn't very keen to spend too much work on it. If you want me to add another link just write me an e-mail.
A. Elitzur and
L. Vaidman, the inventors of
A. Zeilinger worked on experimental realizations
P. Kwiat also invented new experimental schemes
Wave-particle duality lies at the heart of quantum physics and therefore provides one of the deepest mysteries that mankind has discovered up to now. The basic principle is quickly stated: any elementary quantum of matter or radiation behaves both like a particle (e.g., a billiard ball) and like a wave (e.g., like a sea wave). This behaviour has been observed with light, electrons, atoms, molecules, and elementary "particles" (a somewhat unfortunate name in our context) and thus applies to all phenomena of the microscopic world.
What is the difference between a wave and a particle? A particle like a billiard ball has a fixed size and shape. Its motion is completely characterized by its trajectory: at each instant of time we know exactly the position and the velocity of the billiard ball. A wave, on the other hand, is an extended object. A sea wave, for instance, is extended over many hundredths of meters (or miles) and can change its shape. There is nothing like a trajectory of a wave (at best one may define something like an average trajectory). A wave can be characterized instead by its amplitude (how high the sea wave is) and its phase (whether there is a peak or a valley at a given position and time).
An even more profound difference between particles and waves is their behaviour when they meet each other. If a billiard ball meets another one they are simply scattered. They change their direction but do not change their shape. On the other hand, two waves can be superposed: if one sea wave meets anbother one their amplitudes are added. If a peak of the first wave meets a peak of the second wave the combined amplitude is twice as high as that of the individual waves. If a peak meets a valley the combined amplitude is zero. Thus, waves change their shape when they meet each other.
Now you will probably ask the question how one can imagine an object which simultaneously behaves like a particle and a wave. The answer is: to visualize such an object is probably impossible. In this sense physicists haven't understood their most successful theory, quantum mechanics (this observation is due to nobel laureate Richard Feynman). However, physicists have very well understood the mathematical structure which describes these objects. We can observe this weird behaviour and build machines with it (a laser, for instance).
I guess the last paragraph wasn't very satisfying since I only told you what we haven't understood. Now I will try to give you a basic idea of how wave-particle duality is actually achieved in quantum mechanics. The postulates of quantum theory distinguish between the dynamics of a quantum system (for instance, how a photon evolves in time) and the observation of the system. The dynamics is described by a type of wave equation, i.e., the dynamics of a system obeys similar laws as a sea wave. However, it is not the amplitude of a water wave which evolves in quantum mechanics, but a probability amplitude. The larger this probability amplitude is, the larger is the probability to find the system at a particular position. If two probability-amplitude waves meet each other they can be combined in the same way as sea waves are combined.
The observation of a quantum system follows entirely different
laws. It is the kind of question you pose (the kind of apparatus
which you use to observe the system) which decides whether the system
appears to you as particle-like or as wave-like. For instance, if
you ask the question whether the system is at a particular position,
you will get either the answer that it is there or that it isn't there.
This sounds trivial, but remember that for a wave it is not possible
to tell where exactly it is. A wave is always spread over some area,
and only for a particle you will get a definite answer (there or
not there) for your question. Thus, the quantum system will appear to you
as a particle.
On the other hand, if you ask in the initial configuration of Quantum Mine Sweeper whether the photon will leave the upper right mirror upwards or rightwards, the answer will depend on the phase shift of the mirrors. As a phase is a typical wave-phenomenon which has no meaning for a particle one can see that this kind of question lets the system appear as a wave.
This was a basic explanation of what wave-particle duality means. In Quantum Mine Sweeper, both aspects are of importance. Since at the end we ask whether a photon leaves the upper right mirror rightwards we observe a wave phenomenon. The bomb, on the other hand, is essentially a kind of measurement whether the photon is at the position of the bomb and therefore corresponds to a particle phenomenon.
I finally have to admit that the operation of the "single photon" button is slightly (but not crucially) misleading since it appears that the photon is taking a definite path and thus behaves completely as a particle would do. However, it is not possible to assign a photon a definite path since it is only deceided at the time of the measurement whether we observe a particle-like or wave-like behaviour (not during the dynamical evolution). The mathematical model behind Quantum Mine Sweeper is correct, however, since the evolution of the photon obeys a wave equation. The definite path only represents the measurement whether the photon is at some time at the position of the bomb or not.