The information contained in a system is registered in its physics and the way of accessing this information is in itself a physical process. Thus, information theory is an excellent way of describing both the physical world and the form in which we know it. This turns out to be crucial when we try to describe the microscopic world, since the observer-world relationship is fundamental in quantum physics.
Every one of us has a favourite “beautiful idea”. Mine is the definition of the natural number by Bertrand Russell: “number three is the collection of all the sets of three things”. The basis of the beauty of this idea is that the bigger the part of the universe involved, the greater its definition, the more precisely number 3 is defined. If we put three socks in the collection, we will obtain a clumsy stain in the form of a B or a 3, and only if we gradually add three cherries, three free throws, three little pigs, is the outline gradually more and more defined, up to the point of maximum definition, which is reached when all possible trios are considered. This way of proceeding has something of El Aleph by Borges, in which a very well defined point, a very specific idea contains, nonetheless, the entire universe. And it also has something of the perfect story, one in which the ending is the result of all the actions, of all the characters, of all the scenes, the years and the landscapes that have appeared in it.
The idea is beautiful because it rejects the existence of the abstract concept of 3 – it is also entertaining to consider that one of the 20th century’s most brilliant mathematicians needed the entire universe to be able to define something as simple as a 3. It is also beautiful because it rejects the separation between matter and form and invites us to think that any idea, however abstract it may be, is constructed through sets of things. And it is beautiful because it represents an imaginary invitation from Bertrand Russell, the mathematician who undertook the task of writing the foundations of mathematical reasoning, to Rolf Landauer, the physicist who observed in a non-trivial way that information is not an abstract entity.
Information is physical
In the very words of Landauer, and as a second “beautiful idea”: “Information is physical”. Information is inevitably registered in a physical medium, such as the colours of a traffic light, the grooves of a record, radio waves… Rolf Landauer did not restrict himself to observing the intrinsic relationship between information and the physical but he stated it in the form of a principle of physics: “Any operation that implies a loss of information in a system, for example, erasing a file from a computer or adding up 3 plus 3 on the calculator (adding up means losing information because we obtain 6 but we lose the summands), increases the entropy (the disorder of a system) and dissipates heat”: Getting rid of information is not without cost. The information that abandons the system must do so in some physical way, with physical consequences, and it does so in the form of heat. (Part of the heat currently burning my legs underneath my laptop obeys this principle of Landauer’s. It must be because for every word that I write, I erase thirty. It sounds paradoxical, yes, but writing means continually eliminating information.)
A charming thought experiment illustrates how conventional science tended to consider information as an abstract entity, and the gaining of information in a system as a harmless operation. James Clerk Maxwell devised this at the end of the 19th century as a way of challenging the laws of physics (specifically the second law of thermodynamics), suggesting that a train can be moved by thought alone. Well, this is an exaggeration, a literary licence; such a provocative way of summarising the famous demon imagined by Maxwell, that describing it in detail is inescapable. The second law of thermodynamics says that when the steam of a locomotive cools down, the train stops. In other words, it is not possible to obtain work (move a train) from a system in thermal equilibrium with the environment. So, Maxwell proposed a way of questioning this principle. He imagined the engine of the stopped train, with the piston stopped between two steam compartments, both at the same temperature. The temperature is the average of the speeds of the gas molecules. Some move very fast, others very slowly, but they are totally mixed (disordered) and the average speed in any volume of gas is the same and, therefore, the temperature of the gas is uniform. In the piston that separates the compartments, Maxwell imagined a small door and a little demon with the capacity to know the speed of each molecule, and to open or close that door at will to order the molecules: fast to the right, slow to the left. Every time a fast molecule approaches from the left compartment to the right, he lets it pass. Every time a slow molecule approaches from right to left, he lets it pass. The result is that the right compartment increases in temperature, while the left one cools down. The resulting difference in pressure between the two compartments would be capable of moving the piston and the train’s wheels would give another turn. In other words, a disordered system can be ordered, which implies that work can be obtained from a gas in equilibrium, simply by inserting a demon with the capacity to know things!
Landauer’s response to defend the second law is that, in effect, Maxwell’s demon can order the molecules to generate work based on the difference in temperature, but it does not do so gratuitously. Its brain must register the information on the speed of each molecule and then erase it in order to accommodate in its space that of the following molecule. Every time it does this, it increases the entropy compensating what it has helped to reduce by ordering the molecules, and dissipates the heat out of the system, preventing it from being converted into work.
That the information contained in a system is registered in its physics, and that the way of accessing this information is in itself a physical process, implies that the information theory is an excellent way of describing both the physical world and the form in which we know it, which has an extra bonus if the aim is to describe the microscopic world, since the relationship between observer and the world – the measurement – is a fundamental issue in quantum physics.
One of the advantages of the information theory to grant a certain sense to the quantum world is that the emitter, receiver, code and message are clearly identified. Furthermore, while the transition from classical physics to quantum physics brings with it some traumas for intuition, the transition from classical information to quantum information softens the anti-intuitive shocks and evidences various ways in which we can make use of the paradoxical behaviour of the world at a microscopic level to carry out tasks that are impossible for classical physics. I am referring to the technologies of the second quantum revolution, such as quantum computing, communication, cryptography and metrology.
Bits and Qubits
The basic units of quantum information are qubits (quantum bits). We can expect two possible values when accessing the information contained in a qubit. They have that in common with classical bits. From there on, everything else is different. Let’s explain, then, the differences between classical physics and quantum physics through the differences between classical information and quantum information.
An example of a classical bit is the colour of the flag that indicates whether bathing is allows on a beach. While we are eating breakfast and preparing our beach bags with towels and spades, the emitter, who is responsible for the safety of bathers, evaluates the state of the sea and decides whether to put out the green flag, which allows us to bathe, or the red one, which prohibits bathing. We, the receivers, do not know that we cannot bathe until the time when we arrive at the beach, even if the red flag has been flying for at least an hour.
An example of a qubit or quantum bit is the direction of polarisation of a photon (associated with the direction of oscillation of the electric light field). The polarisation may be horizontal H or vertical V. The photon as a qubit contains information in the sense that, following a measurement, the receiver always obtains two complementary answers: H or V. This is the binary code of the photon, analogous to the two-coloured code of the flag. An observation which it is worth considering, both in the case of the flag and that of the photon, is that we don’t ask them “what colour are you?” or “what direction is your polarisation in” but we ask “are you red or green?” or “are you H or V?”
Flag and photon are analogous communication systems, but there is one fundamental difference: a flag is a macroscopic object, the emitter only has to choose the colour of the flag and raise it and the receiver will only need to look at for the information to be transmitted. A photon, however, is a microscopic object, and emitter and receiver, as manipulators of quantum information, play a more active role than their classical counterparts. The emitter has to prepare the state of the photon, and the receiver has to perform a measurement on it. Preparation and measurement go beyond the classical tasks, and include a certain freedom of choice in the procedures. A freedom that is offered precisely by the quantum nature of the scenario.
Freedom of the quantum emitter: superposition
The emitter’s task is to prepare the photon. It can be prepared in H, it can be prepared in V and also it can be prepared in any combination of these two directions: an oblique direction with a certain horizontal and a certain vertical component. This is what we call a “quantum superposition”. When the receiver performs a measurement on the HV basis, in other words, asks the photon whether it is H or V, he obtains H if the emitter has prepared it in H, and he obtains V if the emitter has prepared it in V. But if the emitter has prepared the photon in an H and V superposition, in an oblique direction, the receiver will obtain H with a certain probability p and V with a certain probability 1-p. The probability p is larger the more the oblique approaches the horizontal, and indicates how uncomfortable the photon is, by responding that its real polarisation address is distant from the options, which are proposed as two sole possible responses. Before the HV measurement, the photon is in a superposition of H and V. In other words, “before reaching the beach, the flag is flying at neither H nor V”.
Freedom of the receiver: the measurement
The receiver also has freedom when measuring the polarisation of the photon. He can make a measurement at the HV base but can also use any other base, any other pair of H’V’ axes, perpendicular to each other, but rotated with respect to HV. The possible results in this case will be H’ or V’, and the probability of obtaining H’ will be different from that of obtaining H. We can understand HV and H’V’ as different properties of the photon: in an HV measurement the photon is asked whether it is H or V, in an H’V’ measurement, it is asked whether it is H’ or V’. Let’s say that measuring HV is like “looking at the colour of the flag” and measuring H’V’ is like “looking at the size of the flag”. And the quantum peculiarity lies in the fact that, while a single glimpse lets us know the colour and the size of the flag, the quantum receiver cannot do the two measurements at the same time.
Consequences of the freedom of the receiver: collapse and destruction
If the receiver makes an HV measurement and the photon responds H, the photon stays polarised at H. In other words, the state of the photon collapses to a specific state. Successive measurements in HV will have the same result H, despite the fact that prior to the first measurement, the probability existed of obtaining V. This means that measuring has devastating consequences for a quantum system. Let’s see to what point this is so.
If HV and H’V’ are rotated 45 degrees – the maximum separation between the two systems of perpendicular axes – the consequences of the choice of measurement for the system are drastic. In effect, if the emitter prepares a photon in H and the receiver measures in HV, the photon will respond H. If the receiver measures this same photon again and this time asks it H’V’ – which is equivalent to asking a horizontal photon whether it is diagonal or anti-diagonal – the photon will be extremely uncomfortable responding either of the two, but it will have to decide on one of them at random – remember that H and H’ form 45 degrees and therefore the probability of H’ is equal to that of V’. Let’s say that it decides on H’. If the receiver measures a third time, now in HV, the photon, diagonal now in HV and extremely uncomfortable once more, could respond – since it would respond at random – that it is in V. But in the first HV measurement it had responded that it was in H!
The impression is given that the photon is changing opinion. But it is not the fault of the photon; it is the fault of the measurement. It is not possible to know simultaneously the result of what the photon would say if the receiver asks HV and what it would say if asked H’V’. Because of course, if the receiver asks HV and the photon responds H, and subsequently the receiver asks H’V’, whatever the photon responds, we cannot guarantee that if asked HV once more, the photon is going to respond H again! It is “as if we could not know the colour and size of the flag with certainty at the same time”.
What is described in the preceding paragraph is an example of the Heisenberg principle of uncertainty, which in its original formulation limited the possibility of knowing with infinite precision the speed (analogous to HV) and the position (analogous to H’V’) of a particle. Only that in this context it is more digestible, because in some way we understand that the properties of the quantum particles do not accompany them, such as the red colour accompanying the red flag, but that they are choices of the observer upon measuring them.
In conclusion, in quantum physics, not only is the result of the measurement not necessarily defined before the receiver carrying out the measurement (principle of superposition), but the property itself is not defined before the receiver decides that it is precisely that property that he is going to measure.
The emitter can be nature itself, and the receiver identifies clearly with the observer, but also the environment of a quantum system behaves as a “receiver”, in other words, as a “collapser of quantum systems”.
What use is all of this?
Superposition, the capacity of a qubit to be a zero and a one at the same time, is fundamental for quantum computing. A qubit can calculate as if it were a 0 and as if it were a 1. A computer essentially computes a function f of a certain piece of information, coded in a number: given the coded information for example in the input 01101, of 5 bits, the computer computes f (01101). If we want to compute f for n inputs – n different pieces of information of 5 bits each – we have to apply n times the function f. However, if we manage to maintain the superposition of 5 qubits – all the possible inputs of 5 bits – with a single application of f, we will manage to get a superposition of all the possible results. Considering all of its possible states, we will obtain a computer operating simultaneously – with a single application of f– with 2^5 inputs. With 300 qubits, we would obtain a quantum computer operating with 2^300 inputs, a greater number than the number of all the atoms of the universe. Emerging from this capacity of maintaining and operating with this superposition is the so-called quantum parallelism. With this architecture, tasks can be carried out of such complexity that they would take a classical computer thousands of millions of years – the complexity of a task gives an idea of the execution time of the algorithm that provides its solution, and it is related with the number of operations necessary to implement that algorithm. A very complex task in that sense is, for example, factorising a large number. The RSA system of encryption in the Internet is based precisely on the complexity of factorising large numbers, in such a way that a quantum computer of a certain size would threaten world security at all scales. The quantum algorithm for quickly factorizing large numbers has existed since 1994 – formulated by Peter Shor – all that is lacking is to build a quantum computer that is big enough. But that is only a question of time… The good news is that quantum physics also offers a method of secure communication, and in this field even commercial solutions already exist.
The design of 100% secure cryptography systems is based on the choice of bases for preparation and measurement (choice of HV, H’V’, etc.). The principle of functioning of quantum cryptography is based on the fact that if the emitter codes in the HV base, and the receiver also measures in the HV base (the choice of measurement bases is the “secret code”), then any intromission by a spy with an H’V’ measuring base, different to that shared by emitter and receiver, will be detected because, as we have seen before, measuring on different bases alters the state of the photon. The protocol that permits emitter and receiver to create and share a secure cryptographic key following this principle was designed by Charles Bennett and Giles Brassard in 1984. It is called BB84.
There are many ways of making use of the peculiarities of quantum systems: teleporting, ultra-precise measurements, etc. Many of these with impressive experimental results, but they will have to be the subject of a different post.
To add a further trio to the 3 collection, here is the third “beautiful idea” of today, from the hands of Richard Feynman and Seth Lloyd. Since information is physical, and the physics that governs the universe is quantum physics, we can conceive all the events of the universe as information being processed by a giant quantum computer, which is the universe itself. The universe is, therefore, a quantum computer that computes itself, which implies that no smaller computer would have the capacity to compute the entire universe and therefore the best way of finding out the future is… waiting to see what will happen!
“Beautiful ideas”… what must they be made of?