Pictures of lines

Thus far I have introduced the uncertainty principle and used it to demonstrate that quantum mechanics behaves quite oddly, to say the least, from the point of view of classical logic. What makes the uncertainty principle highly interesting for quantum theory is its almost complete independence of any physical details; it is indeed an inherent property of mathematics used to describe the quantum world, and is thus necessarily present in all physical systems. This, in fact, suggests that the uncertainty principle is a derivative property of something more fundamental and perhaps more elementary. Here we will leave the uncertainty principle behind and explore this idea by considering the geometry of the quantum state space; it turns out that a useful approach to shed some light on the logic of quantum mechanics is to draw lines.

Quantum states are represented by vectors in a vector space, or more precisely, by linear subspaces of the state space1. For the sake of visualisation, we will concentrate on a two-dimensional Euclidean space, which is a fancy name for the familiar coordinate plane. So in our case each state corresponds to a line in the plane passing through the origin (it could also be the case that the state is the origin or the entire plane). Here is an example of two states, let’s call them red and blue:


We should immediately ask, what are the suitable logical operations for ‘and’ and ‘or’? Since the lines represent the physical states, we want to be able to say something like ‘The system is in the blue and red state’ as well as ‘The system is in the blue or red state’2. We will start with ‘and’ as it is perhaps more intuitive.

For the states represented by lines, ‘and’ is defined simply as the intersection of the lines3. Recall that ‘and’ is represented by the symbol \land. That is, to say that the system is in ‘blue and red’ state is to find where on our plane the state is simultaneously blue and red. Clearly, there are two possibilities here; firstly, if the lines happen to coincide (i.e. they were in fact the same line) the ‘and’ operation is redundant and the intersection is just the original line; in general, however, the lines will be different and will intersect only at the origin. In our example case this looks like:


One could think that the definition of ‘or’ is as simple as that of ‘and’; it is tempting to define ‘or’ as the union of the two lines, the system is either in the blue state or in the red one. This naive definition is, however, implausible both mathematically and physically. Mathematically, the union of two lines is no longer a linear subspace of the plane; indeed, we agreed that each state is either a line, the origin or the entire plane, but the union of two lines is some kind of skewed cross. Thus this definition of ‘or’ does not preserve the structure we started with. Physically, the definition has even more catastrophic consequences; it implies that measuring the state should yield either ‘blue result’ or ‘red result’, depending on which state we happen to measure. We know, however, that this is not the case, instead the measurement will result in some linear combination (i.e. some mixture of both coordinates) of ‘blue’ and ‘red’, this is the so called ‘principle of superposition’. This principle, in fact, already suggests the appropriate definition of ‘or’.

‘Or’ in our linear representation is defined as ‘the smallest linear subspace of the plane containing both lines’. Recall that or is represented by the symbol \lor. Again, two things could happen, the first one being the redundant case when the two lines coincide and nothing happens, and for two distinct lines their ‘or’ is the entire plane. This definition overcomes both problems mentioned above; it guarantees that the resulting state is a linear subspace and it accounts for superposition. Thus for the lines we started with, the ‘or’ operation looks like:


Equipped with these logical operations, let us reconsider the distributive law once again. For that, let’s add a third line, called green, to our plane:


Recall that the classical distributive law asserts:

\textrm{green} \land (\textrm{blue} \lor \textrm{red}) = (\textrm{green} \land \textrm{blue}) \lor (\textrm{green} \land \textrm{red}) 4

Now, using the definitions of ‘and’ and ‘or’ as above, let us figure out visually how both sides of this equality look like. These will turn out to be quite different.

We start from the left-hand side: green and (blue or red). We already know what (blue or red) is from one of the pictures above, it is simply the entire plane. Thus we need the intersection of the green line with the plane, which is just the green line itself:


For the right-hand side, both terms in brackets are intersections of two distinct lines; (green and blue) and (green and red). Thus they are both equal to the origin, as before. Now the result is the smallest linear subspace containing the origin, hence just the origin itself:


We have therefore managed to show that the distributive law does not hold for these definitions of ‘and’ and ‘or’. What is fascinating about this example is its similarity to the one with ‘quantum cyclist’ discussed last time, which required a complicated construction and the uncertainty principle. This visual approach demonstrates that non-distributivity of quantum logic is really a consequence of geometry of vector spaces.

1 mathematically state space is a Hilbert space

2 or the probabilities of these in the case of quantum physics

3 more precisely, intersection of the linear subspaces of the Hilbert space

4 For the intuition behind this, see the beginning of the previous post or the very end of the first post


Quantum physics against intuition – Part II

In the first post we discussed the fact that classical first-order logic is distributive, that is, pizza and (lemonade or water) is the same as (pizza and lemonade) or (pizza and water); or symbolically,

x \land (y \lor z) = (x \land y) \lor (x \land z) .

This time the aim will be to come up with an example demonstrating that this very intuitive identity does not always hold in quantum mechanics. To do that, we will need the uncertainty principle discussed in the previous post.

Quantum cyclist

We are going to use the uncertainty principle for position and momentum to construct a system which does not obey the distributive law. To make the numbers a bit simpler, we take \hbar=1, so the uncertainty relation looks like:

\Delta X \Delta P\geq \frac{1}{2} .

Recall the example with a cyclist from the first post, we observed that the cyclist being in some interval and having some velocity is the same as the cyclist being in the first half of the interval with the same velocity or the cyclist being in the second half of the interval with the same velocity. Now consider a (tiny!) quantum cyclist; for concreteness, suppose the cyclist is in the interval [0,1] and has the momentum in the interval [0,\frac{1}{2}]. For simplicity, we take the uncertainty to be the length of the interval1, so we are saying that the cyclist is equally likely to be anywhere between 0 and 1 and is equally likely to have any momentum between 0 and \frac{1}{2}. Hence we have \Delta X = 1 and \Delta P = \frac{1}{2}. Now let x, y and z be the following statements about our system (i.e. about the cyclist):
x = ‘cyclist has the momentum in [0,\frac{1}{2}]
y = ‘cyclist is in [0,\frac{1}{2}]
z = ‘cyclist is in [\frac{1}{2},1]‘.
The distributive law is:

x \land (y \lor z) = (x \land y) \lor (x \land z) .

Note that the left-hand side of this identity is precisely what we have described above; the cyclist is in [0,1] with momentum in [0,\frac{1}{2}]. We calculate \Delta X\Delta P = 1\cdot \frac{1}{2} = \frac{1}{2} , which satisfies the uncertainty condition, and so the system is physically possible. On the right-hand side, however, we have (x \land y), that is, the cyclist is in [0,\frac{1}{2}] with momentum in [0,\frac{1}{2}], giving both \Delta X and \Delta P as \frac{1}{2}. But this violates the uncertainty bound, since \Delta X\Delta P = \frac{1}{2}\cdot\frac{1}{2} = \frac{1}{4}, which is certainly smaller than \frac{1}{2}! Since (x \land z) gives the same uncertainties, we must conclude that both terms on the right-hand side are physically impossible, and thus false. This makes all of the right-hand side false; we must, therefore, conclude that this identity cannot hold in this case, as it equates a true statement about the physical system with a false one.

The example above raises many questions for classical logic. Must we conclude that its axioms and rules of inference don’t always hold? If yes, what would be the axioms, and how would they account for the fact that classical logic is distributive? If no, how do we account for the anomaly described above? It is not even clear if there should be one formal system of reasoning flawlessly applicable in all situations to all possible systems. No matter the answers to these questions, the example certainly opens up the space for development of a formal system correctly describing the logic of quantum mechanics.2

1This is actually not quite correct, e.g.  \Delta X should really be \frac{1}{\sqrt{12}}. We can, however, get the uncertainties we want by scaling the intervals accordingly, but this doesn’t really contribute to the understanding, and so we drop the scaling for clarity.

2For further reading, see

A very brief introduction to logic

Imagine you are walking to a friend’s house. When you reach the right street, you realise you are not sure about the house number; you know that it is either 23 or 24, but can’t remember which one. You also remember that the house is on the left side of the road. Fortunately, you notice that the houses to the right of you all have even numbers, and the ones to the left correspondingly odd. Hence, provided that what you remember is correct, you have enough information to ring the right doorbell without having to guess. Let us break down the possible reasoning going on here:

(1) The friend’s house number is either 23 or 24.
(2) The friend’s house is on the left.
(3) All the houses on the left have an odd number.
(4) The friend’s house has an odd number. (By (3) and (2))
(5) 24 is not odd.
(6) The friend’s house is not 24. (By (4) and (5))
(7) The friend’s house is 23. (By (1) and (6))

What is remarkable about this reasoning is that if (1), (2), (3) and (5) are all true, then so must be (7), that is, the reasoning is truth-preserving. There is, of course, nothing special about houses and numbers, we could equally replace all the words by something which doesn’t even make much sense, for instance:

(1) My foot is either pink or ultramarine.
(2) My foot is stolen.
(3) All stolen feet are liked by the Holy Frog.
(4) My foot is liked by the Holy Frog. (By (3) and (2))
(5) No ultramarine foot is liked by the Holy Frog.
(6) My foot is not ultramarine. (By (4) and (5))
(7) My foot is pink. (By (1) and (6))

Formally, this is still a perfectly valid piece of reasoning. This, of course, by no means implies that the conclusion ‘My foot is pink’ is true; in this case at least (1) and (2) are certainly false, so the conclusion need not to be true. If we learn anything at all from this exercise, it is the following crucial observation; what makes the reasoning correct is its form rather that the content. This is the basic description of philosophical logic, it tries to capture the correct forms of reasoning, by correct we mean truth-preserving here. Formalisation of these rules for reasoning leads to the so called first-order logic.

While the field of mathematical logic is not limited to the first-order logic1, it is the most common one to consider in mathematics, philosophy and computer science. The reason for this is simple; first-order logic captures the reasoning we are familiar with in our everyday life. This kind of logic is intuitive and understandable for us; we are, in fact, as illustrated by the example above, constantly using this kind of inference rules without putting in much effort or paying attention to it. This is the reason why a person who starts learning programming doesn’t need to learn the ‘rules of logic’ first, they are more or less hard-wired in our interaction with the environment.

First-order logic consists of terms, sentential formulas, logical operators and quantifiers. In addition to these, one has to define a well-formed sentence and the inference rules2.  The terms can be thought of as elements of a set denoted by small letters a, b, c, ... ; a term is made into a sentential formula by specifying a property it has, that is, the set it belongs to, these are denoted by capital letters. For example, Hx means ‘x has the property H‘. The logical operators are \neg (not), \land (and), \lor (or) and \Rightarrow (implication). Finally, the quantifiers of first-order logic are \forall (for all) and \exists (there is). Using this formal language, we can now express symbolically the argument given in the example above:

(1) Ty \lor Fy
(2) Ly
(3) \forall x[Lx\Rightarrow Ox]
(4) Oy
(5) \forall x[Fx\Rightarrow \neg Ox]
(6) \neg Fy
(7) Ty

Where we have denoted: y = ‘the friend’s house’, T = ‘has number 23’, F = ‘has number 24’, L = ‘house is on the left’, O = ‘has an odd number’. Thus for example, (3) reads as ‘All houses on the left have an odd number’. To infer (4) from (3) and (2) we first use the inference rule \forall x P(x) \rightarrow P(y) to get Ly\Rightarrow Oy, which together with (2) and the inference rule A, A\Rightarrow B \rightarrow B gives (4). We can now explicitly see that it doesn’t matter what the letters above stand for, the given argument is true because we can justify each step by the inference rules of first-order logic.

The metaphysical status of first-order logic is an interesting question in philosophy, it can be summarised as ‘What makes logical truths true?’ The suggested answers to this include logical realism, asserting that logic is a property of the reality and the logical truths thus tell us something meaningful about the reality itself; and logical formalism, according to which logical truths do not as such provide any new information about how things are in the world, rather, it is their form which makes them necessarily true.3 While the latter view may sound appealing in the light of the previous example, as we are about to see, there is something about logic which seems to carry along some of our assumptions about the physical reality.

Since the inference rules of first-order logic are truth-preserving, if we start with a set of true statements about a physical system, everything we infer from this set of statements using those inference rules will also be true about the physical system in question. Or at least if this is not the case, we have a serious problem with either our physical understanding of the system, or with our logic. To illustrate this, consider the following example. We are told that a cyclist is somewhere on a road, and has a speed of 30 km/h. It is immediately obvious to us that this is equivalent to: ‘either the cyclist is on the first half of the road with the speed of 30 km/h, or the cyclist is on the second half of the road with the speed of 30 km/h’. This rephrasing is in fact so trivial that you could (rightfully) complain that this is pointless and I am just doing it to overcomplicate things. It does, however, illustrate a more general property of classical logic called distributivity. Using the formal notation defined above, distributivity can be expressed as:

\displaystyle x \land (y \lor z) = (x \land y) \lor (x \land z)

What makes this obvious property of classical logic extremely fascinating is that it no longer holds in general for quantum mechanical systems.

1One can indeed come up with an entire zoo of exotic logics in mathematics, the Wikipedia entry has a good overview of these,
2For a full list of properties defining first-order logic, see
3For further reading, see