## 4.11 Quantum theory, formally

Even though multiplying and adding probability amplitudes is essentially all there is to quantum theory, we hardly ever multiply and add amplitudes in a pedestrian way. Instead, as we have seen, we neatly tabulate the amplitudes into vectors and matrices and let the matrix multiplication take care of multiplication and addition of amplitudes corresponding to different alternatives. Thus vectors and matrices appear naturally as our bookkeeping tools: we use vectors to describe quantum states, and matrices (operators) to describe quantum evolutions and measurements. This leads to a convenient mathematical setting for quantum theory: a complex vector space with an inner product (which is exactly a Hilbert space, since we only work in finite dimension). It turns out, somewhat miraculously, that this pure mathematical construct is exactly what we need to formalise quantum theory. It gives us a precise language which is appropriate for making empirically testable predictions. At a very instrumental level, quantum theory is a set of rules designed to answer questions such as “given a specific preparation and a subsequent evolution, how can we compute probabilities for the outcomes of such-and-such measurement”. Here is how we represent preparations, evolutions and measurements in mathematical terms, and how we get probabilities.

Note that we have already said much of the below, but we are summarising it again now in a more precise way, formally defining the mathematical framework of quantum theory that we use.

We also need to point out that a vital part of the formalism of quantum theory is missing from the following description, namely the idea of **tensor products**.
To talk about this, we need to introduce the notion of **entanglement**, and this will be the subject of the next chapter.

It is a very reasonable question to ask *why* this formalism (Hilbert spaces, unitary operators, the Born rule) is “the good one”.
One answer is that “it just works” — the calculations that we do in this framework give us answers which are in agreement with the results of physical experiments — but this can be rather unsatisfying as an answer.

Quite beautifully, it turns out that if we start from just *five* axioms, then we can prove that our choice of formalism is actually the only one that makes sense.
This is the result of L. Hardy’s “Quantum Theory From Five Reasonable Axioms”, arXiv:quant-ph/0101012.
We start by saying that a quantum system should be characterised by two integers: the number of degrees of freedom

*Probabilities.*Relative frequencies of observed outcomes from measuring an ensemble ofn systems tend to a well defined value, called the**probability**, whenn tends to infinity.*Simplicity.*The integerK is a function ofN , and takes the minimum possible value consistent with these axioms for eachN .*Subspaces.*If a system is such that its states all lie within anM -dimensional subspace (for someM<N ), then it behaves exactly like a system of dimensionM .*Composite systems.*Composite systems behave multiplicatively, i.e. if a system is a composite of two subsystemsA andB , thenN=N_AN_B andK=K_AK_B .*Continuity.*Given any two pure^{84}states of a system, there exists a continuous reversible transformation of the system that sends one to the other.

What is particularly nice, as a bonus result, is that if we make one tiny change to these axioms — just dropping the word “continuous” from the fifth axiom — then the result is exactly *classical* probability theory.

### 4.11.1 Quantum states

With any isolated quantum system which can be prepared in

### 4.11.2 Quantum evolutions

Any physically admissible evolution of an isolated quantum system is represented by a unitary operator.

Unitary operators describing evolutions of quantum systems are usually derived from the **Schrödinger equation**^{85}

This equation contains a complete specification of all interactions both within the system and between the system and the external potentials.
For time-independent Hamiltonians, the formal solution of the Schrödinger equation reads

### 4.11.3 Quantum circuits

In this course we will hardly refer to the Schrödinger equation.
Instead we will assume that our clever colleagues — experimental physicists — are able to implement certain unitary operations, and we will use these unitaries, like lego blocks, to construct other, more complex, unitaries.
We refer to pre-selected elementary quantum operations as **quantum logic gates** and we often draw diagrams, called **quantum circuits**, to illustrate how they act on qubits.
For example, two unitaries,

This diagram should be read from left to right, and the horizontal line represents a qubit that is inertly carried from one quantum operation to another (maybe through space, down a physical wire, but maybe through some other physical implementation — we don’t particularly mind!)

### 4.11.4 Measurements

A complete measurement in quantum theory is determined by the choice of an orthonormal basis

In general, for any decomposition of the identity