## 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, which is a complex vector space with an inner product, often referred to as a Hilbert space. 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 Chapter 5.

### 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**.^{66}

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
^{67} as the exponential of some Hermitian matrix

### 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 preselected 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.

### 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