1.1.1 Basic definitions and notations

We let 𝔽 denote either the real field ℝ or the complex one ℂ. For a complex number z = x+iy ∈ℂ, with x,y ∈ℝ, we write the conjugate z^∗ := x−iy. We let ℳ_m,n(𝔽) denote the space of matrices of dimension m × n with entries in 𝔽 and ℳ_n(𝔽) whenever m = n. For A := (a_ij)_{1≤i≤m; 1≤j≤n} ∈ℳ_m,n(𝔽), A^∗ := (a_ij^∗)_{1≤i≤m; 1≤j≤n} is the complex conjugate. If A ∈ℳ_n(𝔽), we write A^⊤ for its transpose and A^† := (A^∗)^⊤ for its Hermitian conjugate. We finally denote I the identity matrix and write I_n whenever we wish to emphasise the dimension, and 0_m,n the null matrix in ℳ_m,n(𝔽). Recall that a matrix A ∈ℳ_n(𝔽) is invertible (or non-singular) if there exists B ∈ℳ_n(𝔽) such that AB = BA = I_n. Given two matrices A ∈ℳ_p,m(𝔽) and B ∈ℳ_q,n(𝔽), we define their tensor product as

Since a vector is a particular case of a matrix, for u ∈𝔽^m and v ∈𝔽ⁿ, we can write

1.1.2 Inner products

A vector space V over the field 𝔽 is a set endowed with

• a commutative, associative addition operation,
• an operation of multiplication by a scalar.

The addition and the multiplication by a scalar have the following properties (for scalars α,β ∈𝔽 and vectors u,v ∈V):

• v + 0 = v;
• v + (−v) = 0;
• α(βv) = (αβ)v;
• (α + β)v = αv + βv;
• α(u + v) = αu + αv;
• 1 ⋅ v = v.

Armed with this, we can now define an inner product on V:

For example, the following spaces carry a natural inner product:

• The vector space ℂⁿ with the inner product ⟨u,v⟩ := u^†v = ∑ _i=1ⁿu_i^∗v_i;
• The space of complex-valued continuous functions on [0,1] with ⟨f,g⟩ := ∫ ₀¹f(t)^∗g(t)dt;
• If X,Y ∈ℳ_m,n(ℝ), then ⟨X,Y⟩ := Tr(X^⊤Y) = ∑ _i=1^m ∑ _j=1ⁿX_ijY_ij defines an inner product on the space of (real) matrices.

Projection matrices are particularly useful for geometric purposes:

In particular, if W is a vector subspace of 𝔽ⁿ with some orthonormal basis (w₁,…,w_d), it is then easy to check that the map 𝒫_W : 𝔽ⁿ →𝔽ⁿ onto W satisfying

defines an orthogonal projection.

1.1.3 From linear operators to matrices

Let V be a finite-dimensional vector space over 𝔽 and ⟨⋅,⋅⟩ a non-degenerate inner product on V. Given a linear operator 𝒜 : V →V, then, by the Riesz representation theorem  [309, Section III-6], there exists a unique linear operator 𝒜^† : V →V, called the adjoint operator, such that

Indeed, for any v ∈V, the map u ∈V⟨𝒜u,v⟩ is a linear functional, hence an element of the dual space V^† (the space of bounded linear functionals on V), therefore for each v ∈V, there exists v′∈V such that ⟨𝒜u,v⟩ = ⟨u,v′⟩. It is then easy to show that the map vv′ is linear, proving that the adjoint operator is uniquely defined. In the particular case where 𝒜 = 𝒜^†, the operator 𝒜 is called Hermitian, a key requirement in quantum mechanics:

For a Hermitian operator 𝒜, we then have, for any u ∈V,

by conjugate symmetry (Definition 1), and therefore ⟨𝒜u,u⟩ is real. Conversely, if ⟨𝒜u,u⟩ is real, then

Therefore, = 0; since this is true for all u ∈V, then 𝒜 = 𝒜^†.

The following property of operators shall be useful to ensure that systems driven by operators preserve distances, or norms:

Definition 4. The linear operator 𝒜 : V → V is called unitary if it is surjective and

Recall that a linear operator between two finite-dimensional normed spaces is bounded, and therefore continuous. For any u ∈V, this implies that ∥𝒜u∥ = ∥u∥, so that a unitary operator 𝒜 preserves the norm. In that case, 𝒜 is an isometry, therefore injective. Being also surjective, it is bijective and therefore its inverse exists. For a unitary operator 𝒜 and any u,v ∈V, we have

by definition of the adjoint, implying that

where ℐ is the identity operator.

Example (Real Matrices): If V = ℝⁿ with inner product ⟨u,v⟩ := u^⊤v for u,v ∈ℝⁿ, the linear operator 𝒜 can now be viewed as a matrix A in ℳ_n(ℝ). Its adjoint is nothing other than the transpose A^⊤, and therefore A is self-adjoint if and only if it is symmetric. In this case, if A is unitary (or orthogonal), then it is invertible with A⁻¹ = A^⊤. Rotation matrices in ℝ², which will play an important role later when constructing quantum circuits, are the only unitary maps of ℝ² onto itself and are of the form

for 𝜃 ∈ [0,2π) and δ ∈{−1,+1}.

Example (Complex Matrices): If V = ℂⁿ with inner product ⟨u,v⟩ := v^†u for u,v ∈ℂⁿ, the linear operator 𝒜 can now be viewed as a matrix in ℳ_n(ℂ). The adjoint of such a matrix is then the Hermitian conjugate A^† and A is called Hermitian if A = A^† and unitary if A^†A = I_n. We shall denote by 𝒰_n(ℂ) the set of unitary matrices in ℳ_n(ℂ). We will discuss Hermitian matrices over ℂ in more detail in Section 1.1.6.

1.1.4 Condition number

In order to manipulate matrices and measure them, we require matrix norms:

The condition number of a matrix is an important tool to understand the stability of linear equations of the form Ax = b, for A ∈ℳ_n(𝔽), b ∈𝔽ⁿ. Assuming A to be non-singular, the true solution is clearly x_∗ := A⁻¹b. Suppose, however, that the vector b is only known up to some (not necessarily quantum) measurement error, and one observes instead b + Δ_b. The solution is then A⁻¹(b + Δ_b) = x_∗ + Δ_x, with Δ_x := A⁻¹Δ_b. In particular, we can write, for any (sub-multiplicative) matrix norm ∥⋅∥,

From this inequality, we see that the quantity ∥A⁻¹∥∥A∥ bounds the relative error in the solution with respect to the relative error in the measurement of the input vector b. This leads to the following terminology:

Definition 6. Given a matrix A ∈ ℳ_n(𝔽) and a sub-multiplicative norm ∥⋅∥, we call

the condition number (with respect to the norm ∥⋅∥) of the matrix A (and assign to it infinite value if A is singular).

Remark: The definition of the condition number above holds for any matrix norm ∥⋅∥, but admits a more explicit representation in the particular case of the spectral norm ∥⋅∥₂, defined as

where ∥x∥₂ := is the L ₂ norm for vectors. If the matrix A is not singular, then

where λ_max(A) and λ_min(A) denote the largest and smallest eigenvalues of A.

1.1.5 Matrix decompositions and spectral theorem

Having defined essential properties of (complex) matrices, we now introduce several essential tools that allow us to gain a better understanding of their properties.

The Singular Value Decomposition is a key tool to analyse the properties and behaviours of matrices. It is ubiquitous in applied statistics and machine learning and allows us to reduce the explanatory dimension of a large matrix into a small number of meaningful components.

The numbers {σ₁,…,σ_p} are called the singular values of A and are uniquely defined. The columns of U and V are the left-singular and right-singular vectors of A, in the sense that, if σ ∈{σ₁,…,σ_p}, then there exist a column u of U and a column v of V such that Av = σu and A^†u = σv. Recall that the rank of a matrix is defined as the dimension of the span of its columns. As a corollary of the Singular Value Decomposition theorem, the rank of a matrix is therefore equal to the number of non-zero singular values. The Singular Value Decomposition is general in the sense that it holds for any matrix. In the particular case of square matrices, the Schur decomposition and the Spectral Theorem provide refinements.

The Spectral Theorem is a cornerstone result in the theory of linear operators, and in particular for (finite-dimensional) matrices. Recall that an operator 𝒜 : V →V is called normal if it commutes with its adjoint, namely if 𝒜𝒜^† = 𝒜^†𝒜. Self-adjoint (or Hermitian) operators are clearly normal, yet the converse is not true in general. Recall further that an eigenvector of 𝒜 is a non-zero vector u ∈V such that 𝒜u = λu for some λ ∈ℂ, and we denote by σ(𝒜) the set of eigenvalues of 𝒜.

The following result, which is more general than the subsequent spectral theorem, allows us to decompose any arbitrary complex square matrix.

Note that since U is unitary, then U⁻¹ = U^†. We call the matrix T the Schur transform of A and the identity in the theorem means that A and T are similar, so in particular, possess the same eigenvalues, all located on the diagonal of T. If A is a normal matrix, then so is T, and therefore T must be diagonal and we write T = D for clarity. In this case, we say that the matrix A is diagonalisable with A = UDU^†, where the diagonal entries of D are the eigenvalues of A and the column vectors of U are the orthonormal eigenvectors of A.

For each eigenvalue λ ∈ σ(𝒜), denote the corresponding eigenspace

Since the vector space V is the orthogonal direct sum of the eigenspaces (indexed by the eigenvalues of 𝒜), we can then write the spectral decomposition

where 𝒫_λ is the orthogonal projection operator onto 𝒱_λ. Note that such an operator is naturally self-adjoint  [309, Theorem 2, Section III-1].

1.1.6 Hermitian matrices

We introduced above Hermitian matrices as the set of matrices A over the complex field ℂ such that A = A^†. As fundamental building blocks of quantum computing, we investigate their properties further. Clearly, a real matrix is Hermitian if and only if it is symmetric, in which case A^⊤ = A.

The Singular Value Decomposition (Theorem 1) takes a particular flavour in the case of Hermitian matrices:

Proof. The Spectral Theorem shows that there exist a unitary matrix U ∈𝒰_n(ℂ) and a diagonal matrix Σ ∈ℳ_n(ℂ) such that A = UΣU^†, where the diagonal elements of Σ are the eigenvalues of A. Assuming (i), we can define B = UU^†∈ℳ_n(ℂ). Then clearly

since U is unitary. The equality A = B^†B is also obvious. The latter implies that

Finally, assume (iv) and let λ be an eigenvalue of A with eigenvector u. Then

Since the latter is strictly positive, then clearly λ ≥ 0.

The following property lies at the core of Hamiltonian simulation of quantum systems:

Recall that for a matrix A ∈ℳ_n(ℂ), its exponential is given by

In practice, though, given a Hermitian matrix A, finding the corresponding unitary matrix U is not easy. The Hamiltonian simulation problem is defined as follows.

Hamiltonian Problem: Given a Hermitian matrix A ∈ℳ_n(ℂ), a time t > 0, a tolerance level 𝜀 > 0, and some matrix norm ∥⋅∥, find a unitary matrix U such that ≤ 𝜀.

1.1.7 Rotation matrices

Rotation matrices, and later their quantum gate equivalents will play a key role in building quantum circuits. Let us start with the following lemma:

Lemma 1. If a matrix A ∈ℳ_n(ℂ) is such that A² = I, then for any 𝜃 ∈ℝ,

Proof. This follows directly from the series expansion

which has an infinite radius of convergence. □

Lemma 1 will prove essential for computational purposes. As simple examples, consider the following:

Exercise: Compute e^i𝜃A for A ∈{X,Y,Z} and 𝜃 ∈ℝ, where

For any α ∈ [0,2π), consider now the map ℛ_α : ℝ² →ℝ² such that

which is basically a rotation of angle α and does not affect the norm of the input vector. To the map ℛ_α, we can associate the (rotation) matrix R_α such that ℛ_α(u) = R_αu for any u ∈ℝ². It is easy (exercise) to show the following:

Lemma 2. The matrix R_α has the form

This representation is the general form of a rotation matrix in ℝ² (introduced in (1.1.3)).

Exercise: Write the matrices e^i𝜃A for A ∈{X,Y,Z} from the previous exercise as rotation matrices.

1.1.8 Polar coordinates

Recall that a point z = x + iy, with x,y ∈ℝ, lying on the unit circle can be written as z = e^i𝜃 with 𝜃 ∈ [0,2π). Indeed, simply let x = r cos(𝜃), y = r sin(𝜃) and add the constraint r = 1. Consider now a general vector u ∈ℂ² of the form

with α,β ∈ℂ such that |α|² + |β|² = 1. Here, (e₁,e₂) forms a basis of ℝ²:

In polar coordinates, we can then write

Note that arbitrary multiplication phases have no influence – a fact of key importance in quantum mechanics – because, for any γ ∈ℝ,

so that in fact, multiplying u by the global phase e^−i𝜃_α and letting 𝜃 := 𝜃_β −𝜃_α, we consider

Write temporarily r_βe^i𝜃 = x + iy. Insisting on u being on the unit sphere further imposes ∥u∥² = 1, namely

1	= ∥u∥2 = rαe1 + (x + iy)e2†rαe1 + (x + iy)e2
	= rαe1⊤ + (x − iy)e 2⊤rαe1 + (x + iy)e2
	= rα2 + x2 + y2,

since (e₁,e₂) is orthonormal. This is nothing more than the equation of the unit sphere. In polar coordinates, we can write

and clearly r = 1 since we are on the unit sphere. Therefore

u	= cos(𝜃)e1 + sin(𝜃)cos(ϕ) + isin(𝜃)sin(ϕ)e2
	= cos(𝜃)e1 + sin(𝜃)eiϕe 2.

1.1.9 Dirac notations

Given a vector v ∈ℂⁿ, Dirac’s ket and bra notations read

With these notations, the operation := ⟨u|v⟩ defines an inner product on ℂⁿ. The notation for the standard orthonormal basis in  ℂⁿ is ()_{i=0,…,n−1}, i.e.,

In coordinates, we can write, for any u,v ∈ℂⁿ,

and therefore,

1.1.10 Quantum operators

In the language of Dirac’s notations, we can define the outer product ⟨v| (for u ∈U and v ∈V) as a linear operator from V to U, two vector spaces, as

In particular, ⟨v| is the projection on the one-dimensional space generated by . Any linear operator can be expressed as a linear combination of outer products as

where and are the standard basis vectors (1.1.9).

Similarly to the linear algebra setting above, we can define an eigenvector of a linear operator 𝒜 : V →V as a non-zero vector such that

for some complex eigenvalue λ. Associated with any linear operator 𝒜, the adjoint operator 𝒜^† satisfies

Indeed, in the language of linear operators above, we have

by definition of the inner product (Definition 1).

1.1.11 Tensor product

Given two vector spaces U and V of dimensions m and n, the tensor product U ⊗V is a vector space of dimension mn. For u ∈U and v ∈V, we can form the vector := ⊗∈U ⊗V with the following properties:

• = + , for any u′∈U;
• = + , for any v′∈V;
• α = = , for any α ∈ℂ.

Given the linear operators 𝒜 : U →U and ℬ : V →V, we can then define their tensor product as an operator 𝒜⊗ℬ on U ⊗V:

which can be represented in matrix form as A ⊗ B ∈ℳ_mn,mn(ℂ).

This Dirac formalism, fully anchored in (classical) linear algebra, now opens the gates to a proper dive into the foundations of quantum mechanics.

1.2.1 First postulate – Statics

Postulate 1. Associated to any physical system is a complex inner product space known as the state space of the system. The system is completely described at any given point in time by its state vector, which is a unit vector in its state space.

What is the importance of the first postulate from the quantum computing point of view? The answer is that quantum mechanics offers us a straightforward generalisation of the classical binary digit (bit). The classical bit is a two-state system with controlled transitions between them. As an example, we can use an electrical switch that can exist in one of the two discrete, stable states ("on" and "off"). Although electrical switches may seem an odd physical realisation of bits in the age of transistors, they illustrate an important point about computation in general: it is substrate independent. Exactly the same computational results can be obtained using electrical relays and CMOS transistors.

The quantum mechanical version of a bit, called a quantum binary digit (qubit), is a quantum mechanical two-state system. The first postulate of quantum mechanics tells us that the state of such a system can be represented mathematically by a unit vector in the two-dimensional complex vector space. This also means that such a system can exist in a superposition of basis states. Indeed, any vector in the two-dimensional complex vector space,

can be represented as a linear combination of the standard basis vectors:

Since the state vector is a unit vector, the coefficients α and β must satisfy

The coefficients α and β are probability amplitudes. Even though a qubit can exist in a superposition of basis states, once measured (see Postulate 3), its state collapses to one of the basis states: |α|² and |β|² give us the probability of finding the qubit, respectively, in states  and  after measurement.

One can draw an analogy with how the space of natural numbers, ℕ, can be extended to the space of real numbers, ℝ, and then to the space of complex numbers, ℂ. We have a much wider range of functions that can operate on and take values in ℝ and ℂ than in ℕ. Similarly, allowing the two-state system to exist in a superposition of states significantly extends the range of possible operators that can transform such states (i.e., perform computation).

For example, there is no Boolean function f that, when applied twice to a classical bit, would result in a NOT gate: f(f(0)) = 1 and f(f(1)) = 0. But there is such an operator in quantum computing. We can easily verify by direct calculations that the matrix

applied twice to the basis vector would transform it to the basis vector , and applied twice to the basis vector would transform it to the basis vector . M is an example of a quantum logic gate – an operator that transforms the state of a qubit, thus implementing the computation.

Remark: The state space of a physical system can be infinite-dimensional. The quantum computing paradigm based on infinite-dimensional Hilbert spaces is called continuous-variable quantum computing, which is realised in, e.g., some photonic quantum computing systems. However, in the context of digital quantum computing, we will restrict our analysis to finite-dimensional state spaces.

1.2.2 Second postulate – Dynamics

Postulate 2. The time evolution of a closed quantum system is described by the Schrödinger equation

where ℏ is Planck’s constant and ℋ is a time-independent Hermitian operator known as the Hamiltonian of the system.

The Hamiltonian of a quantum system is an operator corresponding to the total energy of that system, and its eigenvalues are the possible energy levels of the system. The knowledge of the Hamiltonian provides all the necessary information about system dynamics.

In the Schrödinger equation (1.2.2), the state of a closed quantum system at time t₁ is related to the state at time t₂ by a unitary operator 𝒰(t₁,t₂) that depends only on t₁ and t₂ via

where 𝒰(t₁,t₂) is obtained from the Hamiltonian ℋ as

Unitary operators preserve the inner product (and therefore norms, lengths, and distances), which means that for two vectors  and , if 𝒰 is a unitary operator, then the inner product between 𝒰 and 𝒰 is the same as the inner product between and :

A unitary operator is a complex generalisation of a rotation: unitary operators take an orthonormal basis to another orthonormal basis, and any operator with this property is unitary. In quantum mechanics, physical transformations such as rotations, translations and time evolution correspond to maps that take quantum states to other quantum states. These maps should be linear and preserve the inner product. This allows us to look at the unitary operators as the quantum logic gates implementing quantum computation protocols. Furthermore, unitary operators are invertible, a key property that ensures that quantum computing is reversible.

1.2.3 Third postulate – Measurement

Given a Hermitian operator 𝒜, the spectral theorem implies that the state of a system can be written as a superposition

where the coefficients (α_i)_i=1,…,N are complex probability amplitudes, assumed to be normalised with ∑ _i=1^N|α_i|² = 1, and where ()_i=1,…,N are eigenfunctions of 𝒜. The measurement postulate then reads as follows:

Postulate 3. If we measure the Hermitian operator 𝒜 in the state given in (1.2.3), the possible outcomes for the measurement are the eigenvalues (λ_i)_i=1,…,N of 𝒜, and the probability p_i to measure λ_i is given by p_i = |α_i|². After the outcome λ_i, the state of the system becomes

An immediate measurement in the same computational basis will deliver the same result without any uncertainty.

The quantum measurements are described by measurement operators (𝒫_i)_i=1,…,N, acting on the state space of the system with N possible outcomes. If the state of the system is before the measurement, then the probability of outcome i is

The measurement operators should also satisfy the completeness condition

where ℐ is the identity operator. This ensures that the sum of the probabilities of all outcomes adds up to 1.

These measurement operators are linear but not unitary. From the quantum computing perspective, we are interested in measurement operators that are projections (Definition 2) onto the computational basis, such as the standard orthonormal basis given by (1.1.9).

For example, the measurement operators for a single qubit can be defined as

We can easily verify that 𝒫₀² = 𝒫₀ and 𝒫₁² = 𝒫₁, as should be the case for projection operators, and that the completeness condition (1.2.3) is satisfied. If the qubit is in state = α + β, then the measurement operator 𝒫₀ will give us with probability |α|², and the measurement operator 𝒫₁ will give us with probability |β|². Indeed,

𝒫0	= ⟨0|α + β = α⟨0| + β⟨0| = α,
𝒫1	= ⟨1|α + β = α⟨1| + β⟨1| = β.

The measurement postulate of quantum mechanics states that an immediate measurement in the same computational basis will deliver the same result without any uncertainty. The key words here are "the same computational basis". What would happen if the subsequent measurement is performed in another basis (the basis specified by another set of linearly independent unit vectors from the state space)? For example, assume that the qubit is in state

Measuring in {,} computational basis will result in observing states  and  with equal probability 1∕2. Let us assume that we measured . The qubit state is now

If we repeat the measurement in the same {,} computational basis, we obtain state  with probability 1 in accordance with the measurement postulate. However, had we measured state in the Hadamard basis {,}, given by

we would have equal probabilities of and outcomes. Let us assume that we measured and the state of the qubit is now

If we repeat the measurement of state in the Hadamard basis {,}, we obtain state with probability 1. But the state of the qubit is an equal superposition of states and from the {,} computational basis perspective and we have an equal chance of measuring either or in this basis.

Remark: The basis vectors and that form the standard computational basis can be transformed into the basis vectors and that form the Hadamard basis by applying the following unitary operator (rotation), called the Hadamard gate:

Chapters 6, 10 and 11 provide examples of applications of the Hadamard gate.

The measurement plays a crucial role in quantum computing. This is the process of collapsing a quantum state and reading out the classical information: measuring qubits encoding a quantum state will produce a classical bit string. The measurement process generates probabilistic outcomes. Therefore, we need to perform measurements on the same quantum state multiple times to generate a sufficiently large number of classical bit strings to produce reliable statistics.

1.2.4 Fourth postulate – Observable

Postulate 4. For every measurable property of a physical system, there exists a corresponding Hermitian operator. The values of the physical observables correspond to the expectation values of Hermitian operators. The expectation value of the Hermitian operator 𝒜 in the normalised state  is given by

Let us consider the general case where the expectation value of a Hermitian operator 𝒜 is calculated in state , which is not an eigenfunction of 𝒜. By the Spectral Theorem 3 (see also (1.2.3)), the state  of a system can be represented as the superposition

where ()_i=1,…,N are the eigenfunctions of 𝒜 and (α_i)_i=1,…,N the corresponding probability amplitudes.

Therefore, the expectation value of 𝒜 in state , given in (1.2.4), is calculated as

where (λ_i)_i=1,…,N are the eigenvalues of 𝒜. The only terms that survive in the expression for  are those with i = j due to the orthogonality of the eigenfunctions, so that

Therefore, the value of the observable is a weighted average of the eigenvalues of the corresponding Hermitian operator. The weights are the coefficients (|α_i|²)_i=1,…,N, which are the probabilities of measuring the corresponding eigenstate of 𝒜.

1.2.5 Fifth postulate – Composite System

Postulate 5. The state space of a composite physical system is the tensor product of the state spaces of the individual component physical systems.

If the first component physical system is in state and the second component physical system is in state , then the state of the combined system, , is given by the tensor product

Not all states of a combined system can be separated into the tensor product of states of individual components. If the state of a system cannot be separated into component parts, we say that the component parts are entangled.

The entanglement of quantum systems is one of the major sources of computational power of quantum computing. It allows us to store exponentially more information in the correlations between the states of individual subsystems (in the limit – individual qubits) than directly in the states of individual subsystems.

To illustrate this point, we can look at the number of probability amplitudes needed to describe the state of an n-qubit system. An individual qubit can be found in one of the two possible states after measurement – one of the two basis states, or . This means that we need to specify two probability amplitudes to fully describe the state of the qubit before measurement. If all our qubits are independent and the state of the system can be represented as a tensor product of individual qubit states,

then we need to specify 2n probability amplitudes (two for each individual quantum states) to describe the state of the system. If, however, all individual qubits are entangled and the tensor product representation of the system state does not exist, we need to specify 2ⁿ probability amplitudes – this is an effective measure of useful information that can be stored in the system.

1.3.1 Density matrix

Let us start with the state of a combined two-component physical system given by (1.2.5). Let ()_i=1,...,N and ()_j=1,...,M denote, respectively, the standard orthonormal bases of the Hilbert spaces of systems A and B:

where (α_i)_i=1,...,N and (β_j)_j=1,...,M are some probability amplitudes. The states that allow the state vector representation (1.3.1) are called pure states. In this case, the state of the combined system is

However, in general, the state of the combined system would look like

where γ_ij are probability amplitudes that may not necessarily be factorised as the product of probability amplitudes (α_i)_i=1,...,N and (β_j)_j=1,...,M. If γ_ij cannot be factorised as α_iβ_j, then the component systems A and B are entangled and their states cannot be represented by the state vectors (1.3.1). Such states of systems A and B are called mixed states.

The more general setup is that of an ensemble of states of the form {p_k,}_k=1,…,N, where each  is a quantum state whose wavefunction is known with certainty (although this does not necessarily provide full knowledge of the measurement statistics), and each p_k is the associated probability (not amplitude) in [0,1]. In order to define properly pure and mixed states, introduce the density operator as follows:

Definition 7. A density operator ρ is a positive semidefinite Hermitian operator with unit trace and takes the form

where ∑ _k=1^Np_k = 1 and equals 1 if k = l and zero otherwise.

Mathematically, such a density operator ρ corresponds to a density matrix (ρ_kl)_k,l=1,…,N such that

1.3.2 Pure state

A pure state is one that can be represented by a state vector

where (α_i)_i=1,...,N are probability amplitudes in ℂ such that ∑ _i=1^N|α_i|² = 1. In the ensemble setup above, this means that there exists k^∗∈{1,…,N} such that p_k^∗ = 1 and hence = and therefore ρ = ⟨ψ|. The density matrix also allows us to compute expectations of the form (1.2.4):

Lemma 3. Let ρ be the density matrix associated to the pure state (1.3.2) and let 𝒜 be an observable (Hermitian operator), then

Proof. The lemma follows from the immediate computation

⟨ψ|𝒜	= ⟨ψ|𝒜∑ i=1Nα i
	= ∑ i=1Nα i ⟨ψ|𝒜
	= ∑ i=1N⟨ψ|𝒜
	= ∑ i=1N ⟨i|ρ𝒜 = Tr(ρ𝒜).

□

With the state given by (1.3.2), we obtain

At the same time we have

Comparison of (1.3.2) and (1.3.2) yields the following expression for the density matrix of a pure state:

Example: An example of a pure state is the Hadamard state

with corresponding density matrix

1.3.3 Mixed state

A mixed state is one that cannot be represented by a single pure state vector, and is therefore represented as a statistical distribution of pure states in the form of an ensemble of quantum states {p_k,}_k=1,…,N, where ∑ _k=1^Np_k = 1 and p_k ∈ [0,1] for each k. The density of a mixed state therefore reads

Similarly to Lemma 3, we can write expectations of observables with respect to mixed states using the density matrix:

Lemma 4. Let ρ be the density matrix associated to the mixed state (1.3.3) and let 𝒜 be an observable (Hermitian operator), then

Proof. The lemma follows from the immediate computation

Tr(ρ𝒜)	= ∑ i=1N ⟨i|ρ𝒜
	= ∑ i=1N ⟨i|𝒜
	= ∑ k=1Np k
	= ∑ k=1Np k ⟨ψk|𝒜.

□

Let us see now how the density matrix formalism can help us describe the state of a combined system. Consider an entangled state of two systems, A and B, given by (1.3.1), and a Hermitian operator 𝒜 that only acts within the Hilbert space of system A. What would be the expectation value of 𝒜 in this state? Starting with (1.2.4), we obtain

Since only terms with l = j survive in (1.3.3) due to the orthogonality of the basis states, we have

Thus, the density matrix that describes the mixed state of system A is

Note that in the case where the probability amplitudes γ_ij can be factorised as the product of probability amplitudes (α_i)_i=1,...,N and (β_j)_j=1,...,M, we obtain

which describes a pure state.

A simple criterion to distinguish a pure state from a mixed state is the following:

Proof. Consider an ensemble of pure states {p_i,}_i=1,…,N, with density matrix given by (1.3.3). Therefore

Tr(ρ2)	= Tr
	= Tr
	= Tr = ∑ i=1Np i2Tr⟨ψi| = ∑ i=1Np i2 = ∑ i=1Np i2,

which is smaller than 1 since the p_i are probabilities in [0,1] summing up to 1. Assume now that Tr(ρ²) equals one, then so does ∑ _i=1^Np_i². If p_i ∈ (0,1) for all i = 1,…,N, then

which is a contradiction, and therefore there exists i^∗∈{1,…,N} such that p_i^∗ = 1, so that ρ = ⟨ψ_i^∗| is a pure state. Conversely, if ρ = ⟨ψ_i| for some i ∈{1,…,N} represents a pure state, then

□

Example: An example of a mixed state is a statistical ensemble of states and . If a physical system is prepared to be either in state or state with equal probability, it can be described by the mixed state

Note that this is different from the density matrix of the pure state

which reads