Postulates of Quantum Mechanics
The basic postulate of quantum mechanics is about the Hilbert space formalism.
(A0) To each quantum mechanical system a complex Hilbert spaceH is associ- ated.
The pure physical states of a system are represented by unit vectors in Hilbert space, but this representation is not one-to-one Two unit vectors, f1 and f2, represent the same physical state if f1 equals zf2, where z is a complex number with a modulus of 1, commonly referred to as phase Thus, the pure physical state of the system defines a corresponding state vector, allowing for variations in phase.
The two-dimensional Hilbert space C² represents a 2-level quantum system known as a qubit, with canonical basis vectors denoted as |↑ (1,0) and |↓ (0,1) Alternatively, these vectors can be represented as |0 for (1,0) and |1 for (0,1) A key example of a qubit is the polarization of a photon, where the state |↑ signifies that the polarization is vertical.
| ↓means that the “polarization is horizontal”.
To specify a state of a qubit we need to give a real number x 1 and a complex number z such that x 2 1 +|z| 2 =1 Then the state vector is x 1 | ↑+z| ↓.
By applying an appropriate phase to the unit vector \( z = z_1 | \uparrow \rangle + z_2 | \downarrow \rangle \), we can ensure that the coefficient of \( | \uparrow \rangle \) is real while maintaining the state unchanged When we express \( z \) in terms of its real and imaginary components as \( z = x_1 + ix_2 + ix_3 \), we encounter the constraint \( x_1^2 + x_2^2 + x_3^2 = 1 \) for the parameters \( (x_1, x_2, x_3) \in \mathbb{R}^3 \).
Therefore, the space of all pure states of a qubit is conveniently visualized as the sphere in the three-dimensional Euclidean space; it is called the Bloch sphere
Traditional quantum mechanics differentiates between pure states and mixed states, with mixed states represented by density matrices A density matrix, or statistical operator, is a positive operator with a trace of 1 on the Hilbert space, ensuring that its eigenvalues sum to 1 Pure states, represented by unit vectors in the Hilbert space, can also be viewed as density matrices through an appropriate identification; for example, if \( x \) is a unit vector, then \( |xx| \) represents a density matrix Geometrically, \( |xx| \) serves as the orthogonal projection onto the linear subspace generated by \( x \), and it is important to note that \( |xx| = |yy| \) when the vectors \( x \) and \( y \) differ only by a phase factor.
(A1) The physical states of a quantum mechanical system are described by statis- tical operators acting on the Hilbert space.
Example 2.2 A state of the spin (of 1/2) can be represented by the 2×2 matrix
This is a density matrix if and only if x 2 1 +x 2 2 +x 2 3 ≤1 (Fig 2.1)
The second axiom is about observables.
(A2) The observables of a quantum mechanical system are described by self- adjoint operators acting on the Hilbert space.
A self-adjoint operator A on a Hilbert spaceH is a linear operatorH →H which satisfies
Ax,y=x,Ay for x,y∈H Self-adjoint operators on a finite dimensional Hilbert space C n are n×n self-adjoint matrices A self-adjoint matrix admits a spectral decomposition
A 2 × 2 density matrix is represented as 1/2 (I + x1σ1 + x2σ2 + x3σ3), where the condition x1² + x2² + x3² ≤ 1 holds true The vectors (x1, x2, x3) have a maximum length of 1, forming a unit ball known as the Bloch ball within three-dimensional Euclidean space Pure states are located on the surface of this Bloch ball.
The equation A = ∑ i λ i E i represents a matrix A expressed as a sum of its eigenvalues λ i and their corresponding orthogonal projections E i onto the subspace spanned by the eigenvectors The multiplicity of each eigenvalue λ i is defined as the rank of the projection E i.
Example 2.3 In case of a quantum spin (of 1/2) the matrices σ1 0 1
0 −1 are used to describe the spin of direction x,y,z (with respect to a coordinate system). They are called Pauli matrices Any 2×2 self-adjoint matrix is of the form
A ( x 0 , x ):=x 0σ0+x 1σ1+x 2σ2+x 3σ3 ifσ0stands for the unit matrix I We can also use the shorthand notation x 0σ0+xãσ. The density matrix (2.1) can be written as
The Bloch vector, represented as \( x \) in the equation \( 2(σ_0 + x \cdot σ) \) where \( x ≤ 1 \), forms the basis of the Bloch ball, establishing an affine relationship between 2×2 density matrices and the unit ball in three-dimensional Euclidean space The extreme points of this ball correspond to pure states, while mixed states can be expressed as convex combinations of pure states in infinitely diverse ways However, the complexity increases significantly in higher dimensions.
Any density matrix can be expressed as ρ = ∑ i λ i |x i x i |, where the unit vectors |x i and non-negative coefficients λ i satisfy the condition ∑ i λ i = 1 This representation arises from the spectral theorem, confirming that the density matrix ρ is self-adjoint.
|x i may be chosen pairwise orthogonal eigenvectors andλ i are the corresponding eigenvalues The decomposition is unique if the spectrum ofρ is non-degenerate, that is, there is no multiple eigenvalue.
Lemma 2.1 The density matrices acting on a Hilbert space form a convex set whose extreme points are the pure states.
The set of density matrices, denoted by Σ, is established as a convex set since any convex combination of density matrices remains positive and has a trace of one A density matrix ρ is classified as an extreme point if it can only be expressed in a trivial convex decomposition, meaning that ρ1 and ρ2 must be identical The Schmidt decomposition indicates that extreme points correspond to pure states To demonstrate that a pure state p, where p = p², is indeed an extreme point, we assume it can be represented as p = λρ1 + (1 − λ)ρ2, leading to the conclusion that ρ1 and ρ2 must be equal.
2.1 Postulates of Quantum Mechanics 7 p=λpρ1 p+ (1−λ)pρ2 p and Tr pρ i p=1 must hold Remember that Tr pρ i p=p,ρ i , whilep,p=1 and ρ i ,ρ i ≤1 In the Schwarz inequality
|e,f| 2 ≤ e,ef,f the equality holds if and only if f =ce for some complex number c Therefore, ρ i =c i p must hold Taking the trace, we get c i =1 andρ1=ρ2=p
The next result, obtained by Schr¨odinger [105], gives relation between different decompositions of density matrices.
Lemma 2.2 Let ρ=∑ k i=1 |x i x i |=∑ k j=1 |y j y j | be decompositions of a density matrix Then there exists a unitary matrix(U i j ) k i, j=1 such that
The Schmidt decomposition of a density matrix ρ is expressed as ∑ n i=1 λ i |z i z i |, where λ i > 0 and |z i are orthogonal unit vectors, with n representing the rank of ρ and satisfying n ≤ k For indices n < i ≤ k, we set |z i := 0 and λ i := 0 To achieve the transformation of vectors √ λ i |z i to |y i, it suffices to construct a unitary operator If two different decompositions can be related to an orthogonal decomposition via a unitary, a new unitary can be created to connect both decompositions effectively.
The vectors|y i are in the linear span of{|z i : 1≤i≤n}; therefore
|y i =∑ n j=1 z j |y i |z j is the orthogonal expansion We can define a matrix(U i j )by the formula
We can easily compute that
8 2 Prerequisites from Quantum Mechanics and this relation shows that the n column vectors of the matrix(U i j )are orthonormal.
If n0 Hence the next example gives a channeling transformation
Example 2.13 (Depolarizing channel) This channel is given by the matrix T from (2.26), where t=0 and T 3=pI Assume that 0H There exists a sequence of(2 nR n ,n)block codes with error probability P e (n) such that P e (n) →0 and R n →R.
Shannon's theorem indicates that any transmission rate equal to or greater than the channel capacity (H) plus a small margin (ε) can be achieved with a minimal probability of error Conversely, rates below H are not feasible under the same error constraints.
Before we enter the proof let me give an outline of the method of types Let x ∈
X n The type of x= (x 1 , x 2 , , x n )∈X n is a probability mass function onX. The mass of x∈X is the relative frequency of x in the sequence(x 1 ,x 2 , , x n ):
The term "type of a sequence" refers to the empirical distribution, represented by P n, which encompasses all types P x for x belonging to X n The individual elements within P n are referred to as n-types, and the total count of possible n-types is a crucial aspect of this concept.
The upper estimate is useful in estimations.
For P∈P n the type class of P is defined as the set of all sequences of type P:
The cardinality of a type class Type(P)is a multinomial coefficient n!
∏ x (nP x (x))! (P x ∈P(T)) but the following exponential bounds are good enough:
1 (n+1) #(X ) 2 nH(P) ≤#(Type(P))≤2 nH(P) (3.12) (A proof could be based on Stirling’s formula on factorial functions, see [26] p 282 or [30] p 430 for other proofs.)
In this example, the method of types is applied to a probability measure p defined on a set X with cardinality d For each natural number n and a given δ > 0, the set Δ(p; n, δ) consists of all sequences x in X^n for which the probability P_x(x) is within δ of p(x) for every x in X Thus, Δ(p; n, δ) encompasses all δ-typical sequences according to the measure p.
Letμ n ,δ be the maximizer of the Shannon entropy on the set of all types P x , x ∈
X n , such that|P x (x)−p(x)|