# Quantum Mechanics, Hilbert Space and Qubits

# Introduction

With Quantum computing booming the internet and giants investing heavily in quantum AI, we are eager to know more about. I am sure in this quest, we all came across Quantum mechanics, complex numbers, Hilbert spaces, qubits etc. and here will discuss more about this.

Quantum computing leverages on quantum mechanical phenomena of superposition and entanglement to create states to enable complex computations effectively.

In the below sections I have detailed the mathematical elements, however the knowledge of linear algebra would be helpful in understanding Quantum computing.

Let’s starts with Quantum mechanics postulates in first section and the latter sections will follow the mathematical elements used in these postulates.

# Quantum Mechanics Postulates

Postulate is an assumption, that is a statement or proposition that is assumed to be true without any proof.

# Postulate 1(States)

The state of a quantum mechanical system is specified by a wave functions in a complex Hilbert space. Wave function can be thought of as vectors in in Hilbert space denoted in Dirac notation as |ψ˃ called ket. Wave function ψ(r,t), is related to probability of state of a mechanical system at position r, at the time t.

e.g.: spin state of an electron

# Postulate 2 (Mixed States)

For every observable classical observable there exists a positive, self adjoint quantum mechanical operator having trace one. This operator is called density operator and we denote the set of density operators on a HILBERT space H by D(H).

A measurable physical quantity of a mechanical system is called observable, e.g. Position, Momentum, Energy.

# Postulate 3 (Measurement Probability 1)

The only possible measurement of an observable A is the eigen value of its operator Â.

# Postulate 4 (Measurement Probability 2)

The probability of obtaining the eigen value of the operator Â is defined by square of inner product of |ψn˃ with its eigen state λn. States are assumed to be normalized.

# Postulate 5 (Projection Postulate)

If the measurement of an observable A has yielded a value λn, then state of the system is the normalized eigenstate |λn>. This is also called the collapse of the wave packet

# Postulate 6 (Time Evolution)

In a quantum system every change of state over time that has not being caused by the measurement is described by the time evolution operator û(t,t0). The time evolved state |ψt> originating from |ψt0> is given by

The time evolution operator is the solution to the initial value problem

And is also equivalent to the Schrodinger equation

Hamilton Operator corresponds to the sum of kinetic and potential energies of all the particles in a system.

# Superposition

Superposition principle states that any normalized linear combination of states is again a state. According to postulate one a physical state corresponds to vector (wave function) in a complex Hilbert Space, conversing this means every unit vector in a Hilbert Space corresponds to a state, Since Hilbert Space is a vector space any linear combination of states is a state as well.

If |ψ>, |φ> of complex Hilbert Space are states then a|ψ>+b|φ>, where a, b € ℂ also belong to complex Hilbert space

# Entanglement

Quantum entanglement is a quantum mechanical phenomenon in which the quantum states of two or more particles are entangled with each other.

A state ρ ∈ D (HA ⊗HB) in a composite system HA ⊗HB, where HA HB are sub systems is said to be separable, if there exist states ρA j ∈ D(HA )and ρB j ∈ D(HB), j ∈ I and a positive real numbers pj satisfying

∑ pj = 1, such that ρ = ∑ pj ρA j ⊗ρB j .

Otherwise ρ is called entangled.

⊗ Operator is the tensor product of 2 vector spaces and D() is the self adjoint operator from postulate 2.

# Qubits

A bit is the smallest unit of information in classical computing, likewise qubit or quantum bit is the smallest unit in quantum mechanics. Bit has only 2 defined state either 1 or 0, but qubit can be a superposition of both the states.

# Complex numbers

What is the solution to the polynomial equation below? Let’s try to solve it

√(-1) has no roots in real number and hence the need to have imaginary number. √(-1) = i is known as the imaginary number.

Complex numbers denoted by ℂ, takes the form a+ib, where a and b are real numbers. ‘a’ is also called the real part and ‘ib’ , the imaginary part.

Thus, if we have complex numbers we have the solution for all the polynomial equations. This property of complex numbers is known as algebraically closed and is widely used various fields including quantum mechanics.

# Hilbert Space and Dirac Notation

Hilbert space is a linear vector space that is complete and has an inner product denoted by <|>satisfying the properties below. For all ϕ, ψ, ϕ1, ϕ2 ∈ H and a, b ∈ C

This inner product also has a norm in which H is complete. Norm, denoted by ǁ ǁ is defined as

Dirac notations or bra-ket notations are used to represent vectors in Hilbert space. Vectors are called kets and denoted by the symbol |>, and linear functionals on vector space is denoted by <f|.

These are very interesting topics and do learn more about them from books like “Mathematics of Quantum Computing” and “Introduction to quantum computing”. Thank You.