<aside> 📌 We’ve studied quite a bit regarding the representation of quantum states (qubits) & observables (operators). Let’s now look at a set of axioms, which will define how these (qubits & operators) fit together into the framework of a workable physical theory.

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Postulate 1 : The State of a System

• The state of a quantum system, as defined earlier, is a vector $\ket{\psi(t)}$ in a Hilbert space (notice that this is same as $\ket{\psi}$ except for the fact that the state is now time dependent).

• We always work with normalized states (called state vectors), such that:

  $\braket{\psi|\psi} = 1$

  In other words, a qubit is a state vector in a 2D Vector Space:

  $\ket{\psi} = \alpha\ket{0} + \beta\ket{1}$ normalized such that, $|\alpha|^2 + |\beta|^2 = 1$

Postulate 2 : Observable Quantities Represented by Operators

• As, earlier defined in the Matrices & Operators chapter, to every dynamical variable $A$ that is a physically measurable quantity, there corresponds an operator $A$.
• The operator $A$ is:
  • Hermitian and,
  • its eigenvectors form a complete orthonormal basis of the vector space.

Postulate 3: Measurements

<aside> 📌 READ THIS ONLY AFTER COVERING QUANTUM MEASUREMENT THEORY !!!

</aside>

• BORN RULE

  consider an observable set of projection operators, denoted by $A$. Let, the eigenvectors of $A$ be denoted by $\ket{u_i}$ each with eigenvalue $a_i$. Then, by Spectral Decomposition Theorem:

  $A = \sum_{i=1}^{n}a_i\ket{u_i}\bra{u_i} = \sum_{i=1}^{n}a_iP_i$

  here, the projection operator corresponding to outcome $a_i$ is given by:

  $P_i=\ket{u_i}\bra{u_i}$

• The probability of obtaining measurement result $a_i$, $Pr(a_i)$ can be written in either of the two ways:

  1. $Pr(i) = \braket{\psi | P_i | \psi} = Tr(P_i\ket{\psi}\bra{\psi})$
  2. $Pr(i) = \left\{\begin{matrix}|\braket{u_i|\psi}|^2 & \textrm{if } \psi \textrm{ is normalized} \\\\ \large\frac{|\braket{u_i|\psi}|^2}{\braket{\psi|\psi}} & \textrm{otherwise}\end{matrix}\right.$

  The case for degenerate eigenvalues can be referred to from the Quantum Measurement Theory chapter.

• The post measurement state of a system can be written in terms of the projection operator as:

  $\large\ket{\psi} \xrightarrow{\text{measure}}\ket{\psi’} = \frac{P_i\ket{\psi}}{\sqrt{\braket{\psi|P_i|\psi}}}$

Postulate 4: Time Evolution of the System

• Time evolution of a closed(physically isolated) Quantum System is governed by the Schrödinger equation:

  $i\hbar \large\frac{d}{dt}\small\ket{\psi(t)} = H\ket{\psi(t)}$

  here,

  $H$ - Hamiltonian of the system

  • Is a Hermitian operator (i.e. $H = H^\dagger$).
  • Its eigen values represent the possible energy values (Potential + Kinetic) of the system.

• DERIVING THE VALUE OF $\ket{\psi(t)}$

  we have,

  $i\hbar \large\frac{d}{dt}\small\ket{\psi(t)} = H\ket{\psi(t)}$

  ⇒ $\large\frac{d\ket{\psi(t)}}{\ket{\psi(t)}} = \frac{Hdt}{i\hbar} = \frac{-iHdt}{\hbar}$ ——(1)

  integrating both sides:

  ⇒ $\large\int\frac{d\ket{\psi(t)}}{\ket{\psi(t)}} = \int\frac{-iHdt}{\hbar}$

  ⇒ $log_e(\ket{\psi(t)}) = \large\frac{-iHt}{\hbar} \small+ c$ ——(2)

  studying equation 2 at time t=0, we obtain:

  $c = log_e(\ket{\psi(0)})$ ——(3)

  hence, we can now write equation 2 as follows:

  ⇒ $log_e(\ket{\psi(t)}) = \large\frac{-iHt}{\hbar} \small+ log_e(\ket{\psi(0)})$

  ⇒ $log_e(\ket{\psi(t)}) - log_e(\ket{\psi(0)}) = \large\frac{-iHt}{\hbar} \small$

  applying the result $\small log(\frac{m}{n}) = log(m) - log(n)$, we get:

  ⇒ $log_e(\large\frac{\ket{\psi(t)}}{\ket{\psi(0)}})\small = \large\frac{-iHt}{\hbar}$

  ⇒ $\large\frac{\ket{\psi(t)}}{\ket{\psi(0)}} = e^{\frac{-iHt}{\hbar}}$

  ⇒ $\small{\ket{\psi(t)}} = \large e^{\frac{-iHt}{\hbar}}\small\ket{\psi(0)}$ ——(final result)

• THE OPERATOR GOVERNING THE TIME EVOLUTION OF A QUANTUM STATE

  From the result obtained above, we can say that the state of quantum system (wrt time) is governed by the following Unitary Operator (refer to the chapter Matrices & Operators for the definition):

  $U = \large e^{-i(\frac{Ht}{\hbar})}$

  now, let $a = -i$ & $A = \large\frac{Ht}{\hbar}$, where $a \in \mathbb{C}$ & $A \in \mathbb{C}^{n \times n}$, then the above operator can be written as:

  $U = \large e^{aA} = \small I +aA + \large \frac{a^2A^2}{2!} \small + \large \frac{a^3A^3}{3!}\small +\dots$ ——expansion of $e^x$