Projection Operator

• Defined by $P$, it is obtained by taking the outer product of a given state $\ket{\psi}$, i.e.

  $P = \ket{\psi}\bra{\psi}$

PROPERTIES

• A projection operator is Hermitian.

  $P = P^\dagger$

• If the state $\ket{\psi}$ (via which the operator $P$ is obtained) is normalized, then the projection operator is equal to its own square:

  $P = P^2$

• Product of two commutative operators is also a projection operator, i.e. if,

  $P_1P_2 = P_2P_1$, then $P_1*P_2$ is also a projection operator.

• Given a vector space that has $n$ dimensions and a basis given by $\ket{1},\ket{2},\dots,\ket{n}$, the operator given by

  $P = \sum_{i=1}^{m}\ket{i}\bra{i}$,

  projects on to the subspace (of the $n$ dimensional space we just defined) spanned by the set $\ket{1}, \ket{2}, \dots \ket{m}$ .

REWRITING SPECTRAL DECOMPOSITION THEOREM

• Recalling the result of the spectral decomposition theorem, any operator $A$ can be represented as follows in terms of its eigenvalues and (some) basis:

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

  *here, $a_i$ ***turn out to be the various eigenvalues of the operator & $\ket{u_i}$ represents the basis set.

• The very same result, can now be rewritten in terms of projection operators:

  $A = \sum_{i=1}^{n}a_iP_i$ , here $P_i = \ket{u_i}\bra{u_i}$

• Further, since the basis states satisfy the completeness relation (recall from the matrices and operators chapter), we have :

  $\sum_{i=1}^{n}\ket{u_i}\bra{u_i} = \sum_{i=1}^{n}P_i = I$

  clearly, $I$ being the identity matrix

• Example

Ensemble - Collection of Quantum Systems

• In most of the scenarios, while studying quantum computing we observe(study) a large number (say $N$) or collection of quantum systems (rather than a single) called an ensemble.

• The members(quantum systems) of this ensemble can themselves be found in one of two or more different quantum states.

  ---

PROBABILITY DISTRIBUTION

Consider a 2D Hilbert space with a basis vector set $\{\ket{x}, \ket{y}\}$. Further, let’s construct $N$ quantum systems from this basis, where each system may exist in one of the two vector states (normalized):

1. $\ket{a} = \alpha\ket{x} + \beta\ket{y}$ $\alpha^2 + \beta^2 = 1$
2. $\ket{b} = \gamma\ket{x} + \delta\ket{y}$ $\gamma^2 + \delta^2 = 1$

TWO LEVELS OF PROBABILITY:

→ At the level of single quantum system:

<aside> 📌 Here, the probabilites are calculated via the Born Rule, which states that the probability of obatining a measurement is proportional to the square of its amplitude.

</aside>

Clearly, as studied previously:

• For a system in state $\ket{a}$, the probability of finding $\ket{x}$ on measurement is $|\alpha|^2$ and that of $\ket{y}$ is $|\beta|^2$.
• Similarly for a system in state $\ket{a}$, the probability of finding $\ket{x}$ on measurement is $|\gamma|^2$ and that of $\ket{y}$ is $|\delta|^2$.

→ On the ensemble level:

<aside> 📌 Here, the probabilites are calculated in a classical manner as at the ensemble level we have a statistical mixture.

</aside>

Now, let’s assume that out of the $N$ systems we prepared, $n_a$ quantum systems have the state $\ket{a}$ and the remaining $n_b$ systems have the state $\ket{b}$. ($n_a + n_b = N$). This gives us that the probability of finding a system in state:

• $\ket{a}$ in this ensemble = $\large\frac{n_a}{N}$
• $\ket{b}$ in this ensemble = $\large\frac{n_b}{N}$