Measurement (Quantum Computing)

quantum-computing

Definition

Measurement (Quantum Computing)

Let $\{|e_i⟩{\}}_i$ be an orthonormal measurement basis of a finite-dimensional complex Hilbert space and let

$$|\psi ⟩=\sum _i{\alpha }_i|e_i⟩.$$

A measurement in the basis $\{|e_i⟩{\}}_i$ returns outcome $i$ with probability

$$\mathrm{Pr}(i)=|{\alpha }_i{|}^2=|⟨e_i\mid \psi ⟩{|}^2.$$

Equivalently, with the orthogonal projection

$$P_i=|e_i⟩⟨e_i|,$$

the probability is

$$\mathrm{Pr}(i)=\Vert P_i|\psi ⟩{\Vert }^2.$$

After observing outcome $i$, the state updates to the normalised projected state

$$|\psi ⟩\mapsto \frac{P_i|\psi ⟩}{\sqrt{\mathrm{Pr}(i)}}=|e_i⟩,$$

whenever $\mathrm{Pr}(i)>0$.

Intuition

Measurement as projection and selection

A measurement basis is a choice of orthonormal directions. Before measurement, the state may have components in many of those directions:

$$|\psi ⟩=\sum _i{\alpha }_i|e_i⟩.$$

The projector

$$P_i=|e_i⟩⟨e_i|$$

asks: “how much of $|\psi ⟩$ lies in the $|e_i⟩$ direction?” It extracts exactly that component:

$$P_i|\psi ⟩={\alpha }_i|e_i⟩.$$

In this sense, measurement first maps the state into an outcome subspace. For a basis measurement, the outcome subspace for $i$ is the one-dimensional space

$$U_i=\mathrm{span}(|e_i⟩).$$

The projection $P_i$ keeps the part of $|\psi ⟩$ that lies in $U_i$ and removes the parts in the orthogonal complement of that outcome subspace. If this kept part is non-zero, it is the unnormalised post-measurement state for outcome $i$.

The probability of outcome $i$ is the squared length of that extracted component:

$$\mathrm{Pr}(i)=\Vert {\alpha }_i|e_i⟩{\Vert }^2=|{\alpha }_i{|}^2.$$

After outcome $i$ is observed, all other components are discarded and the remaining component is normalised back to length $1$:

$${\alpha }_i|e_i⟩⟼|e_i⟩.$$

Difference in Basis

The selection of the orthonormal measurement basis matters, because measurement probabilities are computed from the coordinates of the state in the measurement basis.

A state is one vector $|\psi ⟩$ but its coordinates depend on the basis. If you measure in basis

$$B=\{|e_i⟩\},,$$

you first express the state as

$$|\psi ⟩=\sum _i{\alpha }_i|e_i⟩.$$

Then:

$$\mathrm{Pr}(i)=|{\alpha }_i{|}^2.$$

So changing the basis changes the amplitudes ${\alpha }_i$, therefore it can change the measurement probabilities.

Example

$|0⟩$ measured in the computational basis

$$\{|0⟩,|1⟩\}$$

has expansion

$$|0⟩=1|0⟩+0|1⟩.$$

So:

$$\mathrm{Pr}(0)=1,\mathrm{Pr}(1)=0.$$

But define the plus/minus basis:

$$|+⟩=\frac{1}{\sqrt{2}}(|0⟩+|1⟩),|-⟩=\frac{1}{\sqrt{2}}(|0⟩-|1⟩).$$

Then the same state $|0⟩$ can be written as:

$$|0⟩=\frac{1}{\sqrt{2}}|+⟩+\frac{1}{\sqrt{2}}|-⟩.$$

So measuring $|0⟩$ in the $\{|+⟩,|-⟩\}$ basis gives:

$$\mathrm{Pr}(+)=\frac{1}{2},\mathrm{Pr}(-)=\frac{1}{2}.$$

Same quantum state. Different measurement basis. Different probability distribution.