Measurement (Quantum Computing)

quantum-computing

Definition

Measurement (Quantum Computing)

Let be an orthonormal measurement basis of a finite-dimensional complex Hilbert space and let

A measurement in the basis returns outcome with probability

Equivalently, with the orthogonal projection

the probability is

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

whenever .

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:

The projector

asks: “how much of lies in the direction?” It extracts exactly that component:

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

The projection keeps the part of that lies in 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 .

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

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

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 but its coordinates depend on the basis. If you measure in basis

you first express the state as

Then:

So changing the basis changes the amplitudes , therefore it can change the measurement probabilities.

Example

measured in the computational basis

has expansion

So:

But define the plus/minus basis:

Then the same state can be written as:

So measuring in the basis gives:

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