Notes › PHYS 3571: Quantum Computing Lecture 2

Measurement of a Qubit

calendar_today January 27, 2026 article 560 words access_time 4 min replay Modified January 29, 2026

Table of Contents

Measurement and Basis

Definition 02.1 (Measurement).

A measurement is defined relative to a chosen orthonormal basis of the qubit Hilbert space.

Computational basis
- Z basis $\{\ket{0}, \ket{1}\}$

State Representation

General pure qubit state

$$ \ket{\psi} = \alpha \ket{0} + \beta \ket{1}\ \ \ \alpha, \beta \in \mathbb{C} $$

Normalization (always check this first!)

$$ |\alpha|^2 + |\beta|^2 = 1 $$

Z Basis Measurement

Definition 02.2 (Projection Operators).

$$ \hat{P}(0) = \ket{0}\bra{0} $$$$ \hat{P}(1) = \ket{1}\bra{1} $$

Proposition 02.3 (Measurement Probabilities).

$$ P(0) = \bra{\psi}\hat{P}(0)\ket{\psi} = |\bra{0}\ket{\psi}|^2 $$$$ P(1) = \bra{\psi}\hat{P}(1)\ket{\psi} = |\bra{1}\ket{\psi}|^2 $$

Explicitly:

$$ P(0) = |\alpha|^2 $$$$ P(1) = |\beta|^2 $$

Bloch Sphere Parameterization

Any pure qubit state may be written as

$$ \ket{\psi} = \cos\left(\frac{\theta}{2}\right)\ket{0} + e^{i\phi}\sin\left(\frac{\theta}{2}\right)\ket{1} $$

with the corresponding measurement probabilities:

$$ P(\ket{0}) = \cos^2\left(\frac{\theta}{2}\right) $$$$ P(\ket{1}) = \sin^2\left(\frac{\theta}{2}\right) $$

Repeated Measurements

Proposition 02.4 (Repeatability).

If a qubit is measured in a basis and immediately remeasured in the same basis, the outcome is identical with probability 1.

Post-Measurement (State Collapse)

Theorem 02.5 (State Collapse).

If outcome $i$ is observed, the post measurement state is

$$ \ket{\psi_{\text{new}}}

\frac{\hat{P}(i)\ket{\psi_{\text{old}}}} {|\hat{P}(i)\ket{\psi_{\text{old}}}|} $$

In other words, the new wavefunction is the projection of the observed outcome onto the old wavefunction.

For the Z basis:

$$ \ket{\psi_{\text{new}}} = \ket{i} $$

The projection formula for state collapse would really only be useful if you were applying a change-of-basis for subsequent measurements, since if you already observed the outcome in a particular basis, the outcome is never going to change if you observe it again

Measurement in Other Bases

X Basis

Definition 02.6 (X Basis States).

$$ \ket{+} = \frac{1}{\sqrt{2}}(\ket{0} + \ket{1}) $$$$ \ket{-} = \frac{1}{\sqrt{2}}(\ket{0} - \ket{1}) $$

Definition 02.7 (X Basis Projectors).

$$ \hat{P}(+) = \ket{+}\bra{+} $$$$ \hat{P}(-) = \ket{-}\bra{-} $$

Proposition 02.8 (X Basis Measurement).

$$ P(\pm) = |\bra{\pm}\ket{\psi}|^2 $$

Inner product

$$\bra{+}\ket{\psi} = \frac{1}{\sqrt{2}} \left(\cos\left(\frac{\theta}{2}\right) + e^{i\phi}\sin\left(\frac{\theta}{2}\right)\right) $$

Probability

$$ P(+)

\frac{1}{2} \left| \cos\left(\frac{\theta}{2}\right) e^{i\phi}\sin\left(\frac{\theta}{2}\right) \right|^2 $$

Y Basis

Definition 02.9 (Y Basis States).

$$ \ket{+i} = \frac{1}{\sqrt{2}}(\ket{0} + i\ket{1}) $$$$ \ket{-i} = \frac{1}{\sqrt{2}}(\ket{0} - i\ket{1}) $$

Definition 02.10 (Y Basis Projectors).

$$ \hat{P}(+i) = \ket{+i}\bra{+i} $$$$ \hat{P}(-i) = \ket{-i}\bra{-i} $$

Proposition 02.11 (Y Basis Measurement).

$$ P(\pm i) = |\bra{\pm i}\ket{\psi}|^2 $$

General Projective Measurement

Theorem 02.12 (Projective Measurement Rule).

Let $\{\ket{\phi_k}\}$ be an orthonormal basis

Projection operators

$$ \hat{P}_k = \ket{\phi_k}\bra{\phi_k} $$

Probability of outcome $k$

$$ P(k) = |\bra{\phi_k}\ket{\psi}|^2 $$

Post measurement state

$$ \ket{\psi_{\text{new}}}

\frac{\hat{P}_k\ket{\psi}} {\sqrt{P(k)}} $$

Completeness of Basis Sets

If you let $\ket{0}\bra{0} + \ket{1}\bra{1} = \hat{I}$, then $\hat{I}\ket{0} = \ket{0}$ and $\hat{I}\ket{1} = \ket{1}$
For any arbitrary point on the Bloch Sphere:

$$ \begin{align} \ket{\psi} &= \cos \left( \frac{\theta}{2} \right) \ket{0} + e^{i \psi} \sin \left( \frac{\theta}{2} \right) \ket{1} \\ \ket{\psi'} &= \cos \left( \frac{\pi - \theta}{2} \right) \ket{0} + e^{i(\pi+\phi)}\sin \left( \frac{\pi-\theta}{2} \right) \ket{1} \\ &=\sin\left( \frac{\theta}{2} \right) \ket{0} - e^{i \psi} \sin \left( \frac{\theta}{2} \right) \ket{1} \end{align} $$

Letting $\hat{u} = \ket{\psi}\bra{\psi} + \ket{\psi'}\bra{\psi'}$ and letting $\cos{\left( \frac{\theta}{2} \right), \sin{\left( \frac{\theta}{2} \right)} = c, s}$, we have:

$$ \begin{align} \hat{u} &= (c\ket{0} + e^{i\psi} s\ket{1} )(c \bra{0} + e^{-i \psi} s \bra{1} ) + (s\ket{0}-e^{i \psi} c\ket{1})(s \bra{0} - e^{-i \psi} c \ket{1}) \\ &= (s^2 + c^2)\ket{0}\bra{0} + (s^2 + c^2)\ket{1}\bra{1} \\ &= \ket{0}\bra{0} + \ket{1}\bra{1} \\ &= \hat{I} \end{align} $$

You can use this property to perform a change-of-basis, since $\ket{\psi} = \hat{I}\ket{\psi}$, and plug in the values from the other basis

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