Qubit Measurements

In order to classically access information about the state of qubits we use the measurement operation $\mathbf{meas}$. This is an intrinsically probabilistic process that can be applied to any extant qubit. For information processing, one can think of $\mathbf{meas}$ as a subroutine or function that returns either $\mathfrak{0}$ or $\mathfrak{1}$ as output. The output is called the ``measurement outcome''. The probabilities of the measurement outcomes are determined by the current state. The state of the qubit being measured is ``collapsed'' to the logical state corresponding to the outcome. Suppose we have just one qubit, currently in the state $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\psi}\mbox{$\rangle\hspace... ...t$}{\mathfrak{1}}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}$. Measurement of this qubit has the effect

$$\mathbf{meas}\Big(\alpha\mbox{$\vert\hspace*{-3pt}\vert\hspa... ...ox{with probability $\vert\beta\vert^2$}. \end{array}\right.$$ (33)

The classical output is given before the new state for each possible outcome. This measurement behavior explains why the amplitudes have to define unit length vectors: Up to a phase, they are associated with square roots of probabilities.

For two qubits the process is more involved. Because of possible correlations between the two qubits, the measurement affects the state of the other one too, similar to conditioning for pbits after one ``looks'' at one of them. As an example, consider the state

$$\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\psi}\m... ...pt}\rangle\hspace*{-4.3pt}\rangle$}_{{}_{\!\!{\mathsf {AB}}}}.$$ (34)

To figure out what happens when we measure qubit $\mathsf {A}$, we first rewrite the current state in the form $\alpha\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\mathfrak{0}}\mbox{$... ...rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}_{{}_{\!\!{\mathsf {B}}}}$, where $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\phi_0}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}_{{}_{\!\!{\mathsf {B}}}}$ and $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\phi_1}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}_{{}_{\!\!{\mathsf {B}}}}$ are pure states for qubit $\mathsf {B}$. It is always possible to do that. For the example of Eq. 34:

$$\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\psi}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}_{{}_{\!\!{\mathsf {AB}}}} = {2\over 3}\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\m... ...rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}_{{}_{\!\!{\mathsf {B}}}}$$

$$= \mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\mathfrak{0}... ...rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}_{{}_{\!\!{\mathsf {B}}}}$$

$$= {\sqrt{5}\over 3}\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\ve... ...\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}_{{}_{\!\!{\mathsf {B}}}}\Big),$$ (35)

so $\alpha={\sqrt{5}\over 3}$, $\beta={i2\over 3}$, $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\phi_0}\mbox{$\rangle\hspa... ...rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}_{{}_{\!\!{\mathsf {B}}}}$ and $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\phi_1}\mbox{$\rangle\hspa... ...rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}_{{}_{\!\!{\mathsf {B}}}}$. The last step required pulling out the factor of ${\sqrt{5}\over 3}$ to make sure that $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\phi_0}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}_{{}_{\!\!{\mathsf {B}}}}$ is properly normalized for a pure state. Now that we have rewritten the state, the effect of measuring qubit $\mathsf {A}$ can be given as follows:

$${\mathbf{meas}}^{({\mathsf {A}})}\Big(\alpha\mbox{$\vert\hsp... ...mbox{with probability $\vert\beta\vert^2$.} \end{array}\right.$$ (36)

For the example, the measurement outcome is $\mathfrak{0}$ with probability ${5\over 9}$, in which case the state collapses to $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\mathfrak{0}}\mbox{$\rangl... ...e\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}_{{}_{\!\!{\mathsf {B}}}}\Big)$. The outcome is $\mathfrak{1}$ with probability ${4\over 9}$, in which case the state collapses to $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\mathfrak{1}}\mbox{$\rangl... ...rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}_{{}_{\!\!{\mathsf {B}}}}$. The probabilities add up to $1$ as they should.

The same procedure works for figuring out the effect of measuring one of any number of qubits. Say we want to measure qubit $\mathsf {B}$ among qubits $\mathsf {A,B,C,D}$, currently in state $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\psi}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}_{{}_{\!\!{\mathsf {ABCD}}}}$. First rewrite the state in the form $\alpha\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\mathfrak{0}}\mbox{$... ...ngle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}_{{}_{\!\!{\mathsf {ACD}}}}$, making sure that the $\mathsf {ACD}$ superpositions are pure states. Then the outcome of the measurement is $\mathfrak{0}$ with probability $\vert\alpha\vert^2$ and $\mathfrak{1}$ with probability $\vert\beta\vert^2$. The collapsed states are $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\mathfrak{0}}\mbox{$\rangl... ...ngle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}_{{}_{\!\!{\mathsf {ACD}}}}$ and $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\mathfrak{1}}\mbox{$\rangl... ...ngle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}_{{}_{\!\!{\mathsf {ACD}}}}$, respectively.

Probabilities of the measurement outcomes and the new states can be calculated systematically. For example, to compute the probability and state for outcome $\mathfrak{0}$ of ${\mathbf{meas}}^{({\mathsf {A}})}$ given the state $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\psi}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}_{{}_{\!\!{\mathsf {AB}}}}$, one can first obtain the unnormalized ket expression $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\phi'_0}\mbox{$\rangle\hsp... ...angle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}_{{}_{\!\!{\mathsf {AB}}}}$ by using the rules for multiplying kets by bras. The probability is given by $p_0={}^{\scriptscriptstyle\mathsf { B}}\!\mbox{$\langle\hspace*{-4.3pt}\langle\... ...rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}_{{}_{\!\!{\mathsf {B}}}}$, and the collapsed, properly normalized pure state is

$$\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\mathfr... ...hspace*{-4.3pt}\rangle$}_{{}_{\!\!{\mathsf {AB}}}}/\sqrt{p_0},$$ (37)

The operator $P_{\mathfrak{0}}=\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\mathfrak... ...3pt}\langle$}{\mathfrak{0}}\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}$ is called a ``projection operator'' or ``projector'' for short. If we perform the same computation for the outcome $\mathfrak{1}$, we find the projector $P_{\mathfrak{1}}=\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\mathfrak... ...3pt}\langle$}{\mathfrak{1}}\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}$. The two operators satisfy ${P_{\mathfrak {a}}}^2=P_{\mathfrak {a}}$, ${P_{\mathfrak {a}}}^\dagger = P_{\mathfrak {a}}$ and $P_{\mathfrak{0}}+P_{\mathfrak{1}}={\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l} {\rm 1\mskip-4.5mu l} {\rm 1\mskip-5mu l}}$. In terms of the projectors, the measurement's effect can be written as follows:

$${\mathbf{meas}}^{({\mathsf {A}})}\mbox{$\vert\hspace*{-3pt}\... ...\sqrt{p_1} & \mbox{with probability $p_1$}, \end{array}\right.$$ (38)

where $p_0={}^{\scriptscriptstyle\mathsf { AB}}\!\mbox{$\langle\hspace*{-4.3pt}\langle... ...angle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}_{{}_{\!\!{\mathsf {AB}}}}$ and $p_1 = {}^{\scriptscriptstyle\mathsf { AB}}\!\mbox{$\langle\hspace*{-4.3pt}\lang... ...angle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}_{{}_{\!\!{\mathsf {AB}}}}$. In quantum mechanics, any pair of projectors satisfying the properties given above is associated with a potential measurement whose effect can be written in the same form. This is called a binary ``von Neumann'', or ``projective'', measurement.