Measurement in Quantum mechanics

Positive operator-valued measure (POVM)

A quantum measurement can be described by a set of measurement operators ${M_{m}}$ acting on the Hilbert space. The probability of getting result $m$ when measuring a state $|\psi\rangle$"

$$p(m) = \langle \psi|M_{m}^{\dagger} M_{m}|\psi\rangle$$

and measurement operators must satisfies

$$\sum_{m}M_{m}^{\dagger}M_{m} = I.$$

Thus, $1 = \sum_{m}p_{m} = \langle \psi|M_{m}^{\dagger} M_{m}|\psi\rangle$. After the measurement, our state becomes

$$\frac{M_{m}|\psi\rangle}{\sqrt{\langle \psi|M_{m}^{\dagger}M_{m}|\psi\rangle}}.$$

Projective measurement

Projective measurement is a special case of general quantum measurement, where the set of measurement operators $\{P_{m}\}$ consists entirely of projectors.

Given a set of projectors $\{P_{m}\}$, we can define a corrsponding obervable (measurable physical quantity):

$$M = \sum_{m} m P_{m}$$

where $m$ are the eigenvalue (measurement result) and $P_{m}$ the projection opreator. A projector $P$ can be expressed using an orthonotmal basis $\{|i\rangle\}$ spanning some subspace $W \subset \mathcal{H}$

$$P = \sum_{i=1}^{k}|i\rangle\langle i|$$

This means that $P$ projects any state vector onto the subspace $W$ spanned by the basis vectors $\{|i\rangle\}$.

The porjection operator has the foloowing properties

$$P_{m}^{\dagger} = P_{m}, \ P_{m}^{2} = P_{m}.$$

The probability of obtaining outcome $m$ when measuring state $|\psi\rangle$ is:

$$p(m) = \langle \psi |P_{m}|\psi\rangle$$

After the measurement, the system state collapses to

$$|\psi'\rangle = \frac{P_{m}|\psi\rangle}{\sqrt{p(m)}}$$

This is a normalized projection of the original state onto the eigenspace corresponding to $P_{m}$. The expected value of observable $M$ is

$$\langle \psi \bigg(\sum_{m}mP_{m}\bigg)\psi\rangle = \langle \psi |M|\psi\rangle$$

Two qubits example

Suppose we have two qubits. If these were two classical bits, then there would be four possible states and four computational basis state denote $|00\rangle$, $|01\rangle$, $|10\rangle$, and $|11\rangle$. A pair of qubits can also exist in superpositions of these four states, so that quantum state of two qubits involves associating a complex coefficient (ampllitude). We can describe this two qubtis system as

$$\lvert\psi\rangle = a_{00}\lvert00\rangle+a_{01}\lvert01\rangle+a_{10}\lvert10\rangle+a_{11}\lvert11\rangle.$$

Similiar to the single qubit case, the measurement resutls $x$ (any of the followings: $00$, $01$, $10$, $11$) occurs with probability $|\alpha_{x}|^{2}$, with the state of the qubits after the measurement being $|x\rangle$. For a two-qubit system, we could measure jsut a subset of the qubits, for example, we want to measure the first qubit alone gives $0$ with probability $|\alpha_{00}|^{2}+|\alpha_{01}|^{2}$, leaving the post-measurement state

$$|\psi'\rangle = \frac{\alpha_{00}|00\rangle+\alpha_{01}|01\rangle}{\sqrt{|\alpha_{00}|^{2}+|\alpha_{01}|^{2}}}.$$

Note that the post-measurement state is re-normalized by the factor $\sqrt{|\alpha_{00}|^{2}+|\alpha_{01}|^{2}}$ so that it still satisfies the normalization condition.

References

[1] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, 10th Anniversary Ed., Cambridge: Cambridge University Press, 2010.

[2]. https://en.wikipedia.org/wiki/Measurement_in_quantum_mechanics

[3]. https://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanics

[4]. https://en.wikipedia.org/wiki/Born_rule

[5]. https://en.wikipedia.org/wiki/POVM