Weak measurement

Consider a system that starts in the quantum state ${\displaystyle |\psi \rangle }$ and an ancilla that starts in the state ${\displaystyle |\phi \rangle }$, the combined initial state is ${\displaystyle |\Psi \rangle =|\psi \rangle \otimes |\phi \rangle }$. These two systems interact via the Hamiltonian ${\displaystyle H=A\otimes B}$, which generates the time evolutions ${\displaystyle U(t)=\exp[-ixtH]}$ (in units where ${\displaystyle \hbar =1}$), where ${\displaystyle x}$ is the "interaction strength", which has units of inverse time. Assume a fixed interaction time ${\displaystyle t=\Delta t}$ and that ${\displaystyle \lambda =x\Delta t}$ is small, such that ${\displaystyle \lambda ^{3}\approx 0}$. A series expansion of ${\displaystyle U}$ in ${\displaystyle \lambda }$ gives

${\displaystyle {\begin{aligned}U&=I\otimes I-i\lambda H-{\frac {1}{2}}\lambda ^{2}H^{2}+O(\lambda ^{3})\\&\approx I\otimes I-i\lambda A\otimes B-{\frac {1}{2}}\lambda ^{2}A^{2}\otimes B^{2}.\end{aligned}}}$

Because it was only necessary to expand the unitary to a low order in perturbation theory, we call this is a weak interaction. Further, the fact that the unitary is predominately the identity operator, as ${\displaystyle \lambda }$ and ${\displaystyle \lambda ^{2}}$ are small, implies that the state after the interaction is not radically different from the initial state. The combined state of the system after interaction is

${\displaystyle |\Psi '\rangle =\left(I\otimes I-i\lambda A\otimes B-{\frac {1}{2}}\lambda ^{2}A^{2}\otimes B^{2}\right)|\Psi \rangle .}$

Now we perform a measurement on the ancilla to find out about the system, this is known as an ancilla-coupled measurement. We will consider measurements in a basis ${\displaystyle |q\rangle }$ (on the ancilla system) such that ${\displaystyle \sum _{q}|q\rangle \langle q|=I}$. The measurements action on both systems is described by the action of the projectors ${\displaystyle \Pi _{q}=I\otimes |q\rangle \langle q|}$ on the joint state ${\displaystyle |\Psi '\rangle }$. From quantum measurement theory we know the conditional state after the measurement is

${\displaystyle {\begin{aligned}|\Psi _{q}\rangle &={\frac {\Pi _{q}|\Psi '\rangle }{\sqrt {\langle \Psi '|\Pi _{q}|\Psi '\rangle }}}\\&={\frac {I\langle q|\phi \rangle -i\lambda A\langle q|B|\phi \rangle -{\frac {1}{2}}\lambda ^{2}A^{2}\langle q|B^{2}|\phi \rangle }{\mathcal {N}}}|\psi \rangle \otimes |q\rangle ,\end{aligned}}}$

where ${\displaystyle {\mathcal {N}}={\sqrt {\langle \Psi '|\Pi _{q}|\Psi '\rangle }}}$ is a normalization factor for the wavefunction. Notice the ancilla system state records the outcome of the measurement. The object ${\displaystyle M_{q}:=I\langle q|\phi \rangle -i\lambda A\langle q|B|\phi \rangle -{\frac {1}{2}}\lambda ^{2}A^{2}\langle q|B^{2}|\phi \rangle }$ is an operator on the system Hilbert space and is called a Kraus operator.

With respect to the Kraus operators the post-measurement state of the combined system is

${\displaystyle |\Psi _{q}\rangle ={\frac {M_{q}|\psi \rangle }{\sqrt {\langle \psi |M_{q}^{\dagger }M_{q}|\psi \rangle }}}\otimes |q\rangle .}$

The objects ${\displaystyle E_{q}=M_{q}^{\dagger }M_{q}}$ are elements of what is called a POVM and must obey ${\displaystyle \sum _{q}E_{q}=I}$ so that the corresponding probabilities sum to unity: ${\displaystyle \sum _{q}\Pr(q|\psi )=\sum _{q}\langle \psi |E_{q}|\psi \rangle =1}$. As the ancilla system is no longer correlated with the primary system, it is simply recording the outcome of the measurement, we can trace over it. Doing so gives the conditional state of the primary system alone:

${\displaystyle |\psi _{q}\rangle ={\frac {M_{q}|\psi \rangle }{\sqrt {\langle \psi |M_{q}^{\dagger }M_{q}|\psi \rangle }}},}$

which we still label by the outcome of the measurement ${\displaystyle q}$. Indeed, these considerations allow one to derive a quantum trajectory.

We will use the canonical example of Gaussian Kraus operators given by Barchielli, Lanz, Prosperi;[13] and Caves and Milburn.[17] Take ${\displaystyle H=x\otimes p}$, where the position and momentum on both systems have the usual Canonical commutation relation ${\displaystyle [x,p]=i}$. Take the initial wavefunction of the ancilla to have a Gaussian distribution

${\displaystyle |\Phi \rangle ={\frac {1}{(2\pi \sigma ^{2})^{1/4}}}\int dq'\exp[-q'^{2}/(4\sigma ^{2})]|q'\rangle .}$

The position wavefunction of the ancilla is

${\displaystyle \Phi (q)=\langle q|\Phi \rangle ={\frac {1}{(2\pi \sigma ^{2})^{1/4}}}\exp[-q^{2}/(4\sigma ^{2})].}$

The Kraus operators are (compared to the discussion above, we set ${\displaystyle \lambda =1}$)

${\displaystyle {\begin{aligned}M(q)&=\langle q|\exp[-ix\otimes p]|\Phi \rangle \\&={\frac {1}{(2\pi \sigma ^{2})^{1/4}}}\exp[-(q-x)^{2}/(4\sigma ^{2})],\end{aligned}}}$

while the corresponding POVM elements are

${\displaystyle {\begin{aligned}E(q)&=M_{q}^{\dagger }M_{q}\\&={\frac {1}{\sqrt {2\pi \sigma ^{2}}}}\exp[-(q-x)^{2}/(2\sigma ^{2})],\end{aligned}}}$

which obey ${\displaystyle \int dq\,E(q)=I}$. An alternative representation is often seen in the literature. Using the spectral representation of the position operator ${\displaystyle x=\int x'dx'|x'\rangle \langle x'|}$, we can write

${\displaystyle {\begin{aligned}M(q)&={\frac {1}{(2\pi \sigma ^{2})^{1/4}}}\int dx'\exp[-(q-x')^{2}/(4\sigma ^{2})]|x'\rangle \langle x'|,\\E(q)&={\frac {1}{\sqrt {2\pi \sigma ^{2}}}}\int dx'\exp[-(q-x')^{2}/(2\sigma ^{2})]|x'\rangle \langle x'|.\end{aligned}}}$

Notice that ${\displaystyle \lim _{\sigma \to 0}E(q)=|x=q\rangle \langle x=q|}$.[17] That is, in a particular limit these operators limit to a strong measurement of position; for other values of ${\displaystyle \sigma }$ we refer to the measurement as finite-strength; and as ${\displaystyle \sigma \to \infty }$, we say the measurement is weak.