Generalised quantum measurements

Last updated Aug 5, 2023

• Rework

The Ideal quantum measurements can be generalised:

Let $\mathcal{M}$ be a set of possible measurement outcomes (for simplicity discrete)

1. The probability $P(m)$ for a measurement result $m\in\mathcal{M}$ in a system $\rho$ is given by $$P(m) = \text{tr}(F_m \rho),$$ where $F_m$ defines the effect

Definition: Effect

The effect is a positive operator with $$\sum_m F_m = I$$

2. For a selective measurement, after measuring the value $m$, the system takes the state $$\rho^\prime_m = P(m)^{-1} \Phi_m(\rho),$$ where $\Phi_m(\rho)$ defines the operation

Definition: Operation

The Operation is a positive super-operator (operator acting on space of operators) with $$\text{tr}\Phi_m(\rho) = \text{tr}(F_m \rho)$$

3. For a non-selective measurement, the density matrix becomes $$\rho^\prime = \sum_{m\in\mathcal{M}}P(m)\rho^\prime_m = \sum_{m\in\mathcal{M}}\Phi_m(\rho)$$

Example: Ideal measurement

Ideal quantum measurements are a special case of this general formulation, with the effect $F_m=\Delta E_\alpha$ and the operation $\Phi(\rho)=\Delta E_\alpha\rho\Delta E_\alpha$. Here $\Delta E_\alpha$ is a decomposition of unity, which means $\sum_\alpha \Delta E_\alpha =I$ is a part of the spectral family for a unique interval of the real axis $\Delta r_{\alpha} = (r_{\alpha-1}, r_\alpha]$: $$\Delta E_\alpha = E_{r_\alpha} - E_{r_{\alpha - 1}}$$

We consider an arbitrary operation $\Phi_m(\rho)$. We want to identify $P(m)=\text{tr}\Phi_m(\rho)$. $\Phi_m$ is supposed to be an operation, associated with a measurement outcome $m$ which can be rewritten as effect through the cyclic trace. For the identification as probability to be possible the operator needs to satisfy three conditions:

1. Normalization for probability: $$0 \leq \text{tr}\Phi_m(\rho) \leq 1$$
2. Convex linear (to fulfill Bayes): for different $p_i$ and $\rho_i$ $$\Phi_m\left(\sum_i p_i\rho_i\right) = \sum_i p_i\Phi_m(\rho_i)$$
3. Completely positive (For partial traces to be physical): For a space $\mathcal{H}\otimes\tilde{\mathcal{H}}$ $\Phi_m\otimes I$ is positive for $\Phi_m \in \mathcal{H}$ and $I\in\tilde{\mathcal{H}}$ for any size of $\tilde{\mathcal{H}}$-

Representation theorem

An operator fulfills the three conditions normalization, convex linearity, and complete positivity iff it can be written as $$\Phi_m(\rho) = \sum_k \Omega_{mk}\rho\Omega^\dagger_{mk}$$ and the Effect $$F_m = \sum_k \Omega_{mk}^\dagger\Omega_{mk} \leq I,$$ where the ordering $\leq$ is understood with respect to $I-F_m$ beeing a Positive operator

Note:

• two operations $\Phi_m$ and $\tilde \Phi_m$ are equal iff $\exists u: u^{-1}u=1 \land \Omega_{mk} = \sum_l u_{kl}\tilde{\Omega}_{ml}$

In general a quantum measurement changes the Von Neumann entropy. One can show by inserting the representation $\rho^\prime = \sum_{m\in\mathcal{M}}\Phi_m(\rho)$ that for a ideal non-selective measurement the change of entropy $$\Delta S = S(\rho) - S(\rho^\prime) \geq 0$$ Note:

• $\Delta S = 0$ iff $\rho = \rho^\prime$
• Generalized measurements can decrease the entropy

Definition: Approximate measurement

Given we are measuring an operator $$\hat{R} = \sum_m r_m\ket{\psi_m}\bra{\psi_m}$$ an approximate measurement arises if the eigenvalues $r_m$ are too closely spaced for the resolution of the measurement apparatus.

• This is accounted for by a conditional probability $W(m|m^\prime)$, which gives the probability of measuring $r_m$ if the system is in the state $\ket{\psi_m}$.
• For an exact measurement we would of course have $W(m|m^\prime) = \delta_{mm^\prime}$ and the normalization is given by $\sum_m W(m|m^\prime)=1$.

The probability distribution for the measurement outcome $r_m$ becomes

$$\boxed{P(m) = \sum_{m^\prime} W(m|m^\prime)\braket{\psi_{m^\prime}|\hat{R}|\psi_{m^\prime}}}$$

And the effect operator

$$\boxed{F_m = \sum_{m^\prime} \ket{\psi_{m^\prime}}W(m|m^\prime)\bra{\psi_{m^\prime}}}$$

The most general expression for the operation is found to be

$$\boxed{\Omega_m = U_m F^{1/2}m},$$ where $U_m$ is some unitary matrix determinded by the measurement device and $$F^{1/2}m := \sum{m^\prime}\ket{\psi{m^\prime}}\sqrt{W(m|m^\prime)}\bra{\psi_{m^\prime}},$$ leading to the operation $$\boxed{\Phi_m(\rho)=U_mF^{1/2}_m\rho F^{1/2}_mU^\dagger_m}$$

The process can be viewed as consisting of two steps, first $F^{1/2}_m$ measurement, then $U_m$ measruement. Since $\hat{R}$ and $F_m$ have the same spectral family (why?) $[\hat{R},F_m]=0$.

In an indirect measurement we:

1. Have a object $\mathcal{H}_O$ to be measured in state $\rho_O$
2. Have a probe $\mathcal{H}_P$ as auxillary system in state $\rho_P$
3. The probe is brought into interaction with the system until a fixed time $t=\tau$
4. An ideal measurement is performed on the probe to obtain information about the object

Think of a thermometer brought into a bath and the temperature beeing read of from it.

1. At $t=0$ the probe and object did not interact yet so the total system is in a seperatable state $\rho_O\otimes\rho_P$

2. The evolution is given by the hamiltonian $$H(t) = H_O + H_P + H_I(t),$$ where the interaction $H_I(t)$ vanishes for $t\notin [0,\tau]$. The evolution of the full system is given by the time evolution operator $$U:=U(\tau,0)=T\exp\left[-i\int^\infty_0dt H(t)\right]$$ with the full density matrix at $\tau$ $$\rho(\tau)=U(\rho_O\otimes\rho_P)U^\dagger$$

3. At $\tau$ we measure an operator $\hat{R}$, which acts on $\mathcal{H}P$. With $\hat{R}$ given as ($\ket{\psi_m}$ are states of the probe!) $$\hat{R}=\sum_m r_m\ket{\psi_m}\bra{\psi_m}$$ the probability for an outcome $r_m$ is $$P(m)=\text{tr}(\ket{\psi_m}\bra{\psi_m}\rho(\tau)) = \text{tr}(U^\dagger\ket{\psi_m}\bra{\psi_m}U(\rho_O\otimes\rho_P))$$ This can be written in terms of two partial traces of subsystems: $$P(m)=\text{tr}O(F_m\rho_O)$$ Where the effect of the object is given by the partial trace over the probe $$F_m\rho_O = \text{tr}P(U^\dagger\ket{\psi_m}\bra{\psi_m}U(\rho_O\otimes\rho_P))$$ From the fundamental measurement postulate for the Ideal quantum measurements we get the density matrix after the measurement $\rho^\prime_m$ by just identifying the projection operator $E(m)=\ket{\psi_m}\bra{\psi_m}$ how this form? $$\rho^\prime_m = P(m)^{-1}\braket{\psi_m|U(\rho_O\otimes\rho_P)U^\dagger|\psi_m}$$ We can then write the operation on the object in a form like given in the representation theorem: $$ \Phi_m(\rho_O) = \sum_k \Omega{mk}\rho_O\Omega^\dagger{mk} $$ $$F_m = \sum_k \Omega^\dagger{mk}\Omega_{mk}.$$ with $$\Omega_{mk} = \sqrt{p_k}\braket{\psi_m|U|\phi_k}$$ where $\rho_P = \sum_k p_k\ket{\phi_k}\bra{\phi_k}$ is the spectral decomposition of the prope system $\rho_P$.

Example: Indirect momentum measurement

We want to indirectly measure the observable $A$ $$A=\sum_k a_k \ket{\chi_k}\bra{\chi_k}$$ of an object by measureing the momentum $P$ of a probe. For the indirect measurement, the position $Q$ of the probe is coupled to the object by the interaction $$H_I(t) = g(t)AQ$$ Neglecting the free evolution of the object and the probe, the time evolution is given by $$U = \exp(-i\int^t_0dt g(t)AQ)$$. Then for a pure probe $\rho_P = \ket{\phi}\bra{\phi}$ we have the object operation for the momentum measurement $p$ follow with the representation theorem from $$\Omega_p = \braket{p|U|\phi} = \sum_\nu \ket{\chi_\nu}\braket{p|\exp\left[-iG a_\nu Q\right]|\phi}\bra{\chi_\nu}$$ where we chose momentum eigenstates $\ket{p}$ as basis for $\ket{\psi_m}$. Then $$\Omega_p = \sum_\nu \ket{\chi_\nu}\phi(p+Ga_\nu)\bra{\chi_\nu} = \phi(p+GA),$$ with $\phi(p) = \braket{p|\phi}$ and measurement probabilites for the object $$P(p) = \text{tr}(\Omega^\dagger_p\Omega_p \rho_O) = \sum_\nu |\phi(p+Ga_\nu)|^2\braket{\chi_\nu|\rho_O|\chi_\nu}$$ This is the probability of measurements of $P$ and now we want to infer some information about $A$ from that. We can do this by interpreting the measurement as an approximate measurement.

The operation which follows from $\Omega_p$ projects the state of the object on the eigenstates of $A$, $\chi_\nu$. The eigenvalues satisfy (why?) $$p + Ga_\nu \approx \braket{p}$$ The distribution of eigenvalues of $A$ $$a = \frac{1}{G}(\braket{p}-p)$$ is then (derivation??) $$f(a)=\sum_\nu W(a - a_\nu)\braket{\chi_\nu|\rho_O|\chi_nu}$$ with the conditional probability $W$ like in the approximate measurement $$W(a-a_\nu)=|G|\cdot |\phi(\braket{p} - G(a-a_\nu))|^2$$ ?? so I measure p of the probe and get a value, how to obtain an a ??