Generalised quantum measurements

Last updated Dec 29, 2024

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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 $$\rho^\prime = \sum_{m\in\mathcal{M}}P(m)\rho^\prime_m = \sum_{m\in\mathcal{M}}\Phi_m(\rho)$$ So after the measurement, the state is in any eigenstate of the measured operator..

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 $\tau$ $$\rho(\tau)=U(\rho_O\otimes\rho_P)U^\dagger$$

4. 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 $\rho_P$. Note $\Omega{mk}$ is an operator in the Hilberstapce of the object, the probe d.o.f. are removed by the probe states $\bra{\psi_m}$ and $\ket{\phi_k}$.

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 $$U = \exp(-iGAQ),$$ with $G=\int^\tau_0dt g(t)$ Then for a pure probe $\rho_P = \ket{\phi}\bra{\phi}$ we have the object operation for the momentum measurement $p$ follow by inserting the decomposition of $A$ $$\Omega_p = \braket{p|U|\phi} = \sum_\nu \ket{\chi_\nu}\braket{p|\exp\left[-iG a_\nu Q\right]|\phi}\bra{\chi_\nu}$$ 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 $\phi(p) = \braket{p|\phi}$ the wave function of the probe in momentum space. The measurement probabilites for the object momenta becomes (using the effect $F=\Omega^\dagger\Omega$) $$P(p) = \text{tr}(\Omega^\dagger_p\Omega_p \rho_O) = \sum_\nu |\phi(p+Ga_\nu)|^2\braket{\chi_\nu|\rho_O|\chi_\nu}$$ 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 $A$, $\chi_\nu$. which satisfy (assuming $\phi(p)$ is peaked around its expectation value $\braket{p}$) $$p + Ga_\nu \approx \braket{p}$$. Which states exactly contribute depends also on the width of $\phi$ The distribution of eigenvalues of $A$ (defined as the $a$ inferred from the measurement of $p$) $$a = \frac{1}{G}(\braket{p}-p)$$ is then (by plugging $p(a)$ as defined above into $P(p)$) $$f(a)=\sum_\nu W(a - a_\nu)\braket{\chi_\nu|\rho_O|\chi_nu}$$ 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 the direct measurement of the probe momentum $P$ gives an approximate measurement of the object observable $A$.

The estimator of $a$ is unbiased, however $$\text{Var}(a)=\text{Var}(A)+\frac{1}{G^2}\text{Var}(P).$$

One can also write down the Operation for $a$.

• If an observable $A$ commutes with the density matrix $\rho$, a non-selective measurement will not change the state of the system $\rho$.

Definition: Quantum non-demolition measurements

A generalized measurement with operations $\Phi_m$ is defined as a measurement of an observable $A$ that leaves the probability distribution for an ideal measurement of $A$ the same.

This means that the distribution of eigenvalues of $A$ are the same for the density matrices before and after the measurement.

Mathematically the probability distributions are the same if all moments of the are the same. In QM the moments of the eigenvalue distribution are given by $$\text{tr}(A^k\rho)$$.

Therefore a Quantum non-demolition measurement is defined by $$\text{tr}(A^k\rho)=\text{tr}(A^k\rho\prime)=\sum_m\text{tr}(A^k\Omega_m\rho\Omega^\dagger_m).$$ This is equivalent to $$[A,\Omega_m]=0$$

So a measurement is a QND measurement if the measured operator $A$ and the operator inducing the change of the density matrix $\Omega_m$ commute.

• Ideal measurements are always QND measurements.
• Indirect measurements are always QND measurements.