Measurement Operator

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The following discussion was largely taken from Howard Wiseman's thesis, section 2.2. We describe both the system and environment as pure states for simplicity. The environment is prepared in the state $|\psi_{E}\rangle$ . The total state, which we assume to be initially separable, evolves into the generally entangled form

$|\psi_{T}(t+dt)\rangle=U(dt)|\psi_{E}\rangle|\psi_{S}(t)\rangle$ .

Through measurement, the environment is projected into a particular state $α$ via the projector

$P_{\alpha}=1_{S}\otimes|\psi_{E,\alpha}\rangle\langle\psi_{E,\alpha}|$ .

So the resulting unnormalized total state is

$|\psi_{T,\alpha}(t+dt)\rangle=P_{\alpha}|\psi_{T}(t+dt)\rangle=|\psi_{E,\alpha}\rangle|\psi_{S,\alpha}(t+dt)\rangle$

where

$\begin{matrix} |\psi_{S,\alpha}(t+dt)\rangle&=&\Omega_{\alpha}(dt)|\psi_{S}(t)\rangle\\ \Omega_{\alpha}(dt)&=&\langle\psi_{E,\alpha}|U(dt)|\psi_{E}\rangle \end{matrix}$

Notice that the measurement operator $Ω α (d t)$ which acts on only the system subspace is not necessarily a projector. Also note that this reduction is only useful if the environment has certain dynamical properties. If we are to use the same measurement operators for every time step, the environment must quickly return to its prepared state (equilibrium approximation) and never return the measured information to the system (Markovian approximation). Of course, we may explicitly return information to the system via Hamiltonian feedback, but here we are considering the open loop dynamics alone.

Dynamics of Continuous Measurement

Now we assume the existence of valid measurement operators $Ω α (d t)$ and work out the dynamics. The continuous measurement operators are usually specified as follows, with the ${L i}$ being the so-called jump operators

$\begin{matrix} \Omega_{i}(dt)&=&\sqrt{dt}L_{i}\\ \Omega_{0}(dt)&=&1-iH_{eff}dt=1-i(H_{S}-iK)dt \end{matrix}$

with $i = 1,2,3,...$ and $K=\frac{1}{2}\sum_{i>0} L_{i}^{\dagger}L_{i}$ from the Kraus Normalization Condition.

Conditional Evolution

Let us define an effect $F i$ to be associated with a measurement operator through $F_{i}=\Omega^{\dagger}_{i}\Omega_{i}$ . In the simulation of conditional evolution, we imagine flipping a coin weighted according to $P i = T r (F i ρ)$ to get a result $i$ . In experiment, measure for some period of time to get a result $i$ (including the null measurement of $i = 0$ ). In either case, given result $i$ , apply:

$\begin{matrix} \rho(t+dt)= \frac{\Omega_{i}(dt)\rho(t)\Omega^{\dagger}_{i}(dt)}{P_{i}} \end{matrix}$

The random evolution described by this formalism is referred to as a quantum trajectory.

Unconditional Evolution

In the case where the measurement record is ignored, our best estimate of the system state must be given by the evolution

$\begin{matrix} \rho(t+dt)=\sum_{i} \Omega_{i}(dt)\rho(t)\Omega^{\dagger}_{i}(dt) \end{matrix}$

$\frac{d\rho}{dt}=\mathcal{L}\rho$

where $\mathcal{L}$ is the Liouvillian (i.e. Lindbladian). The average of the conditional trajectories should reproduce this unconditional behavior.

Transforming the Measurement

Formally, one can unitarily rearrange the measurement operators without changing the unconditional evolution, thus creating what is called a different unravelling of the master equation. Thus the transformation

$\begin{matrix} \Omega_{i}'=\sum_{j} U_{i,j} \Omega_{j} \end{matrix}$

results in no change of the unconditional evolution

$\begin{matrix} \rho(t+dt)&=&\sum_{j} \Omega_{j}(dt)\rho(t)\Omega^{\dagger}_{j}(dt)\\ &=&\sum_{i} \Omega'_{i}(dt)\rho(t)\Omega^{'\dagger}_{i}(dt)\end{matrix}$

where $\sum_{r}U_{r,s}U^{*}_{r,q}=\delta_{s,q}$ .

Whether or not a particular unraveling is physically realizable is another question. For the decaying mode of a cavity, different unravelings correspond to different means of detection. Direct detection results in conditional evolution with jump like behavior. But by adding a local oscillator to the output field one can perform a homodyne measurement, which in the large oscillator limit results in diffusive motion of the conditional state. The switch from direct detection to homodyne (or heterodyne) corresponds to a unitary re-arrangement of the direct detection measurement operators.

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Categories: Quantum Measurement | Concept | Quantum Mechanics