Projection Measurement

Traditionally, a measurement, in the sense of Von Neumann, is a series of projection operators. By performing spectral decomposition on the self-adjoint operator corresponding to the observable, denoted as $ O = \sum_i \lambda_i |\varphi_i\rangle\langle\varphi_i| $, we obtain these projection operators $ |\varphi_i\rangle\langle\varphi_i| $. This part is well-known to students who have studied elementary quantum mechanics.

Besides the Von Neumann measurement, there is a more general type of measurement called Generalized Measurements or Positive Operator Valued Measures (POVMs).

Generalized Measurements (POVMs)

Definition: POVM

A POVM is a mapping $ \mathsf{E}: X \rightarrow \mathcal{L}(\mathcal{H}) $, satisfying

$ \mathsf{E}(x) \ge 0, \quad \forall x \in X $ $ \sum_ {x \in X} \mathsf{E}(x) = \mathbb{I}_{\mathcal{H}} $ where $ X $ represents the set of possible measurement outcomes, and $ \mathcal{L}(\mathcal{H}) $ represents the set of all bounded operators on $ \mathcal{H} $. Born’s Rule

The probability of obtaining result $ x $ when measuring a quantum state $ \rho $ with a POVM $ \mathsf{E} $ is given by $ p_{\rho}^{\mathsf{E}}(x) = \operatorname{tr}[\rho \mathsf{E}(x)] $.

When I first encountered POVMs, I was not clear about how they are implemented. This is because traditional measurements involve projection operators satisfying $ \mathsf{E}(x)^2 = \mathsf{E}(x), \quad \forall x \in X $. However, general POVMs do not satisfy this condition.

In fact, traditional projection measurements, also known as PVMs (Projection Valued Measures), are a special type of POVM.

PVMs are to POVMs what pure states are to mixed states. In other words, a POVM is a statistical mixture of PVMs, just as a mixed state is a statistical mixture of pure states.

Implementing POVMs

So, how do we implement POVMs? We can actually use PVMs and composite systems to implement POVMs.

Consider a system with a Hilbert space $ \mathcal{H} $ and a state $ \rho $.

Next, we couple system $ \mathcal{H} $ with another system $ \mathcal{K} $. Suppose the initial state of system $ \mathcal{K} $ is $ \sigma $. Then, the initial state of the composite system is $ \rho \otimes \sigma $ in $ \mathcal{H} \otimes \mathcal{K} $.

After that, we let the composite system evolve for a period of time, resulting in the system’s state becoming $ U(\rho \otimes \sigma) U^\dag $.

Finally, we perform a PVM $ \mathsf{Z} $ measurement on subsystem $ \mathcal{K} $. According to Born’s rule, the probability of obtaining result $ x $ is $ \operatorname{tr}[U(\rho\otimes \sigma) U^\dag(\mathbb{I}_{\mathcal{H}}\otimes \mathsf{Z}(x))] $.

Thus, we have implemented a POVM $ \mathsf{E} $ such that $ \operatorname{tr}[\rho \mathsf{E}(x)] = \operatorname{tr}[U(\rho\otimes \sigma) U^\dag(\mathbb{I}_{\mathcal{H}}\otimes \mathsf{Z}(x))] $.

Measurement Model

Let’s summarize the physical objects involved in implementing POVMs: the ancilla system $ \mathcal{K} $, the initial state $ \sigma $ of system $ \mathcal{K} $, the evolution $ U $ of the composite system, and the projection measurement $ \mathsf{Z} $ on the ancilla system. These physical objects together realize a POVM measurement.

So, we can define $ \mathfrak{M} = (\mathcal{K},\sigma,U,\mathsf{Z}) $ as a Measurement Model.

In actual experiments, the ancilla system $ \mathcal{K} $ can be considered as a probe of the instrument, and $ \mathsf{Z} $ is the readout of the probe.

Post-Measurement State POVMs themselves cannot determine the post-measurement state. The actual determination of the post-measurement state depends on the specific implementation of the measurement, that is, the measurement model.

For a measurement model $ \mathfrak{M} = (\mathcal{K},\sigma,U,\mathsf{Z}) $, the post-measurement state of system $ \mathcal{H} $ is $ \rho_x = \operatorname{tr}_{\mathcal{K}}[U(\rho\otimes \sigma) U^\dag(\mathbb{I}_{\mathcal{H}}\otimes \mathsf{Z}(x))] $, where $ \operatorname{tr}_{\mathcal{K}}[\cdot] $ denotes the partial trace.

Quantum Instruments

POVMs provide probabilities for each measurement outcome but do not determine the post-measurement state. Therefore, we aim to find a mathematical object that includes both the measurement probabilities and the post-measurement states. This mathematical object is called a Quantum Instrument.

Definition: Quantum Instrument

An operator $ \mathcal{I}_x: \mathcal{I}_x(\rho) = \rho_x $ is called a Quantum Instrument if $ \operatorname{tr}\left[\sum_{x\in X} \rho_x \right] = \operatorname{tr}[\rho] = 1 $.

It can be seen that $ \mathcal{I}_x $ not only provides the measurement probability $ p_x = \operatorname{tr}[\rho_x] $, but also gives the post-measurement state $ \frac{\rho_x}{\operatorname{tr}[\rho_x]} $.

Obviously, the same POVM corresponds to countless Quantum Instruments.

So, for any Quantum Instrument, can we always implement it? The answer is yes.

Theorem

Given any Quantum Instrument, we can always find a Measurement Model to implement it (in fact, there are countless implementations). Proof omitted.