Projective measurements

See also

This content is adapted from the work of Timothée Dao; “Informationally Complete Generalized Measurements for Estimating Quantum Expectation Values” [Unpublished master’s thesis]; ETH Zürich (2023).

Measurement in an arbitrary basis

Consider a $d$-dimensional quantum system with a computational basis $\{|k⟩{\}}_{k=1}^d$. The corresponding projective measurement in this basis is described by $\mathbf{P}=\{|k⟩⟨k|{\}}_{k=1}^d$. Suppose we want to perform a projective measurement in another orthonormal basis $\{|{\psi }_k⟩{\}}_{k=1}^d$. As a unitary transformation is equivalent to a change of basis, there exists a unitary $U$ such that $|{\psi }_k⟩=U|k⟩$ for all $k=1,2,\dots ,d$. This implies that the new projective measurement is described by the PVM $\{|{\psi }_k⟩⟨{\psi }_k|{\}}_{k=1}^d=\{U|k⟩⟨k|U^{†}{\}}_{k=1}^d$. The probability of obtaining the outcome $k$ is then given by

(5)$$p_k=\mathrm{Tr}[|{\psi }_k⟩⟨{\psi }_k|\rho ]=\mathrm{Tr}[U|k⟩⟨k|U^{†}\rho ]=\mathrm{Tr}[|k⟩⟨k|U^{†}\rho U],$$

where we used the invariance of the trace under cyclic permutations in the last equality. It becomes now clear that the two procedures described below are equivalent:

Procedure 1A	Procedure 1B
Prepare state $\rho$	Prepare state $\rho$
2. Measure in the basis $\{|{\psi }_k⟩{\}}_k=\{U|k⟩{\}}_k$	2. Let the state evolve as $\rho \mapsto U^{†}\rho U$
	3. Measure in the computational basis

This equivalence is relevant in many practical situations. In experiments, one can often only perform measurements in a single, fixed (computational) basis but can apply various unitary transformations to the state before the measurement. Therefore, through this equivalence, one can emulate other projective measurements.

Example:

Consider a qubit system and suppose we only have an apparatus performing measurements in the computational basis, ${\mathbf{M}}_Z=\{Z_{+},Z_{-}\}=\{|0⟩⟨0|,|1⟩⟨1|\}$. We can still perform an $X$ measurement, ${\mathbf{M}}_X=\{X_{+},X_{-}\}=\{|+⟩⟨+|,|-⟩⟨-|\}$, by applying the Hadamard transformation

(6)$$\begin{array}{r}H=\frac{1}{\sqrt{2}}\left[\begin{array}{cc}1& 1\\ 1& -1\end{array}\right]\end{array}$$

to the state and then performing a measurement in the computational basis. Indeed, we have $|+⟩=H|0⟩$ and $|-⟩=H|1⟩$.

PM-simulable measurements

We can extend the Procedures 1A and 1B to PM-simulable POVMs, which can always be achieved by a randomization technique. Suppose we want to perform the measurement associated with the POVM $\mathbf{M}=\underset{i}{⨄}{q}_i{\mathbf{P}}_i=\{q_i|{\psi }_k^i⟩⟨{\psi }_k^i|{\}}_{(i,k)}$, where $\{q_i{\}}_i$ is a probability distribution and ${\mathbf{P}}_i=\{|{\psi }_k^i⟩⟨{\psi }_k^i|{\}}_k$ are rank-1 PVMs. The outcomes are labeled by the pair $(i,k)$. Let $\{U_i{\}}_i$ be the set of unitary operators such that $|{\psi }_k^i⟩=U_i|k⟩$ for all $k,i$. Then, the two procedures described below are equivalent:

Procedure 2A	Procedure 2B
1. Prepare state $\rho$	Prepare state $\rho$
2. Randomly pick $i$ with probability $q_i$	2. Randomly pick $i$ with probability $q_i$
3. Measure in the basis $\{|{\psi }_k^i⟩{\}}_k=\{U_i|k⟩{\}}_k$	3. Let the state evolve as $\rho \mapsto U_i^{†}\rho U_i$
	4. Measure in the computational basis

Note that usually, not all unitary operations are achievable in practice. Therefore, instead of starting from the POVM we would ideally like to perform, we usually first define the set of achievable unitary operations $\mathcal{U}=\{U_i{\}}_i$. We then determine the corresponding set of achievable PVMs $\mathcal{S}=\{{\mathbf{P}}_i{\}}_i$, where ${\mathbf{P}}_i=\{U_i|k⟩⟨k|U_i^{†}{\}}_k,\mathrm{\forall }i$. Finally, we choose the POVM to be performed from the convex hull ${\mathcal{S}}^{\mathrm{conv}}$.