Introduction¶

Generalized measurements¶

The most general class of measurements in quantum mechanics are described by the POVM formalism. An \(n\)-outcome POVM is a set of \(n\) positive semi-definite Hermitian operators \(\mathbf{M} = \{M_k\}_{k \in \{1, \dots, n \}}\) that sum to the identity. Mathematically, this means the set of operators satisfies the following properties:

\(M_k^\dagger = M_k\) for all \(k \in \{1,2 \dots, n\}\),

\(\langle \psi | M_k | \psi \rangle \geq 0\) for all \(k \in \{1,2 \dots, n\}\) and all states \(|\psi \rangle\),

\(\sum_{k=1}^n M_k = \mathbb{I}\).

Given a \(d\)-dimensional state \(\rho\), the probability of observing outcome \(k\) is given by Born’s rule as \(p_k = \mathrm{Tr}[\rho M_k]\). Standard projective measurements (PMs) are a special case of POVMs, where each POVM operator is a projector such that \(M_k = \ketbra{\phi_k}{\phi_k}\) for some pure states \(\phi_k\). A POVM is said to be informationally complete (IC) if it spans the space of Hermitian operators [1]. Then, for any observable \(\mathcal{O}\), there exist \(\omega_k \in \mathbb{R}\) such that

(1)¶\[\mathcal{O}= \sum_{k=1}^{n} \omega_k M_k .\]

Given such a decomposition of \(\mathcal{O}\), the expectation value \({\langle\mathcal{O}\rangle}_\rho\) can be written as

(2)¶\[{\langle\mathcal{O}\rangle}_\rho = \mathrm{Tr}[\rho O] = \sum_k \omega_k \mathrm{Tr}[\rho M_k] = \mathbb{E}_{k \sim \{p_k\}}[\omega_k].\]

In other words, \({\langle\mathcal{O}\rangle}_\rho\) can be expressed as the mean value of the random variable \(\omega_k\) over the probability distribution \(\{p_k\}\). Given a sample of \(S\) measurement outcomes \(\{ k^{(1)}, \dots, k^{(S)} \}\), we can thus construct an unbiased Monte-Carlo estimator of \({\langle\mathcal{O}\rangle}_\rho\) as

(3)¶\[\hat{o} : \{k^{(1)},\dots, k^{(S)}\} \mapsto \frac{1}{S} \sum_{s=1}^{S} \omega_{k^{(s)}}.\]

The expected value of the estimator is given by \(\mathbb{E}[\hat{o}] = {\langle\mathcal{O}\rangle}_\rho\) and its variance is given by

(4)¶\[\mathrm{Var}[\hat{o}] = \frac{1}{S}\left(\sum_k p_k \omega_k^2 - ( \sum_k p_k \omega_k )^2\right) \propto \sum_k p_k \omega_k^2 - {\langle\mathcal{O}\rangle}_\rho^2 \, ,\]

which depends on both the choice of the POVM \(\{M_k\}_{k}\) and the decomposition weights \(\{\omega_k\}_{k}\), when they are not unique. Ideally we would like to choose the POVM and the decomposition weights that minimize the variance.

PM-simulable POVMs¶

Digital quantum computers typically only give access to projective measurements (PMs) in a specified computational basis. More general POVMs can be implemented through additional quantum resources, e.g., by coupling to a higher-dimensional space in a Naimark dilation [2] with ancilla qubits [3] or qudits [4] [5] or through controlled operations with mid-circuit measurements and classical feed-forward [6]. While these techniques have been demonstrated in proof-of-principle experiments, their full-scale high-fidelity implementation remains a challenge for current quantum devices [4]. Of particular interest are thus POVMs that can be implemented without additional quantum resources, i.e., only through projective measurements in available measurement bases.

More complex POVMs can be built from available projective measurements through convex combinations of POVMs: For two \(n\)-outcome POVMs \(\mathbf{M}_1\) and \(\mathbf{M}_2\) acting on the same space, their convex combination with elements \(M_k = p M_{1,k} + (1-p) M_{2,k}\) for some \(p \in [0,1]\) is also a valid POVM. This can be achieved in practice by a randomization of measurements procedure, which simply consists of the following two steps for each measurement shot. First, randomly pick \(\mathbf{M}_1\) or \(\mathbf{M}_2\) with probability \(p\) or \(1-p\), respectively, then perform the measurement associated with the chosen POVM. We call POVMs that can be achieved by randomization of projective measurements PM-simulable. On digital quantum computers the easiest basis transformations are single-qubit transformations of the computational basis. POVMs that consist of single-qubit PM-simulable POVMs are thus the most readily accessible class of generalized measurements and have found widespread application. These include classical shadows and most of their derivatives.

Importantly, PM-simulable informationally-complete POVMs are overcomplete [7]. The decomposition of observables from Eq. (2) is thus not unique. In this toolbox, we leverage these additional degrees of freedom with frame theory.

References