Introduction¶
See also
This content is adapted from the work of Laurin E. Fischer, Timothée Dao, Ivano Tavernelli, and Francesco Tacchino; “Dual-frame optimization for informationally complete quantum measurements”; Phys. Rev. A 109, 062415; DOI: https://doi.org/10.1103/PhysRevA.109.062415
Generalized measurements¶
The most general class of measurements in quantum mechanics are
described by the POVM formalism. An
for all ,
for all and all states ,
.
Given a
Given such a decomposition of
In other words,
The expected value of the estimator is given by
which depends on both the choice of the POVM
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
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