Frame theory and dual space¶
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
POVMs as frames¶
As a reminder, given an IC-POVM
We will now outline a formal approach to obtain the coefficients
Simply speaking, a frame is a generalization
of the notion of the basis of a vector space, where the basis elements may
be linearly dependent. The set of POVM operators
for any Hermitian operator
Notably, dual operators generalize the concept of classical shadows of a quantum state [2] (see section below for details), thus providing a direct connection to the popular randomized measurement toolbox [3].
Constructing dual frames¶
For a minimally IC POVM, only one dual frame exists. It can be constructed from the POVM elements as
with the canonical frame superoperator
where we have used the widespread vectorized ‘double-ket’ notation. Thus, the frame superoperator can be used to transform between the POVM space and the dual space.
For an overcomplete POVM, the canonical frame superoperator creates one of infinitely many possible dual frames. Other valid dual frames can be obtained through a parametrized frame superoperator as follows:
for real parameters
Relation to classical shadows¶
We now show the explicit connection to the technique of classical shadows
[2]. The technique consists of rotating the state
is actually an
In other words, the classical shadows technique
consists of performing a PM-simulable POVM and choosing a specific
dual frame. However, nothing prevents us from choosing another dual
frame. Any dual frame defines an unbiased estimator
of the state. More precisely, for any dual frame
which follows from the reciprocity of duality. That is, if
References