Randomization of Quantum 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work of Timothée Dao; “Informationally Complete Generalized Measurements for Estimating Quantum Expectation Values” [Unpublished master’s thesis]; ETH Zürich (2023).
Convexity¶
Following Ref. [1], the set of
- Example:
Consider a
-outcome POVM on a single-qubit system : . It can be considered as a 4-outcome POVM with a null effect, . Note that is an equivalent POVM up to the re-ordering of the effects.
Consider two POVMs
respectively. Crucially, these operations do not preserve the set of
POVMs. That is, the effects resulting from these operations do not form
POVMs in general. Indeed, it is easy to check that the resulting effects
do not sum to the identity (except for the scalar multiplication by
Lemma [2]:
The set
is convex. That is, for any and any , we have .
- Proof:
The elements of
, given by for , are positive semi-definite by the convexity of the set of positive semi-definite operators . Moreover,(8)¶which concludes the proof. ◻
Such convex combinations can be extended to more than two POVMs.
Consider a set of POVMs
as a corollary of Lemma [2].
The set of all convex combinations generated by a set
In practice¶
These convex combinations can be achieved in practice by the so-called randomization of quantum measurements procedure, which simply consists of two steps for each measurement shot:
Randomly pick
with probability ,Perform the measurement associated with
.
Indeed, the probability of obtaining the outcome
which is exactly equivalent to directly performing the measurement
associated with
- Example:
To illustrate the usefulness of this procedure, imagine one has two measurement apparatuses associated with the POVMs
and respectively. To perform the measurement associated with , one does not need to build a new apparatus but only needs to use each of the two existing apparatuses with probability and respectively.- Example:
Consider two POVMs
. We have(11)¶and also
(12)¶where in the second example
and are considered as 4-outcome POVMs with 2 null effects: respectively.
We will use the latter construction very often. Therefore, given two
POVMs
where
where the outcomes are now denoted by the pair
References
Footnotes
Convexity LEMMA
The concept of multiset is an extension of the notion of set, where each element can be represented multiple times.