Fix small typos and update explanantions (#88) (#92)

* typo * update explanations * update remaining mentions of "canonical dual frame" * simplify "ketbra" in explanations * update explanations (cherry picked from commit 83ffc87) Co-authored-by: Timothée Dao <[email protected]>
qiskit-community · Aug 14, 2024 · 43b82e8 · 43b82e8
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diff --git a/docs/explanations/frame.rst b/docs/explanations/frame.rst
@@ -115,7 +115,7 @@ unitary :math:`U_i`, sampled from a set :math:`\mathcal{U}`, and then
 performing a measurement in the computational basis. We show in the section :ref:`pm-simulable` that
 this protocol is equivalent to performing the PM-simulable POVM
 :math:`\mathbf{M} = \biguplus_i q_i \mathbf{P}_i = \{q_i P_{i,k}\}_{(i,k)}`,
-where :math:`P_{i,k} = U_i^\dagger \ketbra{k} U_i` and the outcomes are labeled by
+where :math:`P_{i,k} = U_i^\dagger \ketbra{k}{k} U_i` and the outcomes are labeled by
 :math:`(i,k)`. It now appears that the measurement channel
 
 .. math::

diff --git a/docs/explanations/introduction.rst b/docs/explanations/introduction.rst
@@ -46,7 +46,8 @@ Given such a decomposition of :math:`\mathcal{O}`, the expectation value
 .. math::
    :label: expectation_value_decomp
 
-   {\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].
+   {\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, :math:`{\langle\mathcal{O}\rangle}_\rho` can be expressed as the mean
 value of the random variable :math:`\omega_k` over the probability
@@ -59,6 +60,19 @@ unbiased Monte-Carlo estimator of :math:`{\langle\mathcal{O}\rangle}_\rho` as
 
    \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 :math:`\mathbb{E}[\hat{o}] = {\langle\mathcal{O}\rangle}_\rho`
+and its variance is given by
+
+.. math::
+   :label: estimator_variance
+
+   \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 :math:`\{M_k\}_{k}` and the decomposition weights
+:math:`\{\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
 ------------------

diff --git a/docs/how_tos/dual_optimizer.ipynb b/docs/how_tos/dual_optimizer.ipynb
diff --git a/povm_toolbox/post_processor/median_of_means.py b/povm_toolbox/post_processor/median_of_means.py
@@ -86,9 +86,9 @@ def __init__(
             povm_sample: a result from a POVM sampler run.
             dual: the Dual frame that will be used to obtain the decomposition weights of an
                 observable when computing its expectation value. For more details, refer to
-                :meth:`get_decomposition_weights`. When this is ``None``, the canonical Dual frame
-                will be constructed from the POVM stored in the ``povm_sample``'s
-                :attr:`.POVMPubResult.metadata`.
+                :meth:`get_decomposition_weights`. When this is ``None``, the default
+                "state-average" Dual frame will be constructed from the POVM stored in the
+                ``povm_sample``'s :attr:`.POVMPubResult.metadata`.
             num_batches: number of batches, i.e. number of samples means, used in the median-of-means
                 estimator. This value will be overridden if a ``delta_confidence`` argument is supplied.
             upper_delta_confidence: an upper bound for the confidence parameter :math:`\delta` used to

diff --git a/povm_toolbox/post_processor/povm_post_processor.py b/povm_toolbox/post_processor/povm_post_processor.py
@@ -65,9 +65,9 @@ def __init__(
             povm_sample: a result from a POVM sampler run.
             dual: the Dual frame that will be used to obtain the decomposition weights of an
                 observable when computing its expectation value. For more details, refer to
-                :meth:`get_decomposition_weights`. When this is ``None``, the canonical Dual frame
-                will be constructed from the POVM stored in the ``povm_sample``'s
-                :attr:`.POVMPubResult.metadata`.
+                :meth:`get_decomposition_weights`. When this is ``None``, the standard
+                "state-average" Dual frame will be constructed from the POVM stored in the
+                ``povm_sample``'s :attr:`.POVMPubResult.metadata`.
 
         Raises:
             ValueError: If the provided ``dual`` is not a dual frame to the POVM stored in the