Compatibility of Generalized Noisy Qubit Measurements

TL;DR

This work identifies the critical white-noise visibility at which all qubit POVMs become jointly measurable, establishing the threshold $\eta=1/2$. It constructs a protocol that represents noisy POVMs $A^{\eta}_{i|a}$ as coarse-grainings of a parent POVM $G_{\vec{\lambda}}=\frac{1}{4\pi}(\mathds{1}+\vec{\lambda}\cdot\vec{\sigma})$ with conditional probabilities $p(i|a,\vec{\lambda})$, exploiting a suitable Bloch-frame to guarantee joint measurability for $\eta\le 1/2$. The results are then leveraged to derive a tight local hidden state model for the two-qubit Werner state $\rho_W^\eta=\eta|\Psi^-\rangle\langle\Psi^-|+(1-\eta)\mathds{1}/4$, showing unsteerability for $\eta\le 1/2$ and precluding Bell-nonlocality with POVMs for the same range; these findings show POVMs offer no advantage over projective measurements for steering in this regime. The appendix discusses the non-constructive nature of frame existence and provides explicit constructions in special cases, as well as numerical search methods via public code.

Abstract

It is a crucial feature of quantum mechanics that not all measurements are compatible with each other. However, if measurements suffer from noise they may lose their incompatibility. Here, we consider the effect of white noise and determine the critical visibility such that all qubit measurements, i.e. all positive operator-valued measures (POVMs), become compatible, i.e. jointly measurable. In addition, we apply our methods to quantum steering and Bell nonlocality. We obtain a tight local hidden state model for two-qubit Werner states of visibility $1/2$. This determines the exact steering bound for two-qubit Werner states and also provides a local hidden variable model that improves on previously known models. Interestingly, this proves that POVMs are not more powerful than projective measurements to demonstrate quantum steering for these states.

Figures (5)

• Figure 1: A measurement device can perform different measurements (labeled with $a$) that produce an outcome $i$. If the measurements are too noisy they can be simulated by a device that just performs a single measurement. In this work, we address the question of how much white noise can be tolerated before all qubit measurements become jointly measurable.
• Figure 2: An illustration for a SIC-POVM SIC: a) The different outcomes $i$ are represented with different colors and the colored vectors represent $\vec{a}_i$ (note also $p_i=1/2$ for $i=1,2,3,4$). b) The opacity of the colors represents the probability to output $i$ given that $\vec{\lambda}$ lies in that region of the sphere, hence $p(i|a, \vec{\lambda})$. This function is constant in each octant of the chosen frame, which is simply the standard coordinate frame in this case. For the $\vec{\lambda}$ shown in the left sphere ($s_x=-1$, $s_y=s_z=+1$), the outcome is most likely blue ($50\%$) or green ($49\%$). c) Collecting all results $\vec{\lambda}$ from one octant behaves like the operator $G_{s_xs_ys_z}$ represented by the cyan arrows for the blue outcome. The sum of these operators simulates the desired (blue) operator $A^{1/2}_{1|a}$. (more details in Appendix \ref{['SICexample']})
• Figure 3: Construction for the two-outcome POVM with operators $A^{1/2}_{i|a}=(\mathds{1}+ \vec{a}_i\cdot \vec{\sigma}/2)/2$ (with $\vec{a}_2=-\vec{a}_1$): a) Here, $\vec{a}_1$ can be an arbitrary direction in the Bloch sphere. b) We can choose the rotated frame such that the $x'$-axis is aligned with $\vec{a}_1$. We also show the corresponding cube here. c) The conditional probabilities $p(i|a,\vec{\lambda})$ reduce precisely to $p(1|a,\vec{\lambda})=1$ if $\vec{\lambda}\cdot \vec{a}_1\geq 0$ and $p(1|a, \vec{\lambda})=0$ if $\vec{\lambda}\cdot \vec{a}_1< 0$ as indicated with the two colors. Hence, if the outcome $\vec{\lambda}$ of the parent POVM lies in the hemisphere centered around $\vec{a}_1$ (blue region) the outcome is always $i=1$ and if it lies in the hemisphere centered around $\vec{a}_2$ (red region), the outcome will be $i=2$.
• Figure 4: An illustration of $p(i|a,\vec{\lambda})$ for a three-outcome POVM. Here the conditional probabilities do not depend on $z'$ due to the choice of the coordinate frame. If $\vec{\lambda}$ is close to one of the colored vectors it is also more likely that this color is produced as an output.
• Figure 5: (similar figure as in the main text) The coloured arrows denote the vectors $p_i\cdot \vec{a}_i$ according to $p_i=1/2$ and the vectors $\vec{a}_i$ given above. The right part of the figure represents the conditional probabilities given in the table above. Here, $\vec{\lambda}$ lies in the octant that corresponds to "-++" ($s_x=-1$, $s_y=s_z=+1$) the outcome is $i=1$ (blue) with $p(1|a,\vec{\lambda})=0.5$, $i=2$ (red) with $p(2|a,\vec{\lambda})=0.003$, $i=3$ (green) with $p(3|a,\vec{\lambda})=0.487$, and $i=4$ (yellow) with $p(4|a,\vec{\lambda})=0.01$. Therefore, the outcome is most likely to be either $i=1$ or $i=3$.

Theorems & Definitions (4)

• Lemma 1
• proof
• Lemma 2
• proof