# Hardware Limitations of Dynamic Metasurface Antennas in the Uplink: A
Comparative Study

Maryam Rezvani1, Raviraj Adve1, Amr El-Keyi2, Akram bin Sediq2  1Dept. of
Electrical and Computer Engineering, University of Toronto, Toronto, Canada
2Ericsson Canada, Ottawa, Canada,
emails: {mrezvani<EMAIL_ADDRESS>{amr.el-keyi,
<EMAIL_ADDRESS>
###### Abstract

Dynamic Metasurface Antennas (DMAs) have emerged as promising candidates for
basestation deployment in the next generation of wireless communications.
While overlooking the practical and hardware limitations of DMA, previous
studies have highlighted DMAs’ potential to deliver high data rates while
maintaining low power consumption. In this paper, we address this oversight by
analyzing the impact of practical hardware limitations such as antenna
efficiency, power consumed in required components, processing limitations,
etc. Specifically, we investigate DMA-assisted wireless communications in the
uplink and propose a model which accounts for these hardware limitations. To
do so, we propose a concise model to characterize the power consumption of a
DMA. For a fair assessment, we propose a wave-domain combiner, based on
holography theory, to maximize the achievable sum rate of DMA-assisted
antennas. We compare the achievable sum rate and energy efficiency of DMA
antennas with that of a partially connected hybrid phased array. Our findings
reveal the true potential of DMAs when accounting for the limitations of both
designs.

###### Index Terms:

Hybrid multiple-input multiple output (MIMO), Dynamic metasurface antenna
(DMA), Holography theory, Uplink wireless communications.

##  I Introduction

High throughput, energy efficiency, and low power consumption are key demands
for the next generation of wireless communications [1]. While extra-large
multiple-input multiple-output (XL-MIMO) [2] with each antenna connected to a
radio frequency (RF) chain, i.e., digital phased arrays (DPAs), can provide
high throughput, they would result in excessive power usage at basestations
(BSs).

In contrast, hybrid MIMO antennas [3] and dynamic metasurface antennas (DMAs)
[4] connect a large number of antennas/unit-cells to fewer RF chains reducing
power consumption. Specifically, while an $N$-element DPA uses $N$ RF chains,
in hybrid MIMO systems, i.e., hybrid phased arrays (HP), using an analog
combiner/precoder enables connecting $N$ antennas to $M$ RF chains, where
$N\gg M$ [3]. One variant of HPs is a partially connected HP (PCHP). In PCHPs,
each antenna is connected to a phase shifter (PS), and the captured signal
from $N_{s}$ antenna-PS pairs, is combined via a Wilkinson combiner [5] before
being fed to an RF chain.

On the other hand, DMAs introduce a groundbreaking design by incorporating
metasurfaces into the antenna structure, offering a wave-domain (WD)
precoder/combiner. In DMAs, a large number of unit-cells are loosely coupled
[4] to a few microstrip lines (or other waveguide types). Each unit-cell
influences the impinging wave through a weighting factor, $w$. This factor can
follow either Lorentzian phase modulation (LPM), expressed as
$w=\frac{\jmath+e^{\jmath\theta}}{2}\ \theta\in[0,2\pi)$ or binary amplitude
modulation (BAM), denoted as $w\in\\{0,1\\}$ [6]. Notably, each unit-cell
couples its signal to the neighboring microstrip line, resembling the
structure of PCHPs. In DMAs, one end of each microstrip line connects to an RF
chain while the other end is terminated to prevent standing waves.

Since their introduction, several aspects of DMA have been studied. For
example, [6] studied the performance of a DMA in uplink wireless
communications and showed that a LPM-constrained DMA can achieve a sum rate
comparable to that of an unconstrained DMA. Furthermore, in [7], the authors
show that a DMA offers higher energy efficiency than a DPA or fully-connected
HP.

When comparing antenna types, it is crucial to consider the hardware
limitations across designs. Examples of these limitations are the antenna
efficiency, the power consumed by the device components, and limited beam
control due to the LPM constraint. While the DMA literature often overlooks
antenna efficiency and limitations, recent work [8] has considered the
efficiency of DMAs and the effect of sampling from the guided mode. The
authors propose a tri-hybrid architecture where a DMA is used in front of a
PCHP in a downlink scenario. However, the full-wave simulation used in [8]
provides accurate modeling for the specific simulated DMA design but lacks
generalizability to other DMA variants.

Notably, the impact of DMA’s hardware limitations on uplink performance
remains unexplored. In the uplink, where users contend with limited
transmitted power, it is imperative to account for antenna efficiency and
hardware limitations since these factors significantly impact achieving
optimal performance and reliability. Equivalently, for a fair comparison, one
must design effective signal processing techniques to maximize the potential
performance of the chosen system.

The primary limitation of using a DMA in the uplink stems from the loose
coupling between unit-cells and the underlying microstrip line [7]. This loose
coupling ensures that, in transmit mode, sampling from the guided mode and the
introduced decay rate do not prevent the guided mode signal from reaching all
unit-cells. The decay rate is controlled by the geometry and weighting factor
of each unit-cell [7], respectively. However, the static control of the decay
factor significantly impacts DMA performance in the uplink. It restricts the
amount of power coupled to the microstrip line, subsequently affecting the
signal power at the RF chain and reducing the received signal-to-noise ratio
(SNR).

Contributions: In this paper, we aim to precisely describe the hardware
limitations of DMAs. Our goal is to compare the achievable sum rate when using
a DMA with when using a PCHP in uplink wireless communications. Additionally,
we propose a method to account for DMA’s limitations independently of its
specific design. Notably, our approach requires characterizing a single unit-
cell, rather than the entire DMA structure, through full-wave simulations. We
propose a model to calculate the power consumption of different designs while
accounting for the power consumed by the RF chain as well as the required
driving circuitry for the unit-cell’s configuration.

For the signal processing, we propose an algorithm based on holography theory
to design a WD combiner which maximizes achievable sum rate in the uplink of
DMA-assisted antenna systems. We compare the achievable sum rate and energy
efficiency of using DMAs with using PCHP under equal aperture area and equal
number of RF chains. This comparison is in single- and multi-user scenarios in
rich scattering and realistic 3GPP channels while accounting for hardware
limitations in both designs. Our results reveal the shortcomings of DMA-
assisted antennas in comparison with using PCHP.

Notation: Italics denote scalars. We use lowercase and uppercase boldface
letters to denote vectors and matrices, respectively. $(\cdot)^{\dagger}$ and
$(\cdot)^{H}$ denote pseudo inverse and conjugate transpose, respectively. We
represent the vector fields by $\vec{\mathbf{A}}$. Also, $\jmath$ is defined
as $\sqrt{-1}$ and $\operatorname{sinc}(x)=\frac{\sin(\pi x)}{\pi x}$.
$\mathcal{CN}(\boldsymbol{\mu},\mathbf{R})$ represents the complex Gaussian
distribution with mean $\boldsymbol{\mu}$ and covariance matrix $\mathbf{R}$.

##  II System and DMA Model

System Model: We consider a BS equipped with a DMA comprising $N_{\mu}$
microstrip lines with $N_{e}$ unit-cells on each microstrip line, i.e., a
total number of unit-cells $N=N_{\mu}N_{e}$. The BS serves $K$ users. The
number of RF chains, $M$, is equal to $N_{\mu}$, since each microstrip line
connects to an RF chain at one end. The decoded signal at the BS is given by

$\mathbf{x}=g\mathbf{W}^{D}(\mathbf{G}^{\text{DMA}}(\sqrt{P_{T}}\mathbf{Hs}+\mathbf{z}_{\text{ant}})+\mathbf{z}_{\text{RF}})$
(1)

where $\mathbf{x}\in\mathbb{C}^{K\times 1}$ is the received symbol vector at
the BS, $P_{T}$ is the transmitted signal power of each user (we assume equal
power transmission), $\mathbf{H}\in\mathbb{C}^{N\times K}$ is channel matrix,
$\mathbf{z}_{\text{ant}}\in\mathbb{C}^{N\times 1}$ denotes the noise captured
by the antenna from the environment [9]. Furthermore, $g$ and
$\mathbf{z}_{\text{RF}}\in\mathbb{C}^{M\times 1}$ are gain and noise terms
introduced by the RF chain where
$z_{\text{RF}}\sim\mathcal{CN}(0,\sigma^{2}_{\text{RF}})$. Also, via
$\mathbf{G}^{\text{DMA}}\in\mathbb{C}^{M\times N}$ and
$\mathbf{W}^{D}\in\mathbb{C}^{K\times M}$ we, respectively, model the effect
of the DMA’s WD and digital combiner on the received signals.

DMA model: To model the DMA, we begin by pointing out its hardware limitations
and the maximum weighting factor of each unit-cell.

Figure 1: A schematic of a dynamic metasurface antenna

As shown in Fig.1, a DMA comprises $N_{\mu}$ microstrip lines separated by a
half-wavelength. Each line feeds the signal to (in transmit mode) or extracts
the signal from (in receive mode) $N_{e}$ unit-cells. Each unit-cell is
loosely coupled to the microstrip line to prevent extensive decay of the
propagating wave in the microstrip line [10] in transmitting mode. This
requirement leads to the loss of a large portion of signal power transmitted
by users in receiving mode. The coupling factor of each unit-cell is dictated
by the dynamic state of the cell, average coupling factor, dielectric losses
of unit-cells as well as their geometry and spacing [7]. Therefore, while one
can maximize the dynamic component of the coupling factor of each unit-cell by
setting it “on” (in BAM mode) or by setting its weighting factor to $1\jmath$
(in LPM mode), there is a maximum coupling value dictated by the losses and
geometry of the unit-cell. In what follows, we model this aspect of the
coupling factor.

In a DMA, each unit-cell is modeled as a magnetic dipole with a magnetic
polarizability defined by Lorentzian resonance [11] described as

$\alpha_{m}(\omega)=\frac{F_{m}\omega^{2}}{\omega_{0}^{2}-\omega^{2}+\jmath\Gamma_{m}\omega},$
(2)

where $\omega_{0}$ and $\omega$ are the resonance and working angular
frequencies, $F_{m}$ is the coupling factor, and ${\Gamma}_{m}$ is the damping
factor $=\frac{\omega_{0}}{2Q_{m}}$ where $Q_{m}$ is the quality factor of the
unit-cell.

By ignoring the coupling between unit-cells (well-accepted as an accurate
approximation [11]) the magnetic dipole formed by the $i$-th unit-cell is
described as $\vec{\mathbf{m}}_{i}=\alpha_{m,i}\vec{\mathbf{H}}_{i}$ where
$\vec{\mathbf{H}}_{i}$ is the impinging magnetic field at the $i$-th unit-cell
[4]. In the uplink, the transmitted signal by the users excites each unit-cell
and hence, the $i$-th unit-cell magnetic dipole is proportional to the
received electromagnetic (EM) plane-wave described as [12]111Here we used the
reciprocity theorem and described the user’s transmitted EM field to be equal
to the EM field radiated by the DMA as if the system is in the downlink. This
ensures that the EM field coupled to the microstrip line is the dominant
quasi-transverse EM (TEM) mode.. For a single user

$\vec{\mathbf{H}}_{i}=\sqrt{P_{T}}k_{0}^{2}\eta_{0}f(\vec{r})\hat{\phi}$ (3)

where $P_{T}$ is the transmitted signal power, and $k_{0}$ and $\eta_{0}$ are
free space wavenumber and impedance, respectively. In (3) $f(\vec{r})$
captures the dependency of the magnetic field on the channel between the user
and BS where $\vec{r}$ is their radial distance, for example in free space
$f(\vec{r})=\frac{e^{-\jmath\vec{k}\cdot\vec{r}}}{4\pi\absolutevalue{\vec{r}}}$.

Combining (2)-(3), the maximum magnetic dipole moment,
$m^{\text{DMA}}_{\text{max}}$, occurring at the unit-cell’s resonance
frequency, is equal to
$m^{\text{DMA}}_{\text{max}}=2Q_{m}F_{m}k_{0}^{2}\eta_{0}$. For example, in
[11], for a DMA with 30 unit-cells on each microstrip line and designed to
operate at $3.5\ \text{GHz}$, $F_{m}=3.0\times 10^{-9}\ [\text{m}^{3}]$ and
$Q_{m}=10.0$. In this case, the maximum amplitude of the magnetic dipole
formed by each unit-cell would be $0.09$ [J/T]. which ensures the feed signal
would reach all $30$ unit-cells in the underlying microstrip line. This value
is consistent with the maximum value of a DMA’s weighting factor in BAM mode
used in [6]. Note that $F_{m}$ and $Q_{m}$ can be extracted from the unit-cell
design using the method introduced in [13].

Note that considering the maximum magnetic dipole moment of DMA is crucial
when designing signal processing techniques or comparing it with other antenna
designs. This accounts for the practicality of the proposed technique and
ensures a fair comparison.

Microstrip Model: As in [14], we capture the effect of EM field propagation in
the microstrip line and weighting factor of each unit-cell in matrices
$\mathbf{P}^{\text{DMA}}$ and $\mathbf{W}^{\text{DMA}}$, respectively. The DMA
weighting matrix $\mathbf{W}^{\text{DMA}}$ is given by
$\mathbf{W}^{\text{DMA}}=\text{blkdiag}(\mathbf{w}^{1},\dots,\mathbf{w}^{M})$,
where $\mathbf{W}^{\text{DMA}}$ is of size $N_{\mu}\times N$, $N=N_{\mu}N_{e}$
is the total number of unit-cells in DMA,
$\mathbf{w}^{i}=[w^{i}_{1},\dots,w^{i}_{j},\dots,w^{i}_{N_{e}}]$ is a row
vector representing the weight factors of unit-cells on the $i$-th microstrip
line, and $w^{i}_{j}$ is the weight factor of its $j$-th unit-cell. Each non-
zero element of $\mathbf{W}^{\text{DMA}}$ follows either a BAM or LPM
constraint:

$w^{i}_{j}\in\mathcal{W}^{\text{DMA}}\triangleq\begin{cases}\\{m^{\text{DMA}}_{\text{max}}\
\frac{\jmath+e^{\jmath\phi}}{2},\phi\in[0,2\pi]\\}&\text{LPM}\\\
\\{0,m^{\text{DMA}}_{\text{max}}\\}&\text{BAM.}\end{cases}$ (4)

The propagation matrix $\mathbf{P}^{\text{DMA}}$ which captures the changes in
EM fields due to the propagation in the microstrip line, is described by a
diagonal matrix defined as

$\displaystyle\begin{split}\mathbf{P}&{}^{\text{DMA}}_{(i-1)N_{e}+j}=e^{-(\alpha_{w}+\jmath\beta_{w})d_{i,j}},\\\
&\forall(i,j)\in\left\\{(i,j)|i\in\\{1,\dots,N_{\mu}\\},j\in\\{1,\dots,N_{e}\\}\right\\},\end{split}$
(5)

where $d_{i,j}$ is the distance of the $j$-th element on the $i$-th microstrip
line from the RF chain, $\alpha_{w}$ is the waveguide attenuation coefficient,
and $\beta_{w}$ is the microstrip line wavenumber.

Combining the two matrices, the WD combiner of DMA is described as
$\mathbf{G}^{\text{DMA}}=\mathbf{W}^{\text{DMA}}\mathbf{P}^{\text{DMA}}$.

Power consumption model: The consumed power by the DMA comprises two terms
related to the RF chains and the needed power to configure unit-cells. We
begin with the RF chains. Accounting for the main components in RF chains,
i.e., low noise amplifier (LNA), mixer (Mix), clock distribution network
(Clk), in-phase and quadrature demodulator (IQD), analog-to-digital converter
(ADC), and its driver (drv), as shown in Fig. 1, the power consumed, $P_{RF}$
for one RF chain is given by222A complete block diagram of RF chain can be
found in [15].

$\small
P_{\text{RF}}=P_{\text{LNA}}+P_{\text{Mix}}+P_{\text{Clk}}+P_{\text{IQD}}+2(P_{\text{ADC}}+P_{\text{drv}})$
(6)

Note that while $P_{LNA}$ is usually considered to be fixed [16], it contains
a static term due to LNA’s internal circuitry, $P_{\text{sLNA}}$, and a
dynamic term depending on the input power, $P_{\text{in}}$, and power-added-
efficiency, $\eta_{\text{LNA}}$ [17]. Therefore, we model an LNA’s power
consumption as
$P_{\text{LNA}}=\max(P_{\text{sLNA}},\frac{G_{\text{LNA}}-1}{\eta_{\text{LNA}}}P_{\text{in}})$
where $G_{\text{LNA}}$ is the LNA gain.

To calculate the needed power to configure the DMA unit-cells, we need to
carefully account for components in driving circuitry. The properties of each
unit-cell are controlled via the voltage applied to the implemented varactor
diode. The digital-to-analog converter (DAC) driven varactor diodes provide
modulation of the unit-cell’s weighting factor by translating the applied
voltage to an adjustable capacitance. However, more than the power consumed by
the DAC, $P_{\text{DAC}}$, the data line driving the DAC, $P_{\text{Ctrl}}$
determines the power consumed. Finally, there is a static power consumption by
the core of the DMA’s field programmable gate array (FPGA), $P_{\text{FPGA}}$.
We assume that all the processing is handled in the main FPGA/processor and
the DMA’s FPGA only drives the varactors (via the DAC) consuming limited
power. Hence, the total power consumption of the configuration is

$P_{\text{DMA}}=N(P_{\text{DAC}}+P_{\text{Ctrl}})+P_{\text{FPGA}}$ (7)

where $N$ is the number of unit-cells.

We note that the power consumed by the RF chains and configuring circuitry of
the DMA, as defined in (6) and (7) is independent of the DMA state, and only
depends on the received power by the antenna via $P_{\text{LNA}}$.

##  III Maximizing The Achievable sum rate

The achievable sum rate, $R$, for a DMA-assisted antenna in the uplink is
defined as $R=\sum_{k=1}^{K}\log_{2}(1+\gamma_{k})$ where $\gamma_{k}$ is the
signal-to-interference-plus-noise ratio (SINR) for user $k$, i.e.,

$\gamma_{k}=\frac{P_{T}\absolutevalue{\mathbf{G}_{k,:}\mathbf{h}_{:,k}}^{2}}{\sum_{j\neq
k}^{K}P_{T}\absolutevalue{\mathbf{G}_{k,:}\mathbf{h}_{:,j}}^{2}+z},$ (8)

where we defined $\mathbf{G}=\mathbf{W}^{D}\mathbf{G}^{\text{DMA}}$ and
$z=\sigma_{\text{ant}}^{2}(\mathbf{G}_{k,:}\mathbf{\Sigma_{\text{rx}}}(\mathbf{G}_{k,:})^{H})+\sigma_{\text{RF}}^{2}\|\mathbf{W}^{D}_{k,:}\|^{2}$
represents the noise terms comprising external noise with power
$\sigma_{\text{ant}}^{2}$, colored by the WD combiner, and the thermal noise
added by the RF chain of power $\sigma_{\text{RF}}^{2}$ [14].

To maximize the achievable sum rate, we are interested in solving the
following optimization problem:

$\displaystyle\max_{\mathbf{W}^{D},\mathbf{W}^{\text{DMA}}}R$ (9a) s.t.
$\displaystyle\mathbf{W}^{\text{DMA}}=\text{blkdiag}(\mathbf{w}^{1},\dots,\mathbf{w}^{M})$
(9b) $\displaystyle w^{i}_{j}\in\mathcal{W}^{\text{DMA}}.$ (9c)

where $\mathcal{W}^{\text{DMA}}$ is defined in (4). Solving (9) is not trivial
due to the non-convexity of objective function and constraints. To obtain an
effective solution, adopting holography theory [18], we first find a desired
combiner, $\mathbf{G}^{\text{des}}$ assuming the existence of a virtual DPA at
the location of the DMA. This is a trivial task using the massive MIMO
literature; for example, in single-user and multi-user scenarios, a good
choice for $\mathbf{G}^{\text{des}}$ is maximum ratio combining (MRC) and
zero-forcing (ZF), respectively, when noise in the system is white. Knowing
$\mathbf{G}^{\text{des}}$, we are interested in finding
$\mathbf{W}^{\text{DMA}}$ and $\mathbf{W}^{D}$ that are solutions to the
following optimization problem.

$\begin{split}\min_{\mathbf{W}^{D},\mathbf{W}^{\text{DMA}}}&\norm{\mathbf{G}^{\text{des}}-\mathbf{W}^{D}\mathbf{W}^{\text{NW}}\mathbf{W}^{\text{DMA}}\mathbf{P}^{\text{DMA}}}\\\
\text{s.t. }&\eqref{OP:cnstr1},\ \eqref{OP:cnstr2}\end{split}$ (10)

where, since $\mathbf{G}^{\text{des}}$ is a good choice for systems with added
white noise and in DMA we are dealing with colored noise,
$\mathbf{W}^{\text{NW}}=\mathbf{\Sigma}_{z}^{-\frac{1}{2}}$ is introduced as a
noise whitening filter where $\mathbf{\Sigma}_{z}$ is the noise correlation
matrix. While (10) is still non-convex jointly in $\mathbf{W}^{\text{D}}$ and
$\mathbf{W}^{\text{DMA}}$, adopting an alternative optimization approach,
similar to [14], we find an optimum answer for $\mathbf{W}^{D}$ and
$\mathbf{W}^{\text{DMA}}$.

WD combiner: Given $\mathbf{W}^{D}$, various solutions have been proposed to
obtain $\mathbf{W}^{\text{DMA}}$. Here, we use the non-zero mapper (NZM) and
the closed-form mapper (CFM) as introduced in [6] and [14]. Specifically,
defining $\mathbf{G}^{D}=\mathbf{W}^{D}\mathbf{W}^{\text{NW}}$, NZM finds the
closest WD combiner to $\mathbf{G}^{\text{des}}$ by normalizing the Frobenious
norm between
$(\mathbf{G}^{D})^{\dagger}\mathbf{G}^{\text{des}}(\mathbf{P}^{\text{DMA}})^{-1}$
and $\mathbf{W}^{\text{DMA}}$ entry-wise. On the other hand, in CFM, knowing
$\mathbf{G}^{D}$ a closed-form solution for elements of
$\mathbf{W}^{\text{DMA}}$ that minimizes Frobenious distance between
$\mathbf{G}=\mathbf{G}^{\text{des}}(\mathbf{P}^{\text{DMA}})^{-1}$ and
$\mathbf{G}^{D}\mathbf{W}^{\text{DMA}}$ is calculated as
$w^{i}_{j}=\frac{(\mathbf{G}^{D})_{i}^{H}(\mathbf{G})_{i,j}}{\|(\mathbf{G}^{D})_{i}\|^{2}}$
where $(\mathbf{G})_{i,j}$ and $\mathbf{G}^{D}_{i}$ denote the $j$-th column
of the $i$-th $M\times N_{e}$ block of $\mathbf{G}$ and $i$-th column of
$\mathbf{G}^{D}$. Then $w^{i}_{j}$ is first mapped to the unit-circle followed
by projection to the Lorentzian circle to obtain its final value.

Conversely, given $\mathbf{W}^{\text{DMA}}$ to calculate the digital combiner,
we adopt the following approaches:

  * •

Single-user case: In this case, with $\mathbf{W}^{\text{DMA}}$ known, the best
solution is applying a noise whitening filter $\mathbf{W}^{\text{NW}}$
followed by match filtering defined as
$\mathbf{W}^{D}=(\mathbf{W}^{\text{NW}}\mathbf{W}^{\text{DMA}}\mathbf{P}^{\text{DMA}}\mathbf{H})^{H}$

  * •

Multi-user case: Knowing $\mathbf{W}^{\text{DMA}}$, an optimal choice for
$\mathbf{W}^{D}$ is the least-squares solution, i.e.,
$\mathbf{W}^{D}=\mathbf{G}^{\text{des}}(\mathbf{W}^{\text{NW}}\mathbf{W}^{\text{DMA}}\mathbf{P}^{\text{DMA}})^{\dagger}$.

The process of alternatively optimizing between $\mathbf{W}^{\text{DMA}}$ and
$\mathbf{W}^{D}$ continues until convergence.

##  IV Numerical Study

(a)

(b)

(c)

Figure 2: $K=1,M=4$. Achievable sum rate (a) ignoring, (b) factoring hardware
limitations in rich scattering environment. (c) Achievable sum rate in
realistic 3GPP channels factoring hardware limitations.

(a)

(b)

(c)

Figure 3: $K=2,M=4$. Achievable sum rate (a) ignoring (b) factoring hardware
limitations in rich scattering environment. (c) Achievable sum rate in
realistic 3GPP channels factoring hardware limitations.

In this section, we compare the sum rate when using a DMA with using a PCHP,
considering the hardware issues described above. We examine equal aperture
areas, RF chains $M$, single-user and multi-user scenarios, in an outdoor
wireless communication system where $K$ users communicate with a base station
at $3$ GHz carrier frequency with $20$ MHz bandwidth. As the BS is in the far-
field of users, it receives a plane wave with Poynting vector magnitude equal
to $P_{T}\frac{\absolutevalue{\vec{\mathbf{E}}_{\text{user}}}^{2}}{2\eta_{0}}$
$[\frac{W}{m^{2}}]$ where $P_{T}$ is the power transmitted by users.

We assume both PCHP and DMA have physical apertures equal to
$2\lambda_{0}\times 8\lambda_{0}$ and $M=4$ RF chains. The PCHP is an array of
patch antennas separated by $\lambda_{0}/2$ with $90\%$ efficiency [19]. Also,
the unit-cells of the DMA are of size $\lambda_{0}/2\times\lambda_{0}/6$ [20].
This configuration translates to $N_{s}$ and $N_{e}$ equal to $16$ and $48$,
respectively. Furthermore, the RF chain is at room temperature, i.e.,
$T_{\text{BS}}=290\degree\ [K]$. For a microstrip line as described in [11]
$\alpha_{w}$ and $\beta_{w}$ are set $0.13$ and $113.8\ [\text{m}^{-1}]$,
respectively. Finally, $g$ and the noise figure of the RF chain are set to be
$12.5$ [dB] and $18.8$ [dB], respectively. Also, to form a clear picture of
the effect of hardware limitations, we report the performance of DPA with
$N=M=4$. We repeat the simulation for 10 instances of user placements and 100
channel realizations per instance.

The power captured by each antenna is equal to $P_{T}A_{\text{eff}}$ where
$A_{\text{eff}}$ is the antenna effective area [21, Ch. 2]. Each unit-cell is
approximated by a continuous current sheet, hence, $A_{\text{eff,UC}}$ is
equal to its physical area [21, Ch. 2]. In the case of PA, using Matlab’s
Antenna Designer app [22], a typical patch antenna is designed and its
effective area is calculated.

The noise power captured by the antenna is given by
$\sigma_{\text{ant}}^{2}=KT_{E}B\frac{A_{\text{eff}}}{\lambda_{0}^{2}}\Delta\Omega$
[23, Ch. 13] where $K$, $T_{E}$, and $\Delta\Omega$ are Boltzmann’s constant,
environment temperature ($290\degree\ K$) [24], and the solid angle of noise
sources equal to $2\pi$.

For rich scattering environments, we use the Kronecker Rayleigh fading model
where $\mathbf{H}=\mathbf{\Sigma}_{rx}\hat{\mathbf{H}}\mathbf{\Sigma}_{tx}$,
$\hat{\mathbf{H}}\sim\mathcal{CN}(\mathbf{0},\mathbf{I}_{NK})$, and
$\mathbf{\Sigma}_{tx}$ and $\mathbf{\Sigma}_{rx}$ are transmitters’, i.e.,
users’, and receivers’, i.e., antennas’, correlation matrices, respectively.
We assume independent channel between users, i.e.,
$\mathbf{\Sigma}_{tx}=\text{diag}(\beta_{1},\dots,\beta_{K})$ where
$\beta_{k}$ is channel path loss of $k$-th user computed using the Winner II
urban microcell (UMi) non-line of sight (NLoS) scenario [25]. To calculate
$\mathbf{\Sigma}_{rx}$, we assume scatterers are distributed in 3D space and
hence,
$(\mathbf{\Sigma}_{\text{rx}})_{(i,j)}=\operatorname{sinc}(\frac{2d_{i,j}}{\lambda_{0}})$
[26] where $d_{i,j}$ is the distance between $i$-th and $j$-th unit-cells.
Also, to evaluate the achievable sum rate of different antenna designs in
realistic 3GPP channels, we use channels generated by QuadRiGa [27]. In
realistic channels the external noise correlation matrix is equal to that in a
rich scattering environment, i.e.,
$\mathbf{z}_{\text{ant}}\sim\mathcal{(}0,\sigma_{\text{ant}}^{2}\mathbf{\Sigma}_{\text{rx}})$.

In single-user and multi-user scenarios, MRC and ZF combiners are used for the
DPA. The PCHP’s combiner is obtained using the algorithm in [28]. Finally, to
account for hardware limitations in PCHP, we assume a $k$-way Wilkinson
combiner is formed by cascading $\lceil\frac{N}{M}\rceil$ $2$-way Wilkinson
combiners [5]. The insertion loss of the Wilkinson power combiner and phase
shifter (PS) is $3.9\ [\text{dB}]$ [29] and $5\ [\text{dB}]$ [30],
respectively.

Single-user scenario: Depicted in Fig. 2 is the achievable sum rate of
different designs in the single-user case. As shown in Fig. 2(a), when
hardware limitations are not factored in, the PCHP and DMA achievable sum rate
surpasses that of DPA with equal $M$. In PCHP, PSs can fully compensate for
the phase differences in signal captured by different antennas, while in DMA,
due to LPM (or BAM) constraints, signals captured by different unit-cells are
not completely in phase. This explains the performance gap between DMA and
PCHP. In contrast, when hardware limitations are accounted for in rich
scattering channels, as illustrated in Fig. 2(b), the achievable sum rate of
the PCHP is close to that of the DPA and the DMA’s performance drops
dramatically. The superior performance of PCHP with respect to DPA is mainly
due to its larger physical aperture. which leads to higher signal power
captured from the environment.

Due to the small coupling factor between unit-cells and microstrip lines in
the DMA structure, most of the signal power illuminating the antenna’s
aperture is not coupled to microstrip lines leading to poor performance of the
DMA in comparison to DPA and PCHP.

Finally, the achievable sum rate of different designs in realistic 3GPP
channels is demonstrated in Fig. 2(c). In these selective channels, while the
performance of all designs drops, DPA experiences the most degradation in its
achievable sum rate due to its smaller aperture area. Furthermore, the
achievable sum rate of the BAM-constrained DMA case is extremely low which
makes this design inappropriate.

(a)

(b)

(c)

(d)

Figure 4: Energy efficiency in rich scattering environments: single-user
scenario (a) ignoring, (b) factoring hardware limitations, multi-user scenario
(b) ignoring, (c) factoring hardware limitations.

Multi-user scenario: The performance of different antenna designs in a multi-
user scenario where BS serves $2$ users is depicted in Fig. 3. Similar to the
single-user case, when hardware limitations are not accounted for, as shown in
Fig. 3(a), both PCHP and the DMA achieve higher sum rates than the DPA due to
their larger number of antennas and available flexibility to form more complex
patterns. Also, the DMA’s performance in LPM mode using NZM surpasses that of
PCHP mainly due to dense unit-cells in the DMA structure and the flexibility
it provides to nullify inter-user interference. Moreover, the achievable sum
rate of the DMA in BAM mode has the same trajectory as that of PCHP. In
contrast, when accounting for hardware limitations, in rich scattering
channels, the DPA’s achievable sum rate surpasses other designs due to its
lower signal losses and, hence, higher SINR. On the other hand, in realistic
3GPP channels, PCHP’s performance is better than the DPA due to its larger
effective aperture and increased signal power. Here, the DMA’s sum rate is
close to that of the DPA, due to its large aperture area but lower than PCHP
due to higher losses in the system.

Energy Efficiency: Energy efficiency (EE) is defined as
$\text{EE}=\frac{R}{P}$. The power consumption of the DPA and PCHP is due to
$M$ RF chains calculated using (6). In contrast, DMA power consumption is due
to RF chains and unit-cell’s configuring circuitry as described in (7). To
calculate consumed power in each case, $P_{\text{sLNA}}$ [31],
$P_{\text{Mix}}$ [32], $P_{\text{IQD}}$ [33], $P_{\text{drv}}$ [34],
$P_{\text{ADC}}$ [35], $P_{\text{Clk}}$ [36], $P_{\text{DAC}}$ [37],
$P_{\text{Ctrl}}$ [38] and $P_{\text{FPGA}}$ [38] are set as $0.75$ [W], $0.4$
[W], $2.2$ [W], $0.15$ [W], $0.725$ [W], $0.1$ [W], $0.002$ [W], $0.0006$ [W],
and $0.1$ [W], respectively. Finally, $\eta_{\text{LNA}}$ is 0.12 [31]. Note
that in calculating $P_{\text{LNA}}$, we assume the power received by each
antenna is equal to $\beta_{A}P_{T}A_{\text{eff}}$ where $\beta_{A}$ is the
average path-loss of the users. Also, in a DMA/PCHP, the total power captured
by $N_{s}$/$N_{e}$ antennas enters each LNA while in DPA, the power captured
by one antenna forms $P_{in}$ of the LNA.

Depicted in Fig. 4, is the EE in rich scattering channels. As seen, the EE
shows the same trend as the achievable sum rate. This is because the major
contributor to the power consumption of the antenna is the number of RF
chains, $M$, which is equal across all compared designs. Also, the DMA’s power
consumption is slightly larger than the other two antenna designs due to the
required circuitry to configure the state of unit-cells. Note that combining a
DMA with HPs, as in [8], would result in a more energy-efficient design.

##  V Conclusion

In this paper, we studied the impact of the hardware limitations of a DMA in
uplink wireless communications. Our results show that the hardware limitations
of DMA drastically affect its achievable sum rate in comparison with PCHP and
DPA with an equal number of RF chains. This underlines the importance of
accounting for the practical limitations of DMA, indeed any other antenna
design, when used in a wireless communication scenario.

## Acknowledgments

This work was supported by the Ericsson Canada.

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# Measurement uncertainty relation for three observables

Sixia Yu  Hefei National Laboratory for Physical Sciences at Microscale and
Department of Modern Physics, University of Science and Technology of China,
Hefei, Anhui 230026, China  Ya-Li Mao  Shenzhen Institute for Quantum Science
and Engineering and Department of Physics, Southern University of Science and
Technology, Shenzhen, 518055, China  Chang Niu  Hefei National Laboratory for
Physical Sciences at Microscale and Department of Modern Physics, University
of Science and Technology of China, Hefei, Anhui 230026, China  Hu Chen
Shenzhen Institute for Quantum Science and Engineering and Department of
Physics, Southern University of Science and Technology, Shenzhen, 518055,
China  Guangdong Provincial Key Laboratory of Quantum Science and Engineering,
Southern University of Science and Technology, Shenzhen, 518055, China  Zheng-
Da Li  Shenzhen Institute for Quantum Science and Engineering and Department
of Physics, Southern University of Science and Technology, Shenzhen, 518055,
China  Guangdong Provincial Key Laboratory of Quantum Science and Engineering,
Southern University of Science and Technology, Shenzhen, 518055, China
Jingyun Fan  Shenzhen Institute for Quantum Science and Engineering and
Department of Physics, Southern University of Science and Technology,
Shenzhen, 518055, China  Guangdong Provincial Key Laboratory of Quantum
Science and Engineering, Southern University of Science and Technology,
Shenzhen, 518055, China  Center for Advanced Light Source, Southern University
of Science and Technology, Shenzhen, 518055, China

###### Abstract

In this work we establish rigorously a measurement uncertainty relation (MUR)
for three unbiased qubit observables, which was previously shown to hold true
under some presumptions. The triplet MUR states that the uncertainty, which is
quantified by the total statistic distance between the target observables and
the jointly implemented observables, is lower bounded by an incompatibility
measure that reflects the joint measurement conditions. We derive a necessary
and sufficient condition for the triplet MUR to be saturated and the
corresponding optimal measurement. To facilitate experimental tests of MURs we
propose a straightforward implementation of the optimal joint measurements.
The exact values of incompatibility measure are analytically calculated for
some symmetric triplets when the corresponding triplet MURs are not saturated.
We anticipate that our work may enrich the understanding of quantum
incompatibility in terms of MURs and inspire further applications in quantum
information science. This work presents a complete theory relevant to a
parallel work [Y.-L. Mao, et al., Testing Heisenberg’s measurement uncertainty
relation of three observables, arXiv:2211.09389] on experimental tests.

##  I Introduction

One of the most distinguishing features of quantum theory, from a modern point
of view, is the incompatibility [1, 2] of quantum measurements whose outcomes
cannot be read out simultaneously. From a practical point of view the quantum
incompatibility, just like entanglement [3, 4], can be established as a non-
classical resource and has found numerous applications in quantum informations
science including Bell’s nonlocality, quantum steering, quantum contextuality,
and quantum state discrimination. From a fundamental point of view, the
quantum incompatibility quantifies Bohr’s complementarity and lies at the
heart of all kinds of Heisenberg’s uncertainty relations [5].

By employing different measures for uncertainty (or errors or disturbance)
such as mean square errors, trace distance, or entropies, various kinds of
uncertainty relations have been established [6, 7, 8, 9, 10, 11, 12, 13, 14,
15]. These uncertainty relations can be classified into two different kinds,
namely, the preparation uncertainty relations (PUR, also known as the
Heisenberg-Robertson uncertainty relation) [16, 17] and the measurement
uncertainty relations (MUR) [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29,
30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45]. While the
PURs prohibit us from preparing quantum states with definite values for
incompatible observables, the MURs capture the essence of quantum
incompatibility, namely, quantum measurements may disturb each other, which
was the main concern in the original Heisenberg’s gedanken experiment of
microscope [29, 30]. Recent years have witnessed a number of MURs of two
observables being conceived and verified in experiments [19, 21, 22, 23, 24,
25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43,
44, 46].

The quantum incompatibility can be quantified most naturally in the framework
of joint measurement. For example, we can approximately measure two
incompatible observables by performing a pair of compatible, i.e., jointly
measurable, quantum observables by introducing some errors. The resulting MUR
indicates that the total error, which may be quantified in, e.g., a worst case
senario to characterize the performance of the measuring device in the spirit
of Bush, Lahti, and Werner (BLW) [30], is lower bounded by some measure of
incompatibility. The optimal errors, however, are in generally achieved on
different states.

An operational and meaningful lower bound for the total error was proposed
independently in Refs.[40, 47], via the distance, e.g., relative entropy or
statistic distance, from the given pair of incompatible measurements to the
set of all the jointly measurable pairs, calculated on the same state. For
relative entropy approach the lower bounds as well as optimal measurements are
in general not analytically tractable. Moreover a relation of the lower bound
to the joint measurability condition is missing. For the statistic distance
measure of total error, an elegant form of MUR is proposed with a presumption
that approximate joint measurements are restricted to the unbiased triplets
[48, 49]. Thus a MUR for a triplet of observables is still missing and how to
find the optimal measurements to attain the lower bound and their experimental
implementations remain to be outstanding problems.

In this paper we at first re-establish the MUR for triplet of unbiased qubit
observables by considering the most general jointly measurable triplet as
approximations. And then we provide a necessary and sufficient condition of
attainability as well as the optimal measurement. Third, we propose a
straightforward implementation of the optimal joint measurement by measuring
randomly four ideal qubit observables. Lastly, we derive analytically the
incompatibility measure for two symmetric triplets after showing that the
optimal jointly measurable triplet shares the same graded symmetries with the
original triplet of incompatible observables.

##  II Triplet MUR and an implementation

For a qubit the most general measurement or observable with two outcomes is
represented by positive operator valued measure (POVM) $\\{N_{\pm}\\}$ with

$N_{\pm}=\frac{{I}\pm(x+{\vec{n}}\cdot\vec{\sigma})}{2}\mathrel{\mathop{\mathchar
58\relax}}=N_{\pm}(x,\vec{n}),$ (1)

where we have denoted explicitly its dependence on the biasedness $x$ and
Bloch vector $\vec{n}$ satisfying $|x|+|\vec{n}|\leq 1$. The observables with
vanishing biasedness, i.e. $x=0$, are referred to as unbiased. In general,
some POVMs are called jointly measurable if there exists a parent POVM with
multiple outcomes such that each observable in the given set arises as a
marginal measurement or equivalently from a post-measurement processing [50].
The exact joint measurement conditions are known in a few special cases [51,
52, 53, 54] and in general the problem can be cast into semidefinite
programmings [55, 56].

In particular, a set ${\mathcal{N}}=\\{N_{j}\\}_{j=1}^{3}$ of three general
two-outcome qubit observables
$N_{j}=\\{N_{\pm|j}=N_{\pm}(x_{j},\vec{n}_{j})\\}\\}$ is jointly measurable
iff there exists a parent measurement $\\{M({\mu_{1},\mu_{2},\mu_{3}})\\}$
with 8 outcomes, labelled with binary vector $\mu=(\mu_{1},\mu_{2},\mu_{3})$
for $\mu_{j}=\pm 1$, such that the given three general observables arise as
marginals, i.e., for each $j=1,2,3$ it holds

$N_{\mu_{j}|j}=\prod_{k\not=j}\sum_{\mu_{k}=\pm}M({\mu_{1},\mu_{2},\mu_{3}}).$
(2)

Equivalently, ${\mathcal{N}}$ is jointly measurable iff there exists a parent
measurement $\\{M_{\omega}\\}$ together with a set of post-measurement
processing $\\{p_{j}(\pm|\omega)\\}$, i.e., a set of probability distributions
for each $j$ and outcome $\omega$, such that

$N_{\pm|j}=\sum_{\omega}p_{j}(\pm|\omega)M_{\omega}.$ (3)

The necessary and sufficient conditions for a triplet of unbiased qubit
observables to be compatible is explicitly given by [53]

$\sum_{k=0}^{3}|\vec{q}_{k}-\vec{q}_{f}|\leq 4,$ (4)

where

$\vec{q}_{k}=\sum_{j=1}^{3}\gamma_{jk}\vec{n}_{j},\quad\gamma_{jk}=(-1)^{j\lfloor\frac{k}{2}\rfloor+k\lfloor\frac{j}{2}\rfloor}$

with $\vec{q}_{f}$ being the Fermat-Torricelli (FT) point of
$\\{\vec{q}_{k}\\}^{3}_{k=0}$, the vector that minimize the distance sum as
given in the left-hand-side of inequality Eq.(4). We note that
$\\{\vec{q}_{k}\\}_{k=0}^{3}=\\{\mu\vec{n}_{1}+\nu\vec{n}_{2}+\mu\nu\vec{n}_{3}\mid\mu,\nu=\pm\\}$.

Lemma  If a triplet $\\{N_{\pm}(x_{j},\vec{n}_{j})\\}_{j=1}^{3}$ is jointly
measurable, then the corresponding unbiased triplet
$\\{N_{\pm}(0,\vec{n}_{j})\\}_{j=1}^{3}$ is also jointly measurable.

Proof  As the triplet $\\{N_{\pm}(x_{j},\vec{n}_{j})\\}$ is jointly
measurable, there exists a joint measurement

$\displaystyle
8M({\mu_{1},\mu_{2},\mu_{3}})=1+\sum_{j=1}^{3}\mu_{j}(x_{j}+\vec{n}_{j}\cdot\vec{\sigma})$
$\displaystyle+\sum_{j>k}\mu_{j}\mu_{k}(z_{jk}+\vec{z}_{jk}\cdot\vec{\sigma})-\mu_{1}\mu_{2}\mu_{3}(z+\vec{z}\cdot\vec{\sigma})\hskip-34.14322pt$

for some real $\\{z,z_{jk}\\}$ and vectors $\\{\vec{z}_{jk},\vec{z}\\}$
$(j>k=1,2,3)$ with the given triplet as marginals. The positivity requirements
$M({\mu_{1},\mu_{2},\mu_{3}})\geq 0$, i.e., demand that

$\displaystyle
1+\sum_{j=1}^{3}\mu_{j}x_{j}+\sum_{j>k}\mu_{j}\mu_{k}z_{jk}-\mu_{1}\mu_{2}\mu_{3}z$
$\displaystyle\geq\left|\textstyle\sum_{j=1}^{3}\mu_{j}\vec{n}_{j}+\sum_{j>k}\mu_{j}\mu_{k}\vec{z}_{jk}-\mu_{1}\mu_{2}\mu_{3}\vec{z}\right|.\hskip-34.14322pt$

By summing up over all $\mu_{j}=\pm$ we obtain

$\displaystyle 8$ $\displaystyle\geq$
$\displaystyle\sum_{\mu_{1},\mu_{2},\mu_{3}}\left|\textstyle\sum_{j=1}^{3}\mu_{j}\vec{n}_{j}+\sum_{j>k}\mu_{j}\mu_{k}\vec{z}_{jk}-\mu_{1}\mu_{2}\mu_{3}\vec{z}\right|$
$\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{\mu_{1},\mu_{2},\mu_{3},\pm}\left|\textstyle\sum_{j=1}^{3}\mu_{j}\vec{n}_{j}\pm\sum_{j>k}\mu_{j}\mu_{k}\vec{z}_{jk}-\mu_{1}\mu_{2}\mu_{3}\vec{z}\right|$
$\displaystyle\geq$
$\displaystyle\sum_{\mu_{1},\mu_{2},\mu_{3}}\left|\textstyle\sum_{j=1}^{3}\mu_{j}\vec{n}_{j}-\mu_{1}\mu_{2}\mu_{3}\vec{z}\right|$
$\displaystyle=$ $\displaystyle
2\sum_{\mu_{1}\mu_{2}=\mu_{3}}\left|\textstyle\sum_{j=1}^{3}\mu_{j}\vec{n}_{j}-\vec{z}\right|\geq
2\sum_{\mu_{1}\mu_{2}=\mu_{3}}\left|\textstyle\sum_{j=1}^{3}\mu_{j}\vec{n}_{j}-\vec{q}_{f}\right|$

where the first equality is due to an average over $\vec{\mu}$ and
$-\vec{\mu}$ while the first inequality is because of triangle inequality and
the last inequality comes from the definition of FT point $\vec{q}_{f}$ of
four vectors
$\\{\mu_{1}\vec{n}_{1}+\mu_{2}\vec{n}_{2}+\mu_{1}\mu_{2}\vec{n}_{3}\\}_{\mu_{1},\mu_{2}=\pm}=\\{\vec{q}_{k}\\}_{k=0}^{3}$.
This means that the unbiased triplet $\\{N_{\pm}(0,\vec{n}_{j})\\}$ is also
jointly measurable. $\square$

Let ${\mathcal{M}}=\\{M_{j}\\}_{j=1}^{3}$ with
$M_{j}=\\{M_{\pm|j}\mathrel{\mathop{\mathchar
58\relax}}=N_{\pm}(0,\vec{m}_{j})\\}$ be a triplet of unbiased qubit
observables. A jointly measurable triplet of general observables
${\mathcal{N}}=\\{N_{\pm|j}=N_{\pm}(x_{j},\vec{n}_{j})\\}$ is performed and
the total uncertainty measured by the statistics distance defines the
incompatibility of the triplet

$\Delta_{\mathcal{M}}\mathrel{\mathop{\mathchar
58\relax}}=\min_{{\mathcal{N}}}\max_{\rho}\sum_{i=1}^{3}d_{\rho}(M_{i};N_{i})$
(5)

where

$d_{\rho}(M_{i};N_{i})=2\sum_{\pm}\left|{\rm Tr}\rho M_{\pm|j}-{\rm Tr}\rho
N_{\pm|j}\right|$ (6)

An elegant lower bound of $\Delta_{\mathcal{M}}$ was proposed in [48] which is
intimately related to the joint measurability condition. However, in deriving
their result a strong presumption that the optimal measurements are unbiased
was introduced. We strengthen this measurement uncertainty relation by
considering the most general form of jointly measurable triplet by proving
that the optimal measurement is actually unbiased.

Theorem 1  (Triplet MUR) For an unbiased triplet
${\mathcal{M}}=\\{M_{j}\\}_{j=1}^{3}$, by performing the most general
measurements $\\{N_{j}\\}_{j=1}^{3}$ that are jointly measurable, it holds MUR

$\Delta_{\mathcal{M}}\geq\frac{1}{2}\sum_{k=0}^{3}|\vec{p}_{k}-\vec{p}_{f}|-2\mathrel{\mathop{\mathchar
58\relax}}=2\delta$ (7)

where $\\{\vec{p}_{k}=\sum_{j}\gamma_{jk}\vec{m}_{j}\\}$ with $\vec{p}_{f}$
being its FT point. The lower bound is saturated if and only if

$\delta\leq\min_{k}|\vec{p}_{k}-\vec{p}_{f}|.$ (8)

If the condition is met, the optimal set of jointly measurable triplet reads

$\vec{n}_{j}=\vec{m}_{j}+\frac{\delta}{4}\sum_{k=0}^{3}\gamma_{jk}\frac{\vec{p}_{f}-\vec{p}_{k}}{|\vec{p}_{f}-\vec{p}_{k}|}\quad(k=1,2,3).$
(9)

Proof  To lower bound the incompatibility we calculate

$\displaystyle\frac{1}{2}\Delta_{\mathcal{M}}$ $\displaystyle=$
$\displaystyle\min_{{\mathcal{N}}}\max_{\rho}\textstyle\sum_{j=1}^{3}\sum_{\pm}\left|{\rm
Tr}\rho(M_{\pm|j}-N_{\pm|j})\right|$ (10a) $\displaystyle=$
$\displaystyle{\min_{\\{x_{j},\vec{n}_{j}\\}}\max_{\vec{r}}}\textstyle\sum_{j=1}^{3}|\vec{r}\cdot(\vec{m}_{j}-\vec{n}_{j})-x_{j}|$
(10b) $\displaystyle=$
$\displaystyle\min_{\\{x_{j},\vec{n}_{j}\\}}\max_{\vec{r}}\max_{\mu_{1},\mu_{2},\mu_{3}=\pm
1}\textstyle\sum_{j=1}^{3}\mu_{j}\left(\vec{r}\cdot(\vec{m}_{j}-\vec{n}_{j})-x_{j}\right)$
(10c) $\displaystyle=$
$\displaystyle\min_{\\{x_{j},\vec{n}_{j}\\}}\max_{\mu_{1},\mu_{2},\mu_{3}=\pm
1}\max_{\vec{r}}\vec{r}\cdot\left(\textstyle\sum_{j}\mu_{j}\vec{m}_{j}-\sum_{j}\mu_{j}\vec{n}_{j}\right)-\textstyle\sum_{j}\mu_{j}x_{j}$
(10d) $\displaystyle=$
$\displaystyle\min_{\\{x_{j},\vec{n}_{j}\\}}\max_{\mu,\nu=\pm}\left|\textstyle\mu\vec{m}_{1}+\nu\vec{m}_{2}+\mu\nu\vec{m}_{3}-(\mu\vec{n}_{1}+\nu\vec{n}_{2}+\mu\nu\vec{n}_{3})\right|+\left|\mu
x_{1}+\nu x_{2}+\mu\nu x_{3}\right|$ (10e) condition 0 $\displaystyle\geq$
$\displaystyle\min_{\\{\vec{n}_{j}\\}}\max_{k=0,1,2,3}|\vec{p}_{k}-\vec{q}_{k}|$
(10f) condition 1 $\displaystyle\geq$
$\displaystyle\min_{\\{\vec{n}_{j}\\}}\frac{1}{4}\textstyle\sum_{k=0}^{3}|\vec{p}_{k}-\vec{q}_{k}|$
(10g) condition 2 $\displaystyle\geq$
$\displaystyle\min_{\\{\vec{n}_{j}\\}}\frac{1}{4}\textstyle\sum_{k=0}^{3}\big{|}|\vec{p}_{k}-\vec{q}_{f}|-|\vec{q}_{k}-\vec{q}_{f}|\big{|}$
(10h) condition 3 $\displaystyle\geq$
$\displaystyle\min_{\\{\vec{n}_{j}\\}}\frac{1}{4}\textstyle\sum_{k=0}^{3}\big{(}|\vec{p}_{k}-\vec{q}_{f}|-|\vec{q}_{k}-\vec{q}_{f}|\big{)}$
(10i) condition 4 $\displaystyle\geq$
$\displaystyle\frac{1}{4}{\textstyle\sum_{k=0}^{3}|\vec{p}_{k}-\vec{p}_{f}|}-\frac{1}{4}\max_{\\{\vec{n}_{j}\\}}\textstyle\sum_{k=0}^{3}|\vec{q}_{k}-\vec{q}_{f}|$
(10j) condition 5 $\displaystyle\geq$
$\displaystyle\frac{1}{4}{\textstyle\sum_{k=0}^{3}|\vec{p}_{k}-\vec{p}_{f}|}-1=\delta$
(10k)

Here we have used the fact that the absolute value can be rewritten as
$|A|=\max_{\pm}\\{\pm A\\}$ in deriving Eq.(10c) and the fact that
$\sum_{j}\mu_{j}\vec{m}_{j}=\mu_{1}\mu_{2}\mu_{3}\sum_{(i,j,k)\ \rm
cyclic}\mu_{i}\mu_{j}\vec{m}_{k}$ and redefine
$\mu=\mu_{2}\mu_{3},\nu=\mu_{1}\mu_{3}$ in deriving Eq.(10e). In the first
inequality yielding condition 0 we have used Lemma to take $x_{j}=0$ while
keeping the resulting unbiased set still jointly measurable with a no larger
total uncertainty. In derive condition 2 we have taken $\vec{q}_{f}$ to be the
FT point of $\\{\vec{q}_{k}\\}$. In condition 4 we have used the property of
the FT point $\vec{p}_{f}$ for $\\{\vec{p}_{k}\\}$ while the joint measurement
condition is used in deriving condition 5.

In order to saturate the MUR in Eq.(7) we have only to let all the
inequalities become equalities in the above derivation. It is straightforward
to see that in order to saturate Eq.(10f), i.e., the condition 0, the optimal
measurement has to be unbiased. For the other inequalities we have,
respectively,

  1. 1.

$|\vec{p}_{k}-\vec{q}_{k}|$ is independent of $k$

  2. 2.

$\vec{p}_{k},\vec{q}_{k},\vec{q}_{f}$ are linearly dependent for each $k$

  3. 3.

$|\vec{p}_{k}-\vec{q}_{f}|\geq|\vec{q}_{k}-\vec{q}_{f}|$ for each $k$

  4. 4.

$\vec{q}_{f}$ coincides with $\vec{p}_{f}$

  5. 5.

$\sum_{k}|\vec{q}_{k}-\vec{q}_{f}|=4$.

From conditions 2 and 4 it follows that we can have linear expansions
$\vec{q}_{k}=(1-\alpha_{k})\vec{p}_{k}+\alpha_{k}\vec{p}_{f}$ with some real
coefficient $\alpha_{k}$ (arbitrary for now) for each $k=0,1,2,3$. As a result
of condition 1 we have

$\delta=|\vec{p}_{k}-\vec{q}_{k}|=|\alpha_{k}|\cdot|\vec{p}_{k}-\vec{p}_{f}|$

is independent of $k$. Therefore we can assume $|\vec{p}_{k}-\vec{p}_{f}|>0$
for all $k$ because otherwise $|\vec{p}_{k}-\vec{q}_{k}|$ would become zero
for all $k$ (condition 1) so that the triplet ${\mathcal{M}}$ is also joint
measurable, a contradiction. Therefore from conditions 3 and 4, i.e.,

$|\vec{q}_{k}-\vec{q}_{f}|=|\vec{q}_{k}-\vec{p}_{f}|=|1-\alpha_{k}|\cdot|\vec{p}_{k}-\vec{p}_{f}|\leq|\vec{p}_{k}-\vec{p}_{f}|$

we obtain $|1-\alpha_{k}|\leq 1$ so that we have $0\leq\alpha_{k}\leq 2$. Thus

$\alpha_{k}=\frac{\delta}{|\vec{p}_{k}-\vec{p}_{f}|}.$

From condition 5 it follows

$\displaystyle 4$ $\displaystyle=$
$\displaystyle\sum_{k}|\vec{q}_{k}-\vec{q}_{f}|=\sum_{k}|\vec{q}_{k}-\vec{p}_{f}|$
$\displaystyle=$
$\displaystyle\sum_{k}|1-\alpha_{k}|\cdot|\vec{p}_{k}-\vec{p}_{f}|$
$\displaystyle=$
$\displaystyle\sum_{k}\Big{|}|\vec{p}_{k}-\vec{p}_{f}|-\delta\Big{|}$
$\displaystyle\geq$
$\displaystyle\sum_{k}\big{(}|\vec{p}_{k}-\vec{p}_{f}|-\delta\big{)}$
$\displaystyle=$ $\displaystyle 4(\delta+1)-4\delta=4$

That is, the inequality is in fact an equality, meaning that for all
$k=0,1,2,3$ it holds

$\frac{1}{4}\sum_{k}|\vec{p}_{k}-\vec{p}_{f}|-1=\delta\leq|\vec{p}_{k}-\vec{p}_{f}|.$

Sufficiency. Suppose the condition is satisfied we have
$|\vec{p}_{k}-\vec{p}_{f}|>0$ for all $k$ and need to show that the unbiased
triplet defined in Eq.(9) is jointly measurable and saturate the MUR. By
construction we have

$\vec{q}_{k}=\sum_{j=1}^{3}\gamma_{jk}\vec{n}_{j}=\vec{p}_{k}+\alpha_{k}(\vec{p}_{f}-\vec{p}_{k}),\quad\alpha_{k}=\frac{\delta}{|\vec{p}_{f}-\vec{p}_{k}|}.$

Condition ensures that $\alpha_{k}\leq 1$ so that

$\sum_{k}\frac{\vec{p}_{f}-\vec{q}_{k}}{|\vec{p}_{f}-\vec{q}_{k}|}=\sum_{k}\frac{1-\alpha_{k}}{|1-\alpha_{k}|}\cdot\frac{\vec{p}_{f}-\vec{p}_{k}}{|\vec{p}_{f}-\vec{p}_{k}|}=\sum_{k}\frac{\vec{p}_{f}-\vec{p}_{k}}{|\vec{p}_{f}-\vec{p}_{k}|}=0$

showing that $\vec{q}_{f}=\vec{p}_{f}$. As a result we have the joint
measurement condition

$\sum_{k=0}^{3}|\vec{q}_{k}-\vec{p}_{f}|=\sum_{k=0}^{3}(1-\alpha_{k})|\vec{p}_{k}-\vec{p}_{f}|=4.$

Furthermore, we actual have qubit observables, i.e., $|\vec{n}_{k}|\leq 1$
which follows from joint measurement condition

$|\vec{n}_{j}|=\frac{1}{4}\left|{\textstyle\sum_{k}}\gamma_{jk}(\vec{q}_{k}-\vec{p}_{f})\right|\leq\frac{1}{4}\sum_{k}|\vec{q}_{k}-\vec{p}_{f}|=1$

And the optimal lower bound can be attained in the pure state with Bloch
vector $\vec{r}=\vec{e}_{0}$ where

$\vec{e}_{k}\mathrel{\mathop{\mathchar
58\relax}}=\frac{\vec{p}_{f}-\vec{p}_{k}}{|\vec{p}_{f}-\vec{p}_{k}|},\quad(k=0,1,2,3)$

satisfying $\sum_{k}\vec{e}_{k}=0$ by the definition of the FT point. In fact
we have the total uncertainty

$\displaystyle\sum_{j=1}^{3}|\vec{r}\cdot(\vec{m}_{j}-\vec{n}_{j})|=\frac{\delta}{4}\sum_{j=1}^{3}\left|\textstyle\vec{e}_{0}\cdot\sum_{k=0}^{3}\gamma_{jk}\vec{e}_{k}\right|$
$\displaystyle=\frac{\delta}{2}\sum_{j=1}^{3}|\vec{e}_{0}\cdot(\vec{e}_{0}+\vec{e}_{j})|=\frac{\delta}{2}\sum_{j=1}^{3}(1+\vec{e}_{0}\cdot\vec{e}_{j})=\delta\hskip-31.2982pt.$

which means that the MUR is saturated. $\square$

As an optimal measurement always lies on the boundary, i.e., saturating the
joint measurement condition, we may accomplish the optimal joint measurement
in a single-qubit experiment [31].

Theorem 2 (Implementation) A jointly measurable triplet of unbiased qubit
observables $\\{N_{i}\\}$ that saturates the joint measurement condition
Eq.(4) can be implemented by the following parent measurement
$\\{P_{k}O_{k}\\}$ with $O_{k}=\\{O_{\mu_{k}|k}\\}$ where

$P_{k}=\frac{|\vec{q}_{k}-\vec{q}_{f}|}{4},\quad
O_{\mu_{k}|k}=\frac{1}{2}\left(1+\mu_{k}\frac{\vec{q}_{k}-\vec{q}_{f}}{|\vec{q}_{k}-\vec{q}_{f}|}\cdot\vec{\sigma}\right),$
(11)

with outcome labeled with $\mu_{k}=\pm 1$ for each $k=0,1,2,3$ and post-
measurement processing $p_{j}(\mu|k,\mu_{k})=\frac{1+\mu\ \gamma_{jk}\
\mu_{k}}{2}.$

Proof  The measurement can be implemented by first randomly choose $k=0,1,2,3$
according to distribution $P_{k}=\frac{|\vec{q}_{k}-\vec{q}_{f}|}{4}$ and then
perform a measurement $\\{O_{k}=\\{O_{\pm|k}\\}\\}$ with

${{O}}_{\pm|k}=\frac{{I}\pm\vec{g}_{k}\cdot\vec{\sigma}}{2},\quad\vec{g}_{k}=\frac{\vec{q}_{k}-\vec{q}_{f}}{|\vec{q}_{k}-\vec{q}_{f}|}.$

In fact by using post-measurement processing

$p_{j}(\mu|k,\mu_{k})=\frac{1+\mu\ \gamma_{jk}\ \mu_{k}}{2}$

we obtain the desired marginals

$\sum_{k,\mu_{k}}p_{j}(\mu|k,\mu_{k})M(k,\mu_{k})=\sum_{k}P_{k}\frac{1+\mu\gamma_{jk}\vec{g}_{k}\cdot\vec{\sigma}}{2}=N_{\mu|j}$

The standard parent measurement with 8 outcomes can be constructed as

$\displaystyle M(\omega_{1},\omega_{2},\omega_{3})$ $\displaystyle=$
$\displaystyle\sum_{k,\mu_{k}}P_{k}\prod_{j=1}^{3}p_{j}(\omega_{j}|k,\mu_{k}){{O}}_{\mu_{k}|k}$
$\displaystyle=$
$\displaystyle\frac{1+\sum_{j}\omega_{j}\vec{n}_{j}\cdot\vec{\sigma}-\omega_{1}\omega_{2}\omega_{3}\vec{q}_{f}\cdot\vec{\sigma}}{8}.$

with $\omega_{j}=\pm$, which coincides with the joint measurement given in
Ref. [53]. $\square$

As an immediate result, when the MUR is attained the optimal joint measurement
Eq.(9), can be readily implemented by performing 4 ideal qubit measurements
along directions $\vec{e}_{k}$ with probability
$\frac{1}{4}(|\vec{p}_{k}-\vec{p}_{f}|-\delta)$ for $k=0,1,2,3$.

##  III Symmetry

Symmetry is considered to be the main characteristics of the laws of physics
according to Feynman [57] and plays an essential role in determining
incompatibility noise robustness [58, 59]. Some physically relevant sets of
measurements may admit certain symmetries among measurement directions. For
example in the case of an ideal triplet ${\mathcal{M}}_{\perp}$ as shown in
Fig.1 with one observable ($\vec{m}_{3}$ along $\vec{x}$ direction) being
orthogonal to the other two ($\vec{m}_{1,2}$ on the $YZ$ plane) the reflection
$\tau_{YZ}$ over $YZ$ plane and the reflections $\tau_{\pm}$ over planes
passing through $\vec{z}$ and angle bisectors $\vec{m}_{1}\pm\vec{m}_{2}$ of
$\vec{m}_{1,2}$ are a kind of graded symmetry. By a graded symmetry here we
mean a reflection such that three directions are permuted among themselves up
to some inversions (which correspond to relabeling two outcomes of a
measurement). For example, we have $\tau_{YZ}\cdot\vec{m}_{3}=-\vec{m}_{3}$
while preserving directions $\vec{m}_{1,2}$ and
$\tau_{\pm}\cdot\vec{m}_{k}=\pm\vec{m}_{3-k}$ with $k=1,2$ while preserving
$\vec{m}_{3}$. It turns out that the optimal measurement must share the same
graded symmetry.

Figure 1: Target triplet ${\mathcal{M}}_{\perp}$ together with its symmetries
and the optimal triplet ${\mathcal{N}}_{\perp}$ with the same symmetries.

Theorem 3 (Symmetry) If the ideal triplet ${\mathcal{M}}$ along directions
$\\{\vec{m}_{j}\\}_{j=1}^{3}$ admits a graded symmetry $g$, i.e., a reflection
such that $g\cdot\vec{m}_{j}={\omega_{j}}\vec{m}_{\sigma(j)}$ with
$\omega_{j}=\pm$ and $\sigma$ being a permutation of $\\{1,2,3\\}$, then the
optimal jointly measurable triplet ${\mathcal{N}}=\\{\vec{n}_{j}\\}_{j=1}^{3}$
for the incompatibility $\Delta_{\mathcal{M}}$ shares the same graded
symmetry, i.e., $g\cdot\vec{n}_{j}={\omega_{j}}\vec{n}_{\sigma(j)}.$

Proof  Firstly, we note that as $g$ is a reflection we have $g^{2}=1$ so that
$\vec{m}_{j}=g\cdot
g\cdot\vec{m}_{j}=g\cdot\omega_{j}\vec{m}_{\sigma(j)}=\omega_{j}\omega_{\sigma(j)}\vec{m}_{\sigma^{2}(j)}$
from which it follows that $\sigma^{2}(j)=j$ and
$\omega_{j}=\omega_{\sigma(j)}$. Under the graded symmetry $g$ we have

$\displaystyle g\cdot\vec{m}_{\mu}$ $\displaystyle=$ $\displaystyle
g\cdot\sum_{j=1}^{3}\mu_{j}\vec{m}_{j}=\sum_{j=1}^{3}\mu_{j}g\cdot\vec{m}_{j}$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{3}\mu_{j}\omega_{j}\vec{m}_{\sigma(j)}=\sum_{j=1}^{3}\mu_{\sigma(j)}\omega_{\sigma(j)}\vec{m}_{j}=\vec{m}_{\mu_{\sigma}\omega_{\sigma}}.$

In order to show that the optimal joint measurement also has the same symmetry
it suffices to show for each joint measurement there exists another joint
measurement having the symmetry and in the mean time with a no larger
uncertainty. In fact, if $\\{\vec{n}_{j}\\}$ is an arbitrary jointly
measurable triplet of unbiased observables, then we have

$\displaystyle 2\max_{\mu}|\vec{m}_{\mu}-\vec{n}_{\mu}|$ (12) $\displaystyle=$
$\displaystyle\max_{\mu}|\vec{m}_{\mu}-\vec{n}_{\mu}|+\max_{\mu}|g\cdot(\vec{m}_{\mu}-\vec{n}_{\mu})|$
$\displaystyle=$
$\displaystyle\max_{\mu}|\vec{m}_{\mu}-\vec{n}_{\mu}|+\max_{\mu}|\vec{m}_{\mu_{\sigma}\omega_{\sigma}}-g\cdot\vec{n}_{\mu}|$
$\displaystyle=$
$\displaystyle\max_{\mu}|\vec{m}_{\mu}-\vec{n}_{\mu}|+\max_{\mu}|\vec{m}_{\mu}-g\cdot\vec{n}_{\mu_{\sigma}\omega}|$
$\displaystyle\geq$ $\displaystyle
2\max_{\mu}\left|\vec{m}_{\mu}-\frac{\vec{n}_{\mu}+g\cdot\vec{n}_{\mu_{\sigma}\omega}}{2}\right|.$

Therefore if we consider another triplet of unbiased observables along
directions

$\vec{n}^{\prime}_{j}=\frac{\vec{n}_{j}+\omega_{j}g\cdot\vec{n}_{\sigma(j)}}{2}$

we have $g\cdot\vec{n}^{\prime}_{j}=\omega_{j}\vec{n}^{\prime}_{\sigma(j)}$
(by noting $\sigma^{2}(j)=j$ and $\omega_{j}=\omega_{\sigma(j)}$), i.e., the
averaged triplet $\\{\vec{n}^{\prime}_{j}\\}$ shares the same symmetry with
triplet ${\mathcal{M}}$. In order to show that $\\{\vec{n}^{\prime}_{j}\\}$ is
also jointly measurable we note that the joint measurement condition for
$\\{\vec{n}_{j}\\}$ can be equivalently written as

$\sum_{\mu}|\vec{n}_{\mu}-\mu_{0}\vec{q}_{f}|\leq
8\Leftrightarrow\sum_{\mu}|g\cdot\vec{n}_{\mu_{\sigma}\omega}-\omega_{0}\mu_{0}g\cdot\vec{q}_{f}|\leq
8,$ (13)

where we have denoted $\mu=(\mu_{1},\mu_{2},\mu_{3})$ and
$\mu_{0}=\mu_{1}\mu_{2}\mu_{3}$ and
$\omega_{0}=\omega_{1}\omega_{2}\omega_{3}$. By summing up these two
inequalities and using triangle inequality we have

$8\geq\sum_{\mu}\left|\vec{n}_{\mu}^{\prime}-\mu_{0}\frac{\vec{q}_{f}+\omega_{0}g\cdot\vec{q}_{f}}{2}\right|\geq\sum_{\mu}\left|\vec{n}_{\mu}^{\prime}-\mu_{0}\vec{q}^{\prime}_{f}\right|,$

where

$\vec{n}^{\prime}_{\mu}=\sum_{j}\mu_{j}\vec{n}_{j}^{\prime}=\frac{\vec{n}_{\mu}+g\cdot\vec{n}_{\mu_{\sigma}\omega}}{2}$

with $\vec{q}_{f}^{\prime}$ being the FT point of
$\\{\vec{q}^{\prime}_{k}\\}=\\{\vec{n}^{\prime}_{\mu}|\mu_{0}=1\\}$, which
gives the desired joint measurability of $\\{\vec{n}^{\prime}_{j}\\}$. In sum,
starting form an arbitrary jointly measurable triplet $\\{\vec{n}_{j}\\}$ the
newly introduced triplet $\\{\vec{n}_{j}^{\prime}\\}$ is also jointly
measurable, shares the same symmetry as ${\mathcal{M}}$, and, due to Eq.(12),
has a no larger total uncertainty. Therefore the optimal measurement can be
taken to share the same symmetry as ${\mathcal{M}}$ without loss of
generality. $\square$

We note that Eq.(13) actually proves that if the jointly measurable triplet
$\\{\vec{n}_{j}\\}$ possesses some graded symmetry $g$, i.e.,
$g\cdot\vec{n}_{j}=\omega_{j}\vec{n}_{\sigma(j)}$, then the FT point also
shares the same graded symmetry in the sense that
$g\cdot\vec{p}_{f}=\omega_{0}\vec{p}_{f}$.

##  IV Two analytical examples

Some triplets of unbiased observables might be determined completely by their
symmetry. For example the triplet of three orthogonal observables, e.g.,
$\\{\sigma_{k}\\}$, is completely determined by 3 reflections over planes
$XY,YZ,ZX$ (upto some scalings). In this case, the symmetry theorem above
therefore enables us to determine the optimal measurements completely (upto
some scalings). In other cases however the symmetry might determine partially
the triplet. And in this case the symmetry also help to simplify the
optimization. Our first example is the triplet ${\mathcal{M}}_{\perp}$ along
directions

$\vec{m}_{1}=(\sin\theta,0,\cos\theta),\quad\vec{m}_{2}=(1,0,0),\quad\vec{m}_{3}=(0,1,0)$

The reflection $\tau_{YZ}$ over $YZ$ plane and the reflections $\tau_{\pm}$
over planes passing $\vec{x}$ and angle bisectors
$\vec{m}_{\pm}=\vec{m}_{1}\pm\vec{m}_{2}$ of $\vec{m}_{1,2}$ generate the
graded symmetry group of this triplet. In fact, we have
$\tau_{YZ}\cdot\vec{m}_{3}=-\vec{m}_{3}$ while preserving directions
$\vec{m}_{1,2}$ and $\tau_{\pm}\cdot\vec{m}_{k}=\pm\vec{m}_{3-k}$ with $k=1,2$
while preserving $\vec{m}_{3}$. Using theorem above we see that the optimal
joint measurement $\\{\vec{n}_{j}\\}$ should also have such a symmetry. From
$\tau_{XY}\cdot\vec{n}_{3}=-\vec{n}_{3}$ it follows that $\vec{n}_{3}$ should
be also along the direction given by the eigenvector of $\tau_{YZ}$
corresponding eigenvalue -1, which infers $\vec{n}_{3}\propto\vec{m}_{3}$.
Similarly from $\tau_{YZ}\cdot\vec{n}_{k}=\vec{n}_{k}$ it follows that
$\vec{n}_{1,2}$ also lie on the $YZ$ plane. Finally, from symmetry
$\tau_{\pm}\cdot\vec{n}_{k}=\pm\vec{n}_{3-k}$ we see that $\vec{n}_{1,2}$ have
the same length and are also symmetric about the angle bisect $\vec{m}_{\pm}$.
As a result the optimal joint measurement must be of form

$\vec{n}_{3}=n_{3}\hat{m}_{3},\
\vec{n}_{1}=\frac{\beta_{+}\hat{m}_{+}+\beta_{-}\hat{m}_{-}}{2},\
\vec{n}_{2}=\frac{\beta_{+}\hat{m}_{+}-\beta_{-}\hat{m}_{-}}{2},$

where

$\hat{m}_{\pm}=\frac{\vec{m}_{1}\pm\vec{m}_{2}}{|\vec{m}_{1}\pm\vec{m}_{2}|},\quad\hat{m}_{3}=\frac{\vec{m}_{3}}{|\vec{m}_{3}|}$

with 3 suitable constants $n_{3},\beta_{\pm}$ satisfying joint measurement
condition $|\beta_{+}|+|\beta_{-}|\leq 2\sqrt{1-n_{3}^{2}}.$ Now the
incompatibility $\Delta_{\perp}$ can be calculated as

$\displaystyle\min_{n_{3},\beta_{\pm}}\max_{\pm}\sqrt{(1-n_{3})^{2}+(\beta_{\pm}-m_{\pm})^{2}}$
$\displaystyle=$
$\displaystyle\min_{n_{3},\beta_{\pm}}\max_{\pm}\sqrt{(1-n_{3})^{2}+(|\beta_{\pm}|-m_{\pm})^{2}}$
$\displaystyle=$
$\displaystyle\min_{n_{3},\kappa}\sqrt{(1-n_{3})^{2}+(d_{+}-\sqrt{1-n_{3}^{2}}+|\kappa|)^{2}}$
$\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}\displaystyle\min_{n_{3}}\sqrt{(1-n_{3})^{2}+(d_{+}-\sqrt{1-n_{3}^{2}})^{2}}&d_{-}\leq\sqrt{1-n_{3}^{2}}\\\
\displaystyle\min_{n_{3}}\sqrt{(1-n_{3})^{2}+(m_{+}-2\sqrt{1-n_{3}^{2}})^{2}}&d_{-}>\sqrt{1-n_{3}^{2}}\end{array}\right.$

In the above we have denoted

$m_{\pm}=|\vec{m}_{\pm}|,\quad d_{\pm}=\sqrt{1\pm\cos\theta}$

so that the joint measurement condition gives

$|\beta_{\pm}|=\sqrt{1-n_{3}^{2}}\pm(d_{-}-\kappa)$

for some real $\kappa$ satisfying

$d_{-}-\sqrt{1-n_{3}^{2}}\leq\kappa\leq d_{-}+\sqrt{1-n_{3}^{2}}.$

In the first case $d_{-}\leq\sqrt{1-n_{3}^{2}}$ we have to minimize

$\displaystyle\sqrt{(1-n_{3})^{2}+(d_{+}-\sqrt{1-n_{3}^{2}})^{2}}\hskip
31.2982pt$
$\displaystyle=\sqrt{2+d_{+}^{2}-2(n_{3}+d_{+}\sqrt{1-n_{3}^{2}})}\hskip-31.2982pt$

on the condition $d_{-}\leq\sqrt{1-n_{3}^{2}}$. By Cauchy-Schwarz inequality
we have minimal value $\sqrt{1+d^{2}_{+}}-1$ as long as the optimal
$n_{3}^{*}=\frac{1}{\sqrt{1+d^{2}_{+}}}$ satisfies the constraint which reads
$1\leq(1-d_{-}^{2})(1+d^{2}_{+})$ giving rise to condition
$\cos\theta\geq\sqrt{2}-1$, i.e., $\theta\leq\theta_{0}\approx 65.53^{\circ}$.
Beyond this range, CS inequality cannot be attained so that the minimal value
is taken by boundary value $d_{-}=\sqrt{1-n_{3}^{2}}$ which gives the second
bound

$\displaystyle\sqrt{(1-\sqrt{1-d_{-}^{2}})^{2}+(d_{+}-d_{-})^{2}}\hskip
31.2982pt$
$\displaystyle=\sqrt{3+\cos\theta-2\sqrt{\cos\theta}-2\sin\theta}\hskip-31.2982pt$

While the first case gives the first two bounds, the second case gives the
third bound Let $n_{3}=\cos t$ for the minimization of the second case we have
(by derivative)

$\sin\theta=\frac{1}{8}\tan^{2}t(1+3\cos t)^{2}-1\geq\sqrt{1-\cos^{4}t}$

with equality holding for $\theta=\theta_{1}\approx 71.53^{\circ}$. The third
bound therefore can be implicitly given by a parametric curve
$\Delta_{opt}(\theta)$

$\displaystyle\Delta_{opt}$ $\displaystyle=$ $\displaystyle\frac{1-\cos
t}{\cos t}\sqrt{1+3\cos^{2}t}$ $\displaystyle\theta$ $\displaystyle=$
$\displaystyle\arcsin\left(\frac{1}{8}\tan^{2}t(1+3\cos t)^{2}-1\right)$

To summarize the compatibility reads

$\Delta_{\perp}=\left\\{\begin{array}[]{ll}{2\sqrt{2+\cos\theta}-2}&\theta\leq\theta_{0}\\\
{2\sqrt{3+\cos\theta-2\sqrt{\cos\theta}-2\sin\theta}}&\theta_{0}\leq\theta\leq\theta_{1}\\\
{\Delta_{opt}(\theta)}&\theta\geq\theta_{1}\end{array}\right..$ (15)

Respectively, there are three optimal measurements for three ranges of
$\theta$:

  * M1

$\theta\leq\theta_{0}$. In this case the condition of Theorem 2 is met and the
MUR can be saturated so that the optimal measurement, as given by Theorem 2,
reads

$\vec{n}_{3}=\frac{\vec{m}_{3}}{\sqrt{d^{2}_{+}+1}},\
\vec{n}_{1,2}=\frac{\beta_{+}^{*}\hat{m}_{+}\pm\beta_{-}^{*}\hat{m}_{-}}{2},$

where $\beta_{\pm}^{*}=\frac{d_{+}}{\sqrt{d^{2}_{+}+1}}\pm d_{-}.$

  * M2

$\theta_{1}\geq\theta\geq\theta_{0}$. In this case MUR cannot be saturated and
the optimal measurement reads

$\vec{n}_{3}=\sqrt{1-d_{-}^{2}}\
\vec{m}_{3},\quad\vec{n}_{1}=\vec{n}_{2}=d_{-}\ \hat{m}_{+}$

  * M3

$\theta\geq\theta_{1}$. The optimal joint measurement reads

$\vec{n}_{3}=\sqrt{1-(d_{-}-\kappa)^{2}}\
\vec{m}_{3},\quad\vec{n}_{1}=\vec{n}_{2}=(d_{-}-\kappa)\hat{m}_{+}$

with $\kappa\geq 0$ minimizing

$\sqrt{(1-\sqrt{1-(d_{-}-\kappa)^{2}})^{2}+(m_{-}+2\kappa)^{2}}.$

If we introduce $\cos 2\gamma=\vec{m}_{1}\cdot\vec{m}_{2}=\sin\theta$ and
three corresponding regions for different optimal measurements are determined
by which one of the following three intervals

$0<\Gamma_{0}<\Gamma_{1}<90^{\circ}$

that $|\gamma-45^{\circ}|$ falls into, where

$\Gamma_{j}=45^{\circ}-\frac{180^{\circ}}{2\pi}{\arcsin\cos\theta_{j}},\quad(j=0,1).$

Figure 2: Target triplet ${\mathcal{M}}_{Y}$ together with its symmetries and
the optimal triplet ${\mathcal{N}}_{Y}$ with the same symmetries.

Our second example is the following highly symmetric triplet
${\mathcal{M}}_{Y}$ of unbiased observables along directions

$\vec{m}_{j}=\hat{z}\cos\gamma+\hat{e}_{j}\sin\gamma,\quad\hat{e}_{j}\cdot\hat{e}_{k}=-\frac{1}{2},\quad\hat{z}\cdot\hat{e}_{j}=0,$

with $0\leq\gamma\leq\frac{\pi}{2}$ and $\hat{z}$ is the unit vector along $Z$
direction, as illustrated in Fig.2. Firstly, let us see under what conditions
the MUR $\Delta_{Y}\geq 2\delta_{\perp}$ for this triplet can be saturated.
Four FT anchors are $\vec{p}_{0}=3\cos\gamma\hat{z}$ and
$\vec{p}_{j}=-\cos\gamma\hat{z}+2\sin\gamma\vec{e}_{j}$ for $j=1,2,3$ with FT
point reads $\vec{p}_{f}=\alpha\hat{z}$ (by symmetry) where

$\alpha=\left\\{\begin{array}[]{lcl}\frac{\sin\gamma}{\sqrt{2}}-\cos\gamma&&0\leq\tan\gamma\leq
4\sqrt{2}\\\ 3\cos\gamma&&|\tan\gamma|\geq 4\sqrt{2}\\\
-\frac{\sin\gamma}{\sqrt{2}}-\cos\gamma&&0\geq\tan\gamma\geq-4\sqrt{2}\end{array}\right.$

(a detailed derivation given later) giving rise to the lower bound
$\Delta_{\perp}\geq 2\delta_{\perp}$ where

$\delta_{\perp}=\left\\{\begin{array}[]{cc}|\cos\gamma|+\sqrt{2}\sin\gamma-1&|\tan\gamma|\leq
4\sqrt{2}\\\
\frac{3}{2}\sqrt{4\cos^{2}\gamma+\sin^{2}\gamma}-1&|\tan\gamma|\geq
4\sqrt{2}\end{array}\right.$

In the case of $\gamma\leq\gamma_{0}=\arctan 2\sqrt{2}\approx 70.53^{\circ}$,
which is determined by the condition
$\delta=\min_{k}|\vec{p}_{k}-\vec{p}_{f}|$, namely,
$\sin\gamma/\sqrt{2}-\cos\gamma=1/3$, the attainability condition of Theorem 1
is satisfied so that the MUR can be attained.

In the case of $\gamma>\gamma_{0}$ MUR cannot be attained. In order to find
the optimal measurement for the incompatibility we consider the symmetry of
triplet ${\mathcal{M}}_{Y}$. Let $T_{j}$ be the reflection over the plane
orthogonal to $\hat{z}\times\hat{e}_{j}$ passing the origin. Then we have
$T_{j}\cdot\vec{m}_{j}=\vec{m}_{j}$ while $T_{j}\cdot\vec{m}_{k}=\vec{m}_{l}$
with $(j,k,l)$ cyclic. This symmetry is obeyed by the optimal joint
measurement triplet according to Theorem 3. As a result the optimal triplet
${\mathcal{N}}_{Y}$ should be along directions

$\vec{n}_{j}=x\hat{z}+y\hat{e}_{j},\quad j=1,2,3.$ (16)

Next we shall prove that the triplet above is jointly measurable if and only
if

$y\leq\left\\{\begin{array}[]{ll}\frac{1-x}{\sqrt{2}}&|x|\geq\frac{1}{9}\\\
\frac{2}{3}\sqrt{1-9x^{2}}&|x|\leq\frac{1}{9}\end{array}\right.$ (17)

In fact four FT anchors for this triplet are $\vec{q}_{0}=3x\hat{z},\
\vec{q}_{j}=-x\hat{z}+2y\hat{e}_{j}$ with FT point being
$\vec{q}_{f}=\alpha\hat{z}$ (by symmetry) so that we have total distance

$\sum_{k}|\vec{q}_{k}-\vec{q}_{f}|=|3x-\alpha|+3\sqrt{(x+\alpha)^{2}+4y^{2}}$

which is minimized by (via derivative as a function of $\alpha$)

$\alpha=\left\\{\begin{array}[]{ll}\frac{y}{\sqrt{2}}-x&4\sqrt{2}x\geq y\\\
3x&4\sqrt{2}|x|\leq y\\\
-\frac{y}{\sqrt{2}}-x&4\sqrt{2}x\leq-y\end{array}\right.$

giving rise to the joint measurement condition

$4\geq\sum_{k=0}^{3}|\vec{q}_{k}-\vec{q}_{f}|=\left\\{\begin{array}[]{cc}4|x|+4\sqrt{2}y&4\sqrt{2}|x|\geq
y\\\ 6\sqrt{4x^{2}+y^{2}}&4\sqrt{2}|x|\leq y\end{array}\right.,$

which leads to the desired condition Eq.(17) for $x,y$.

According to Theorem 3 the optimal incompatibility can be obtained by
considering only triplet of from Eq.(16), which now becomes

$\frac{\Delta_{Y}}{2}=\min_{x,y}\max\\{3|\cos\gamma-x|,\sqrt{(\cos\gamma-x)^{2}+4(\sin\gamma-y)^{2}}\\}$

with minimum taken over region determined by Eq.(17). By min-max inequality we
have

$\frac{\Delta_{Y}}{2}\geq\min_{x,y}\sqrt{(\cos\gamma-x)^{2}+4(\sin\gamma-y)^{2}}$

whose minimum is attained at (by derivative over the boundary
$y=\frac{2}{3}\sqrt{1-9x^{2}}$)

$(\cos\gamma-x)y=16(\sin\gamma-y)x.$ (18)

Furthermore in order for $\Delta_{Y}$ to attain this lower bound we must also
have

$3|\cos\gamma-x|\leq\sqrt{(\cos\gamma-x)^{2}+4(\sin\gamma-y)^{2}}$ (19)

which is possible as long as $\gamma\geq\gamma_{1}$ where

$\gamma_{1}=\arctan\frac{1}{79}\sqrt{2}\left(5\sqrt{337}+129\right)\approx
75.80^{\circ}$

is determined by Eq.(18) joint measurability condition
$y=\frac{2}{3}\sqrt{1-9x^{2}}$ together with the equality in Eq.(19).

Figure 3: Exact values of incompatibility for two triplets of ideal qubit
measurements with symmetries, (a) triplet ${\mathcal{M}}_{\perp}$ with $|\cos
2\gamma|=\vec{m}_{1}\cdot\vec{m}_{2}$ and (b) highly symmetric triplet
${\mathcal{M}}_{Y}$ of unbiased observables (see main text for details)
according to analytical results computed with Eqs. (15) and (20),
respectively.

Though not all conditions in Eq.(10) can be satisfied simultaneously, some of
them, e.g., condition 1, can be satisfied, which leads to

${\Delta_{Y}}\geq\min_{x,y}3|\cos\gamma-x|+\sqrt{(\cos\gamma-x)^{2}+4(\sin\gamma-y)^{2}}$

which is attained by $(x,y)$ equalizing Eq.(19) and
$y=\frac{2}{3}\sqrt{1-9x^{2}}$, giving rise to

$\frac{\Delta_{Y}}{2}=\frac{\sin\gamma}{\sqrt{2}}+2|\cos\gamma|-\sqrt{\frac{2}{3}-\big{(}\sin\gamma-\sqrt{2}|\cos\gamma|\big{)}^{2}}$

This happens if $\gamma_{0}\leq\gamma\leq\gamma_{1}$. To summarize, the
incompatibility reads

$\Delta_{Y}=\left\\{\begin{array}[]{ll}{2\cos\gamma+2\sqrt{2}\sin\gamma-2}\\\
{\sqrt{2}{\sin\gamma}+4\cos\gamma-2\sqrt{\frac{2}{3}-\left(\sin\gamma-\sqrt{2}\cos\gamma\right)^{2}}}\\\
{\displaystyle\min_{x^{2}+y^{2}=\frac{1}{9}}2\sqrt{(\cos\gamma-x)^{2}+4(\sin\gamma-2y)^{2}}}\end{array}\right.$
(20)

for three intervals $0<\gamma_{0}<\gamma_{1}<90^{\circ}$ divided by
$\gamma_{0}\approx 70.53^{\circ}$ and $\gamma_{1}\approx 75.80^{\circ}$. In
Fig.3 we plot the exact incompatibility measures of these two symmetric
triplets, which fit perfect with the numerical results.

##  V Conclusions an discussions

Quantum incompatibility reflects the basic fact that some quantum measurements
may disturb each other, preventing us from measuring them with a single
measurement device without introducing errors. The MUR sets the limit to how
well we can perform the joint measurement with the minimal amount of errors
according to quantum mechanics, providing a quantitative characterization of
quantum incompatibility. In establishing rigorously the triplet MUR we also
find out the necessary and sufficient conditions under which the MUR is
attained and, when the attainability condition is met, we also explicitly
construct the optimal jointly measurable triplet. In order to facilitate
experimental tests we design a straightforward implementation of optimal joint
measurements by just performing randomly four ideal qubit measurements.
Lastly, symmetry is found to play an essential role in deriving analytically
the incompatibility measure.

As a critical step in the study of joint-measurement on multiple observables,
this study may deepen our understanding of Heisenberg’s measurement
uncertainty principle and lead to new applications in quantum metrology and
quantum information science. We stress that the demonstrated strategy can be
readily generalized to calculate the incompatibility of four or more qubit
observables and the case of weighted distance as uncertainty measure, which
will be considered elsewhere.

## Acknowledgement

This work is supported by the Key-Area Research and Development Program of
Guangdong Province Grant No.2020B0303010001, Grant No.2019ZT08X324, National
Natural Science Foundation of China Grants No.12004207 and No.12005090,
Shenzhen Science and Technology Program Grant No.RCYX20210706092043065,
Guangdong Provincial Key Laboratory Grant No.2019B121203002 and SIQSE202104.

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