Robustness of incompatibility for quantum devices

We fix the sets ${{{\bf Q}}_{1}}$ and ${{{\bf Q}}_{2}}$ of quantum devices that are either observables or channels, i.e., with fixed Hilbert spaces $\mathcal{H}$, $\mathcal{K}$, and $\mathcal{K}^{\prime}$ and value spaces $(\Omega ,\Sigma )$ and $(\Omega ^{\prime} ,{{\Sigma }^{\prime }})$, ${{{\bf Q}}_{1}}$ is either ${\bf Obs}(\Sigma ,\mathcal{H})$ or ${\bf Ch}(\mathcal{H},\mathcal{K})$ and ${{{\bf Q}}_{2}}$ is either ${\bf Obs}({{\Sigma }^{\prime }},\mathcal{H})$ or ${\bf Ch}(\mathcal{H},\mathcal{K}^{\prime} )$. We denote the set of compatible pairs within ${{{\bf Q}}_{1}}\times {{{\bf Q}}_{2}}$ by ${\bf Comp}$, i.e., ${{\Phi }_{1}}\;\in \;{{{\bf Q}}_{1}}$ and ${{\Phi }_{2}}\;\in \;{{{\bf Q}}_{2}}$ are compatible if and only if $({{\Phi }_{1}},{{\Phi }_{2}})\;\in \;{\bf Comp}$. The set ${{{\bf Q}}_{1}}\times {{{\bf Q}}_{2}}$ is endowed with a natural convex structure by defining for any $({{\Phi }_{1}},{{\Phi }_{2}}),({{\Psi }_{1}},{{\Psi }_{2}})\;\in \;{{{\bf Q}}_{1}}\times {{{\bf Q}}_{2}}$ and $t\;\in \;[0,1]$ the combination $t({{\Phi }_{1}},{{\Phi }_{2}})+(1-t)({{\Psi }_{1}},{{\Psi }_{2}})=\left( t{{\Phi }_{1}}+(1-t){{\Psi }_{1}},t{{\Phi }_{2}}+(1-t){{\Psi }_{2}} \right)$. Whatever the sets of devices involved, the reader may easily check that ${\bf Comp}$ is a convex subset of ${{{\bf Q}}_{1}}\times {{{\bf Q}}_{2}}$.

Denote by L the relative complement of ${\bf Comp}$ with respect to the minimal affine subspace containing ${\bf Comp}$, which coincides with the minimal affine subspace containing ${{{\bf Q}}_{1}}\times {{{\bf Q}}_{2}}$; see remark 2 for a proof of this fact. The product ${{{\bf Q}}_{1}}\times {{{\bf Q}}_{2}}$ we denote by K. We may define the robustness measures ${{w}_{L}}(\cdot |\cdot )$, w_L, and w_L^K introduced in general form in section 2 for the set of compatible pairs ${\bf Comp}$. For simplicity, we denote $w_{L}^{K}(\cdot |\cdot )\;w(\cdot |\cdot )$, ${{w}_{L}}\;w$, and $w_{L}^{K}\;W$. Moreover, for any ${{\Phi }_{1}},{{\Psi }_{1}}\;\in \;{{{\bf Q}}_{1}}$ and ${{\Phi }_{2}},{{\Psi }_{2}}\;\in \;{{{\bf Q}}_{2}}$, we simplify our notations:

$$\begin{eqnarray*}\begin{array}{ccccccccccccccc} w\left( ({{\Phi }_{1}},{{\Phi }_{2}})|({{\Psi }_{1}},{{\Psi }_{2}}) \right) & & w({{\Phi }_{1}},{{\Phi }_{2}}|{{\Psi }_{1}},{{\Psi }_{2}}), \\ w\left( ({{\Phi }_{1}},{{\Phi }_{2}}) \right) & & w({{\Phi }_{1}},{{\Phi }_{2}}), \\ W\left( ({{\Phi }_{1}},{{\Phi }_{2}}) \right) & & W({{\Phi }_{1}},{{\Phi }_{2}}). \\ \end{array}\end{eqnarray*}$$

When ${{{\bf Q}}_{1}}={{{\bf Q}}_{2}}={\bf Ch}(\mathcal{H},\mathcal{K})$, we denote $W(\mathcal{E},\mathcal{E})=W(\mathcal{E})$; this quantity we call as the robustness of self-incompatibility of $\mathcal{E}$.

In the case where ${{{\bf Q}}_{1}}={\bf Obs}(\Sigma ,\mathcal{H})$ and ${{{\bf Q}}_{2}}={\bf Ch}(\mathcal{H},\mathcal{H})$, the quantity ${\bf Obs}(\Sigma ,{\mathcal{H}})\;\ni \;{\sf M} \mapsto W({\sf M} ,{\rm id})$, where ${\rm id}$ stands for the identity channel in ${\bf Ch}(\mathcal{H},\mathcal{H})$, measures how well an approximate version of ${\sf M}$ can be measured while disturbing the system as little as possible. Equivalently, one may think of the number $W({\sf M} ,{\rm id})$ as the measure of how disturbing any measurement of the observable ${\sf M}$ inherently is. In section 5.3, this quantity is calculated in the case of finite-dimensional $\mathcal{H}$ for any rank-1 sharp observable.

Remark 2. In [4], it was shown that, whenever $t\leqslant 1/2$, $({\sf A} ,{\sf B} )\;\in \;{\bf Obs}(\Sigma ,{\mathcal{H}})\times {\bf Obs}({{\Sigma }^{\prime }},{\mathcal{H}})$, and $({\sf S} ,{\sf T} )\;\in \;{\bf Obs}(\Sigma ,{\mathcal{H}})\times {\bf Obs}({{\Sigma }^{\prime }},{\mathcal{H}})$ is a pair of trivial observables, then $t{\sf A} +(1-t){\sf S} {\rm and}t{\sf B} +(1-t){\sf T}$ are jointly measurable, so that

$$\begin{eqnarray*}&&W({\sf A} ,{\sf B} )\geqslant w({\sf A} ,{\sf B} |{\sf S} ,{\sf T} )\geqslant \frac{1}{2}.\end{eqnarray*}$$

Hence, $1/2$ is a global lower bound for the robustness W. Indeed, the same reasoning as that in [4] also applies to the compatibility questions involving other devices than observables as well as to the case of continuous observables not discussed in [4]. Let us study, e.g., the case of observable-channel pairs, i.e., ${{{\bf Q}}_{1}}={\bf Obs}(\Sigma ,\mathcal{H})$ and ${{{\bf Q}}_{2}}={\bf Ch}(\mathcal{H},\mathcal{K})$. Pick any probability measure p on $(\Omega ,\Sigma )$ and any state $\sigma \;\in \;\mathcal{S}(\mathcal{K})$ defining the associated trivial observable ${{{\sf T} }_{p}}=p(\cdot ){\mathbb{1}_{{\mathcal{H}}}}$ and the constant channel ${{\mathcal{T}}_{\sigma }}:\rho \mapsto \sigma$. With any ${\sf M} \;\in \;{\bf Obs}(\Sigma ,{\mathcal{H}})$, $\mathcal{E}\;\in \;{\bf Ch}(\mathcal{H},\mathcal{K})$, and $t\;\in \;[0,1]$,one may set up the instrument $\Gamma \;\in \;{\bf Ins}(\Sigma ,\mathcal{H},\mathcal{K})$,

$$\begin{eqnarray*}&&{{\Gamma }_{X}}(\rho )=t{\rm tr}[\rho {\sf M} (X)]\sigma +(1-t)p(X){\mathcal{E}}(\rho ),\qquad \rho \;\in \;{\mathcal{S}}({\mathcal{H}}).\end{eqnarray*}$$

The observable marginal of Γ is easily found to be $t{\sf M} +(1-t){{{\sf T} }_{p}}$ and the channel marginal is $t{{\mathcal{T}}_{\sigma }}+(1-t)\mathcal{E}$. Especially it follows that fixing any pair $({{{\sf T} }_{p}},{{{\mathcal{T}}}_{\sigma }})$ like that above, one has that $\frac{1}{2}({\sf M} +{{{\sf T} }_{p}})$ and $\frac{1}{2}(\mathcal{E}+{{\mathcal{T}}_{\sigma }})$ are compatible. Similar result is easily proven for channel-channel pairs. It thus follows that there is ${{x}_{0}}\;\in \;{\bf Comp}$ such that $\frac{1}{2}(x+{{x}_{0}})\;\in \;{\bf Comp}$ for all $x\;\in \;{{{\bf Q}}_{1}}\times {{{\bf Q}}_{2}}$ implying $W\geqslant 1/2$ for any device pair

It also follows that the minimal affine subspace F' containing ${\bf Comp}$ coincides with the minimal affine subspace F containing ${{{\bf Q}}_{1}}\times {{{\bf Q}}_{2}}$. Indeed, trivially, $F^{\prime} \subset F$. For the reversed inclusion, let us pick any ${{x}_{0}}\subset {\bf Comp}$ such that $\frac{1}{2}(x+{{x}_{0}})\;\in \;{\bf Comp}$ for all $x\;\in \;{{{\bf Q}}_{1}}\times {{{\bf Q}}_{2}}$. Let $z\;\in \;F$ meaning that there are $x,y\;\in \;{{{\bf Q}}_{1}}\times {{{\bf Q}}_{2}}$ and $\lambda \geqslant 0$ such that $z={{x}_{0}}+\lambda (x-y)$. Defining $x^{\prime} :=\frac{1}{2}(x+{{x}_{0}})\;\in \;{\bf Comp}$ and $y^{\prime} :=\frac{1}{2}(y+{{x}_{0}})\;\in \;{\bf Comp}$, it follows $z={{x}_{0}}+2\lambda (x^{\prime} -y^{\prime} )\;\in \;F^{\prime}$ implying $F\subset F^{\prime}$.

Following [14], in the case of observables operating in a d-dimensional ($d\lt \infty$) Hilbert space one can give an even tighter bound for robustness of incompatibility for observable pairs:

$$\begin{eqnarray*}&&W({\sf A} ,{\sf B} )\geqslant \frac{2+d}{2(1+d)}.\end{eqnarray*}$$

One easily sees, using similar techniques as in [14] that the above inequality holds also for the robustness measures W involving quantum devices other than observables.

In this section we discuss some of the special features of the robustness measures for incompatibility. We will find out that the robustness measure behaves monotonically under certain orderings of device pairs. In the sequel, whenever $(\Phi ,\Psi )$ is a device pair and we write $W(\Phi ,\Psi )$ (or $w(\Phi ,\Psi )$) we implicitly assume that the base set ${{{\bf Q}}_{1}}\times {{{\bf Q}}_{2}}\;\ni \;(\Phi ,\Psi )$ contains compatible pairs, i.e., if Φ and Ψ are observables, they operate in the same Hilbert space, if Φ and Ψ are channels, their input spaces coincides, and if Φ is an observable and Ψ is a channel, then Φ operates in the input space of the channel Ψ.

Let us fix the Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$ and four standard Borel measurable spaces $(\Omega ,\Sigma )$, $(\Omega ^{\prime} ,{{\Sigma }^{\prime }})$, $(\tilde{\Omega },\tilde{\Sigma })$, and $(\tilde{\Omega }^{\prime} ,{{\tilde{\Sigma }}^{\prime }})$. We denote the subset of those pairs $({\sf M} ,{\sf N} )\;\in \;{\bf Obs}(\Sigma ,{\mathcal{H}})\times {\bf Obs}({{\Sigma }^{\prime }},{\mathcal{H}})$ that are jointly measurable by ${\bf JM}(\Sigma ,{{\Sigma }^{\prime }},\mathcal{H})$. We define the corresponding sets of compatible observable pairs also for other pairs of value spaces.

Suppose that ${\sf M} \;\in \;{\bf Obs}(\Sigma ,{\mathcal{K}})$ is a pre-processing of ${\sf A} \;\in \;{\bf Obs}(\Sigma ,{\mathcal{H}})$ and ${\sf N} \;\in \;{\bf Obs}({{\Sigma }^{\prime }},{\mathcal{K}})$ is a pre-processing of ${\sf B} \;\in \;{\bf Obs}({{\Sigma }^{\prime }},{\mathcal{H}})$, both with the same channel $\mathcal{E}\;\in \;{\bf Ch}(\mathcal{H},\mathcal{K})$, i.e., ${\sf M} ={{{\mathcal{E}}}^{*}}\;\circ \;{\sf A}$ and ${\sf N} ={{{\mathcal{E}}}^{*}}\;\circ \;{\sf B}$. In this case, we write $({\sf M} ,{\sf N} ){{\leqslant }_{{\rm prae}}}({\sf A} ,{\sf B} )$. The relation ${{\leqslant }_{{\rm prae}}}$ is a pre-order in the set of observable pairs. We denote $({\sf A} ,{\sf B} ){{=}_{{\rm prae}}}({\sf M} ,{\sf N} )$ if $({\sf M} ,{\sf N} ){{\leqslant }_{{\rm prae}}}({\sf A} ,{\sf B} )$ and $({\sf A} ,{\sf B} ){{\leqslant }_{{\rm prae}}}({\sf M} ,{\sf N} )$ and say that the pairs $({\sf A} ,{\sf B} )$ and $({\sf M} ,{\sf N} )$ are pre-processing equivalent.

Another pre-order in the set of observable pairs is defined by post-processing: If $\tilde{{\sf M} }\;\in \;{\bf Obs}(\tilde{\Sigma },{\mathcal{H}})$ is a post-processing of ${\sf A} \;\in \;{\bf Obs}(\Sigma ,{\mathcal{H}})$ and $\tilde{{\sf N} }\;\in \;{\bf Obs}({{\tilde{\Sigma }}^{\prime }},{\mathcal{H}})$ is a post-processing of ${\sf B} \;\in \;{\bf Obs}({{\Sigma }^{\prime }},{\mathcal{H}})$ with possibly different Markov kernels, we write $(\tilde{{\sf M} },\tilde{{\sf N} }){{\leqslant }_{{\rm post}}}({\sf A} ,{\sf B} )$. We define the post-processing equivalence ${{=}_{{\rm post}}}$ for pairs of observables in the same way as the pre-processing equivalence.

Suppose that $({\sf A} ,{\sf B} )\;\in \;{\bf JM}(\Sigma ,{{\Sigma }^{\prime }},{\mathcal{H}})$ have a joint observable ${\sf G}$. Then ${{{\mathcal{E}}}^{*}}\;\circ \;{\sf G}$ is a joint observable for $({{{\mathcal{E}}}^{*}}\;\circ \;{\sf A} ,{{{\mathcal{E}}}^{*}}\;\circ \;{\sf B} )$ for any channel $\mathcal{E}$. This means that, if the pair $({\sf A} ,{\sf B} )$ is jointly measurable and $(\tilde{{\sf M} },\tilde{{\sf N} }){{\leqslant }_{{\rm prae}}}({\sf A} ,{\sf B} )$, then $(\tilde{{\sf M} },\tilde{{\sf N} })$ is jointly measurable. Moreover, it follows immediately from the definition of joint measurability that, if $({\sf A} ,{\sf B} )$ is jointly measurable and $({\sf M} ,{\sf N} ){{\leqslant }_{{\rm post}}}({\sf A} ,{\sf B} )$, then $({\sf M} ,{\sf N} )$ is jointly measurable.

We may show that the W-robustness measure of incompatibility has the following properties. With a slight modification of the proof given here, one easily shows that these results also hold for w.

Theorem 1. Let $({\sf A} ,{\sf B} )$, $({\sf M} ,{\sf N} )$, and $(\tilde{{\sf M} },\tilde{{\sf N} })$ be observable pairs.

• (a)  
  If $({\sf M} ,{\sf N} ){{\leqslant }_{{\rm prae}}}({\sf A} ,{\sf B} )$, then $W({\sf M} ,{\sf N} )\geqslant W({\sf A} ,{\sf B} )$, and if $({\sf M} ,{\sf N} ){{=}_{{\rm prae}}}({\sf A} ,{\sf B} )$, then $W({\sf M} ,{\sf N} )=W({\sf A} ,{\sf B} )$.
• (b)  
  If $(\tilde{{\sf M} },\tilde{{\sf N} }){{\leqslant }_{{\rm post}}}({\sf A} ,{\sf B} )$, then $W(\tilde{{\sf M} },\tilde{{\sf N} })\geqslant W({\sf A} ,{\sf B} )$, and if $(\tilde{{\sf M} },\tilde{{\sf N} }){{=}_{{\rm post}}}({\sf A} ,{\sf B} )$, then $W(\tilde{{\sf M} },\tilde{{\sf N} })=W({\sf A} ,{\sf B} )$.

Proof. Clearly, if $W({\sf A} ,{\sf B} )=0$, the first claim in item (a) needs no proof. Assume, hence, that $W({\sf A} ,{\sf B} )\gt 0$; in fact, according to remark 2, the robustness measure is always bounded from below by $1/2$. Suppose that $({\sf M} ,{\sf N} ){{\leqslant }_{{\rm prae}}}({\sf A} ,{\sf B} )$, the pre-processing carried out by a channel $\mathcal{E}\;\in \;{\bf Ch}(\mathcal{H},\mathcal{K})$ and $t\lt W({\sf A} ,{\sf B} )$, and let $({{{\sf A} }_{1}},{{{\sf B} }_{1}})\;\in \;{\bf Obs}(\Sigma ,{\mathcal{H}})\times {\bf Obs}({{\Sigma }^{\prime }},{\mathcal{H}})$ and $({{{\sf A} }_{2}},{{{\sf B} }_{2}})\;\in \;{\bf JM}(\Sigma ,{{\Sigma }^{\prime }},{\mathcal{H}})$ be such that

$$\begin{eqnarray*}&&t({\sf A} ,{\sf B} )+(1-t)({{{\sf A} }_{1}},{{{\sf B} }_{1}})=({{{\sf A} }_{2}},{{{\sf B} }_{2}}).\end{eqnarray*}$$

Let now ${{{\sf M} }_{r}}={{{\mathcal{E}}}^{*}}\;\circ \;{{{\sf A} }_{r}}$ and ${{{\sf N} }_{r}}={{{\mathcal{E}}}^{*}}\;\circ \;{{{\sf B} }_{r}}$ for r=1,2. Hence, $({{{\sf M} }_{2}},{{{\sf N} }_{2}})$ is a compatible pair. It follows immediately (as one may check) that one may write

$$\begin{eqnarray*}&&t({\sf M} ,{\sf N} )+(1-t)({{{\sf M} }_{1}},{{{\sf N} }_{1}})=({{{\sf M} }_{2}},{{{\sf N} }_{2}}),\end{eqnarray*}$$

from which the first claim of item (a) follows as one lets $t\lt W({\sf A} ,{\sf B} )$. The second claim of item (a) follows from symmetry.

In the proof of item (b), we may again restrict to the case where $W({\sf A} ,{\sf B} )\gt 0$. Assume now that $(\tilde{{\sf M} },\tilde{{\sf N} }){{\leqslant }_{{\rm post}}}({\sf A} ,{\sf B} )$, so that there are Markov kernels β and γ such that $\tilde{{\sf M} }={{{\sf A} }^{\beta }}$ and $\tilde{{\sf N} }={{{\sf B} }^{\gamma }}$, and $t\leqslant W({\sf A} ,{\sf B} )$, and let $({{{\sf A} }_{1}},{{{\sf B} }_{1}})\;\in \;{\bf Obs}(\Sigma ,{\mathcal{H}})\times {\bf Obs}({{\Sigma }^{\prime }},{\mathcal{H}})$ and $({{{\sf A} }_{2}},{{{\sf B} }_{2}})\;\in {\bf JM}(\Sigma ,{{\Sigma }^{\prime }},{\mathcal{H}})$ be as above. Now, $({\sf A} _{2}^{\beta },{\sf B} _{2}^{\gamma })$ is a compatible pair, and one may write

$$\begin{eqnarray*}&&t(\tilde{{\sf M} },\tilde{{\sf N} })+(1-t)({\sf A} _{1}^{\beta },{\sf B} _{1}^{\gamma })=({\sf A} _{2}^{\beta },{\sf B} _{2}^{\gamma })\end{eqnarray*}$$

proving the first claim of item (b) as one lets $t\uparrow W({\sf A} ,{\sf B} )$. The second claim of item (b) is proven by symmetry again. □

The previous theorem implies, in particular, that the measure $W$ is minimized among pairs $({\sf A} ,{\sf B} )$ of post-processing maximal (i.e. rank-1) observables. In other words, post-processing maximal observable pairs resist joint measurement under added noise the best.

Let now $\mathcal{H},\mathcal{H}^{\prime} ,{{\mathcal{K}}_{1}},{{\mathcal{K}}_{2}},\mathcal{K}_{1}^{\prime }$, and $\mathcal{K}_{2}^{\prime }$ be Hilbert spaces. We denote, e.g., for the spaces $\mathcal{H}$, ${{\mathcal{K}}_{1}}$, and ${{\mathcal{K}}_{2}}$, the set of compatible pairs in ${\bf Ch}(\mathcal{H},{{\mathcal{K}}_{1}})\times {\bf Ch}(\mathcal{H},{{\mathcal{K}}_{2}})$ by ${\bf Comp}(\mathcal{H},{{\mathcal{K}}_{1}},{{\mathcal{K}}_{2}})$.

For channels $\mathcal{E}^{\prime} \;\in \;{\bf Ch}(\mathcal{H}^{\prime} ,{{\mathcal{K}}_{1}})$, $\mathcal{F}^{\prime} \;\in \;{\bf Ch}(\mathcal{H}^{\prime} ,{{\mathcal{K}}_{2}})$, $\mathcal{E}\;\in \;{\bf Ch}(\mathcal{H},{{\mathcal{K}}_{1}})$, and $\mathcal{F}\;\in \;{\bf Ch}(\mathcal{H},{{\mathcal{K}}_{2}})$, we denote $(\mathcal{E}^{\prime} ,\mathcal{F}^{\prime} ){{\leqslant }_{{\rm prae}}}(\mathcal{E},\mathcal{F})$ if there is a channel $\mathcal{G}\;\in \;{\bf Ch}(\mathcal{H}^{\prime} ,\mathcal{H})$ such that $\mathcal{E}^{\prime} =\mathcal{E}\;\;\mathcal{G}$ and $\mathcal{F}^{\prime} =\mathcal{F}\;\;\mathcal{G}$. This gives rise to the partial order ${{\leqslant }_{{\rm prae}}}$ associated to pre-processing of channel pairs. We denote by ${{=}_{{\rm prae}}}$ the corresponding equivalence relation. Suppose that the $(\mathcal{E},\mathcal{F})\;\in \;{\bf Comp}(\mathcal{H},{{\mathcal{K}}_{1}},{{\mathcal{K}}_{2}})$ has the joint channel $\mathcal{M}\;\in \;{\bf Ch}(\mathcal{H},{{\mathcal{K}}_{1}}\otimes {{\mathcal{K}}_{2}})$ and pick $\mathcal{G}\;\in \;{\bf Ch}(\mathcal{H}^{\prime} ,\mathcal{H})$. It follows that also the pair $(\mathcal{E}\;\;\mathcal{G},\mathcal{F}\;\;\mathcal{G})$ is compatible since it has (among others) the joint channel $\mathcal{M}\;\;\mathcal{G}$. This means that,whenever the pair $(\mathcal{E},\mathcal{F})$ is compatible and $(\mathcal{E}^{\prime} ,\mathcal{F}^{\prime} ){{\leqslant }_{{\rm prae}}}(\mathcal{E},\mathcal{F})$, then also $(\mathcal{E}^{\prime} ,\mathcal{F}^{\prime} )$ is compatible.

When $\mathcal{E}^{\prime} \;\in \;{\bf Ch}(\mathcal{H},\mathcal{K}_{1}^{\prime })$, $\mathcal{F}^{\prime} \;\in \;{\bf Ch}(\mathcal{H},\mathcal{K}_{2}^{\prime })$, $\mathcal{E}\;\in \;{\bf Ch}(\mathcal{H},{{\mathcal{K}}_{1}})$, and $\mathcal{F}\;\in \;{\bf Ch}(\mathcal{H},{{\mathcal{K}}_{2}})$, we denote $(\mathcal{E}^{\prime} ,\mathcal{F}^{\prime} ){{\leqslant }_{{\rm post}}}(\mathcal{E},\mathcal{F})$ if there are channels $\mathcal{A}\;\in \;{\bf Ch}({{\mathcal{K}}_{1}},\mathcal{K}_{1}^{\prime })$ and $\mathcal{B}\;\in \;{\bf Ch}({{\mathcal{K}}_{2}},\mathcal{K}_{2}^{\prime })$ such that $\mathcal{E}^{\prime} =\mathcal{A}\;\;\mathcal{E}$ and $\mathcal{F}^{\prime} =\mathcal{B}\;\;\mathcal{F}$. We denote the equivalence relation corresponding to the partial order ${{\leqslant }_{{\rm post}}}$ by ${{=}_{{\rm post}}}$. If $(\mathcal{E},\mathcal{F})\;\in \;{\bf Comp}(\mathcal{H},{{\mathcal{K}}_{1}},{{\mathcal{K}}_{2}})$ has the joint channel $\mathcal{M}\;\in \;{\bf Ch}(\mathcal{H},{{\mathcal{K}}_{1}}\otimes {{\mathcal{K}}_{2}})$ and we choose $\mathcal{A}\;\in \;{\bf Ch}({{\mathcal{K}}_{1}},\mathcal{K}_{1}^{\prime })$ and $\mathcal{B}\;\in \;{\bf Ch}({{\mathcal{K}}_{2}},\mathcal{K}_{2}^{\prime })$ we may define the joint channel $(\mathcal{A}\otimes \mathcal{B})\;\;\mathcal{M}$ for the pair $(\mathcal{A}\;\;\mathcal{E},\mathcal{B}\;\;\mathcal{F})$. Thus, whenever the pair $(\mathcal{E},\mathcal{F})$ is compatible and $(\mathcal{E}^{\prime} ,\mathcal{F}^{\prime} ){{\leqslant }_{{\rm post}}}(\mathcal{E},\mathcal{F})$, then also $(\mathcal{E}^{\prime} ,\mathcal{F}^{\prime} )$ is compatible.

As for observables, we may easily prove the following (the robustness measure w possesses the same properties).

Theorem 2. Let $(\mathcal{E},\mathcal{F})$, $(\mathcal{C},\mathcal{D})$, and $(\mathcal{C}^{\prime} ,\mathcal{D}^{\prime} )$ be channel pairs.

• (a)  
  If $(\mathcal{C},\mathcal{D}){{\leqslant }_{{\rm prae}}}(\mathcal{E},\mathcal{F})$, then $W(\mathcal{C},\mathcal{D})\geqslant W(\mathcal{E},\mathcal{F})$, and if $(\mathcal{C},\mathcal{D}){{=}_{{\rm prae}}}(\mathcal{E},\mathcal{F})$, then $W(\mathcal{C},\mathcal{D})=W(\mathcal{E},\mathcal{F})$.
• (b)  
  If $(\mathcal{C}^{\prime} ,\mathcal{D}^{\prime} ){{\leqslant }_{{\rm post}}}(\mathcal{E},\mathcal{F})$, then $W(\mathcal{C}^{\prime} ,\mathcal{D}^{\prime} )\geqslant W(\mathcal{E},\mathcal{F})$, and if $(\mathcal{C}^{\prime} ,\mathcal{D}^{\prime} ){{=}_{{\rm post}}}(\mathcal{E},\mathcal{F})$, then $W(\mathcal{C}^{\prime} ,\mathcal{D}^{\prime} )=W(\mathcal{E},\mathcal{F})$.

Thus, especially, we have $W(\mathcal{E},\mathcal{F})\geqslant W({\rm id})$ for any $\mathcal{E}\;\in \;{\bf Ch}(\mathcal{H},{{\mathcal{K}}_{1}})$ and $\mathcal{F}\;\in \;{\bf Ch}(\mathcal{H},{{\mathcal{K}}_{2}})$, where ${\rm id}\;\in \;{\bf Ch}(\mathcal{H},\mathcal{H})$ is the identity channel, i.e., the pair $({\rm id},{\rm id})$ is the most incompatible pair of channels with respect to the robustness measures. The robustness measure W (as well as w) attains the same minimal value at any channel pair in the post-processing equivalence class determined by the identity channel pair $({\rm id},{\rm id})$, ${\rm id}\;\in \;{\bf Ch}(\mathcal{H},\mathcal{H})$. Clearly, a pair $(\mathcal{E},\mathcal{F})\;\in \;{\bf Ch}(\mathcal{H},{{\mathcal{K}}_{1}})\times {\bf Ch}(\mathcal{H},{{\mathcal{K}}_{2}})$ is in this equivalence class when they are left-invertible by channels, i.e., there are channels $\mathcal{A}\;\in \;{\bf Ch}({{\mathcal{K}}_{1}},\mathcal{H})$ and $\mathcal{B}\;\in \;{\bf Ch}({{\mathcal{K}}_{2}},\mathcal{H})$ such that $\mathcal{A}\;\;\mathcal{E}=\mathcal{B}\;\;\mathcal{F}={\rm id}$. From now on, we call such channels decodable. As a special case of [17, corollary 1], when $\mathcal{H}$ and $\mathcal{K}$ are finite dimensional, a channel $\mathcal{E}\;\in \;{\bf Ch}(\mathcal{H},\mathcal{K})$ is decodable if and only if there is a Hilbert space ${{\mathcal{K}}_{0}}$, a unitary operator $U:\mathcal{H}\otimes {{\mathcal{K}}_{0}}\to \mathcal{K}$, and a positive trace-1 operator T on ${{\mathcal{K}}_{0}}$ such that $\mathcal{E}(\rho )=U(\rho \otimes T){{U}^{*}}$ for all $\rho \;\in \;\mathcal{S}(\mathcal{H})$. It follows that a channel with unitarily equivalent input and output spaces is decodable if and only if it is a unitary channel, i.e., of the form $\rho \mapsto U\rho {{U}^{*}}$ with a unitary operator U. The decodable channels posses essentially the same properties with respect to the robustness measures as the identity channel.

Let us fix Hilbert spaces $\mathcal{H},\mathcal{H}^{\prime} ,\mathcal{K}$, and $\mathcal{K}^{\prime}$ and standard Borel spaces $(\Omega ,\Sigma )$ and $(\Omega ^{\prime} ,{{\Sigma }^{\prime }})$. We denote, e.g., for $\mathcal{H},\mathcal{K}$, and Σ, by ${\bf Comp}(\Sigma ,\mathcal{H},\mathcal{K})$ the set of compatible pairs in ${\bf Obs}(\Sigma ,\mathcal{H})\times {\bf Ch}(\mathcal{H},\mathcal{K})$.

We denote $({\sf M} ^{\prime} ,{\mathcal{E}}^{\prime} ){{\leqslant }_{{\rm prae}}}({\sf M} ,{\mathcal{E}})$ for ${\sf M} ^{\prime} \;\in \;{\bf Obs}(\Sigma ,{\mathcal{H}}^{\prime} )$, $\mathcal{E}^{\prime} \;\in \;{\bf Ch}(\mathcal{H}^{\prime} ,\mathcal{K})$, ${\sf M} \;\in \;{\bf Obs}(\Sigma ,{\mathcal{H}})$, and $\mathcal{E}\;\in \;{\bf Ch}(\mathcal{H},\mathcal{K})$ if there is $\mathcal{G}\;\in \;{\bf Ch}(\mathcal{H}^{\prime} ,\mathcal{H})$ such that ${\sf M} ^{\prime} ={{{\mathcal{G}}}^{*}}\;\circ \;{\sf M}$ and $\mathcal{E}^{\prime} =\mathcal{E}\;\;\mathcal{G}$. Again, the equivalence relation corresponding to the partial order ${{\leqslant }_{{\rm prae}}}$ is ${{=}_{{\rm prae}}}$. It is easy to see that, if $({\sf M} ,{\mathcal{E}})$ is a compatible pair and $({\sf M} ^{\prime} ,{\mathcal{E}}^{\prime} ){{\leqslant }_{{\rm prae}}}({\sf M} ,{\mathcal{E}})$, then $({\sf M} ^{\prime} ,{\mathcal{E}}^{\prime} )$ is compatible as well.

If ${\sf M} ^{\prime} ={{{\sf M} }^{\beta }}$ and $\mathcal{E}^{\prime} =\mathcal{B}\;\;\mathcal{E}$, where ${\sf M} ^{\prime} \;\in \;{\bf Obs}({{\Sigma }^{\prime }},{\mathcal{H}})$, $\mathcal{E}^{\prime} \;\in \;{\bf Ch}(\mathcal{H},\mathcal{K}^{\prime} )$, ${\sf M} \;\in \;{\bf Obs}(\Sigma ,{\mathcal{H}})$, and $\mathcal{E}\;\in \;{\bf Ch}(\mathcal{H},\mathcal{K})$, for a channel $\mathcal{B}\;\in \;{\bf Ch}(\mathcal{K},\mathcal{K}^{\prime} )$ and a Markov kernel $\beta :\Sigma ^{\prime} \times \Omega \to \mathbb{R}$, we denote $({\sf M} ^{\prime} ,{\mathcal{E}}^{\prime} ){{\leqslant }_{{\rm post}}}({\sf M} ,{\mathcal{E}})$.The equivalence relation associated with the partial order ${{\leqslant }_{{\rm post}}}$ is denoted by ${{=}_{{\rm post}}}$. Suppose that $({\sf M} ,{\mathcal{E}})\;\in \;{\bf Comp}(\Sigma ,{\mathcal{H}},{\mathcal{K}})$ has the joint instrument $\Gamma \;\in \;{\bf Ins}(\Sigma ,\mathcal{H},\mathcal{K})$ and pick $\mathcal{B}\;\in \;{\bf Ch}(\mathcal{K},\mathcal{K}^{\prime} )$ and a Markov kernel $\beta :\Sigma ^{\prime} \times \Omega \to \mathbb{R}$. It is straight-forward to check that $\Gamma ^{\prime} \;\in \;{\bf Ins}({{\Sigma }^{\prime }},\mathcal{H},\mathcal{K}^{\prime} )$,

$$\begin{eqnarray*}&&\Gamma _{Y}^{\prime }(\rho )=\mathcal{B}\left( {{\int }_{\Omega }}\beta (Y,\omega ){{\Gamma }_{d\omega }}(\rho ) \right),\qquad Y\;\in \;{{\Sigma }^{\prime }},\quad \rho \;\in \;\mathcal{S}(\mathcal{H}),\end{eqnarray*}$$

is a joint instrument for $({{{\sf M} }^{\beta }},{\mathcal{B}}\;\circ \;{\mathcal{E}})$ implying that, whenever the pair $({\sf M} ,{\mathcal{E}})$ is compatible and $({\sf M} ^{\prime} ,{\mathcal{E}}^{\prime} ){{\leqslant }_{{\rm post}}}({\sf M} ,{\mathcal{E}})$, then $({\sf M} ^{\prime} ,{\mathcal{E}}^{\prime} )$ is compatible as well.

Again, one easily proves the following properties (which also hold for w).

Theorem 3. Let $({\sf M} ,{\mathcal{E}})$, $({\sf N} ,{\mathcal{F}})$, and $({\sf N} ^{\prime} ,{\mathcal{F}}^{\prime} )$ be observable-channel pairs.

• (a)  
  If $({\sf N} ,{\mathcal{F}}){{\leqslant }_{{\rm prae}}}({\sf M} ,{\mathcal{E}})$, then $W({\sf N} ,{\mathcal{F}})\geqslant W({\sf M} ,{\mathcal{E}})$, and if $({\sf N} ,{\mathcal{F}}){{=}_{{\rm prae}}}({\sf M} ,{\mathcal{E}})$, then $W({\sf N} ,{\mathcal{F}})=W({\sf M} ,{\mathcal{E}})$.
• (b)  
  If $({\sf N} ^{\prime} ,{\mathcal{F}}^{\prime} ){{\leqslant }_{{\rm post}}}({\sf M} ,{\mathcal{E}})$, then $W({\sf N} ^{\prime} ,{\mathcal{F}}^{\prime} )\geqslant W({\sf M} ,{\mathcal{E}})$, and if $({\sf N} ^{\prime} ,{\mathcal{F}}^{\prime} ){{=}_{{\rm post}}}({\sf M} ,{\mathcal{E}})$, then $W({\sf N} ^{\prime} ,{\mathcal{F}}^{\prime} )=W({\sf M} ,{\mathcal{E}})$.

Theorems 1, 2, and 3 together tell that instead of considering robustness measures as functions on individual device pairs, they can be defined on pre- or post-processing equivalence classes. The partial orders invoked by pre- and post-processing in the set of observables and their meaning are studied, e.g., in [3, 11]. The results of this section also tell that the measure $R:=1/W-1$ (as well as $1/w-1$) is an incompatibility monotone from the perspective of [12]. The operations defining the preorders ${{\leqslant }_{{\rm prae}}}$ and ${{\leqslant }_{{\rm post}}}$, common pre-processing and independent bipartite post-processing, can be naturally viewed as compatibility non-decreasing maps and any incompatibility measure should naturally behave monotonously under these operations. Monotonicity under ${{\leqslant }_{{\rm prae}}}$ has been required already in [12, 22].

In this section, we calculate the robustness of incompatibility for a particular pair of incompatible observables: a finite-dimensional Weyl pair. Let us fix a d-dimensional Hilbert space $\mathcal{H}$ ($d\lt \infty$) which has the orthonormal base ${{\{{{\varphi }_{j}}\}}_{j\,\in \,{{\mathbb{Z}}_{d}}}}$. We denote $\langle {\mkern 1mu} j,k\rangle ={{{\rm e}}^{{\rm i}2\pi jk/d}}$ for all $j,k\;\in \;{{\mathbb{Z}}_{d}}$ and define the linear operator $\mathcal{F}\;\in \;\mathcal{L}(\mathcal{H})$ through

$$\begin{eqnarray}&&\mathcal{F}{{\varphi }_{j}}=\frac{1}{\sqrt{d}}\mathop{\sum }\limits_{i\,\in \,{{\mathbb{Z}}_{d}}}\overline{\langle i,j\rangle }{{\varphi }_{i}},\qquad j\;\in \;{{\mathbb{Z}}_{d}}.\end{eqnarray} \tag{ 5.1 }$$

This operator is the Fourier-operator and its adjoint is defined through

$$\begin{eqnarray*}&&{{\mathcal{F}}^{*}}{{\varphi }_{j}}=\frac{1}{\sqrt{d}}\mathop{\sum }\limits_{i\,\in \,{{\mathbb{Z}}_{d}}}\langle i,j\rangle {{\varphi }_{i}},\qquad j\;\in \;{{\mathbb{Z}}_{d}}.\end{eqnarray*}$$

For simplicity, we denote by ${\bf Ob}{{{\bf s}}_{d}}$ the set of observables operating in $\mathcal{H}$ whose value space is ${{\mathbb{Z}}_{d}}$ (equipped with its power set as the σ-algebra). Hence, an observable ${\sf M} \;\in \;{\bf Ob}{{{\bf s}}_{d}}$ is defined by the values ${\sf M} (\{{\mkern 1mu} j\}):={{{\sf M} }_{j}}\;\in \;{\mathcal{L}}({\mathcal{H}})$, $j\;\in \;{{\mathbb{Z}}_{d}}$, and we write ${\sf M} ={{({{{\sf M} }_{j}})}_{j\,\in \,{{\mathbb{Z}}_{d}}}}$. We denote the set of compatible pairs in ${\bf Ob}{{{\bf s}}_{d}}\times {\bf Ob}{{{\bf s}}_{d}}$ by ${\bf J}{{{\bf M}}_{d}}$. We denote the set of ${{\mathbb{Z}}_{d}}\times {{\mathbb{Z}}_{d}}$-valued observables operating in $\mathcal{H}$ (the possible joint observables for the compatible pairs $({\sf M} ,{\sf N} )\;\in \;{\bf J}{{{\bf M}}_{d}}$) by ${\bf Ob}{{{\bf s}}_{d\times d}}$. When ${\sf G} \;\in \;{\bf Ob}{{{\bf s}}_{d\times d}}$, we set ${{{\sf G} }_{j,k}}:={\sf G} (\{({\mkern 1mu} j,k)\})$ for all $j,k\;\in \;{{\mathbb{Z}}_{d}}$.

We fix another orthonormal basis ${{\{{{\psi }_{k}}\}}_{k\,\in \,{{\mathbb{Z}}_{d}}}}$ by setting ${{\psi }_{k}}={{\mathcal{F}}^{*}}{{\varphi }_{k}}$. It follows that $\langle {{\varphi }_{j}}|{{\psi }_{k}}\rangle ={{d}^{-1/2}}$ for all $j,k\;\in \;{{\mathbb{Z}}_{d}}$, so that the bases $\{{{\varphi }_{j}}\}$ and $\{{{\psi }_{k}}\}$ are an example of a pair of mutually unbiased bases. Let us denote

$$\begin{eqnarray*}&&{{{\sf Q} }_{j}}:=|{{\varphi }_{j}}\rangle \langle {{\varphi }_{j}}|,\quad {{{\sf P} }_{k}}:=|{{\psi }_{k}}\rangle \langle {{\psi }_{k}}|,\qquad j,k\;\in \;{{\mathbb{Z}}_{d}}\end{eqnarray*}$$

and define the sharp observables ${\sf Q} :={{({{{\sf Q} }_{j}})}_{j\,\in \,{{\mathbb{Z}}_{d}}}}\;\in \;{\bf Ob}{{{\bf s}}_{d}}$ and ${\sf P} :={{({{{\sf P} }_{k}})}_{k\,\in \,{{\mathbb{Z}}_{d}}}}\;\in \;{\bf Ob}{{{\bf s}}_{d}}$.

For each $q,p\;\in \;{{\mathbb{Z}}_{d}}$, we may define the operators ${{U}_{q}},{{V}_{p}},{{W}_{q,p}}\;\in \;\mathcal{L}(\mathcal{H})$ through

$$\begin{eqnarray}&&{{U}_{q}}{{\varphi }_{j}}={{\varphi }_{j+q}},\end{eqnarray} \tag{ 5.2 }$$

$$\begin{eqnarray}&&{{V}_{p}}{{\varphi }_{j}}=\langle {\mkern 1mu} j,p\rangle {{\varphi }_{j}},\end{eqnarray} \tag{ 5.3 }$$

$$\begin{eqnarray}&&{{W}_{q,p}}={{U}_{q}}{{V}_{p}},\end{eqnarray} \tag{ 5.4 }$$

where the sums and differences are considered as cyclic on ${{\mathbb{Z}}_{d}}$. Thus $(q,p)\mapsto {{W}_{q,p}}$ is a projective unitary representation of ${{\mathbb{Z}}_{d}}\times {{\mathbb{Z}}_{d}}$ in $\mathcal{H}$ which we call as the d-dimensional Weyl representation. It follows that

$$\begin{eqnarray*}&&W_{q,p}^{*}{{{\sf Q} }_{j}}{{W}_{q,p}}={{{\sf Q} }_{j-q}},\quad W_{q,p}^{*}{{{\sf P} }_{k}}{{W}_{q,p}}={{{\sf P} }_{k-p}},\qquad j,k,q,p\;\in \;{{\mathbb{Z}}_{d}},\end{eqnarray*}$$

i.e., ${\sf Q}$ and ${\sf P}$ are Weyl-covariant.

We denote the set of all Weyl-covariant pairs $({\sf A} ,{\sf B} )\;\in \;{\bf Ob}{{{\bf s}}_{d}}\times {\bf Ob}{{{\bf s}}_{d}}$, i.e.,

$$\begin{eqnarray}&&W_{q,p}^{*}{{{\sf A} }_{j}}{{W}_{q,p}}={{{\sf A} }_{j-q}},\quad W_{q,p}^{*}{{{\sf B} }_{k}}{{W}_{q,p}}={{{\sf B} }_{k-p}},\qquad j,k,q,p\;\in \;{{\mathbb{Z}}_{d}},\end{eqnarray} \tag{ 5.5 }$$

by ${\bf Obs}_{d\times d}^{W}$. Any Weyl-covariant pair $({\sf A} ,{\sf B} )$, where ${\sf A}$ and ${\sf B}$ are sharp, is unitarily equivalent with the fixed pair $({\sf Q} ,{\sf P} )$ in the sense that there is a unitary operator U on $\mathcal{H}$ such that ${{{\sf A} }_{j}}={{U}^{*}}{{{\sf Q} }_{j}}U$ and ${{{\sf B} }_{k}}={{U}^{*}}{{{\sf P} }_{k}}U$ for all $j,k\;\in \;{{\mathbb{Z}}_{d}}$. From now on, we call the pair $({\sf Q} ,{\sf P} )$ as the Weyl pair.

If $({\sf M} ,{\sf N} )\;\in \;{\bf Obs}_{d\;\,\times \;\,d}^{W}$, there are probability distributions $\mu ={{({{\mu }_{j}})}_{j\,\in \,{{\mathbb{Z}}_{d}}}}$ and $\nu ={{({{\nu }_{k}})}_{k\,\in \,{{\mathbb{Z}}_{d}}}}$ such that ${\sf M} =\mu *{\sf Q}$ and ${\sf N} =\nu *{\sf P}$, i.e.,

$$\begin{eqnarray}&&{{{\sf M} }_{j}}=\mathop{\sum }\limits_{q\,\in \,{{\mathbb{Z}}_{d}}}{{\mu }_{j-q}}{{{\sf Q} }_{q}},\quad {{{\sf N} }_{k}}=\mathop{\sum }\limits_{p\,\in \,{{\mathbb{Z}}_{d}}}{{\nu }_{k-p}}{{{\sf P} }_{p}}\end{eqnarray} \tag{ 5.6 }$$

for all $j,k\;\in \;{{\mathbb{Z}}_{d}}$. Moreover, such a Weyl-covariant pair is jointly measurable if and only if there is a state $\rho \;\in \;\mathcal{S}(\mathcal{H})$ such that

$$\begin{eqnarray}&&{{\mu }_{j}}={\rm tr}[\rho {{{\sf Q} }_{-j}}],\quad {{\nu }_{k}}={\rm tr}[\rho {{{\sf P} }_{-k}}],\qquad j,k\;\in \;{{\mathbb{Z}}_{d}}.\end{eqnarray} \tag{ 5.7 }$$

The latter condition can also be written using a purification $\eta \;\in \;\mathcal{H}\otimes \mathcal{H}$ of ρ, so that

$$\begin{eqnarray}&&{{\mu }_{j}}=\langle \eta |({{{\sf Q} }_{-j}}\otimes {\mathbb{1}})\eta \rangle ,\quad {{\nu }_{k}}=\langle \eta |({{{\sf P} }_{-k}}\otimes {\mathbb{1}})\eta \rangle ,\qquad j,k\;\in \;{{\mathbb{Z}}_{d}}.\end{eqnarray} \tag{ 5.8 }$$

For proofs of these facts about Weyl-covariant pairs, we refer to [5].

The following lemma is useful for evaluating the robustness of incompatibility for any Weyl-covariant pair.

Lemma 1. Let $({\sf A} ,{\sf B} )\;\in \;{\bf Obs}_{d\;\,\times \;\,d}^{W}$. One has

$$\begin{eqnarray*}&&W({\sf A} ,{\sf B} )=\mathop{{\rm sup} }\limits_{({\sf M} ,{\sf N} )\,\in \,{\bf Obs}_{d\times d}^{W}}w({\sf A} ,{\sf B} |{\sf M} ,{\sf N} ).\end{eqnarray*}$$

Proof. Let us first define a map ${\bf Ob}{{{\bf s}}_{d\times d}}\;\ni \;{\sf G} \mapsto {{{\sf G} }^{W}}\;\in \;{\bf Ob}{{{\bf s}}_{d\times d}}$ by setting

$$\begin{eqnarray*}&&{\sf G} _{j,k}^{W}=\frac{1}{{{d}^{2}}}\mathop{\sum }\limits_{q,p\,\in \,{{\mathbb{Z}}_{d}}}W_{q,p}^{*}{{{\sf G} }_{j+q,k+p}}{{W}_{q,p}},\quad j,k\;\in \;{{\mathbb{Z}}_{d}}.\end{eqnarray*}$$

It follows (as one may easily check) that

$$\begin{eqnarray*}&&W_{q,p}^{*}{\sf G} _{j,k}^{W}{{W}_{q,p}}={\sf G} _{j-q,k-p}^{W}\end{eqnarray*}$$

for all $j,k,q,p\;\in \;{{\mathbb{Z}}_{d}}$. Similarly for any ${\sf M} \;\in \;{\bf Ob}{{{\bf s}}_{d}}$, we define ${{{\sf M} }^{W,1}},{{{\sf M} }^{W,2}}\;\in \;{\bf Ob}{{{\bf s}}_{d}}$ through

$$\begin{eqnarray*}\begin{array}{ccccccccccccccc} {\sf M} _{j}^{W,1} & = & \frac{1}{{{d}^{2}}}\mathop{\sum }\limits_{q,p\,\in \,{{\mathbb{Z}}_{d}}}W_{q,p}^{*}{{{\sf M} }_{j+q}}{{W}_{q,p}},\quad j\;\in \;{{\mathbb{Z}}_{d}}, \\ {\sf M} _{k}^{W,2} & = & \frac{1}{{{d}^{2}}}\mathop{\sum }\limits_{q,p\,\in \,{{\mathbb{Z}}_{d}}}W_{q,p}^{*}{{{\sf M} }_{k+p}}{{W}_{q,p}},\quad k\;\in \;{{\mathbb{Z}}_{d}}. \\ \end{array}\end{eqnarray*}$$

It follows that, when ${\sf G} \;\in \;{\bf Ob}{{{\bf s}}_{d\times d}}$ is a joint observable for $({\sf M} ,{\sf N} )\;\in \;{\bf J}{{{\bf M}}_{d}}$, i.e., ${{\sum }_{k}}{{{\sf G} }_{j,k}}={{{\sf M} }_{j}}$ and ${{\sum }_{j}}{{{\sf G} }_{j,k}}={{{\sf N} }_{k}}$, then $({{{\sf M} }^{W,1}},{{{\sf N} }^{W,2}})\;\in \;{\bf J}{{{\bf M}}_{d}}$ and this pair has (among others) the joint observable ${{{\sf G} }^{W}}$. Furthermore, $({{{\sf M} }^{W,1}},{{{\sf N} }^{W,2}})\;\in \;{\bf Obs}_{d\;\,\times \;\,d}^{W}$ for any ${\sf M} ,{\sf N} \;\in \;{\bf Ob}{{{\bf s}}_{d}}$, and, if $({\sf A} ,{\sf B} )\;\in \;{\bf Obs}_{d\;\,\times \;\,d}^{W}$, then ${{{\sf A} }^{W,1}}={\sf A}$ and ${{{\sf B} }^{W,2}}={\sf B}$.

Suppose now that $t\lt W({\sf A} ,{\sf B} )$ and let $({\sf M} ,{\sf N} )\;\in \;{\bf Ob}{{{\bf s}}_{d}}\times {\bf Ob}{{{\bf s}}_{d}}$, and $(\tilde{{\sf M} },\tilde{{\sf N} })\;\in \;{\bf J}{{{\bf M}}_{d}}$ be such that

$$\begin{eqnarray*}&&t({\sf A} ,{\sf B} )+(1-t)({\sf M} ,{\sf N} )=(\tilde{{\sf M} },\tilde{{\sf N} }).\end{eqnarray*}$$

Since $({\sf A} ,{\sf B} )$ is Weyl covariant, it follows that

$$\begin{eqnarray*}&&t({\sf A} ,{\sf B} )+(1-t)({{{\sf M} }^{W,1}},{{{\sf N} }^{W,2}})=({{\tilde{{\sf M} }}^{W,1}},{{\tilde{{\sf N} }}^{W,2}}),\end{eqnarray*}$$

where $({{\tilde{{\sf M} }}^{W,1}},{{\tilde{{\sf N} }}^{W,2}})\;\in \;{\bf J}{{{\bf M}}_{d}}$, since,if ${\sf G}$ is a joint observable for $(\tilde{{\sf M} },\tilde{{\sf N} })$, then ${{{\sf G} }^{W}}$ is a joint observable for $({{\tilde{{\sf M} }}^{W,1}},{{\tilde{{\sf N} }}^{W,2}})$. Hence, for all $t\lt W({\sf A} ,{\sf B} )$, we find $({\sf M} ,{\sf N} )\;\in \;{\bf Obs}_{d\;\,\times \;\,d}^{W}$ and $(\tilde{{\sf M} },\tilde{{\sf N} })\;\in \;{\bf J}{{{\bf M}}_{d}}\cap {\bf Obs}_{d\;\,\times \;\,d}^{W}$ such that

$$\begin{eqnarray*}&&t({\sf A} ,{\sf B} )+(1-t)({\sf M} ,{\sf N} )=(\tilde{{\sf M} },\tilde{{\sf N} }),\end{eqnarray*}$$

and the claim is proven. □

Theorem 4. The robustness of incompatibility for the sharp Weyl pair is given by

$$\begin{eqnarray}&&W({\sf Q} ,{\sf P} )=\frac{1}{2}\left( 1+\frac{1}{\sqrt{d}} \right).\end{eqnarray} \tag{ 5.9 }$$

Proof. Let $t\lt W({\sf Q} ,{\sf P} )$ and, using lemma 1, suppose that $({\sf M} ,{\sf N} )\;\in \;{\bf Obs}_{d\;\,\times \;\,d}^{W}$ and $({\sf A} ,{\sf B} )\;\in \;{\bf J}{{{\bf M}}_{d}}$ are such that

$$\begin{eqnarray*}&&({\sf A} ,{\sf B} )=t({\sf Q} ,{\sf P} )+(1-t)({\sf M} ,{\sf N} ).\end{eqnarray*}$$

Clearly, $({\sf A} ,{\sf B} )\;\in \;{\bf Obs}_{d\;\,\times \;\,d}^{W}$. Let μ and ν be probability distributions such that ${\sf M}$ and ${\sf N}$ are given by (5.6). Define δ to be the probability distribution having ${{\delta }_{0}}=1$ (and, of course, ${{\delta }_{j}}=0$ for $j\ne 0$). One may write

$$\begin{eqnarray*}&&{\sf A} =(t\delta +(1-t)\mu )\;*\;{\sf Q} ,\quad {\sf B} =(t\delta +(1-t)\nu )\;*\;{\sf P} .\end{eqnarray*}$$

Hence, there has to be $\eta \;\in \;\mathcal{H}\otimes \mathcal{H}$ such that

$$\begin{eqnarray}&&\langle \eta |({{{\sf Q} }_{-j}}\otimes {\mathbb{1}})\eta \rangle =t{{\delta }_{j}}+(1-t){{\mu }_{j}},\quad j\;\in \;{{\mathbb{Z}}_{d}},\end{eqnarray} \tag{ 5.10 }$$

$$\begin{eqnarray}&&\langle \eta |({{{\sf P} }_{-k}}\otimes {\mathbb{1}})\eta \rangle =t{{\delta }_{k}}+(1-t){{\nu }_{k}},\quad k\;\in \;{{\mathbb{Z}}_{d}}.\end{eqnarray} \tag{ 5.11 }$$

We may write $\eta ={{\sum }_{j}}{{\varphi }_{j}}\otimes {{\zeta }_{j}}={{\sum }_{k}}{{\psi }_{k}}\otimes {{\xi }_{k}}$ for some ${{\zeta }_{j}},{{\xi }_{k}}\;\in \;\mathcal{H}$, $j,k\;\in \;{{\mathbb{Z}}_{d}}$. Following the procedure carried out in the proof of [5, Lemma 1], one obtains the (tight) inequalities

$$\begin{eqnarray}&&\sqrt{t(1-{{\nu }_{0}})+{{\nu }_{0}}}\leqslant \frac{1}{\sqrt{d}}\left( \sqrt{t(1-{{\mu }_{0}})+{{\mu }_{0}}}+\alpha \sqrt{1-t} \right),\end{eqnarray} \tag{ 5.12 }$$

$$\begin{eqnarray}&&\sqrt{t(1-{{\mu }_{0}})+{{\mu }_{0}}}\leqslant \frac{1}{\sqrt{d}}\left( \sqrt{t(1-{{\nu }_{0}})+{{\nu }_{0}}}+\beta \sqrt{1-t} \right),\end{eqnarray} \tag{ 5.13 }$$

where $\alpha ={{\sum }_{j\ne 0}}\sqrt{{{\mu }_{j}}}$ and $\beta ={{\sum }_{k\ne 0}}\sqrt{{{\nu }_{k}}}$. It is easy to see that $\alpha \leqslant \sqrt{d-1}\sqrt{1-{{\mu }_{0}}}$ and $\beta \leqslant \sqrt{d-1}\sqrt{1-{{\nu }_{0}}}$, and these bounds are reached when ${{\mu }_{j}}=(1-{{\mu }_{0}})/(d-1)$ and ${{\nu }_{k}}=(1-{{\nu }_{0}})/(d-1)$ for all $j,k\ne 0$. Solving from (5.12), (5.13), one obtains the following inequalities:

$$\begin{eqnarray*}\begin{array}{ccccccccccccccc} t & \leqslant & \frac{{{(\alpha +\sqrt{{\rm d}}\beta )}^{2}}-{{(d-1)}^{2}}{{\mu }_{0}}}{{{(\alpha +\sqrt{{\rm d}}\beta )}^{2}}+{{(d-1)}^{2}}(1-{{\mu }_{0}})} \\ {} & \leqslant & \frac{(d-1){{\left( \sqrt{1-{{\mu }_{0}}}+\sqrt{d(1-{{\nu }_{0}})} \right)}^{2}}-{{(d-1)}^{2}}{{\mu }_{0}}}{(d-1){{\left( \sqrt{1-{{\mu }_{0}}}+\sqrt{d(1-{{\nu }_{0}})} \right)}^{2}}+{{(d-1)}^{2}}(1-{{\mu }_{0}})}. \\ \end{array}\end{eqnarray*}$$

It is easy to see that as one lets ${{\mu }_{0}},{{\nu }_{0}}\downarrow 0$, the latter bound increases and setting ${{\mu }_{0}}={{\nu }_{0}}=0$, one obtains $W({\sf Q} ,{\sf P} )\leqslant \frac{1}{2}(1+1/\sqrt{d})$.

It remains to be shown that the bound obtained above is reachable. Setting $t=\frac{1}{2}(1+1/\sqrt{d})$,

$$\begin{eqnarray*}&&\eta =\sqrt{\frac{\sqrt{d}}{2(\sqrt{d}+1)}}({{\varphi }_{0}}+{{\psi }_{0}})\otimes \xi \end{eqnarray*}$$

for any unit vector $\xi \;\in \;\mathcal{H}$, ${{\mu }_{0}}={{\nu }_{0}}=0$, and ${{\mu }_{j}}={{\nu }_{j}}=1/(d-1)$, for $j\ne 0$, one finds that equations (5.10), (5.11) hold (and in (5.12), (5.13) the inequalities can both be replaced by equalities). Hence, the claim is proven. □

Let us fix a finite-dimensional Hilbert space $\mathcal{H}$, ${\rm dim}\mathcal{H}=d$. We denote ${\bf Ch}(\mathcal{H},\mathcal{H})={\bf C}{{{\bf h}}_{d}}$ and ${\bf Ch}(\mathcal{H},\mathcal{H}\otimes \mathcal{H})={\bf C}{{{\bf h}}_{d\times d}}$. The set of compatible pairs within ${\bf C}{{{\bf h}}_{d}}\times {\bf C}{{{\bf h}}_{d}}$ is denoted by ${\bf Com}{{{\bf p}}_{d}}$. Moreover, ${\rm id}$ stands for the identity channel $\mathcal{S}(\mathcal{H})\to \mathcal{S}(\mathcal{H})$, i.e., ${\rm id}(\rho )=\rho$ for all $\rho \;\in \;\mathcal{S}(\mathcal{H})$. Clearly, the dual ${\rm i}{{{\rm d}}^{*}}$ is the identity map on $\mathcal{L}(\mathcal{H})$ which we denote by ${\rm id}$ as well.

We fix an orthonormal basis $\{|n\rangle \}_{n=1}^{d}$ for the duration of this section and introduce the rank-1 operators ${{\Omega }_{d}}=\sum _{m,n=1}^{d}|mm\rangle \langle nn|\;\in \;\mathcal{L}(\mathcal{H}\otimes \mathcal{H})$ and ${{\Omega }_{d\otimes d}}=\sum _{j,k,m,n=1}^{d}|jkjk\rangle \langle mnmn|\;\in \;\mathcal{L}(\mathcal{H}\otimes \mathcal{H}\otimes \mathcal{H}\otimes \mathcal{H})$, where $|{{m}_{1}}\cdots {{m}_{n}}\rangle :=|{{m}_{1}}\rangle \otimes \cdots \otimes |{{m}_{n}}\rangle$. For any $\mathcal{E}\;\in \;{\bf C}{{{\bf h}}_{d}}$ (respectively $\mathcal{F}\;\in \;{\bf C}{{{\bf h}}_{d\times d}}$) we define the Choi operator $M(\mathcal{E})=({{\mathcal{E}}^{*}}\otimes {\rm id})({{\Omega }_{d}})\;\in \;\mathcal{L}(\mathcal{H}\otimes \mathcal{H})$ (respectively $M(\mathcal{F})=({{\mathcal{F}}^{*}}\otimes {\rm id}\otimes {\rm id})({{\Omega }_{d\times d}})\;\in \mathcal{L}(\mathcal{H}\otimes \mathcal{H}\otimes \mathcal{H})$). We denote the transpose of $B\;\in \;\mathcal{L}(\mathcal{H})$ with respect to the fixed basis by B^T and denote $\bar{B}:={{B}^{T*}}$. Furthermore, we denote the partial transpose restricted to the subsystems 2 and 3 of $C\;\in \;\mathcal{L}(\mathcal{H}\otimes \mathcal{H}\otimes \mathcal{H})$ by ${{C}^{\Gamma }}$, i.e., when ${{C}_{1}},{{C}_{2}},{{C}_{3}}\;\in \;\mathcal{L}(\mathcal{H})$, we have ${{({{C}_{1}}\otimes {{C}_{2}}\otimes {{C}_{3}})}^{\Gamma }}={{C}_{1}}\otimes C_{2}^{T}\otimes C_{3}^{T}$.

If $\mathcal{E}\;\in \;{\bf C}{{{\bf h}}_{d}}$ (respectively $\mathcal{F}\;\in \;{\bf C}{{{\bf h}}_{d\times d}}$) is such that $\mathcal{E}(U\rho {{U}^{*}})=U\mathcal{E}(\rho ){{U}^{*}}$ (respectively $\mathcal{F}(U\rho {{U}^{*}})=(U\otimes U)\mathcal{F}(\rho ){{(U\otimes U)}^{*}}$) for all $\rho \;\in \;\mathcal{S}(\mathcal{H})$ and all unitary $U\;\in \;\mathcal{L}(\mathcal{H})$, we say that $\mathcal{E}$ (respectively $\mathcal{F}$) is fully covariant and denote $\mathcal{E}\in {\bf Co}{{{\bf v}}_{d}}$ (respectively $\mathcal{F}\;\in \;{\bf Co}{{{\bf v}}_{d\times d}}$).Denote by dU the normalized Haar measure of the (compact) unitary group U(d). We may define the map ${\bf C}{{{\bf h}}_{d}}\;\ni \;\mathcal{E}\mapsto {{\mathcal{E}}_{av}}\;\in \;{\bf Co}{{{\bf v}}_{d}}$ by setting

$$\begin{eqnarray*}&&{{\mathcal{E}}_{av}}(\rho )={{\int }_{U(d)}}{{U}^{*}}\mathcal{E}(U\rho {{U}^{*}})U\;{\rm d}U,\qquad \rho \;\in \;\mathcal{S}(\mathcal{H}).\end{eqnarray*}$$

Likewise, one can set up a map ${\bf C}{{{\bf h}}_{d\times d}}\;\ni \;\mathcal{F}\mapsto {{\mathcal{F}}_{av}}\;\in \;{\bf Co}{{{\bf v}}_{d\times d}}$ through

$$\begin{eqnarray}&&{{\mathcal{F}}_{av}}(\rho )={{\int }_{U(d)}}{{(U\otimes U)}^{*}}\mathcal{F}(U\rho {{U}^{*}})(U\otimes U){\rm d}U,\qquad \rho \;\in \;\mathcal{S}(\mathcal{H}).\end{eqnarray} \tag{ 5.14 }$$

The sets ${\bf Co}{{{\bf v}}_{d}}$ and ${\bf Co}{{{\bf v}}_{d\times d}}$ coincide with the sets of the fixed points of these maps.

Suppose that $M\;\in \;\mathcal{L}(\mathcal{H}\otimes \mathcal{H}\otimes \mathcal{H})$ is a positive operator whose partial trace over the subsystems 2 and 3 coincides with ${\mathbb{1}_{\mathcal{H}}}$, i.e., M is a Choi operator of a channel $\mathcal{F}\;\in \;{\bf C}{{{\bf h}}_{d\times d}}$. We may define the operator ${{M}_{av}}\;\in \;\mathcal{L}(\mathcal{H}\otimes \mathcal{H}\otimes \mathcal{H})$ through

$$\begin{eqnarray}&&{{M}_{av}}={{\int }_{U(d)}}(U\otimes \bar{U}\otimes \bar{U})M{{(U\otimes \bar{U}\otimes \bar{U})}^{*}}\;{\rm d}U.\end{eqnarray} \tag{ 5.15 }$$

We have that $(U\otimes \bar{U}\otimes \bar{U})M\ \;=\ \;M(U\otimes \bar{U}\otimes \bar{U})$ for all $U\;\in \;U(d)$ if and only if $M={{M}_{av}}$. It is straightforward to check that

$$\begin{eqnarray*}&&M({{\mathcal{F}}_{av}})=M{{(\mathcal{F})}_{av}},\qquad \mathcal{F}\;\in \;{\bf C}{{{\bf h}}_{d\times d}}.\end{eqnarray*}$$

Lemma 2. Let $\mathcal{E}\;\in \;{\bf Co}{{{\bf v}}_{d}}$. The robustness of self-incompatibility for $\mathcal{E}$ is given by

$$\begin{eqnarray*}&&W(\mathcal{E})=\mathop{{\rm sup} }\limits_{\mathcal{C}\,\in \,{\bf Co}{{{\bf v}}_{d}}}w(\mathcal{E},\mathcal{E}|\mathcal{C},\mathcal{C}).\end{eqnarray*}$$

Proof. Let $t\lt W(\mathcal{E})$ and suppose that $(\mathcal{A},\mathcal{B})\;\in \;{\bf C}{{{\bf h}}_{d}}\times {\bf C}{{{\bf h}}_{d}}$ and $(\mathcal{A}^{\prime} ,\mathcal{B}^{\prime} )\;\in \;{\bf Com}{{{\bf p}}_{d}}$ are such that

$$\begin{eqnarray*}&&t(\mathcal{E},\mathcal{E})+(1-t)(\mathcal{A},\mathcal{B})=(\mathcal{A}^{\prime} ,\mathcal{B}^{\prime} ).\end{eqnarray*}$$

It follows that $(\mathcal{A}_{av}^{\prime },\mathcal{B}_{av}^{\prime })\;\in \;{\bf Com}{{{\bf p}}_{d}}$ as well, since if $\mathcal{F}$ is a joint channel for $(\mathcal{A}^{\prime} ,\mathcal{B}^{\prime} )$, it is immediate that ${{\mathcal{F}}_{av}}$ is a joint channel for $(\mathcal{A}_{av}^{\prime },\mathcal{B}_{av}^{\prime })$. Hence,

$$\begin{eqnarray*}&&t(\mathcal{E},\mathcal{E})+(1-t)({{\mathcal{A}}_{av}},{{\mathcal{B}}_{av}})=(\mathcal{A}_{av}^{\prime },\mathcal{B}_{av}^{\prime }).\end{eqnarray*}$$

Denote $\mathcal{C}=\frac{1}{2}({{\mathcal{A}}_{av}}+{{\mathcal{B}}_{av}})$ and $\mathcal{C}^{\prime} =\frac{1}{2}(\mathcal{A}_{av}^{\prime }+\mathcal{B}_{av}^{\prime })$. Again it easily follows

$$\begin{eqnarray*}&&t(\mathcal{E},\mathcal{E})+(1-t)(\mathcal{C},\mathcal{C})=(\mathcal{C}^{\prime} ,\mathcal{C}^{\prime} ).\end{eqnarray*}$$

Denote by $F\;\in \;\mathcal{L}(\mathcal{H}\otimes \mathcal{H})$ the flip operator, $F(\varphi \otimes \psi )=\psi \otimes \varphi$ for all $\varphi ,\psi \;\in \;\mathcal{H}$. The pair $(\mathcal{C}^{\prime} ,\mathcal{C}^{\prime} )$ is compatible, since if M is the Choi operator of a joint channel of $(\mathcal{A}_{av}^{\prime },\mathcal{B}_{av}^{\prime })$, then $\frac{1}{2}(M+({\mathbb{1}}\otimes F)M({\mathbb{1}}\otimes F))$ is the Choi operator for a joint channel for $(\mathcal{C}^{\prime} ,\mathcal{C}^{\prime} )$. □

Since ${\rm id}$ is fully covariant, the preceding lemma restricts the problem of evaluating the robustness of self-incompatibility of ${\rm id}$ (quite considerably, as we will see). According to lemma 2 (and its proof), $W({\rm id})$ is simply the supremum of those $t\;\in \;[0,1]$ such that $t\;{\rm id}+(1-t)\mathcal{E}$ is self-compatible for some $\mathcal{E}\;\in \;{\bf Co}{{{\bf v}}_{d}}$. Next, we determine the set of self-compatible fully covariant channels which essentially resolves the problem of determining $W({\rm id})$.

Let us fix a self-compatible $\mathcal{A}\;\in \;{\bf Co}{{{\bf v}}_{d}}$ and a joint channel $\mathcal{F}\;\in \;{\bf C}{{{\bf h}}_{d\times d}}$ for the pair $(\mathcal{A},\mathcal{A})$. Since $\mathcal{A}$ is fully covariant, the channel ${{\mathcal{F}}_{av}}$ still has the same marginals. Let $M\;\in \;\mathcal{L}(\mathcal{H}\otimes \mathcal{H}\otimes \mathcal{H})$ be the Choi operator of $\mathcal{F}$ (with respect to our fixed basis). Hence, M_av is the Choi operator of ${{\mathcal{F}}_{av}}$. From (5.15) it follows that

$$\begin{eqnarray*}&&{{M}_{av}}={{\left( \int (U\otimes U\otimes U){{M}^{\Gamma }}{{(U\otimes U\otimes U)}^{*}}\;{\rm d}U \right)}^{\Gamma }}.\end{eqnarray*}$$

This means that $(U\otimes U\otimes U)M_{av}^{\Gamma }=M_{av}^{\Gamma }(U\otimes U\otimes U)$ for all $U\in U(d)$, i.e., $M_{av}^{\Gamma }$ is $U\otimes U\otimes U$-invariant.

For any permutation π of three elements, denote by ${{V}_{\pi }}\in \mathcal{L}(\mathcal{H}\otimes \mathcal{H}\otimes \mathcal{H})$ the unitary operator defined through

$$\begin{eqnarray*}&&{{V}_{\pi }}({{\varphi }_{1}}\otimes {{\varphi }_{2}}\otimes {{\varphi }_{3}})={{\varphi }_{{{\pi }^{-1}}(1)}}\otimes {{\varphi }_{{{\pi }^{-1}}(2)}}\otimes {{\varphi }_{{{\pi }^{-1}}(3)}}\end{eqnarray*}$$

for all ${{\varphi }_{1}},{{\varphi }_{2}},{{\varphi }_{3}}\in \mathcal{H}$. A well-known result from Weyl states that any $U\otimes U\otimes U$-invariant operator is a linear combination of the permutation operators ${{V}_{\pi }}$ from which it follows immediately that our averaged Choi operator can be expressed as a linear combination

$$\begin{eqnarray*}&&{{M}_{av}}=\mathop{\sum }\limits_{\pi }{{\lambda }_{\pi }}V_{\pi }^{\Gamma },\end{eqnarray*}$$

where ${{\lambda }_{\pi }}$ are complex numbers. However, we must make sure that this linear combination is a positive operator whose partial trace over the subsystems 2 and 3 is ${\mathbb{1}_{\mathcal{H}}}$. To this end, we must express the linear combination in a more revealing form.

As the commutant of the set $U\otimes \bar{U}\otimes \bar{U}$, $U\in U(d)$, the operator system spanned by the six operators $V_{\pi }^{\Gamma }$ is a 6-dimensional algebra with the exception in case d = 2, when the algebra is 5-dimensional. In [7], this algebra was shown to have the basis consisting of the operators

$$\begin{eqnarray*}\begin{array}{ccccccccccccccc} {{S}_{\pm }} & = & \frac{1}{2}\left( {\mathbb{1}}\pm V_{(23)}^{\Gamma }-\frac{1}{d\pm 1}(V_{(12)}^{\Gamma }+V_{(13)}^{\Gamma }\pm V_{(123)}^{\Gamma }\pm V_{(132)}^{\Gamma }) \right), \\ {{S}_{0}} & = & \frac{1}{{{d}^{2}}-1}\left( d(V_{(12)}^{\Gamma }+V_{(13)}^{\Gamma })-(V_{(123)}^{\Gamma }+V_{(132)}^{\Gamma }) \right), \\ {{S}_{1}} & = & \frac{1}{{{d}^{2}}-1}\left( d(V_{(123)}^{\Gamma }+V_{(132)}^{\Gamma })-(V_{(12)}^{\Gamma }+V_{(13)}^{\Gamma }) \right), \\ {{S}_{2}} & = & \frac{1}{\sqrt{{{d}^{2}}-1}}(V_{(12)}^{\Gamma }-V_{(13)}^{\Gamma }), \\ {{S}_{3}} & = & \frac{{\rm i}}{\sqrt{{{d}^{2}}-1}}(V_{(123)}^{\Gamma }-V_{(132)}^{\Gamma }), \\ \end{array}\end{eqnarray*}$$

where ${{S}_{\pm }},{{S}_{0}}$ are mutually orthogonal projections summing up to ${\mathbb{1}}$ and ${{S}_{1}},{{S}_{2}}$, and S₃ are selfadjoint operators supported on the eigenspace of S₀ that are interrelated in the same way as the Pauli matrices. Hence, formally ${{S}_{j}}{{S}_{\pm }}={{S}_{\pm }}{{S}_{j}}=0$ and $S_{j}^{2}={{S}_{0}}$ for all $j=0,1,2,3$, and ${{S}_{1}}{{S}_{2}}=i{{S}_{3}}$ with cyclic permutations. Note that, in the case d = 2, ${{S}_{-}}=0$. It follows that a linear combination ${{\mu }_{+}}{{S}_{+}}+{{\mu }_{-}}{{S}_{-}}+{{\mu }_{0}}{{S}_{0}}+{{\mu }_{1}}{{S}_{1}}+{{\mu }_{2}}{{S}_{2}}+{{\mu }_{3}}{{S}_{3}}$ is positive if and only if the multipliers are real, ${{\mu }_{\pm }},{{\mu }_{0}}\geqslant 0$, and $\mu _{1}^{2}+\mu _{2}^{2}+\mu _{3}^{2}\leqslant \mu _{0}^{2}$.

Let us now impose our additional symmetry condition, i.e., the marginals of ${{\mathcal{F}}_{av}}$ must coincide or, equivalently, ${{V}_{(23)}}{{M}_{av}}={{M}_{av}}{{V}_{(23)}}$; note that $V_{(23)}^{\Gamma }={{V}_{(23)}}$. One easily sees that this requirement necessitates that the multipliers of S₂ and S₃ in M be zero. Moreover, through direct calculation, one finds that for the partial traces over the subsystems 2 and 3,

$$\begin{eqnarray*}\begin{array}{ccccccccccccccc} {\rm t}{{{\rm r}}_{23}}[{{S}_{+}}] & = & \frac{1}{2}(d-1)(d+2){\mathbb{1}_{\mathcal{H}}}, & \qquad & {\rm t}{{{\rm r}}_{23}}[{{S}_{-}}] & = & \frac{1}{2}(d+1)(d-2){\mathbb{1}_{\mathcal{H}}}, \\ {\rm t}{{{\rm r}}_{23}}[{{S}_{0}}] & = & 2{\mathbb{1}_{\mathcal{H}}}, & \qquad & {\rm t}{{{\rm r}}_{23}}[{{S}_{1}}] & = & 0. \\ \end{array}\end{eqnarray*}$$

Putting all this together, one finds that M_av is of the form $M({{t}_{+}},{{t}_{-}},{{t}_{0}},{{t}_{1}})$,

$$\begin{eqnarray*}&&M({{t}_{+}},{{t}_{-}},{{t}_{0}},{{t}_{1}})=\frac{2}{(d-1)(d+2)}{{t}_{+}}{{S}_{+}}+\frac{2}{(d+1)(d-2)}{{t}_{-}}{{S}_{-}}+\frac{1}{2}({{t}_{0}}{{S}_{0}}+{{t}_{1}}{{S}_{1}}),\end{eqnarray*}$$

where ${{t}_{\pm }},{{t}_{0}}\geqslant 0$, ${{t}_{+}}+{{t}_{-}}+{{t}_{0}}=1$ and ${{t}_{1}}\in [-{{t}_{0}},{{t}_{0}}]$.

The set of the Choi operators $M({{t}_{+}},{{t}_{-}},{{t}_{0}},{{t}_{1}})$ is a tetrahedron with the extreme points

$$\begin{eqnarray*}\begin{array}{ccccccccccccccc} {{M}_{\pm }} & = & \frac{2}{(d\mp 1)(d\pm 2)}{{S}_{\pm }}, \\ {{\tilde{M}}_{\pm }} & = & \frac{1}{2}({{S}_{0}}\pm {{S}_{1}}), \\ \end{array}\end{eqnarray*}$$

and the partial traces over system 2 (or, equivalently, over system 3) of these are

$$\begin{eqnarray*}\begin{array}{ccccccccccccccc} {\rm t}{{{\rm r}}_{2}}[{{M}_{\pm }}] & = & \frac{1}{{{d}^{2}}-1}(d{\mathbb{1}_{\mathcal{H}\otimes \mathcal{H}}}-{{\Omega }_{d}}), \\ {\rm t}{{{\rm r}}_{2}}[{{\tilde{M}}_{\pm }}] & = & \frac{1}{2(d\pm 1)}\left( {\mathbb{1}_{\mathcal{H}\otimes \mathcal{H}}}+(d\pm 2){{\Omega }_{d}} \right). \\ \end{array}\end{eqnarray*}$$

Hence, the (coinciding) marginals of the channels corresponding to ${{M}_{\pm }}$ are ${{({{d}^{2}}-1)}^{-1}}({{d}^{2}}\mathcal{T}-{\rm id})$ and the (coinciding) marginals of the channels corresponding to ${{\tilde{M}}_{\pm }}$ are ${{(2(d\pm 1))}^{-1}}(d\mathcal{T}+(d\pm 2){\rm id})$, where $\mathcal{T}:\mathcal{S}(\mathcal{H})\to \mathcal{S}(\mathcal{H})$ is the constant channel $\mathcal{T}(\rho )={{d}^{-1}}{\mathbb{1}_{\mathcal{H}}}$. Thus, the set of self-compatible fully covariant channels consists of the elements $\lambda \mathcal{T}+(1-\lambda ){\rm id}$ where $\frac{d}{2(d+1)}\leqslant \lambda \leqslant \frac{{{d}^{2}}}{{{d}^{2}}-1}$.It follows that $W({\rm id})$ is the supremum of the $t\in [0,1]$ such that

$$\begin{eqnarray*}&&\frac{1}{2(1+d)}(d\mathcal{T}+(d+2){\rm id})-t\;{\rm id}\end{eqnarray*}$$

is completely positive or, using the Choi operators, ${\rm t}{{{\rm r}}_{2}}[{{\tilde{M}}_{+}}]-t{{\Omega }_{d}}\geqslant 0$. One easily finds that this condition is satisfied if and only if $t\leqslant \frac{1}{2}(1+1/d)$. Thus, $W({\rm id})=\frac{1}{2}(1+1/d)$. According to the discussion following theorem 2, a pair $(\mathcal{V},\mathcal{W})\in {\bf Ch}(\mathcal{H},{{\mathcal{K}}_{1}})\times {\bf Ch}(\mathcal{H},{{\mathcal{K}}_{2}})$ of decodable channels has the same robustness of incompatibility as the pair of identity channels which is the minimum of the robustness measure, and hence.

Theorem 5. Any pair $(\mathcal{V},\mathcal{W})\in {\bf Ch}(\mathcal{H},{{\mathcal{K}}_{1}})\times {\bf Ch}(\mathcal{H},{{\mathcal{K}}_{2}})$ of decodable channels minimizes W amongst the channel pairs with the input space $\mathcal{H}$, and this minimum value is

$$\begin{eqnarray*}&&W(\mathcal{V},\mathcal{W})=W({\rm id})=\frac{1}{2}\left( 1+\frac{1}{d} \right),\end{eqnarray*}$$

where $d={\rm dim}\mathcal{H}$ and ${\rm id}$ is the identity channel on $\mathcal{S}(\mathcal{H})$.

According to the discussion preceding theorem 5, the self-compatible fully covariant channel $\mathcal{A}$ that has the optimality property

$$\begin{eqnarray}&&\mathcal{A}=\frac{1}{2}\left( 1+\frac{1}{d} \right){\rm id}+\frac{1}{2}\left( 1-\frac{1}{d} \right)\mathcal{E}\end{eqnarray} \tag{ 5.16 }$$

for some $\mathcal{E}\in {\bf C}{{{\bf h}}_{d}}$ is associated with the Choi operator ${\rm t}{{{\rm r}}_{2}}[{{\tilde{M}}_{+}}]$ and is thus given by

$$\begin{eqnarray*}&&\mathcal{A}=\frac{d+2}{2(d+1)}{\rm id}+\frac{d}{2(d+1)}\mathcal{T}.\end{eqnarray*}$$

One joint channel for the optimal pair $(\mathcal{A},\mathcal{A})$ is thus the optimal universal cloner $\rho \mapsto 2{{(d+1)}^{-1}}S(\rho \otimes {\mathbb{1}_{\mathcal{H}}})S$, where $S\in \mathcal{L}(\mathcal{H}\otimes \mathcal{H})$ is the orthogonal projector onto the symmetric subspace (the 1-eigenspace of the flip operator) of $\mathcal{H}\otimes \mathcal{H}$. In fact, the joint channel associated with the Choi operator ${{\tilde{M}}_{+}}$ is exactly this optimal cloner. The Choi operator associated with the channel $\mathcal{E}$ in the decomposition (5.16) is ${\rm t}{{{\rm r}}_{2}}[{{M}_{+}}]={\rm t}{{{\rm r}}_{2}}[{{M}_{-}}]$, so that

$$\begin{eqnarray*}&&\mathcal{E}=-\frac{1}{{{d}^{2}}-1}{\rm id}+\frac{{{d}^{2}}}{{{d}^{2}}-1}\mathcal{T}.\end{eqnarray*}$$

Especially $\mathcal{E}$ is self-compatible so that, in fact

$$\begin{eqnarray*}&&W({\rm id})=w({\rm id},{\rm id})=\frac{1}{2}\left( 1+\frac{1}{d} \right).\end{eqnarray*}$$

The same holds for any pair $(\mathcal{V},\mathcal{W})$ of decodable channels. Moreover the pair

$$\begin{eqnarray*}&&\frac{1}{2}\left( 1+\frac{1}{d} \right)(\mathcal{V},\mathcal{W})+\frac{1}{2}\left( 1-\frac{1}{d} \right)({{\mathcal{E}}_{\mathcal{V}}},{{\mathcal{E}}_{\mathcal{W}}})=\frac{d+2}{2(d+1)}(\mathcal{V},\mathcal{W})+\frac{d}{2(d+1)}(\mathcal{T},\mathcal{T})\end{eqnarray*}$$

is compatible, where

$$\begin{eqnarray*}&&({{\mathcal{E}}_{\mathcal{V}}},{{\mathcal{E}}_{\mathcal{W}}})=-\frac{1}{{{d}^{2}}-1}(\mathcal{V},\mathcal{W})+\frac{{{d}^{2}}}{{{d}^{2}}-1}(\mathcal{T},\mathcal{T}).\end{eqnarray*}$$

In this section, we calculate the robustness of incompatibility for a von Neumann observable, i.e., a rank-1 sharp observable (PVM), and a decodable channel. We start with the (slightly) simpler case where the decodable channel is simply the identity channel from which the more general case follows according to section 4.2.

For the remainder of this section, we fix a d-dimensional Hilbert space ($d\lt \infty$), and we fix an orthonormal basis $\{|n\rangle \}_{n=0}^{d-1}\subset \mathcal{H}$; we treat the index set $\{0,1,\ldots ,d-1\}$ as the cyclic group ${{\mathbb{Z}}_{d}}$ and all sums and differences of these indices are considered cyclic. For simplicity, we denote the set of observables on the power set of ${{\mathbb{Z}}_{d}}$ and operating in $\mathcal{H}$ by ${\bf Ob}{{{\bf s}}_{d}}$ and the set of channels ${\bf Ch}(\mathcal{H},\mathcal{H})$ by ${\bf C}{{{\bf h}}_{d}}$. We denote the set of instruments on the power set of ${{\mathbb{Z}}_{d}}$ and operating within $\mathcal{S}(\mathcal{H})$ by ${\bf In}{{{\bf s}}_{d}}$, i.e., instruments in ${\bf In}{{{\bf s}}_{d}}$ are the possible joint instruments for pairs $({\sf M} ,{\mathcal{E}})\in {\bf Ob}{{{\bf s}}_{d}}\times {\bf C}{{{\bf h}}_{d}}$. We treat any $\Gamma \in {\bf In}{{{\bf s}}_{d}}$ as a sequence ${{({{\Gamma }_{j}})}_{j\in {{\mathbb{Z}}_{d}}}}$ of operations summing up to a trace-preserving operation.

For any $q,p\in {{\mathbb{Z}}_{d}}$, we define the operators ${{U}_{q}},{{V}_{p}},{{W}_{q,p}}\in \mathcal{L}(\mathcal{H})$ in the same way as in equations (5.2)–(5.4) with the basis ${{\{{{\varphi }_{j}}\}}_{j\in {{\mathbb{Z}}_{d}}}}$ replaced with the basis $\{|n\rangle {{\}}_{n\in {{\mathbb{Z}}_{d}}}}$, so that, e.g., ${{U}_{q}}|n\rangle =|n+q\rangle$ for all $q,n\in {{\mathbb{Z}}_{d}}$. We denote the set of those ${\sf M} \in {\bf Ob}{{{\bf s}}_{d}}$ such that $W_{q,p}^{*}{{{\sf M} }_{j}}{{W}_{q,p}}={{{\sf M} }_{j-q}}$ for all $q,p,j\in {{\mathbb{Z}}_{d}}$ by ${\bf Obs}_{d}^{W,1}$. Furthermore, we denote by ${\bf Ch}_{d}^{W,2}$ the set of those $\mathcal{E}\in {\bf C}{{{\bf h}}_{d}}$ such that $\mathcal{E}({{W}_{q,p}}\rho W_{q,p}^{*})={{W}_{q,p}}\mathcal{E}(\rho )W_{q,p}^{*}$ for all $q,p\in {{\mathbb{Z}}_{d}}$ and all $\rho \in \mathcal{S}(\mathcal{H})$. Finally, we denote the set of those $\Gamma \in {\bf In}{{{\bf s}}_{d}}$ such that ${{\Gamma }_{j-q}}({{W}_{q,p}}\rho W_{q,p}^{*})={{W}_{q,p}}{{\Gamma }_{j}}(\rho )W_{q,p}^{*}$ for all $j,q,p\in {{\mathbb{Z}}_{d}}$ and all $\rho \in \mathcal{S}(\mathcal{H})$ by ${\bf Ins}_{d}^{W}$.

The elements in the sets ${\bf Obs}_{d}^{W,1}$, ${\bf Ch}_{d}^{W,2}$, and ${\bf Ins}_{d}^{W}$ have a simple structure: For any ${\sf M} \in {\bf Obs}_{d}^{W,1}$, there is a positive operator $C\in \mathcal{L}(\mathcal{H})$ such that V_pC = CV_p for all $p\in {{\mathbb{Z}}_{d}}$, ${{\sum }_{j}}{{U}_{j}}CU_{j}^{*}={\mathbb{1}_{\mathcal{H}}}$ (so that, especially, ${\rm tr}[C]=1$), and

$$\begin{eqnarray}&&{{{\sf M} }_{j}}={{U}_{j}}CU_{j}^{*},\qquad j\in {{\mathbb{Z}}_{d}}.\end{eqnarray} \tag{ 5.17 }$$

For each $\Gamma \in {\bf Ins}_{d}^{W}$, there is an operation $\mathcal{D}:\mathcal{L}(\mathcal{H})\to \mathcal{L}(\mathcal{H})$ such that $\mathcal{D}({{V}_{p}}\rho V_{p}^{*})={{V}_{p}}\mathcal{D}(\rho )V_{p}^{*}$ for all $p\in {{\mathbb{Z}}_{d}}$ and $\rho \in \mathcal{S}(\mathcal{H})$, and ${{\sum }_{j}}{{U}_{j}}{{\mathcal{D}}^{*}}({\mathbb{1}_{\mathcal{H}}})U_{j}^{*}={\mathbb{1}_{\mathcal{H}}}$, and

$$\begin{eqnarray}&&{{\Gamma }_{j}}(\rho )={{U}_{j}}\mathcal{D}(U_{j}^{*}\rho {{U}_{j}})U_{j}^{*},\qquad j\in {{\mathbb{Z}}_{d}},\quad \rho \in \mathcal{S}(\mathcal{H}).\end{eqnarray} \tag{ 5.18 }$$

For the characterizations given above for W-covariant observables and instruments, see [15, section III]; the conjecture presented in the reference certainly holds in our discrete case. For any covariant channel $\mathcal{E}\in {\bf Ch}_{d}^{W,2}$ there is a positive kernel $(q,p)\mapsto {{\Phi }_{q,p}}\in \mathbb{C}$ such that [16]

$$\begin{eqnarray}&&{{\mathcal{E}}^{*}}({{W}_{q,p}})={{\Phi }_{q,p}}{{W}_{q,p}},\qquad q,p\in {{\mathbb{Z}}_{d}}.\end{eqnarray} \tag{ 5.19 }$$

Since the operators ${{W}_{q,p}}$, $q,p\in {{\mathbb{Z}}_{d}}$, span the whole of $\mathcal{L}(\mathcal{H})$, the kernel Φ completely characterizes the covariant channel. Positivity of the kernel Φ means that the Fourier transform of the kernel is positive, i.e., for any $j,k\in {{\mathbb{Z}}_{d}}$,

$$\begin{eqnarray}&&{{\hat{\Phi }}_{j,k}}:=\frac{1}{d}\mathop{\sum }\limits_{q,p\in {{\mathbb{Z}}_{d}}}\overline{\langle q,k\rangle }\langle j,p\rangle {{\Phi }_{q,p}}\geqslant 0.\end{eqnarray} \tag{ 5.20 }$$

Moreover, when $\mathcal{E}$ is defined as in (5.19), then

$$\begin{eqnarray*}&&\mathcal{E}(\rho )=\frac{1}{d}\mathop{\sum }\limits_{j,k\in {{\mathbb{Z}}_{d}}}{{\hat{\Phi }}_{j,k}}{{W}_{j,k}}\rho W_{j,k}^{*},\qquad \rho \in \mathcal{S}(\mathcal{H}).\end{eqnarray*}$$

When the channel $\mathcal{E}$ arises from a covariant instrument like that in equation (5.18), it follows from straight-forward calculation (utilizing the fact that, for any $B\in \mathcal{L}(\mathcal{H})$, one has ${{\sum }_{q,p}}{{W}_{q,p}}BW_{q,p}^{*}={\rm dtr}[B]{\mathbb{1}}$) that the kernel Φ associated with $\mathcal{E}$ is of the form

$$\begin{eqnarray}&&{{\Phi }_{q,p}}={\rm tr}[W_{q,p}^{*}{{\mathcal{D}}^{*}}({{W}_{q,p}})],\qquad q,p\in {{\mathbb{Z}}_{d}}.\end{eqnarray} \tag{ 5.21 }$$

As earlier, we have the covariantization maps ${\bf Ob}{{{\bf s}}_{d}}\;\ni \;{\sf M} \mapsto {{{\sf M} }^{W,1}}\in {\bf Obs}_{d}^{W,1}$, ${\bf C}{{{\bf h}}_{d}}\;\ni \;\mathcal{E}\mapsto {{\mathcal{E}}^{W,2}}\in {\bf Ch}_{d}^{W,2}$, and ${\bf In}{{{\bf s}}_{d}}\;\ni \;\Gamma \mapsto {{\Gamma }^{W}}\in {\bf Ins}_{d}^{W}$ having the covariant devices as their fixed points. Especially,

$$\begin{eqnarray*}&&\Gamma _{j}^{W}(\rho )=\frac{1}{{{d}^{2}}}\mathop{\sum }\limits_{q,p\in {{\mathbb{Z}}_{d}}}W_{q,p}^{*}{{\Gamma }_{j-q}}({{W}_{q,p}}\rho W_{q,p}^{*}){{W}_{q,p}}\end{eqnarray*}$$

for any $\Gamma \in {\bf In}{{{\bf s}}_{d}}$, $j\in {{\mathbb{Z}}_{d}}$, and $\rho \in \mathcal{S}(\mathcal{H})$. Moreover, by similar arguments as earlier, one can show that, if a pair $({\sf M} ,{\mathcal{E}})\in {\bf Ob}{{{\bf s}}_{d}}\times {\bf C}{{{\bf h}}_{d}}$ is compatible, so is $({{{\sf M} }^{W,1}},{{{\mathcal{E}}}^{W,2}})$, and if a pair $({\sf M} ,{\mathcal{E}})\in {\bf Obs}_{d}^{W,1}\times {\bf Ch}_{d}^{W,2}$ is compatible, it has a joint instrument in ${\bf Ins}_{d}^{W}$. As in preceding analyses, it follows:

Lemma 3. Suppose that $({\sf A} ,{\mathcal{A}})\in {\bf Obs}_{d}^{W,1}\times {\bf Ch}_{d}^{W,2}$. One has

$$\begin{eqnarray*}&&W({\sf A} ,{\mathcal{A}})={\rm sup} \{w({\sf A} ,{\mathcal{A}}|{\sf B} ,{\mathcal{B}})|\;{\sf B} \in {\bf Obs}_{d}^{W,1},{\mathcal{B}}\in {\bf Ch}_{d}^{W,2}\}.\end{eqnarray*}$$

In what follows, we study the robustness of incompatibility of the pair $({\sf A} ,{\rm id})$, where ${{{\sf A} }_{n}}=|n\rangle \langle n|$ for all $n\in {{\mathbb{Z}}_{d}}$. Evidently, ${\sf A} \in {\bf Obs}_{d}^{W,1}$ and ${\rm id}\in {\bf Ch}_{d}^{W,2}$, but this pair is not compatible. According to the preceding lemma, there is a pair $({\sf B} ,{\mathcal{B}})\in {\bf Obs}_{d}^{W,1}\times {\bf Ch}_{d}^{W,2}$ such that the pair

$$\begin{eqnarray*}&&(W{\sf A} +(1-W){\sf B} ,W{\rm id}+(1-W){\mathcal{B}}),\end{eqnarray*}$$

with $W:=W({\sf A} ,{\rm id})$, is compatible, and hence has a joint instrument in ${\bf Ins}_{d}^{W}$. We start determining the value of $W({\sf A} ,{\rm id})$ first by giving a characterization for the instruments of ${\bf Ins}_{d}^{W}$.

Lemma 4. For any $\Gamma \in {\bf Ins}_{d}^{W}$ there is an indexed set $\alpha ={{(\alpha _{r,s}^{n})}_{n,r,s\in {{\mathbb{Z}}_{d}}}}$ of complex numbers such that, for all $n\in {{\mathbb{Z}}_{d}}$, the matrix ${{(\alpha _{r,s}^{n})}_{r,s\in {{\mathbb{Z}}_{d}}}}$ is positive, ${{\sum }_{n,r}}\alpha _{r,r}^{n}=1$, and defining $\mathcal{D}:\mathcal{L}(\mathcal{H})\to \mathcal{L}(\mathcal{H})$,

$$\begin{eqnarray}&&\mathcal{D}(A)=\mathop{\sum }\limits_{n,r,s\in {{\mathbb{Z}}_{d}}}\alpha _{r,s}^{n}\langle n+r|A|n+s\rangle |r\rangle \langle s|,\qquad A\in \mathcal{L}(\mathcal{H}),\end{eqnarray} \tag{ 5.22 }$$

Γ is defined as in equation (5.18).

Proof. Suppose that $\Gamma \in {\bf Ins}_{d}^{W}$ is associated with an operation $\mathcal{D}$ that is covariant with respect to the representation $p\mapsto {{V}_{p}}$ (i.e., is V-covariant), ${\rm tr}[{{\mathcal{D}}^{*}}({\mathbb{1}})]=1$, and Γ is defined as in (5.18). The V-covariance means that, for the Choi operator $M=({{\mathcal{D}}^{*}}\otimes {\rm id})({{\Omega }_{d}})$, where ${{\Omega }_{d}}={{\sum }_{m,n}}|m,m\rangle \langle n,n|$, we have

$$\begin{eqnarray*}&&({{V}_{p}}\otimes {{\bar{V}}_{p}})M=M({{V}_{p}}\otimes {{\bar{V}}_{p}}),\qquad p\in {{\mathbb{Z}}_{d}},\end{eqnarray*}$$

where ${{\bar{V}}_{p}}=V_{p}^{*T}$ with the transpose defined with respect to the basis $\{|n\rangle {{\}}_{n}}$. Since the representation $p\mapsto {{V}_{p}}\otimes {{\bar{V}}_{p}}$ is associated with the spectral measure $q\mapsto {{{\sf R} }_{q}}$,

$$\begin{eqnarray*}&&{{{\sf R} }_{q}}=\mathop{\sum }\limits_{r\in {{\mathbb{Z}}_{d}}}|q+r,r\rangle \langle q+r,r|,\qquad q\in {{\mathbb{Z}}_{d}},\end{eqnarray*}$$

i.e., ${{V}_{p}}\otimes {{\bar{V}}_{p}}={{\sum }_{q}}\langle q,p\rangle {{{\sf R} }_{q}}$, it follows that any operator that commutes with this representation has to be a linear combination of the operators ${{A}_{n,r,s}}=|n+r,r\rangle \langle n+s,s|$, $n,r,s\in {{\mathbb{Z}}_{d}}$. Writing $M={{\sum }_{n,r,s}}\alpha _{r,s}^{n}{{A}_{n,r,s}}$, one finds that $\mathcal{D}$ is defined as in (5.22). The positivity of M is equivalent with ${{{\sf R} }_{n}}M{{{\sf R} }_{n}}\geqslant 0$ for all n which, in turn, is equivalent with ${{(\alpha _{r,s}^{n})}_{r,s}}\geqslant 0$ for all n. Direct calculation shows that ${{\sum }_{j}}{{U}_{j}}{{\mathcal{D}}^{*}}({\mathbb{1}})U_{j}^{*}={\mathbb{1}}$ is equivalent with ${{\sum }_{n,r}}\alpha _{r,r}^{n}=1$. □

Let us pick a compatible pair $({\sf M} ,{\mathcal{E}})\in {\bf Obs}_{d}^{W,1}\times {\bf Ch}_{d}^{W,2}$ that has a joint instrument $\Gamma \in {\bf Ins}_{d}^{W}$ that is associated with α like that in lemma 4. Through simple calculations one finds that the trace-1 positive operator C associated with ${\sf M}$ according to (5.17) and the positive kernel Φ associated with $\mathcal{E}$ according to (5.19) are given by

$$\begin{eqnarray}&&C=\mathop{\sum }\limits_{n,r\in {{\mathbb{Z}}_{d}}}\alpha _{r,r}^{n}|n+r\rangle \langle n+r|,\qquad {{\Phi }_{q,p}}=\mathop{\sum }\limits_{n,r\in {{\mathbb{Z}}_{d}}}\overline{\langle n,p\rangle }\alpha _{r,r-q}^{n},\quad q,p\in {{\mathbb{Z}}_{d}}.\end{eqnarray} \tag{ 5.23 }$$

In the case of our special incompatible pair $({\sf A} ,{\rm id})$, the trace-1 positive operator associated with ${\sf A}$ is $|0\rangle \langle 0|$ and the positive kernel associated with the identity channel is the constant kernel 1. It follows that, if we require the compatible pair like that above to be a convex combination of the form

$$\begin{eqnarray*}&&({\sf M} ,{\mathcal{E}})=t({\sf A} ,{\rm id})+(1-t)({\sf B} ,{\mathcal{B}})\end{eqnarray*}$$

with some $t\in [0,1]$ and some (not necessarily compatible) pair $({\sf B} ,{\mathcal{B}})\in {\bf Ob}{{{\bf s}}_{d}}\times {\bf C}{{{\bf h}}_{d}}$, then we must have $D=C-t|0\rangle \langle 0|\geqslant 0$ and the kernel $\Psi =\Phi -t$ has to be positive, where C and Φ are defined as in (5.23). Since C is diagonalized in the basis $\{|n\rangle {{\}}_{n}}$, it is a straight-forward check that the first condition is equivalent with

$$\begin{eqnarray}&&t\leqslant \mathop{\sum }\limits_{n\in {{\mathbb{Z}}_{d}}}\alpha _{-n,-n}^{n}\;\;{{w}_{1}}(\alpha ).\end{eqnarray} \tag{ 5.24 }$$

Using the characterization of (5.20) for the positivity of the kernel Ψ, the latter condition is equivalent with

$$\begin{eqnarray}&&t\leqslant \frac{1}{d}\mathop{\sum }\limits_{r,s\in {{\mathbb{Z}}_{d}}}\alpha _{r,s}^{0}\;\;{{w}_{2}}(\alpha ).\end{eqnarray} \tag{ 5.25 }$$

The robustness $W({\sf A} ,{\rm id})$ is simply the supremum of ${\rm min} \{{{w}_{1}}(\alpha ),{{w}_{2}}(\alpha )\}$ over all those $\alpha ={{(\alpha _{r,s}^{n})}_{n,r,s}}$ such that ${{(\alpha _{r,s}^{n})}_{r,s}}\geqslant 0$ for all $n\in {{\mathbb{Z}}_{d}}$ and ${{\sum }_{n,r}}\alpha _{r,r}^{n}=1$.

We may simplify the optimization task presented above by a couple of observations: First, we note that, given α, the elements $\alpha _{r,s}^{n}$ where $n\ne 0$, $r\ne -n$, and $s\ne -n$ can be assumed to be zero; this assumption does not affect the value of w₂ and it can only increase the value of w₁. The elements $\alpha _{-n,-n}^{n}$ are, of course, non-negative. Second, the property ${{\sum }_{n,r}}\alpha _{r,r}^{n}=1$ and the positivity requirements are not violated if we replace the α, where the elements $\alpha _{r,s}^{n}$ with $n\ne 0$, $r\ne -n$, and $s\ne -n$ are zero, with $\tilde{\alpha }$, where $\tilde{\alpha }_{0,0}^{0}={{\sum }_{n\in {{\mathbb{Z}}_{d}}}}\alpha _{-n,-n}^{n}$, $\tilde{\alpha }_{r,s}^{n}=0$ for all $n\ne 0$ and $r,s\in {{\mathbb{Z}}_{d}}$, and the rest of the entries in $\tilde{\alpha }$ coincide with their counterparts in α. In this process, the value of w₁ does not change but w₂ may increase. Hence, in our optimization, it suffices to study only those α such that $\alpha _{r,s}^{n}=0$ for any $n\ne 0$ and $r,s\in {{\mathbb{Z}}_{d}}$ and, for the upper block ${{(\alpha _{r,s}^{0})}_{r,s}}\;\;A$, $A\geqslant 0$ and ${\rm tr}[A]=1$. From now on, we forget about the zero blocks and only concentrate on the upper block A. Let us denote the natural basis of ${{\mathbb{C}}^{d}}$ by ${{\{{{e}_{r}}\}}_{r\in {{\mathbb{Z}}_{d}}}}$ and define the Fourier operator $\mathcal{F}\in \mathcal{L}({{\mathbb{C}}^{d}})$ as in (5.1) with the basis ${{\{{{\varphi }_{j}}\}}_{j}}$ replaced by ${{\{{{e}_{n}}\}}_{n}}$. Fix the basis ${{\{{{f}_{n}}\}}_{n}}$, ${{f}_{n}}=\mathcal{F}{{e}_{n}}$, $n\in {{\mathbb{Z}}_{d}}$. We may define ${{w}_{1}}(A)=\langle {{e}_{0}}|A{{e}_{0}}\rangle$, ${{w}_{2}}(A)=\langle {{f}_{0}}|A{{f}_{0}}\rangle$, and ${{w}_{0}}(A)={\rm min} \{{{w}_{1}}(A),{{w}_{2}}(A)\}$. We have found that $W({\sf A} ,{\rm id})$ is the supremum of ${{w}_{0}}(A)$ over the positive trace-1 operators $A\in \mathcal{L}({{\mathbb{C}}^{d}})$.

This optimization task can still be further simplified: For any $A\in \mathcal{L}({{\mathbb{C}}^{d}})$, let us define ${{A}^{\mathcal{F}}}\in \mathcal{L}({{\mathbb{C}}^{d}})$ through

$$\begin{eqnarray*}&&{{A}^{\mathcal{F}}}=\frac{1}{4}\mathop{\sum }\limits_{k=1}^{4}{{\mathcal{F}}^{k}}A{{\mathcal{F}}^{*k}}.\end{eqnarray*}$$

The fixed points of the map $A\mapsto {{A}^{\mathcal{F}}}$ are exactly the Fourier-invariant operators B, i.e., $\mathcal{F}B=B\mathcal{F}$. One finds that ${{w}_{0}}({{A}^{\mathcal{F}}})={{w}_{1}}({{A}^{\mathcal{F}}})={{w}_{2}}({{A}^{\mathcal{F}}})=\frac{1}{2}\left( {{w}_{1}}(A)+{{w}_{2}}(A) \right)\geqslant {{w}_{0}}(A)$ for all positive $A\in \mathcal{L}({{\mathbb{C}}^{d}})$. Thus, our task is simply to optimize the linear functional ${{w}_{0}}(A)=\langle {{e}_{0}}|A{{e}_{0}}\rangle$ over the set of positive trace-1 Fourier-invariant operators A on ${{\mathbb{C}}^{d}}$. The optimal value is reached at one of the extreme points of the set of such operators, and these extreme points coincide with the projections onto the one-dimensional subspaces generated by the eigenvectors of $\mathcal{F}$. The Fourier operator has four eigenvalues, the fourth roots of 1 i^k, $k=1,2,3,4$, and the corresponding eigenprojections are

$$\begin{eqnarray*}&&{{P}_{k}}=\frac{1}{4}\left( {\mathbb{1}}+{{(-i)}^{k}}\mathcal{F}+{{(-1)}^{k}}{{\mathcal{F}}^{2}}+{{i}^{k}}{{\mathcal{F}}^{3}} \right),\quad k=1,2,3,4.\end{eqnarray*}$$

Hence, especially, for an extreme point A of the set of trace-1 positive Fourier-invariant operators, there is a unique $k\in \{1,2,3,4\}$ such that $A={{P}_{k}}A{{P}_{k}}$ and ${{w}_{0}}(A)=\langle {{P}_{k}}{{e}_{0}}|A{{P}_{k}}{{e}_{0}}\rangle$. One finds that ${{P}_{1}}{{e}_{0}}={{P}_{3}}{{e}_{0}}=0$ and ${{P}_{2}}{{e}_{0}}=\frac{1}{2}({{e}_{0}}-{{f}_{0}})$ and ${{P}_{4}}{{e}_{0}}=\frac{1}{2}({{e}_{0}}+{{f}_{0}})$ which means that w is maximized at a projection onto a one-dimensional subspace of the P₂- or P₄-eigenspace. It is obvious that, amongst such operators supported on the P₂-eigenspace, the optimal one is ${{A}_{-}}=|{{v}_{-}}\rangle \langle {{v}_{-}}|$ and, amongst the extreme points supported on the P₄-eigenspace, the highest value for w₀ is given by ${{A}_{+}}=|{{v}_{+}}\rangle \langle {{v}_{+}}|$, where

$$\begin{eqnarray*}&&{{v}_{\pm }}=\sqrt{\frac{\sqrt{d}}{2(\sqrt{d}\pm 1)}}({{e}_{0}}\pm {{f}_{0}}).\end{eqnarray*}$$

Simple check shows that ${{w}_{0}}({{A}_{-}})\lt {{w}_{0}}({{A}_{+}})=\frac{1}{2}(1+1/\sqrt{d})$. Thus, $W({\sf A} ,{\rm id})=\frac{1}{2}(1+1/\sqrt{d})$. Again, for a decodable channel $\mathcal{V}\in {\bf Ch}(\mathcal{H},\mathcal{K})$, one has $({\sf A} ,{\mathcal{V}}){{=}_{{\rm post}}}({\sf A} ,{\rm id})$, so that, according to theorem 3:

Theorem 6. The robustness of incompatibility for any von Neumann observable ${\sf A} \in {\bf Ob}{{{\bf s}}_{d}}$ and any decodable channel $\mathcal{V}\in {\bf Ch}(\mathcal{H},\mathcal{K})$ is

$$\begin{eqnarray*}&&W({\sf A} ,{\mathcal{V}})=\frac{1}{2}\left( 1+\frac{1}{\sqrt{d}} \right).\end{eqnarray*}$$

The optimal positive trace-1 Fourier-invariant operator ${{A}_{+}}$ of the discussion preceding theorem 6 gives rise to an operation $\mathcal{D}$ through equation (5.22) which, in turn, defines an instrument Γ according to (5.18). It follows that

$$\begin{eqnarray*}&&{{\Gamma }_{j}}(\rho )=\frac{\sqrt{d}}{2(\sqrt{d}+1)}\left( \frac{1}{\sqrt{d}}{\mathbb{1}}+{{{\sf A} }_{j}} \right)\rho \left( \frac{1}{\sqrt{d}}{\mathbb{1}}+{{{\sf A} }_{j}} \right).\end{eqnarray*}$$

The optimal decomposition for the compatible pair $({\sf M} ,{\mathcal{E}})=({{\Gamma }_{(1)}},{{\Gamma }_{(2)}})$ is given by

$$\begin{eqnarray*}&&({\sf M} ,{\mathcal{E}})=\frac{1}{2}\left( 1+\frac{1}{\sqrt{d}} \right)({\sf A} ,{\rm id})+\frac{1}{2}\left( 1-\frac{1}{\sqrt{d}} \right)({\sf B} ,{\mathcal{B}}),\end{eqnarray*}$$

where

$$\begin{eqnarray*}&&{\sf B} =-\frac{1}{d-1}{\sf A} +\frac{d}{d-1}{\sf T} ,\qquad {\mathcal{B}}=-\frac{1}{d-1}{\rm id}+\frac{d}{d-1}{{{\mathcal{E}}}_{{\sf A} }},\end{eqnarray*}$$

where ${\sf T}$ is the trivial observable ${{{\sf T} }_{j}}=\frac{1}{d}{\mathbb{1}}$, $j\in {{\mathbb{Z}}_{d}}$, and ${{{\mathcal{E}}}_{{\sf A} }}\in {\bf Ch}_{d}^{W,2}$ is the Lüders channel $\rho \mapsto {{\sum }_{j}}{{{\sf A} }_{j}}\rho {{{\sf A} }_{j}}$ associated with ${\sf A}$. Moreover, the optimal compatible pair $({\sf M} ,{\mathcal{E}})$ is given by

$$\begin{eqnarray*}&&{\sf M} =\frac{\sqrt{d}+2}{2(\sqrt{d}+1)}{\sf A} +\frac{\sqrt{d}}{2(\sqrt{d}+1)}{\sf T} ,\qquad {\mathcal{E}}=\frac{\sqrt{d}+2}{2(\sqrt{d}+1)}{\rm id}+\frac{\sqrt{d}}{2(\sqrt{d}+1)}{{{\mathcal{E}}}_{{\sf A} }}.\end{eqnarray*}$$

Similarly, for a decodable channel $\mathcal{V}$, the pair

$$\begin{eqnarray*}\begin{array}{ccccccccccccccc} \frac{1}{2}\left( 1+\frac{1}{\sqrt{d}} \right)({\sf A} ,{\mathcal{V}})+\frac{1}{2}\left( 1-\frac{1}{\sqrt{d}} \right)({{{\sf B} }_{{\mathcal{V}}}},{{{\mathcal{B}}}_{{\mathcal{V}}}}) & = & \frac{\sqrt{d}+2}{2(\sqrt{d}+1)}({\sf A} ,{\mathcal{V}}) \\ {} & {} & +\frac{\sqrt{d}}{2(\sqrt{d}+1)}({\sf T} ,{\mathcal{V}}\;\circ \;{{{\mathcal{E}}}_{{\sf A} }}) \\ \end{array}\end{eqnarray*}$$

is compatible, where

$$\begin{eqnarray*}&&{{{\sf B} }_{{\mathcal{V}}}}=-\frac{1}{d-1}{\sf A} +\frac{d}{d-1}{\sf T} ,\qquad {{{\mathcal{B}}}_{{\mathcal{V}}}}=-\frac{1}{d-1}{\mathcal{V}}+\frac{d}{d-1}{\mathcal{V}}\;\circ \;{{{\mathcal{E}}}_{{\sf A} }},\end{eqnarray*}$$