${\mathcal{E}}$ fully conserves $N$ while the $\mathsf{Z}$-channel only
conserves $N$ on average, and vice versa. In such cases, the channel
${\mathcal{J}}_{\mathcal{X}}$ will only conserve $N$ on average.

By Lemma 2.6, it holds that if ${\mathcal{J}}_{\mathcal{X}}$ conserves $N$ on
average, and if either $\mathsf{Z}^{\tau}$ is sharp or if
${\mathcal{J}}_{\mathcal{X}}$ also fully conserves $N$, then

$\displaystyle[\mathsf{Z}^{\tau},N]=\mathds{O}.$ (35)

This commutation relation is known as the weak Yanase condition [32]. We note
that if ${\mathcal{E}}$ conserves $N$ on average and if either
$\mathsf{Z}^{\tau}$ is sharp or if ${\mathcal{E}}$ also fully conserves $N$,
then the Yanase condition implies the weak Yanase condition. First, let us
assume that $\mathsf{Z}^{\tau}$ is sharp. Since $\mathsf{Z}(x)$ is an effect
then by two-positivity of CP maps we have
$\mathsf{Z}^{\tau}(x)={\mathcal{E}}^{*}(\mathds{1}_{\\!{\,{\mathcal{S}}}}\otimes\mathsf{Z}(x))\geqslant{\mathcal{E}}^{*}(\mathds{1}_{\\!{\,{\mathcal{S}}}}\otimes\mathsf{Z}(x)^{2})\geqslant{\mathcal{E}}^{*}(\mathds{1}_{\\!{\,{\mathcal{S}}}}\otimes\mathsf{Z}(x))^{2}=\mathsf{Z}^{\tau}(x)$,
and so we have
${\mathcal{E}}^{*}(\mathds{1}_{\\!{\,{\mathcal{S}}}}\otimes\mathsf{Z}(x)^{2})={\mathcal{E}}^{*}(\mathds{1}_{\\!{\,{\mathcal{S}}}}\otimes\mathsf{Z}(x))^{2}$.
On the other hand, if ${\mathcal{E}}$ fully conserves $N$ then
${\mathcal{E}}^{*}(N^{2})={\mathcal{E}}^{*}(N)^{2}=N^{2}$. In either case, by
Corollary 2.1 we have

$\displaystyle[\mathsf{Z}^{\tau}(x),N]=[{\mathcal{E}}^{*}(\mathds{1}_{\\!{\,{\mathcal{S}}}}\otimes\mathsf{Z}(x)),{\mathcal{E}}^{*}(N)]={\mathcal{E}}^{*}([\mathds{1}_{\\!{\,{\mathcal{S}}}}\otimes\mathsf{Z}(x),N])={\mathcal{E}}^{*}(\mathds{1}_{\\!{\,{\mathcal{S}}}}\otimes[\mathsf{Z}(x),{N_{\\!{\,{\mathcal{A}}}}}]),$

and so if $[\mathsf{Z}(x),{N_{\\!{\,{\mathcal{A}}}}}]=\mathds{O}$, then
$[\mathsf{Z}^{\tau}(x),N]=\mathds{O}$. Moreover, if
${\mathcal{E}}(\cdot)=U(\cdot)U^{*}$ is a unitary channel, and ${\mathcal{E}}$
conserves $N$, then
$[\mathsf{Z}^{\tau}(x),N]=[U^{*}(\mathds{1}_{\\!{\,{\mathcal{S}}}}\otimes\mathsf{Z}(x))U,N]=U^{*}(\mathds{1}_{\\!{\,{\mathcal{S}}}}\otimes[\mathsf{Z}(x),{N_{\\!{\,{\mathcal{A}}}}}])U$.
In such a case the weak Yanase condition is equivalent to the Yanase
condition; multiplying both sides of the equality
$U^{*}(\mathds{1}_{\\!{\,{\mathcal{S}}}}\otimes[\mathsf{Z}(x),{N_{\\!{\,{\mathcal{A}}}}}])U=\mathds{O}$
by $U$ from the left and by $U^{*}$ from the right shows that
$[\mathsf{Z}^{\tau}(x),N]=\mathds{O}\iff[\mathsf{Z}(x),{N_{\\!{\,{\mathcal{A}}}}}]=\mathds{O}$.
However, in general it may be the case that the weak Yanase condition is
satisfied but the Yanase condition is violated.

The following proposition shows that if the weak Yanase condition is
satisfied, then the measurability part of the WAY theorem will hold. Moreover,
we see that there are cases where a large coherence of the conserved quantity
in the apparatus is necessary for good measurements even without a full
conservation law—for example, if either the interaction channel
${\mathcal{E}}$ or the $\mathsf{Z}$-channel only conserves $N$ on average, but
$\mathsf{Z}^{\tau}$ is sharp, in which case the weak Yanase condition is
guaranteed to hold.

######  Proposition 4.3.

Let
${\mathcal{M}}:=({\mathcal{H}_{\\!{\,{\mathcal{A}}}}},\xi,{\mathcal{E}},\mathsf{Z})$
be a measurement scheme for an observable acting in
${\mathcal{H}_{\\!{\,{\mathcal{S}}}}}$, and let $\|\epsilon(x)\|$ quantify the
error in measuring the effects of the target observable $\mathsf{E}$. If
${\mathcal{M}}$ satisfies the weak Yanase condition
$[\mathsf{Z}^{\tau},N]=\mathds{O}$, where $\mathsf{Z}^{\tau}$ is the
Heisenberg-evolved pointer observable defined in Eq. (34) and
$N={N_{\\!{\,{\mathcal{S}}}}}\otimes\mathds{1}_{\\!{\,{\mathcal{A}}}}+\mathds{1}_{\\!{\,{\mathcal{S}}}}\otimes{N_{\\!{\,{\mathcal{A}}}}}$,
then for all $x\in{\mathcal{X}}$

$\displaystyle\|[\mathsf{E}(x),{N_{\\!{\,{\mathcal{S}}}}}]\|\leqslant
2\|{N_{\\!{\,{\mathcal{S}}}}}\|\|\epsilon(x)\|+2\mathrm{Var}\left({{N_{\\!{\,{\mathcal{A}}}}},\xi}\right)^{\frac{1}{2}}\big{(}2\|\epsilon(x)\|+\|\mathsf{E}(x)-\mathsf{E}(x)^{2}\|\big{)}^{\frac{1}{2}},$
(36)

and

$\displaystyle\|[\mathsf{E}(x),{N_{\\!{\,{\mathcal{S}}}}}]\|\leqslant
2\|{N_{\\!{\,{\mathcal{S}}}}}\|\|\epsilon(x)\|+\frac{1}{2}{\mathcal{Q}}({N_{\\!{\,{\mathcal{A}}}}},\xi)^{\frac{1}{2}},$
(37)

where $\mathrm{Var}\left({{N_{\\!{\,{\mathcal{A}}}}},\xi}\right)$ is the
variance of ${N_{\\!{\,{\mathcal{A}}}}}$ in $\xi$, and
${\mathcal{Q}}({N_{\\!{\,{\mathcal{A}}}}},\xi)$ is the the quantum Fisher
information of ${N_{\\!{\,{\mathcal{A}}}}}$ in $\xi$. Additionally, if
$\mathsf{E}$ is an extremal observable, and ${\mathcal{M}}$ is a measurement
scheme for $\mathsf{E}$, then for all $x\in{\mathcal{X}}$

$\displaystyle\|[\mathsf{E}(x),{N_{\\!{\,{\mathcal{S}}}}}]\|\leqslant{\mathcal{Q}}({N_{\\!{\,{\mathcal{A}}}}},\xi)^{\frac{1}{2}}\|\mathsf{E}(x)-\mathsf{E}(x)^{2}\|^{\frac{1}{2}}.$
(38)

###### Proof.

By Eq. (28), we may write
$\epsilon(x):=\Gamma_{\xi}^{\mathcal{E}}(\mathds{1}_{\\!{\,{\mathcal{S}}}}\otimes\mathsf{Z}(x))-\mathsf{E}(x)\equiv\Gamma_{\xi}(\mathsf{Z}^{\tau}(x))-\mathsf{E}(x)$.
By additivity of $N$ we have
$\Gamma_{\xi}(N)={N_{\\!{\,{\mathcal{S}}}}}+\mathrm{tr}[{N_{\\!{\,{\mathcal{A}}}}}\xi]\mathds{1}_{\\!{\,{\mathcal{S}}}}$,
and so we may write

$\displaystyle[\mathsf{E}(x),{N_{\\!{\,{\mathcal{S}}}}}]=[{N_{\\!{\,{\mathcal{S}}}}},\epsilon(x)]+[\Gamma_{\xi}(\mathsf{Z}^{\tau}(x)),\Gamma_{\xi}(N)],$

which gives us the bound

$\displaystyle\|[\mathsf{E}(x),{N_{\\!{\,{\mathcal{S}}}}}]\|\leqslant
2\|{N_{\\!{\,{\mathcal{S}}}}}\|\|\epsilon(x)\|+\|[\Gamma_{\xi}(\mathsf{Z}^{\tau}(x)),\Gamma_{\xi}(N)]\|.$

Since $\Gamma_{\xi}$ is a channel, and the weak Yanase condition
$[\mathsf{Z}^{\tau},N]=\mathds{O}$ holds, then by Corollary 2.2 we obtain

$\displaystyle\|[\Gamma_{\xi}(\mathsf{Z}^{\tau}(x)),\Gamma_{\xi}(N)]\|\leqslant
2\|\Gamma_{\xi}(N^{2})-\Gamma_{\xi}(N)^{2}\|^{\frac{1}{2}}\|\Gamma_{\xi}(\mathsf{Z}^{\tau}(x)^{2})-\Gamma_{\xi}(\mathsf{Z}^{\tau}(x))^{2}\|^{\frac{1}{2}}.$

As shown in Lemma 3.1, additivity of $N$ implies that
$\|\Gamma_{\xi}(N^{2})-\Gamma_{\xi}(N)^{2}\|=\mathrm{Var}\left({{N_{\\!{\,{\mathcal{A}}}}},\xi}\right)$.
On the other hand, by Lemma 2.2 we obtain
$\|\Gamma_{\xi}(\mathsf{Z}^{\tau}(x)^{2})-\Gamma_{\xi}(\mathsf{Z}^{\tau}(x))^{2}\|\leqslant
2\|\epsilon(x)\|+\|\mathsf{E}(x)-\mathsf{E}(x)^{2}\|$. As such, we obtain the
bound in Eq. (36). The bounds given in Eq. (37) and Eq. (38) are trivially
obtained by adapting the arguments in Proposition 4.2 to the above. ∎

##  5 Faithful fixed states and measurement disturbance

In Sec. 3 we obtained quantitative bounds giving necessary conditions for non-
disturbance of an observable $\mathsf{F}$ by an $\mathsf{E}$-instrument
${\mathcal{I}}$ implemented in the presence of an additive conservation law,
whether the conservation law is full or average. We saw from Lemma 3.2 that if
$\mathsf{F}^{2}$ is contained in the fixed-point set of the
$\mathsf{E}$-channel ${\mathcal{I}}^{*}_{\mathcal{X}}$—guaranteed by non-
disturbance if $\mathsf{F}$ is either sharp, a rank-1 observable, or a coarse-
graining of a sharp observable—then independently of the amount of coherence
initially present in the apparatus it must hold that $\mathsf{F}$ commutes
with both $\mathsf{E}$ and
$\Delta{N_{\\!{\,{\mathcal{S}}}}}:={\mathcal{I}}_{\mathcal{X}}^{*}({N_{\\!{\,{\mathcal{S}}}}})-{N_{\\!{\,{\mathcal{S}}}}}$.
Now we shall see that if the fixed-point set of the $\mathsf{E}$-channel
${\mathcal{I}}_{\mathcal{X}}^{*}$ is a von Neumann algebra, which is
guaranteed if ${\mathcal{F}}({\mathcal{I}}_{\mathcal{X}})$ contains a faithful
state, then similar constraints on non-disturbance will hold for arbitrary
observables. First, let us prove a useful lemma.

######  Lemma 5.1.

Let
${\mathcal{M}}:=({\mathcal{H}_{\\!{\,{\mathcal{A}}}}},\xi,{\mathcal{E}},\mathsf{Z})$
be a measurement scheme for an $\mathsf{E}$-instrument ${\mathcal{I}}$ acting
in ${\mathcal{H}_{\\!{\,{\mathcal{S}}}}}$. Assume that ${\mathcal{E}}$
conserves an additive quantity
$N=N_{\\!{\,{\mathcal{S}}}}\otimes\mathds{1}_{\\!{\,{\mathcal{A}}}}+\mathds{1}_{\\!{\,{\mathcal{S}}}}\otimes
N_{\\!{\,{\mathcal{A}}}}$ on average, and that
${\mathcal{F}}({\mathcal{I}}_{\mathcal{X}}^{*})$ is a von Neumann algebra.
Then for all $A\in{\mathcal{L}}({\mathcal{H}_{\\!{\,{\mathcal{S}}}}})$, the
following implication holds:
$A\in{\mathcal{F}}({\mathcal{I}}_{\mathcal{X}}^{*})\implies[A,N_{\\!{\,{\mathcal{S}}}}]\in{\mathcal{F}}({\mathcal{I}}_{\mathcal{X}}^{*})\implies[A,N_{\\!{\,{\mathcal{S}}}}]\in\mathsf{E}^{\prime}$.

###### Proof.

Recall that for all $A\in{\mathcal{L}}({\mathcal{H}_{\\!{\,{\mathcal{S}}}}})$,
we have
${\mathcal{I}}^{*}_{\mathcal{X}}(A)=\Gamma_{\xi}^{\mathcal{E}}(A\otimes\mathds{1}_{\\!{\,{\mathcal{A}}}})$,
where $\Gamma_{\xi}^{\mathcal{E}}$ is the channel defined in Eq. (7). Average
conservation of $N$ by ${\mathcal{E}}$ implies that

$\displaystyle{N_{\\!{\,{\mathcal{S}}}}}+\mathrm{tr}[{N_{\\!{\,{\mathcal{A}}}}}\xi]\mathds{1}_{\\!{\,{\mathcal{S}}}}=\Gamma_{\xi}^{\mathcal{E}}({N_{\\!{\,{\mathcal{S}}}}}\otimes\mathds{1}_{\\!{\,{\mathcal{A}}}})+\Gamma_{\xi}^{\mathcal{E}}(\mathds{1}_{\\!{\,{\mathcal{S}}}}\otimes{N_{\\!{\,{\mathcal{A}}}}}).$

Since ${\mathcal{F}}({\mathcal{I}}^{*}_{\mathcal{X}})$ is a von Neumann
algebra, it follows that for all
$A\in{\mathcal{L}}({\mathcal{H}_{\\!{\,{\mathcal{S}}}}})$,
$A\in{\mathcal{F}}({\mathcal{I}}^{*}_{\mathcal{X}})\implies
A^{*}A,AA^{*}\in{\mathcal{F}}({\mathcal{I}}^{*}_{\mathcal{X}})$ which, by
Corollary 2.1, implies that
$A\Gamma_{\xi}^{\mathcal{E}}(B)=\Gamma_{\xi}^{\mathcal{E}}((A\otimes\mathds{1}_{\\!{\,{\mathcal{A}}}})B)$
and
$\Gamma_{\xi}^{\mathcal{E}}(B)A=\Gamma_{\xi}^{\mathcal{E}}(B(A\otimes\mathds{1}_{\\!{\,{\mathcal{A}}}}))$
for all
$B\in{\mathcal{L}}({\mathcal{H}_{\\!{\,{\mathcal{S}}}}}\otimes{\mathcal{H}_{\\!{\,{\mathcal{A}}}}})$.
Therefore, for all $A\in{\mathcal{F}}({\mathcal{I}}^{*}_{\mathcal{X}})$ we
have

$\displaystyle[A,{N_{\\!{\,{\mathcal{S}}}}}]$
$\displaystyle=[A,\Gamma_{\xi}^{\mathcal{E}}({N_{\\!{\,{\mathcal{S}}}}}\otimes\mathds{1}_{\\!{\,{\mathcal{A}}}})]+[A,\Gamma_{\xi}^{\mathcal{E}}(\mathds{1}_{\\!{\,{\mathcal{S}}}}\otimes{N_{\\!{\,{\mathcal{A}}}}})]$
$\displaystyle=\Gamma_{\xi}^{\mathcal{E}}([A\otimes\mathds{1}_{\\!{\,{\mathcal{A}}}},{N_{\\!{\,{\mathcal{S}}}}}\otimes\mathds{1}_{\\!{\,{\mathcal{A}}}}])+\Gamma_{\xi}^{\mathcal{E}}([A\otimes\mathds{1}_{\\!{\,{\mathcal{A}}}},\mathds{1}_{\\!{\,{\mathcal{S}}}}\otimes{N_{\\!{\,{\mathcal{A}}}}}])$
$\displaystyle=\Gamma_{\xi}^{\mathcal{E}}([A,{N_{\\!{\,{\mathcal{S}}}}}]\otimes\mathds{1}_{\\!{\,{\mathcal{A}}}})={\mathcal{I}}^{*}_{\mathcal{X}}([A,{N_{\\!{\,{\mathcal{S}}}}}]).$

Consequently, we see that
$A\in{\mathcal{F}}({\mathcal{I}}^{*}_{\mathcal{X}})\implies[A,{N_{\\!{\,{\mathcal{S}}}}}]\in{\mathcal{F}}({\mathcal{I}}^{*}_{\mathcal{X}})$.
But as shown in Lemma 2.6, if ${\mathcal{F}}({\mathcal{I}}_{\mathcal{X}}^{*})$
is a von Neumann algebra then
${\mathcal{F}}({\mathcal{I}}_{\mathcal{X}}^{*})\subset\mathsf{E}^{\prime}$. It
follows that
$A\in{\mathcal{F}}({\mathcal{I}}^{*}_{\mathcal{X}})\implies[A,{N_{\\!{\,{\mathcal{S}}}}}]\in\mathsf{E}^{\prime}$.

∎

We are now ready to prove the following:

######  Theorem 5.1.

Let
${\mathcal{M}}:=({\mathcal{H}_{\\!{\,{\mathcal{A}}}}},\xi,{\mathcal{E}},\mathsf{Z})$
be a measurement scheme for an $\mathsf{E}$-instrument ${\mathcal{I}}$ acting
in ${\mathcal{H}_{\\!{\,{\mathcal{S}}}}}$. Assume that ${\mathcal{E}}$
conserves an additive quantity
$N=N_{\\!{\,{\mathcal{S}}}}\otimes\mathds{1}_{\\!{\,{\mathcal{A}}}}+\mathds{1}_{\\!{\,{\mathcal{S}}}}\otimes
N_{\\!{\,{\mathcal{A}}}}$ on average, and that
${\mathcal{F}}({\mathcal{I}}_{\mathcal{X}}^{*})$ is a von Neumann algebra. The
following hold:

  1. (i)

${\mathcal{I}}$ does not disturb an observable $\mathsf{F}$ only if
$\mathsf{F}$ commutes with both $\mathsf{E}$ and
$\Delta{N_{\\!{\,{\mathcal{S}}}}}:={\mathcal{I}}_{\mathcal{X}}^{*}({N_{\\!{\,{\mathcal{S}}}}})-{N_{\\!{\,{\mathcal{S}}}}}$,
with $[\mathsf{F},{N_{\\!{\,{\mathcal{S}}}}}]=\mathds{O}$ being sufficient for
$[\mathsf{F},\Delta{N_{\\!{\,{\mathcal{S}}}}}]=\mathds{O}$.

  2. (ii)

${\mathcal{I}}$ is a measurement of the first kind only if $\mathsf{E}$ is a
commutative observable that commutes with $N_{\\!{\,{\mathcal{S}}}}$.

  3. (iii)

${\mathcal{I}}$ is repeatable only if $\mathsf{E}$ is sharp and commutes with
${N_{\\!{\,{\mathcal{S}}}}}$.

###### Proof.

  1. (i):

${\mathcal{I}}$ does not disturb $\mathsf{F}$ only if
$\mathsf{F}\subset{\mathcal{F}}({\mathcal{I}}^{*}_{\mathcal{X}})$. Given that
${\mathcal{F}}({\mathcal{I}}_{\mathcal{X}}^{*})$ is a von Neumann algebra then
it also holds that
$\mathsf{F}^{2}\subset{\mathcal{F}}({\mathcal{I}}^{*}_{\mathcal{X}})$, and so
by Lemma 3.2 $\mathsf{F}$ must commute with both $\mathsf{E}$ and
$\Delta{N_{\\!{\,{\mathcal{S}}}}}:={\mathcal{I}}^{*}_{\mathcal{X}}({N_{\\!{\,{\mathcal{S}}}}})-{N_{\\!{\,{\mathcal{S}}}}}$.
But note that if $[\mathsf{F},{N_{\\!{\,{\mathcal{S}}}}}]=\mathds{O}$ and
$\mathsf{F}$ is non-disturbed, then it is guaranteed that
$[\mathsf{F},\Delta{N_{\\!{\,{\mathcal{S}}}}}]=\mathds{O}$.

  2. (ii):

${\mathcal{I}}$ is a measurement of the first kind only if
$\mathsf{E}\subset{\mathcal{F}}({\mathcal{I}}^{*}_{\mathcal{X}})$. But by
Lemma 2.6
${\mathcal{F}}({\mathcal{I}}^{*}_{\mathcal{X}})\subset\mathsf{E}^{\prime}$,
and so $\mathsf{E}$ must be commutative. Now, let us define
${N_{\\!{\,{\mathcal{S}}}}}(t):=e^{\mathfrak{i}t\mathsf{E}(x)}{N_{\\!{\,{\mathcal{S}}}}}e^{-\mathfrak{i}t\mathsf{E}(x)}$.
We may write

$\displaystyle\frac{d}{dt}{N_{\\!{\,{\mathcal{S}}}}}(t)=\mathfrak{i}[\mathsf{E}(x),{N_{\\!{\,{\mathcal{S}}}}}(t)],$

from which we obtain

$\displaystyle{N_{\\!{\,{\mathcal{S}}}}}(t)$
$\displaystyle={N_{\\!{\,{\mathcal{S}}}}}+\mathfrak{i}\int^{t}_{0}dt_{1}[\mathsf{E}(x),{N_{\\!{\,{\mathcal{S}}}}}(t_{1})]$
$\displaystyle={N_{\\!{\,{\mathcal{S}}}}}+\mathfrak{i}\int^{t}_{0}dt_{1}[\mathsf{E}(x),{N_{\\!{\,{\mathcal{S}}}}}]-\int^{t}_{0}dt_{1}\int^{t_{1}}_{0}dt_{2}\,e^{\mathfrak{i}t_{2}\mathsf{E}(x)}[\mathsf{E}(x),[\mathsf{E}(x),{N_{\\!{\,{\mathcal{S}}}}}]]e^{-\mathfrak{i}t_{2}\mathsf{E}(x)}$
$\displaystyle={N_{\\!{\,{\mathcal{S}}}}}+\mathfrak{i}t[\mathsf{E}(x),{N_{\\!{\,{\mathcal{S}}}}}].$

In the final line we have used the conservation law, together with Lemma 5.1,
which implies that
$[[\mathsf{E}(x),{N_{\\!{\,{\mathcal{S}}}}}],\mathsf{E}(x)]=\mathds{O}$ holds
for all $x$. We thus obtain the inequality

$\displaystyle
2\|{N_{\\!{\,{\mathcal{S}}}}}\|\geqslant\|{N_{\\!{\,{\mathcal{S}}}}}(t)-{N_{\\!{\,{\mathcal{S}}}}}\|=|t|\|[\mathsf{E}(x),{N_{\\!{\,{\mathcal{S}}}}}]\|$

for all $t$. Given that ${N_{\\!{\,{\mathcal{S}}}}}$ is a bounded operator,
this is clearly satisfied only if
$[\mathsf{E}(x),{N_{\\!{\,{\mathcal{S}}}}}]=\mathds{O}$.

  3. (iii):

Since repeatability implies first-kindness, then by (ii) $\mathsf{E}$ must
commute with ${N_{\\!{\,{\mathcal{S}}}}}$. Sharpness of $\mathsf{E}$ follows
from Proposition 4.1 which gives
${\mathcal{I}}_{\mathcal{X}}^{*}(\mathsf{E}(x)^{2})=\mathsf{E}(x)$ for a
repeatable ${\mathcal{I}}$, and the fact that if
${\mathcal{F}}({\mathcal{I}}^{*}_{\mathcal{X}})$ is a von Neumann algebra then
$\mathsf{E}\subset{\mathcal{F}}({\mathcal{I}}^{*}_{\mathcal{X}})$ implies that
$\mathsf{E}^{2}\subset{\mathcal{F}}({\mathcal{I}}^{*}_{\mathcal{X}})$.

∎

We shall now give two examples where
${\mathcal{F}}({\mathcal{I}}^{*}_{\mathcal{X}})$ is a von Neumann algebra, and
so the implications of Theorem 5.1 hold.

######  Lemma 5.2.

Consider the Lüders $\mathsf{E}$-instrument ${\mathcal{I}}^{L}$ acting in
${\mathcal{H}_{\\!{\,{\mathcal{S}}}}}$, defined in Eq. (5). If either (i)
$\dim({\mathcal{H}_{\\!{\,{\mathcal{S}}}}})<\infty$, or (ii) $\mathsf{E}$ is
commutative, then ${\mathcal{F}}({{\mathcal{I}}^{L}_{\mathcal{X}}}^{*})$ is a
von Neumann algebra.

###### Proof.

Let us first consider (i). Define the complete mixture
$\omega:=\mathds{1}_{\\!{\,{\mathcal{S}}}}/\dim({\mathcal{H}_{\\!{\,{\mathcal{S}}}}})$,
which is faithful. It follows trivially that
${\mathcal{I}}^{L}_{\mathcal{X}}(\omega)=\omega$, and so
${\mathcal{F}}({\mathcal{I}}^{L}_{\mathcal{X}})$ contains a faithful state
$\omega$. By Lemma 2.4 ${\mathcal{F}}({{\mathcal{I}}^{L}_{\mathcal{X}}}^{*})$
is a von Neumann algebra. Now let us consider (ii). Recall that
${\mathcal{F}}({\mathcal{I}}^{*}_{\mathcal{X}})$ is a von Neumann algebra if
${\mathcal{F}}({\mathcal{I}}^{*}_{\mathcal{X}})=\\{K_{i},K_{i}^{*}\\}^{\prime}$,
with $\\{K_{i}\\}$ any Kraus representation of ${\mathcal{I}}_{\mathcal{X}}$
[60]. But for a Lüders instrument, we have
$\\{K_{i},K_{i}^{*}\\}^{\prime}=\\{\sqrt{\mathsf{E}(x)}\\}^{\prime}=\mathsf{E}^{\prime}$.
While
$\mathsf{E}^{\prime}\subset{\mathcal{F}}({{\mathcal{I}}^{L}_{\mathcal{X}}}^{*})$
always holds, it was observed that in infinite-dimensional systems there
exists $\mathsf{E}$ for which
${\mathcal{F}}({{\mathcal{I}}^{L}_{\mathcal{X}}}^{*})\not\subset\mathsf{E}^{\prime}$
[82, 83]. However, it was shown in [70] that for two-valued observables, it
always holds that
${\mathcal{F}}({{\mathcal{I}}^{L}_{\mathcal{X}}}^{*})=\mathsf{E}^{\prime}$.
Since two-valued observables are commutative, this led to the conjecture that
the fixed-point set of the Lüders $\mathsf{E}$-channel is the commutant of
$\mathsf{E}$ for all commutative observables [82], making
${\mathcal{F}}({{\mathcal{I}}^{L}_{\mathcal{X}}}^{*})$ a von Neumann algebra,
which was later proven to be the case [84, 85].

∎

Let us highlight an interesting consequence of Lemma 5.2.

######  Corollary 5.1.

Let
${\mathcal{M}}:=({\mathcal{H}_{\\!{\,{\mathcal{A}}}}},\xi,{\mathcal{E}},\mathsf{Z})$
be a measurement scheme for a commutative observable $\mathsf{E}$ with the
Lüders instrument ${\mathcal{I}}^{L}$ acting in
${\mathcal{H}_{\\!{\,{\mathcal{S}}}}}$. If ${\mathcal{E}}$ conserves an
additive quantity
$N=N_{\\!{\,{\mathcal{S}}}}\otimes\mathds{1}_{\\!{\,{\mathcal{A}}}}+\mathds{1}_{\\!{\,{\mathcal{S}}}}\otimes
N_{\\!{\,{\mathcal{A}}}}$ on average, then $\mathsf{E}$ commutes with
${N_{\\!{\,{\mathcal{S}}}}}$.

###### Proof.

If $\mathsf{E}\subset\mathsf{E}^{\prime}$, then
${\mathcal{F}}({{\mathcal{I}}^{L}_{\mathcal{X}}}^{*})=\mathsf{E}^{\prime}$ is
a von Neumann algebra. Moreover, it holds that
$\mathsf{E}\subset{\mathcal{F}}({{\mathcal{I}}^{L}_{\mathcal{X}}}^{*})$, so
that the Lüders instrument for a commutative observable is a measurement of
the first kind. It follows from item (ii) of Theorem 5.1 that $\mathsf{E}$
must commute with ${N_{\\!{\,{\mathcal{S}}}}}$. ∎

######  Lemma 5.3.

Assume that $\dim({\mathcal{H}_{\\!{\,{\mathcal{S}}}}})<\infty$, and let
$\mathsf{G}:=\\{\mathsf{G}(z):z=1,\dots,\dim({\mathcal{H}_{\\!{\,{\mathcal{S}}}}})\\}$
be a rank-1 sharp observable acting in ${\mathcal{H}_{\\!{\,{\mathcal{S}}}}}$.
Let $\mathsf{F}$ be an observable and $M$ be an invertible stochastic matrix
such that

$\displaystyle\mathsf{F}(y)=\sum_{z}M_{y,z}\mathsf{G}(z),$
$\displaystyle\mathsf{G}(z)=\sum_{y}M_{z,y}^{-1}\mathsf{F}(y).$

An instrument ${\mathcal{I}}$ does not disturb $\mathsf{F}$ only if
${\mathcal{F}}({\mathcal{I}}^{*}_{\mathcal{X}})$ is a von Neumann algebra.

###### Proof.

Non-disturbance of $\mathsf{F}$ implies that
$\mathsf{G}\subset{\mathcal{F}}({\mathcal{I}}^{*}_{\mathcal{X}})$. Since
$\mathsf{G}(z)$ is a rank-$1$ projection, it follows that

$\displaystyle\mathrm{tr}[\mathsf{G}(z){\mathcal{I}}_{\mathcal{X}}(\mathsf{G}(z))]=\mathrm{tr}[{\mathcal{I}}_{\mathcal{X}}^{*}(\mathsf{G}(z))\mathsf{G}(z)]=\mathrm{tr}[\mathsf{G}(z)\mathsf{G}(z)]=\mathrm{tr}[\mathsf{G}(z)]=1,$

and so ${\mathcal{I}}_{\mathcal{X}}(\mathsf{G}(z))=\mathsf{G}(z)$.
Consequently, we may construct the faithful state
$\omega=\sum_{z}p_{z}\mathsf{G}(z)$ with $p_{z}>0$ and $\sum_{z}p_{z}=1$, so
that $\omega\in{\mathcal{F}}({\mathcal{I}}_{\mathcal{X}})$. By Lemma 2.4,
${\mathcal{F}}({\mathcal{I}}^{*}_{\mathcal{X}})$ is a von Neumann algebra. ∎

##  6 Non-faithful fixed states and measurement disturbance

In this section we analyse the structure of the fixed-point set of arbitrary
channels, which need not contain a faithful state. From here, the results of
the previous section are generalised. We then provide novel quantitative
bounds for first-kind measurements which, in particular, complement the
impossibility part of WAY, by imposing additional conditions which must be
satisfied for repeatable measurements.

Due to the Schauder–Tychonoff fixed point theorem [86], all channels
$\Phi:{\mathcal{T}}({\mathcal{H}_{\\!{\,{\mathcal{S}}}}})\to{\mathcal{T}}({\mathcal{H}_{\\!{\,{\mathcal{S}}}}})$
have at least one fixed state. However, it may be that none of these are
faithful. In such a case, the fixed-point set of the dual channel is not
necessarily a von Neumann algebra, but rather forms an operator space [87].
This setting has been much less investigated, and its analysis forms the first
part of this section. While the discussion thus far has been applicable for
infinite-dimensional systems—except in some examples—in this section we shall
always assume that $d:=\dim({\mathcal{H}_{\\!{\,{\mathcal{S}}}}})<\infty$.

###  6.1 Fixed-point structure of arbitrary channels

Consider a channel
$\Phi:{\mathcal{T}}({\mathcal{H}_{\\!{\,{\mathcal{S}}}}})\to{\mathcal{T}}({\mathcal{H}_{\\!{\,{\mathcal{S}}}}})$,
and its dual in the Heisenberg picture
$\Phi^{*}:{\mathcal{L}}({\mathcal{H}_{\\!{\,{\mathcal{S}}}}})\to{\mathcal{L}}({\mathcal{H}_{\\!{\,{\mathcal{S}}}}})$.
We may define the channels

$\displaystyle\Phi_{\mathrm{av}}(\cdot):=\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\Phi^{n}(\cdot),$
$\displaystyle\Phi^{*}_{\mathrm{av}}(\cdot):=\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\Phi^{*n}(\cdot),$
(39)

where $\Phi^{n}$ denotes $n$ consecutive applications of $\Phi$. Note that
these limits exists since $d<\infty$. According to the Jordan decomposition
theorem, $\Phi^{*}$ is represented as a summation of projections onto
eigenspaces multiplied by the corresponding eigenvalues, and nilpotent
operators whose eigenspaces are invariant subspaces; $\Phi^{*}_{\mathrm{av}}$
corresponds to the projection onto the subspace with eigenvalue $1$. The
fixed-point set ${\mathcal{F}}(\Phi^{*})$ forms an operator space, i.e., a
norm-closed vector subspace of the codomain of ${\mathcal{F}}(\Phi^{*})$, and
$\Phi^{*}_{\mathrm{av}}$ is a CP projection onto ${\mathcal{F}}(\Phi^{*})$.

######  Lemma 6.1.

Consider the channels $\Phi_{\mathrm{av}}^{*}$ and $\Phi_{\mathrm{av}}$
defined in Eq. (39). These have the following properties:

  1. (i)

$\Phi^{*}\circ\Phi^{*}_{\mathrm{av}}=\Phi^{*}_{\mathrm{av}}\circ\Phi^{*}=\Phi^{*}_{\mathrm{av}}\circ\Phi^{*}_{\mathrm{av}}=\Phi^{*}_{\mathrm{av}}$
and
$\Phi\circ\Phi_{\mathrm{av}}=\Phi_{\mathrm{av}}\circ\Phi=\Phi_{\mathrm{av}}\circ\Phi_{\mathrm{av}}=\Phi_{\mathrm{av}}$.

  2. (ii)

$\Phi^{*}_{\mathrm{av}}({\mathcal{L}}({\mathcal{H}_{\\!{\,{\mathcal{S}}}}}))={\mathcal{F}}(\Phi^{*}_{\mathrm{av}})={\mathcal{F}}(\Phi^{*})$
and
$\Phi_{\mathrm{av}}({\mathcal{T}}({\mathcal{H}_{\\!{\,{\mathcal{S}}}}}))={\mathcal{F}}(\Phi_{\mathrm{av}})={\mathcal{F}}(\Phi)$.

###### Proof.

(i) is trivial, and so we shall only prove (ii). Let us first consider the
Heisenberg picture channel $\Phi_{\mathrm{av}}^{*}$. That
${\mathcal{F}}(\Phi^{*})\subset{\mathcal{F}}(\Phi^{*}_{\mathrm{av}})$ is
trivial. Conversely, for any $A\in{\mathcal{F}}(\Phi^{*}_{\mathrm{av}})$, by
(i) we have
$\Phi^{*}(A)=\Phi^{*}\circ\Phi^{*}_{\mathrm{av}}(A)=\Phi^{*}_{\mathrm{av}}(A)=A$,
and therefore
${\mathcal{F}}(\Phi^{*}_{\mathrm{av}})\subset{\mathcal{F}}(\Phi^{*})$. It
follows that ${\mathcal{F}}(\Phi^{*}_{\mathrm{av}})={\mathcal{F}}(\Phi^{*})$.
Similarly, for all $A\in{\mathcal{L}}({\mathcal{H}_{\\!{\,{\mathcal{S}}}}})$
it holds that
$\Phi^{*}_{\mathrm{av}}\circ\Phi^{*}_{\mathrm{av}}(A)=\Phi^{*}_{\mathrm{av}}(A)$,
and thus
$\Phi^{*}_{\mathrm{av}}({\mathcal{L}}({\mathcal{H}_{\\!{\,{\mathcal{S}}}}}))\subset{\mathcal{F}}(\Phi^{*}_{\mathrm{av}})$.
That
${\mathcal{F}}(\Phi^{*}_{\mathrm{av}})\subset\Phi^{*}_{\mathrm{av}}({\mathcal{L}}({\mathcal{H}_{\\!{\,{\mathcal{S}}}}}))$
is trivial, and so we also have
$\Phi^{*}_{\mathrm{av}}({\mathcal{L}}({\mathcal{H}_{\\!{\,{\mathcal{S}}}}}))={\mathcal{F}}(\Phi^{*}_{\mathrm{av}})$.
The relations in (ii) for the Schrödinger picture channel $\Phi_{\mathrm{av}}$
follow from similar arguments. ∎

Now consider the state

$\displaystyle\rho_{0}:=\Phi_{\mathrm{av}}\left(\frac{1}{d}\mathds{1}_{\\!{\,{\mathcal{S}}}}\right).$
(40)

By Lemma 6.1, it holds that
$\rho_{0}\in{\mathcal{F}}(\Phi_{\mathrm{av}})={\mathcal{F}}(\Phi)$. We define
by $P$ the minimal support projection on $\rho_{0}$:

$\displaystyle P:=\min\\{Q:Q\text{ is a projection, }\rho_{0}=Q\rho_{0}Q\\}.$
(41)

In other words, for all projections $Q$ such that $\rho_{0}=Q\rho_{0}Q$, it
holds that $Q\geqslant P$. Note that if $P=\mathds{1}_{\\!{\,{\mathcal{S}}}}$
then ${\mathcal{F}}(\Phi)$ contains a faithful state. The following lemma
provides some useful properties of $P$.

######  Lemma 6.2.

Consider the state $\rho_{0}$ defined in Eq. (40), with the minimal support
projection $P$ as defined in Eq. (41), and
$P^{\perp}:=\mathds{1}_{\\!{\,{\mathcal{S}}}}-P$ its orthogonal complement.
The following hold:

  1. (i)

$\Phi^{*}_{\mathrm{av}}(P)=\mathds{1}_{\\!{\,{\mathcal{S}}}}$ and
$\Phi_{\mathrm{av}}^{*}(P^{\perp})=\mathds{O}$.

  2. (ii)

For all $A\in{\mathcal{L}}({\mathcal{H}_{\\!{\,{\mathcal{S}}}}})$,
$\Phi^{*}_{\mathrm{av}}(A)=\Phi^{*}_{\mathrm{av}}(PAP)$.

  3. (iii)

$P=\min\\{Q:Q\mbox{ is a projection, }\rho=Q\rho Q\,\
\forall\,\rho\in{\mathcal{F}}(\Phi)\\}$.

  4. (iv)

$P=\min\\{Q:Q\mbox{ is a projection,
}\Phi^{*}_{\mathrm{av}}(Q)=\mathds{1}_{\\!{\,{\mathcal{S}}}}$}.

  5. (v)

$\Phi^{*}(P)\geqslant P$ and $\Phi^{*}(P^{\perp})\leqslant P^{\perp}$.

###### Proof.

  1. (i):

Since $\Phi_{\mathrm{av}}^{*}$ is a channel, and
$\mathds{O}<P\leqslant\mathds{1}_{\\!{\,{\mathcal{S}}}}$, it follows that
$\mathds{O}\leqslant\Phi_{\mathrm{av}}^{*}(P)\leqslant\mathds{1}_{\\!{\,{\mathcal{S}}}}$.
But by Eq. (40)
$\mathrm{tr}[\Phi^{*}_{\mathrm{av}}(P)]=d\,\mathrm{tr}[\rho_{0}P]=d$, and so
$\Phi_{\mathrm{av}}^{*}(P)=\mathds{1}_{\\!{\,{\mathcal{S}}}}$. It trivially
follows that
$\Phi_{\mathrm{av}}^{*}(P^{\perp})=\Phi_{\mathrm{av}}^{*}(\mathds{1}_{\\!{\,{\mathcal{S}}}})-\Phi_{\mathrm{av}}^{*}(P)=\mathds{1}_{\\!{\,{\mathcal{S}}}}-\mathds{1}_{\\!{\,{\mathcal{S}}}}=\mathds{O}$.

  2. (ii):

Since $P$ is positive, then by (i) and Lemma 2.3 it holds that
$\Phi^{*}_{\mathrm{av}}(P^{\perp}A)=\Phi^{*}_{\mathrm{av}}(AP^{\perp})=\mathds{O}$
for all $A$. The claim follows by noting that we may write
$A=(P+P^{\perp})A(P+P^{\perp})$.

  3. (iii):

By Lemma 6.1 and (ii), for all
$A\in{\mathcal{L}}({\mathcal{H}_{\\!{\,{\mathcal{S}}}}})$ and
$\rho\in{\mathcal{F}}(\Phi)$ we have
$\mathrm{tr}[A\rho]=\mathrm{tr}[\Phi_{\mathrm{av}}^{*}(A)\rho]=\mathrm{tr}[\Phi_{\mathrm{av}}^{*}(PAP)\rho]=\mathrm{tr}[AP\rho
P]$, and so $\rho\in{\mathcal{F}}(\Phi)\implies\rho=P\rho P$. Since $P$ is the
minimal support projection on $\rho_{0}\in{\mathcal{F}}(\Phi)$, the claim
follows.

  4. (iv):

By (i), $\Phi^{*}_{\mathrm{av}}(P)=\mathds{1}_{\\!{\,{\mathcal{S}}}}$ holds.
Suppose another projection $Q$ satisfies
$\Phi^{*}_{\mathrm{av}}(Q)=\mathds{1}_{\\!{\,{\mathcal{S}}}}$. Then (ii)
implies that
$\Phi^{*}_{\mathrm{av}}(PQP)=\Phi^{*}_{\mathrm{av}}(Q)=\mathds{1}_{\\!{\,{\mathcal{S}}}}$.
As we have
$1=\mathrm{tr}[(\mathds{1}_{\\!{\,{\mathcal{S}}}}/d)\Phi^{*}_{\mathrm{av}}(PQP)]=\mathrm{tr}[\rho_{0}PQP]=\mathrm{tr}[\rho_{0}Q]$,
it follows that $Q\geqslant P$.

  5. (v):

Since $P$ is the smallest projection satisfying $\rho_{0}=P\rho_{0}P$, while
$\mathrm{tr}[\rho_{0}\Phi^{*}(P)]=\mathrm{tr}[\Phi(\rho_{0})P]=\mathrm{tr}[\rho_{0}P]=1$,
it follows that $\Phi^{*}(P)\geqslant P$, and hence
$\Phi^{*}(P^{\perp})=\Phi^{*}(\mathds{1}_{\\!{\,{\mathcal{S}}}})-\Phi^{*}(P)\leqslant\mathds{1}_{\\!{\,{\mathcal{S}}}}-P=P^{\perp}$.

∎

Let us now define the operations

$\displaystyle\Phi^{*}_{\mathrm{av},P}(\cdot):=P\Phi^{*}_{\mathrm{av}}(\cdot)P,$
$\displaystyle\Phi^{*}_{P}(\cdot):=P\Phi^{*}(\cdot)P.$ (42)

Note that
$\Phi_{\mathrm{av},P}^{*}:{\mathcal{L}}({\mathcal{H}_{\\!{\,{\mathcal{S}}}}})\to{\mathcal{L}}({\mathcal{H}_{\\!{\,{\mathcal{S}}}}})$
is not necessarily unital, since
$\Phi^{*}_{\mathrm{av},P}(\mathds{1}_{\\!{\,{\mathcal{S}}}})=P\leqslant\mathds{1}_{\\!{\,{\mathcal{S}}}}$.
However, the restriction of $\Phi^{*}_{\mathrm{av},P}$ to
${\mathcal{L}}(P{\mathcal{H}_{\\!{\,{\mathcal{S}}}}})\to{\mathcal{L}}(P{\mathcal{H}_{\\!{\,{\mathcal{S}}}}})$,
which is also denoted by $\Phi^{*}_{\mathrm{av},P}$, is unital and hence a
channel, since $P$ is the identity in
${\mathcal{L}}(P{\mathcal{H}_{\\!{\,{\mathcal{S}}}}})$ and
$\Phi^{*}_{\mathrm{av},P}(P)=P$. The same holds for $\Phi_{P}^{*}$.

######  Lemma 6.3.

Consider the operations defined in Eq. (42). The following hold:

  1. (i)

$\Phi^{*}_{P}(A)=\Phi^{*}_{P}(PAP)$ for all
$A\in{\mathcal{L}}({\mathcal{H}_{\\!{\,{\mathcal{S}}}}})$.

  2. (ii)

$\Phi^{*}_{\mathrm{av},P}(A)=\Phi^{*}_{\mathrm{av},P}(PAP)$ for all
$A\in{\mathcal{L}}({\mathcal{H}_{\\!{\,{\mathcal{S}}}}})$.

  3. (iii)

$\Phi^{*}_{\mathrm{av},P}$ is a completely positive projection
${\mathcal{L}}({\mathcal{H}_{\\!{\,{\mathcal{S}}}}})\to{\mathcal{L}}(P{\mathcal{H}_{\\!{\,{\mathcal{S}}}}})$.

  4. (iv)

${\mathcal{F}}(\Phi^{*}_{\mathrm{av},P})=\Phi^{*}_{\mathrm{av},P}({\mathcal{L}}(P{\mathcal{H}_{\\!{\,{\mathcal{S}}}}}))$.

  5. (v)

$\Phi^{*}_{\mathrm{av}}$ is a bijection from
${\mathcal{F}}(\Phi^{*}_{\mathrm{av},P})$ to ${\mathcal{F}}(\Phi^{*})$.

  6. (vi)

The inverse of
$\Phi^{*}_{\mathrm{av}}:{\mathcal{F}}(\Phi^{*}_{\mathrm{av},P})\to{\mathcal{F}}(\Phi^{*})$
is $\mathrm{Ad}_{P}:A\mapsto PAP$.

###### Proof.

  1. (i):

By item (v) of Lemma 6.2, it holds that
$\mathds{O}\leqslant\Phi^{*}_{P}(P^{\perp})=P\Phi^{*}(P^{\perp})P\leqslant
PP^{\perp}P=\mathds{O}$, and so $\Phi^{*}_{P}(P^{\perp})=\mathds{O}$. By Lemma
2.3, it holds that
$\Phi_{P}^{*}(P^{\perp}A)=\Phi_{P}^{*}(AP^{\perp})=\mathds{O}$ for all
$A\in{\mathcal{L}}({\mathcal{H}_{\\!{\,{\mathcal{S}}}}})$. Since
$A=(P+P^{\perp})A(P+P^{\perp})$, the claim follows.

  2. (ii):

By item (ii) of Lemma 6.2,
$\Phi_{\mathrm{av}}^{*}(A)=\Phi_{\mathrm{av}}^{*}(PAP)$. The claim immediately
follows.

  3. (iii):

By item (i) of Lemma 6.1 and (ii) above, we have

$\displaystyle\Phi^{*}_{\mathrm{av},P}\circ\Phi^{*}_{\mathrm{av},P}(A)=\Phi^{*}_{\mathrm{av},P}(P\Phi^{*}_{\mathrm{av}}(A)P)=\Phi^{*}_{\mathrm{av},P}\circ\Phi^{*}_{\mathrm{av}}(A)=P\Phi^{*}_{\mathrm{av}}\circ\Phi^{*}_{\mathrm{av}}(A)P=P\Phi^{*}_{\mathrm{av}}(A)P=\Phi^{*}_{\mathrm{av},P}(A).$

  4. (iv):

By (iii), it follows that for any
$A\in{\mathcal{L}}(P{\mathcal{H}_{\\!{\,{\mathcal{S}}}}})$,
$\Phi^{*}_{\mathrm{av},P}(A)\in{\mathcal{F}}(\Phi^{*}_{\mathrm{av},P})$ and so
$\Phi^{*}_{\mathrm{av},P}({\mathcal{L}}(P{\mathcal{H}_{\\!{\,{\mathcal{S}}}}}))\subset{\mathcal{F}}(\Phi^{*}_{\mathrm{av},P})$.
The converse is trivial.

  5. (v):

For all $A\in{\mathcal{F}}(\Phi^{*})={\mathcal{F}}(\Phi^{*}_{\mathrm{av}})$,
there exists an operator $PAP\in{\mathcal{F}}(\Phi_{\mathrm{av},P}^{*})$ such
that $\Phi_{\mathrm{av}}^{*}(PAP)=\Phi_{\mathrm{av}}^{*}(A)=A$. Therefore,
$\Phi_{\mathrm{av}}^{*}$ is surjective. Now assume that there exists
$A\in{\mathcal{F}}(\Phi_{\mathrm{av},P}^{*})$ such that
$\Phi^{*}_{\mathrm{av}}(A)=\mathds{O}$. This implies that
$A=\Phi_{\mathrm{av},P}^{*}(A)=P\Phi_{\mathrm{av}}^{*}(A)P=\mathds{O}$.
Therefore, $\Phi_{\mathrm{av}}^{*}$ is injective.

  6. (vi):

Follows from above.

∎

The above results have the following useful consequence:

######  Proposition 6.1.

Consider the operations $\Phi_{\mathrm{av},P}^{*}$ and $\Phi^{*}_{P}$ defined
in Eq. (42). It holds that

$\displaystyle{\mathcal{F}}(\Phi^{*}_{\mathrm{av},P})={\mathcal{F}}(\Phi^{*}_{P})=P{\mathcal{F}}(\Phi_{\mathrm{av}}^{*})P=P{\mathcal{F}}(\Phi^{*})P$

is a von Neumann algebra in
${\mathcal{L}}(P{\mathcal{H}_{\\!{\,{\mathcal{S}}}}})$.

###### Proof.

Recall that the operation
$\Phi_{\mathrm{av},P}^{*}:{\mathcal{L}}(P{\mathcal{H}_{\\!{\,{\mathcal{S}}}}})\to{\mathcal{L}}(P{\mathcal{H}_{\\!{\,{\mathcal{S}}}}})$
is unital, where the unit in
${\mathcal{L}}(P{\mathcal{H}_{\\!{\,{\mathcal{S}}}}})$ is $P$. Moreover,
$\rho_{0}$ as defined in Eq. (40) is a faithful fixed point of
$\Phi_{\mathrm{av},P}$ in
${\mathcal{T}}(P{\mathcal{H}_{\\!{\,{\mathcal{S}}}}})$. By Lemma 2.4, the
fixed-point set ${\mathcal{F}}(\Phi^{*}_{\mathrm{av},P})$ is a von Neumann
algebra in ${\mathcal{L}}(P{\mathcal{H}_{\\!{\,{\mathcal{S}}}}})$. Now we need
only show that
${\mathcal{F}}(\Phi^{*}_{\mathrm{av},P})={\mathcal{F}}(\Phi^{*}_{P})=P{\mathcal{F}}(\Phi_{\mathrm{av}}^{*})P=P{\mathcal{F}}(\Phi^{*})P$.
That $P{\mathcal{F}}(\Phi_{\mathrm{av}}^{*})P=P{\mathcal{F}}(\Phi^{*})P$
trivially follows from Lemma 6.1, which gives
${\mathcal{F}}(\Phi^{*})={\mathcal{F}}(\Phi_{\mathrm{av}}^{*})$. That
${\mathcal{F}}(\Phi^{*}_{\mathrm{av},P})=P{\mathcal{F}}(\Phi^{*})P$ follows
from item (vi) of Lemma 6.3, since the map $\mathrm{Ad}_{P}:A\mapsto PAP$ is a
bijection from ${\mathcal{F}}(\Phi^{*})$ to
${\mathcal{F}}(\Phi^{*}_{\mathrm{av},P})$. To show that
${\mathcal{F}}(\Phi^{*}_{\mathrm{av},P})={\mathcal{F}}(\Phi^{*}_{P})$, let us
first define the operation
$\Phi_{P,\mathrm{av}}^{*}:{\mathcal{L}}({\mathcal{H}_{\\!{\,{\mathcal{S}}}}})\to{\mathcal{L}}(P{\mathcal{H}_{\\!{\,{\mathcal{S}}}}})$
as

$\displaystyle\Phi_{P,\mathrm{av}}^{*}(\cdot)$
$\displaystyle:=\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\Phi_{P}^{*n}(\cdot).$

But by item (i) of Lemma 6.3, $\Phi^{*}_{P}(A)=\Phi^{*}_{P}(PAP)$, and so
$\Phi_{P}^{*n}(A)=P\Phi^{*n}(A)P$ for all
$A\in{\mathcal{L}}({\mathcal{H}_{\\!{\,{\mathcal{S}}}}})$ and
$n\in\mathds{N}$. It follows that

$\displaystyle\Phi_{P,\mathrm{av}}^{*}(\cdot)$
$\displaystyle=\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}P\Phi^{*n}(\cdot)P=P\Phi_{\mathrm{av}}^{*}(\cdot)P=:\Phi^{*}_{\mathrm{av},P}(\cdot).$

since $\Phi_{\mathrm{av},P}^{*}=\Phi_{P,\mathrm{av}}^{*}$, it trivially
follows that
$\Phi_{P}^{*}\circ\Phi_{\mathrm{av},P}^{*}=\Phi_{\mathrm{av},P}^{*}\circ\Phi_{P}^{*}=\Phi_{\mathrm{av},P}^{*}\circ\Phi_{\mathrm{av},P}^{*}=\Phi_{\mathrm{av},P}^{*}$.
Therefore,
${\mathcal{F}}(\Phi^{*}_{\mathrm{av},P})={\mathcal{F}}(\Phi^{*}_{P})$ can be
shown by the same arguments as in Lemma 6.1

∎

###  6.2 Measurement disturbance revisited

We are now ready to address the question of measurement disturbance,
generalising the observations of Sec. 5. As before, let
${\mathcal{I}}:=\\{{\mathcal{I}}_{x}:x\in{\mathcal{X}}\\}$ be an
$\mathsf{E}$-compatible instrument, with
${\mathcal{I}}_{\mathcal{X}}(\cdot):=\sum_{x}{\mathcal{I}}_{x}(\cdot)$ the
corresponding $\mathsf{E}$-channel. By Eq. (39) and Eq. (42) we define

$\displaystyle{\mathcal{I}}_{\mathrm{av}}^{*}(\cdot)$
$\displaystyle:=\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}{\mathcal{I}}_{\mathcal{X}}^{*n}(\cdot),\qquad{\mathcal{I}}_{\mathrm{av},P}^{*}(\cdot):=P{\mathcal{I}}_{\mathrm{av}}^{*}(\cdot)P,\qquad{\mathcal{I}}_{P}^{*}(\cdot):=P{\mathcal{I}}_{\mathcal{X}}^{*}(\cdot)P,$

where as in Eq. (41), $P$ is the minimal support projection of
$\rho_{0}:={\mathcal{I}}_{\mathrm{av}}(\frac{1}{d}\mathds{1}_{\\!{\,{\mathcal{S}}}})$,
which corresponds with the minimal projection on the support of
${\mathcal{F}}({\mathcal{I}}_{\mathcal{X}})$. By Proposition 6.1,
${\mathcal{F}}({\mathcal{I}}_{\mathrm{av},P}^{*})={\mathcal{F}}({\mathcal{I}}_{P}^{*})=P{\mathcal{F}}({\mathcal{I}}^{*}_{\mathrm{av}})P=P{\mathcal{F}}({\mathcal{I}}^{*}_{\mathcal{X}})P$
is a von Neumann algebra in
${\mathcal{L}}(P{\mathcal{H}_{\\!{\,{\mathcal{S}}}}})$. We define by
$P\mathsf{E}P:=\\{P\mathsf{E}(x)P:x\in{\mathcal{X}}\\}$ the restriction of
$\mathsf{E}$ to an observable in $P{\mathcal{H}_{\\!{\,{\mathcal{S}}}}}$,
which satisfies $\sum_{x}P\mathsf{E}(x)P=P$, and
$(P\mathsf{E}P)^{\prime}:=\\{A\in{\mathcal{L}}(P{\mathcal{H}_{\\!{\,{\mathcal{S}}}}}):[P\mathsf{E}(x)P,A]=\mathds{O}\,\forall
x\\}$ denotes the commutant of $P\mathsf{E}P$ in
${\mathcal{L}}(P{\mathcal{H}_{\\!{\,{\mathcal{S}}}}})$. $P\mathsf{F}P$ is
similarly defined.

Before generalising Theorem 5.1 for the case where
${\mathcal{F}}({\mathcal{I}}_{\mathcal{X}})$ may not contain any faithful
states, and thus ${\mathcal{F}}({\mathcal{I}}_{\mathcal{X}}^{*})$ may not
necessarily be a von Neumann algebra, let us first prove a generalisation of
Lemma 5.1.

######  Lemma 6.4.

Let
${\mathcal{M}}:=({\mathcal{H}_{\\!{\,{\mathcal{A}}}}},\xi,{\mathcal{E}},\mathsf{Z})$
be a measurement scheme for an $\mathsf{E}$-instrument ${\mathcal{I}}$ acting
in ${\mathcal{H}_{\\!{\,{\mathcal{S}}}}}$. It holds that
${\mathcal{F}}({\mathcal{I}}_{P}^{*})\subset(P\mathsf{E}P)^{\prime}$.
Additionally, if ${\mathcal{E}}$ conserves an additive quantity
$N=N_{\\!{\,{\mathcal{S}}}}\otimes\mathds{1}_{\\!{\,{\mathcal{A}}}}+\mathds{1}_{\\!{\,{\mathcal{S}}}}\otimes
N_{\\!{\,{\mathcal{A}}}}$ on average, then for all
$A\in{\mathcal{L}}(\mathsf{P}{\mathcal{H}_{\\!{\,{\mathcal{S}}}}})$ the
following implication holds:
$A\in{\mathcal{F}}({\mathcal{I}}_{P}^{*})\implies[A,PN_{\\!{\,{\mathcal{S}}}}P]\in{\mathcal{F}}({\mathcal{I}}_{P}^{*})\implies[A,PN_{\\!{\,{\mathcal{S}}}}P]\in(P\mathsf{E}P)^{\prime}$.

###### Proof.

By Eq. (7), let us define the operation
$\Gamma_{\xi,P}^{\mathcal{E}}:{\mathcal{L}}({\mathcal{H}_{\\!{\,{\mathcal{S}}}}}\otimes{\mathcal{H}_{\\!{\,{\mathcal{A}}}}})\to{\mathcal{L}}(P{\mathcal{H}_{\\!{\,{\mathcal{S}}}}}),B\mapsto
P\Gamma_{\xi}^{\mathcal{E}}(B)P$. It is easily verified that
${\mathcal{I}}_{P}^{*}(A)=\Gamma_{\xi,P}^{\mathcal{E}}(A\otimes\mathds{1}_{\\!{\,{\mathcal{A}}}})$
for all $A\in{\mathcal{L}}(P{\mathcal{H}_{\\!{\,{\mathcal{S}}}}})$, and
$P\mathsf{E}(x)P=\Gamma_{\xi,P}^{\mathcal{E}}(\mathds{1}_{\\!{\,{\mathcal{S}}}}\otimes\mathsf{Z}(x))$.
Note that by the same arguments as item (i) of Lemma 6.3, it can easily be
shown that
$\Gamma_{\xi,P}^{\mathcal{E}}(B)=\Gamma_{\xi,P}^{\mathcal{E}}((P\otimes\mathds{1}_{\\!{\,{\mathcal{A}}}})B(P\otimes\mathds{1}_{\\!{\,{\mathcal{A}}}}))$
holds for all
$B\in{\mathcal{L}}({\mathcal{H}_{\\!{\,{\mathcal{S}}}}}\otimes{\mathcal{H}_{\\!{\,{\mathcal{A}}}}})$.
It follows that $\Gamma_{\xi,P}^{\mathcal{E}}$ is unital when restricted to
${\mathcal{L}}(P{\mathcal{H}_{\\!{\,{\mathcal{S}}}}}\otimes{\mathcal{H}_{\\!{\,{\mathcal{A}}}}})\to{\mathcal{L}}(P{\mathcal{H}_{\\!{\,{\mathcal{S}}}}})$,
and we may equivalently write
$P\mathsf{E}(x)P=\Gamma_{\xi,P}^{\mathcal{E}}(P\otimes\mathsf{Z}(x))$.

By Proposition 6.1, ${\mathcal{F}}({\mathcal{I}}_{P}^{*})$ is a von Neumann
algebra in ${\mathcal{L}}(P{\mathcal{H}_{\\!{\,{\mathcal{S}}}}})$, and so for
all $A\in{\mathcal{L}}(P{\mathcal{H}_{\\!{\,{\mathcal{S}}}}})$, if
$A\in{\mathcal{F}}({\mathcal{I}}_{P}^{*})$, then
$A^{*}A,AA^{*}\in{\mathcal{F}}({\mathcal{I}}_{P}^{*})$. By Corollary 2.1 it
holds that for all $A\in{\mathcal{F}}({\mathcal{I}}_{P}^{*})$ and
$B\in{\mathcal{L}}(P{\mathcal{H}_{\\!{\,{\mathcal{S}}}}}\otimes{\mathcal{H}_{\\!{\,{\mathcal{A}}}}})$
we have
$A\Gamma_{\xi,P}^{\mathcal{E}}(B)=\Gamma_{\xi,P}^{\mathcal{E}}((A\otimes\mathds{1}_{\\!{\,{\mathcal{A}}}})B)$
and
$\Gamma_{\xi,P}^{\mathcal{E}}(B)A=\Gamma_{\xi,P}^{\mathcal{E}}(B(A\otimes\mathds{1}_{\\!{\,{\mathcal{A}}}}))$.
It follows that for all $A\in{\mathcal{F}}({\mathcal{I}}_{P}^{*})$ and
$x\in{\mathcal{X}}$ we have

$\displaystyle[A,P\mathsf{E}(x)P]=[A,\Gamma_{\xi,P}^{\mathcal{E}}(P\otimes\mathsf{Z}(x))]=\Gamma_{\xi,P}^{\mathcal{E}}([A\otimes\mathds{1}_{\\!{\,{\mathcal{A}}}},P\otimes\mathsf{Z}(x)])=\mathds{O},$

and so ${\mathcal{F}}({\mathcal{I}}^{*}_{P})\subset(P\mathsf{E}P)^{\prime}$.

Now let us assume that ${\mathcal{E}}$ conserves an additive quantity
$N=N_{\\!{\,{\mathcal{S}}}}\otimes\mathds{1}_{\\!{\,{\mathcal{A}}}}+\mathds{1}_{\\!{\,{\mathcal{S}}}}\otimes
N_{\\!{\,{\mathcal{A}}}}$ on average. This implies that
$P\Gamma_{\xi}(N)P=\Gamma_{\xi,P}^{\mathcal{E}}(N)=\Gamma_{\xi,P}^{\mathcal{E}}((P\otimes\mathds{1}_{\\!{\,{\mathcal{A}}}})N(P\otimes\mathds{1}_{\\!{\,{\mathcal{A}}}}))$,
and so

$\displaystyle
P{N_{\\!{\,{\mathcal{S}}}}}P+\mathrm{tr}[{N_{\\!{\,{\mathcal{A}}}}}\xi]P=\Gamma_{\xi,P}^{\mathcal{E}}(P{N_{\\!{\,{\mathcal{S}}}}}P\otimes\mathds{1}_{\\!{\,{\mathcal{A}}}})+\Gamma_{\xi,P}^{\mathcal{E}}(P\otimes{N_{\\!{\,{\mathcal{A}}}}}).$

Since ${\mathcal{F}}({\mathcal{I}}^{*}_{P})$ is a von Neumann algebra in
${\mathcal{L}}(P{\mathcal{H}_{\\!{\,{\mathcal{S}}}}})$, then by Corollary 2.1
and the arguments above, it follows that for all
$A\in{\mathcal{F}}({\mathcal{I}}_{P}^{*})$ we have

$\displaystyle[A,P{N_{\\!{\,{\mathcal{S}}}}}P]$
$\displaystyle=[A,\Gamma_{\xi,P}^{\mathcal{E}}(P{N_{\\!{\,{\mathcal{S}}}}}P\otimes\mathds{1}_{\\!{\,{\mathcal{A}}}})]+[A,\Gamma_{\xi,P}^{\mathcal{E}}(P\otimes{N_{\\!{\,{\mathcal{A}}}}})]$
$\displaystyle=\Gamma_{\xi,P}^{\mathcal{E}}([A\otimes\mathds{1}_{\\!{\,{\mathcal{A}}}},P{N_{\\!{\,{\mathcal{S}}}}}P\otimes\mathds{1}_{\\!{\,{\mathcal{A}}}}])+\Gamma_{\xi,P}^{\mathcal{E}}([A\otimes\mathds{1}_{\\!{\,{\mathcal{A}}}},P\otimes{N_{\\!{\,{\mathcal{A}}}}}])$
$\displaystyle=\Gamma_{\xi,P}^{\mathcal{E}}([A,P{N_{\\!{\,{\mathcal{S}}}}}P]\otimes\mathds{1}_{\\!{\,{\mathcal{A}}}})={\mathcal{I}}^{*}_{P}([A,P{N_{\\!{\,{\mathcal{S}}}}}P]).$

We thus have
$A\in{\mathcal{F}}({\mathcal{I}}^{*}_{P})\implies[A,P{N_{\\!{\,{\mathcal{S}}}}}P]\in{\mathcal{F}}({\mathcal{I}}^{*}_{P})$,
and since
${\mathcal{F}}({\mathcal{I}}^{*}_{P})\subset(P\mathsf{E}P)^{\prime}$, it
follows that $[A,P{N_{\\!{\,{\mathcal{S}}}}}P]\in(P\mathsf{E}P)^{\prime}$. ∎

We are now ready to generalise Theorem 5.1.

######  Theorem 6.1.

Let
${\mathcal{M}}:=({\mathcal{H}_{\\!{\,{\mathcal{A}}}}},\xi,{\mathcal{E}},\mathsf{Z})$
be a measurement scheme for an $\mathsf{E}$-instrument ${\mathcal{I}}$ acting
in ${\mathcal{H}_{\\!{\,{\mathcal{S}}}}}$, and assume that ${\mathcal{E}}$
conserves an additive quantity
$N=N_{\\!{\,{\mathcal{S}}}}\otimes\mathds{1}_{\\!{\,{\mathcal{A}}}}+\mathds{1}_{\\!{\,{\mathcal{S}}}}\otimes
N_{\\!{\,{\mathcal{A}}}}$ on average. The following hold:

  1. (i)

${\mathcal{I}}$ does not disturb an observable $\mathsf{F}$ only if
$P\mathsf{F}P$ commutes with both $P\mathsf{E}P$ and $\Delta
P{N_{\\!{\,{\mathcal{S}}}}}P:={\mathcal{I}}_{P}^{*}(P{N_{\\!{\,{\mathcal{S}}}}}P)-P{N_{\\!{\,{\mathcal{S}}}}}P$,
with $[P\mathsf{F}P,P{N_{\\!{\,{\mathcal{S}}}}}P]=\mathds{O}$ being sufficient
for $[P\mathsf{F}P,\Delta P{N_{\\!{\,{\mathcal{S}}}}}P]=\mathds{O}$.

  2. (ii)

${\mathcal{I}}$ is a measurement of the first kind only if $P\mathsf{E}P$ is
commutative and commutes with $P{N_{\\!{\,{\mathcal{S}}}}}P$.

  3. (iii)

${\mathcal{I}}$ is repeatable only if $P\mathsf{E}P$ is sharp and commutes
with $P{N_{\\!{\,{\mathcal{S}}}}}P$.

###### Proof.

  1. (i):

Non-disturbance of $\mathsf{F}$ implies that
$\mathsf{F}\subset{\mathcal{F}}({\mathcal{I}}^{*}_{\mathcal{X}})$, and it
follows from Proposition 6.1 that
$P\mathsf{F}P\subset{\mathcal{F}}({\mathcal{I}}^{*}_{P})$. By Lemma 6.4, it
holds that $P\mathsf{F}P$ must commute with $P\mathsf{E}P$, and that
$[P\mathsf{F}(y)P,P{N_{\\!{\,{\mathcal{S}}}}}P]={\mathcal{I}}_{P}^{*}([P\mathsf{F}(y)P,P{N_{\\!{\,{\mathcal{S}}}}}P])$.
The second condition is guaranteed to hold if
$[P\mathsf{F}(y)P,P{N_{\\!{\,{\mathcal{S}}}}}P]=\mathds{O}$. Given that
${\mathcal{F}}({\mathcal{I}}^{*}_{P})$ is a von Neumann algebra, we have
${\mathcal{I}}_{P}^{*}((P\mathsf{F}(y)P)^{2})={\mathcal{I}}_{P}^{*}(P\mathsf{F}(y)P)^{2}=(P\mathsf{F}(y)P)^{2}$.
By Corollary 2.1 it follows that
$[P\mathsf{F}(y)P,P{N_{\\!{\,{\mathcal{S}}}}}P]=[P\mathsf{F}(y)P,{\mathcal{I}}_{P}^{*}(P{N_{\\!{\,{\mathcal{S}}}}}P)]$,
and so $P\mathsf{F}P$ must commute with $\Delta P{N_{\\!{\,{\mathcal{S}}}}}P$.

  2. (ii):

If ${\mathcal{I}}$ is a measurement of the first kind,
$\mathsf{E}\subset{\mathcal{F}}({\mathcal{I}}^{*}_{\mathcal{X}})$, it follows
from Proposition 6.1 that
$P\mathsf{E}P\subset{\mathcal{F}}({\mathcal{I}}^{*}_{P})$. By Lemma 6.4
$P\mathsf{E}P$ must be commutative, and since
${\mathcal{F}}({\mathcal{I}}^{*}_{P})$ is a von Neumann algebra, then for all
spectral projections
$R\in{\mathcal{L}}(P{\mathcal{H}_{\\!{\,{\mathcal{S}}}}})$ of
$P\mathsf{E}(x)P$ it must hold that $[R,P{N_{\\!{\,{\mathcal{S}}}}}P]$
commutes with $P\mathsf{E}P$. But this implies that
$[[R,P{N_{\\!{\,{\mathcal{S}}}}}P],R^{\perp}]=\mathds{O}$, where we define
$R^{\perp}:=P-R$. As such, given that $PR=RP=R$, it holds that
$[R{N_{\\!{\,{\mathcal{S}}}}}P-P{N_{\\!{\,{\mathcal{S}}}}}R,R^{\perp}]=R{N_{\\!{\,{\mathcal{S}}}}}R^{\perp}+R^{\perp}{N_{\\!{\,{\mathcal{S}}}}}R=\mathds{O}$.
Multiplying from the left by $R$, we thus have
$R{N_{\\!{\,{\mathcal{S}}}}}R^{\perp}=R{N_{\\!{\,{\mathcal{S}}}}}(P-R)=\mathds{O}$,
and so $R{N_{\\!{\,{\mathcal{S}}}}}P=R{N_{\\!{\,{\mathcal{S}}}}}R$. Since the
right hand side is self-adjoint, and
$R{N_{\\!{\,{\mathcal{S}}}}}P=RP{N_{\\!{\,{\mathcal{S}}}}}P$, it follows that
$[R,P{N_{\\!{\,{\mathcal{S}}}}}P]=\mathds{O}$. Since this relation holds for
all spectral projections $R$ of all effects of $P\mathsf{E}P$, it follows that
$P\mathsf{E}P$ must commute with $P{N_{\\!{\,{\mathcal{S}}}}}P$.

  3. (iii):

Commutativity of $P\mathsf{E}P$ with $P{N_{\\!{\,{\mathcal{S}}}}}P$ follows
from (ii) and the fact that repeatability implies first-kindness. Sharpness of
$P\mathsf{E}P$ follows from the fact that the fixed points of a repeatable
instrument can only have support in the eigenvalue-1 eigenspaces of
$\mathsf{E}$, as shown in Proposition 4.1.

∎

Note that if $P=\mathds{1}_{\\!{\,{\mathcal{S}}}}$, implying that
${\mathcal{F}}({\mathcal{I}}_{\mathcal{X}})$ contains a faithful state so that
${\mathcal{F}}({\mathcal{I}}_{\mathcal{X}}^{*})$ is a von Neumann algebra,
then the above theorem reduces to Theorem 5.1. Interestingly, in the case of
qubits such an equivalence will always hold, even if
${\mathcal{F}}({\mathcal{I}}_{\mathcal{X}})$ does not contain a faithful
state. We demonstrate this by an alternative proof for Proposition 6 in Ref.
[4].

######  Corollary 6.1.

If $\dim({\mathcal{H}_{\\!{\,{\mathcal{S}}}}})=2$, then Theorem 6.1 reduces to
Theorem 5.1.

###### Proof.

Let us consider the minimal support projection $P$ on the fixed-point set
${\mathcal{F}}({\mathcal{I}}_{\mathcal{X}})$. As all channels must have at
least one fixed state, then when
$\dim({\mathcal{H}_{\\!{\,{\mathcal{S}}}}})=2$ it holds that, for any
instrument ${\mathcal{I}}$, either $P=\mathds{1}_{\\!{\,{\mathcal{S}}}}$ or
$P$ is a rank-1 projection. If $P=\mathds{1}_{\\!{\,{\mathcal{S}}}}$ then
${\mathcal{F}}({\mathcal{I}}_{\mathcal{X}})$ contains a faithful state, so
that by Lemma 2.4 ${\mathcal{F}}({\mathcal{I}}_{\mathcal{X}}^{*})$ is a von
Neumann algebra, and Theorem 6.1 reduces to Theorem 5.1. Now assume that $P$
is a rank-1 projection, so that for all
$A\in{\mathcal{L}}({\mathcal{H}_{\\!{\,{\mathcal{S}}}}})$ it holds that
$PAP=\lambda P$, with some $\lambda\in\mathds{C}$. Recall from Lemma 6.1 that
${\mathcal{F}}({\mathcal{I}}_{\mathcal{X}}^{*})={\mathcal{F}}({\mathcal{I}}_{\mathrm{av}}^{*})$.
By item (ii) of Lemma 6.2, for all
$A\in{\mathcal{F}}({\mathcal{I}}_{\mathcal{X}}^{*})$ it holds that
$A={\mathcal{I}}_{\mathrm{av}}^{*}(A)={\mathcal{I}}_{\mathrm{av}}^{*}(PAP)=\lambda{\mathcal{I}}_{\mathrm{av}}^{*}(P)=\lambda\mathds{1}_{\\!{\,{\mathcal{S}}}}$,
and so ${\mathcal{F}}({\mathcal{I}}_{\mathcal{X}}^{*})$ is a trivial von
Neumann algebra containing only operators proportional to the identity. In
this case, the only non-disturbed observables are trivial, and will clearly
commute with all of ${\mathcal{L}}({\mathcal{H}_{\\!{\,{\mathcal{S}}}}})$. In
such a case we may simply replace $P$ with $\mathds{1}_{\\!{\,{\mathcal{S}}}}$
in items (i)-(iii) of Theorem 6.1, so that it reduces to Theorem 5.1.

∎

###  6.3 Non-disturbance and distinguishability

Here, we present some novel results regarding the structure of non-disturbing
measurements that go beyond those in the preceding sections, indicating an
intimate relationship between non-disturbance and distinguishability. These
results hold for general instruments, are independent of conservation laws,
and do not explicitly depend on the support projection $P$ on the fixed-point
set of the specific measurement channel ${\mathcal{I}}_{\mathcal{X}}$. First,
let us show that if an instrument ${\mathcal{I}}$ does not disturb a non-
trivial observable, then there exists a family of distinguishable states that
remain distinguishable after a non-selective measurement by ${\mathcal{I}}$.

######  Proposition 6.2 (Nondisturbance implies distinguishability).

Consider an instrument ${\mathcal{I}}$ acting in
${\mathcal{H}_{\\!{\,{\mathcal{S}}}}}$, and assume that ${\mathcal{I}}$ does
not disturb a non-trivial observable $\mathsf{F}$. Then there exists a norm-1
observable $\mathsf{G}:=\\{\mathsf{G}(z):z\in{\mathcal{Z}}\\}$ acting in
${\mathcal{H}_{\\!{\,{\mathcal{S}}}}}$ that is non-disturbed by
${\mathcal{I}}$, so that for every family of states
$\\{\rho_{z}:z\in{\mathcal{Z}}\\}$ that are perfectly distinguishable by a
$\mathsf{G}$ measurement,
$\\{{\mathcal{I}}_{\mathcal{X}}(\rho_{z}):z\in{\mathcal{Z}}\\}$ remain
perfectly distinguishable by a $\mathsf{G}$ measurement. Moreover, if
${\mathcal{F}}({\mathcal{I}}_{\mathcal{X}})$ contains a faithful state, then
$\mathsf{G}$ can be taken as a sharp observable.

###### Proof.

Suppose that a non-trivial observable
$\mathsf{F}:=\\{\mathsf{F}(y):y\in{\mathcal{Y}}\\}$ is non-disturbed by
${\mathcal{I}}$, i.e,
$\mathsf{F}\subset{\mathcal{F}}({\mathcal{I}}^{*}_{\mathcal{X}})={\mathcal{F}}({\mathcal{I}}_{\mathrm{av}}^{*})$.
Given that
${\mathcal{F}}({\mathcal{I}}^{*}_{\mathrm{av},P})=P{\mathcal{F}}({\mathcal{I}}^{*}_{\mathrm{av}})P$
(see Proposition 6.1), this implies that
$P\mathsf{F}(y)P\in{\mathcal{F}}({\mathcal{I}}^{*}_{\mathrm{av},P})$. That
$\mathsf{F}$ is non-trivial implies that there must be a $y$ for which
$P\mathsf{F}(y)P$ is not proportional to $P$. If that were the case every
$P\mathsf{F}(y)P$ could be written as $P\mathsf{F}(y)P=c_{y}P$ with some
$c_{y}\geqslant 0$, which would imply that
$\mathsf{F}(y)={\mathcal{I}}^{*}_{\mathrm{av}}(\mathsf{F}(y))={\mathcal{I}}_{\mathrm{av}}^{*}(P\mathsf{F}(y)P)=c_{y}{\mathcal{I}}_{\mathrm{av}}^{*}(P)=c_{y}\mathds{1}_{\\!{\,{\mathcal{S}}}}$,
and so $\mathsf{F}$ would be a trivial observable. Therefore
${\mathcal{F}}({\mathcal{I}}^{*}_{\mathrm{av},P})=P{\mathcal{F}}({\mathcal{I}}^{*}_{\mathrm{av}})P$
is a nontrivial von Neumann algebra in
${\mathcal{L}}(P{\mathcal{H}_{\\!{\,{\mathcal{S}}}}})$, and there exists a
family of projections
$\mathsf{R}:=\\{\mathsf{R}(z):z\in{\mathcal{Z}}\\}\subset{\mathcal{F}}({\mathcal{I}}^{*}_{\mathrm{av},P})$
satisfying $\mathsf{R}(z)\mathsf{R}(y)=\delta_{z,y}\mathsf{R}(z)$ and
$\sum_{z}\mathsf{R}(z)=P$. We may consider $\mathsf{R}$ as a sharp observable
acting in $P{\mathcal{H}_{\\!{\,{\mathcal{S}}}}}$.

Using $\mathsf{R}$, we may define a (generally unsharp) observable
$\mathsf{G}$ acting in ${\mathcal{H}_{\\!{\,{\mathcal{S}}}}}$ by

$\displaystyle\mathsf{G}:=\\{\mathsf{G}(z)={\mathcal{I}}^{*}_{\mathrm{av}}(\mathsf{R}(z)):z\in{\mathcal{Z}}\\},$

where
$\sum_{z}\mathsf{G}(z)={\mathcal{I}}_{\mathrm{av}}^{*}(P)=\mathds{1}_{\\!{\,{\mathcal{S}}}}$.
Given that
${\mathcal{I}}_{\mathcal{X}}^{*}\circ{\mathcal{I}}_{\mathrm{av}}^{*}={\mathcal{I}}_{\mathrm{av}}^{*}$
(see Lemma 6.1), it follows that
$\mathsf{G}\subset{\mathcal{F}}({\mathcal{I}}_{\mathcal{X}}^{*})$, i.e.,
$\mathsf{G}$ is non-disturbed by ${\mathcal{I}}$. Moreover, since
$P\mathsf{G}(z)P={\mathcal{I}}^{*}_{\mathrm{av},P}(\mathsf{R}(z))=\mathsf{R}(z)$
holds, then each (non-zero) effect of $\mathsf{G}$ has at least one
eigenvector with eigenvalue $1$, and so $\|\mathsf{G}(z)\|=1$. Moreover, if
${\mathcal{F}}({\mathcal{I}}_{\mathcal{X}})$ contains a faithful state, then
$P=\mathds{1}_{\\!{\,{\mathcal{S}}}}$, and so we have
$\mathsf{G}(z)=P\mathsf{G}(z)P=\mathsf{R}(z)$, implying that
$\mathsf{G}\equiv\mathsf{R}$ is sharp. Now let us note that the family of
states $\\{\rho_{z}\\}$ are perfectly distinguishable given a measurement of
$\mathsf{G}$ if and only if $\rho_{z}=\mathsf{P}(z)\rho_{z}\mathsf{P}(z)$,
where $\mathsf{P}(z)\geqslant\mathsf{R}(z)$ projects onto the eigenvalue-1
eigenspace of $\mathsf{G}(z)$. In such a case it trivially holds that
$\mathrm{tr}[\mathsf{G}(y)\rho_{z}]=\delta_{z,y}$. But Since
$\mathsf{G}\subset{\mathcal{F}}({\mathcal{I}}^{*}_{\mathcal{X}})$, we also
have

$\displaystyle\mathrm{tr}[\mathsf{G}(y){\mathcal{I}}_{\mathcal{X}}(\rho_{z})]=\mathrm{tr}[{\mathcal{I}}_{\mathcal{X}}^{*}(\mathsf{G}(y))\rho_{z}]=\mathrm{tr}[\mathsf{G}(y)\rho_{z}]=\delta_{z,y},$

and so $\\{{\mathcal{I}}_{\mathcal{X}}(\rho_{z})\\}$ continues to be perfectly
distinguishable by a $\mathsf{G}$ measurement. ∎

In the special case where ${\mathcal{I}}$ is a measurement of the first kind,
we may strengthen the above result as follows:

######  Theorem 6.2.

Let ${\mathcal{I}}$ be an instrument compatible with a non-trivial observable
$\mathsf{E}$ acting in ${\mathcal{H}_{\\!{\,{\mathcal{S}}}}}$. If
${\mathcal{I}}$ is a measurement of the first kind, then $\mathsf{E}$ is
described by a classical post-processing of a norm-1 observable
$\mathsf{G}:=\\{\mathsf{G}(z):z\in{\mathcal{Z}}\\}$ with properties given in
Proposition 6.2, i.e.,

$\displaystyle\mathsf{E}(x)=\sum_{z}p(x|z)\mathsf{G}(z),$ (43)

where $\\{p(x|z)\\}$ is a family of non-negative numbers that satisfy
$\sum_{x}p(x|z)=1$ for each $z$.

###### Proof.

Assume that the $\mathsf{E}$-instrument ${\mathcal{I}}$ is a measurement of
the first kind, that is,
$\mathsf{E}\subset{\mathcal{F}}({\mathcal{I}}^{*}_{\mathcal{X}})$. It follows
that $P\mathsf{E}P\subset{\mathcal{F}}({\mathcal{I}}^{*}_{\mathrm{av},P})$. In
fact, we can show that
$P\mathsf{E}P\subset{\mathcal{F}}({\mathcal{I}}^{*}_{\mathrm{av},P})\cap{\mathcal{F}}({\mathcal{I}}^{*}_{\mathrm{av},P})^{\prime}$.
First, recall from Proposition 6.2 that if $\mathsf{E}$ is non-trivial then
there exists a family of projections
$\mathsf{R}:=\\{\mathsf{R}(z):z\in{\mathcal{Z}}\\}\subset{\mathcal{F}}({\mathcal{I}}^{*}_{\mathrm{av},P})$,
satisfying $\mathsf{R}(z)\mathsf{R}(y)=\delta_{z,y}\mathsf{R}(z)$ and
$\sum_{z}\mathsf{R}(z)=P$. If
$P\mathsf{E}P\not\subset{\mathcal{F}}({\mathcal{I}}^{*}_{\mathrm{av},P})\cap{\mathcal{F}}({\mathcal{I}}^{*}_{\mathrm{av},P})^{\prime}$,
then $\mathsf{R}$ can be chosen so that
$[P\mathsf{E}(x)P,\mathsf{R}(z)]\neq\mathds{O}$ for some $x$ and $z$. But note
that
$\mathsf{R}\subset{\mathcal{F}}({\mathcal{I}}^{*}_{\mathrm{av},P})=P{\mathcal{F}}({\mathcal{I}}^{*}_{\mathcal{X}})P$
implies that the observable
$\\{P{\mathcal{I}}_{x}(\mathsf{R}(z))P,P{\mathcal{I}}_{x}(\mathds{1}_{\\!{\,{\mathcal{S}}}}-\mathsf{R}(z))P:x\in{\mathcal{X}}\\}$
is a joint measurement for $P\mathsf{E}P$ and the sharp observable
$\\{\mathsf{R}(z),P-\mathsf{R}(z)\\}$. By compatibility, it follows that
$[P\mathsf{E}(x)P,\mathsf{R}(z)]=\mathds{O}$ must hold for all $x$ and $z$,
and so $P\mathsf{E}P$ must be contained in the Abelian algebra
${\mathcal{F}}({\mathcal{I}}^{*}_{\mathrm{av},P})\cap{\mathcal{F}}({\mathcal{I}}^{*}_{\mathrm{av},P})^{\prime}$.
Consequently,
$\mathsf{R}\subset{\mathcal{F}}({\mathcal{I}}^{*}_{\mathrm{av},P})\cap{\mathcal{F}}({\mathcal{I}}^{*}_{\mathrm{av},P})^{\prime}$
can be chosen so as to simultaneously diagonalise all $P\mathsf{E}(x)P$, that
is, we may write $P\mathsf{E}(x)P=\sum_{z}p(x|z)\mathsf{R}(z)$. Recalling that
$\mathsf{E}(x)={\mathcal{I}}_{\mathrm{av}}^{*}(\mathsf{E}(x))={\mathcal{I}}_{\mathrm{av}}^{*}(P\mathsf{E}(x)P)$,
then defining the observable $\mathsf{G}$ by
$\mathsf{G}(z)={\mathcal{I}}^{*}_{\mathrm{av}}(\mathsf{R}(z))$ gives us Eq.
(43). As in Proposition 6.2 it holds that
$\mathsf{G}\subset{\mathcal{F}}({\mathcal{I}}_{\mathcal{X}}^{*})$,
$\mathsf{G}$ is a norm-1 observable, if
${\mathcal{F}}({\mathcal{I}}_{\mathcal{X}})$ contains a faithful state then
$\mathsf{G}$ is also sharp, and if $\\{\rho_{z}\\}$ are perfectly
distinguishable by a $\mathsf{G}$ measurement then so are
$\\{{\mathcal{I}}_{\mathcal{X}}(\rho_{z})\\}$. ∎

Finally, we present the following implication of the above theorem:

######  Corollary 6.2.

Let ${\mathcal{I}}$ be an instrument compatible with a non-trivial observable
$\mathsf{E}$ acting in ${\mathcal{H}_{\\!{\,{\mathcal{S}}}}}$, and assume that
${\mathcal{I}}$ is a measurement of the first kind. For any outcome $x$
associated with a non-trivial effect $\mathsf{E}(x)$, and for any pair of unit
vectors $\psi,\phi\in{\mathcal{H}_{\\!{\,{\mathcal{S}}}}}$ satisfying
$\mathsf{E}(x)\psi=\|\mathsf{E}(x)\|\psi$ and
$(\mathds{1}_{\\!{\,{\mathcal{S}}}}-\mathsf{E}(x))\phi=\|\mathds{1}_{\\!{\,{\mathcal{S}}}}-\mathsf{E}(x)\|\phi$,
respectively, it holds that $\psi$ and $\phi$ are orthogonal, and that
$F({\mathcal{I}}_{\mathcal{X}}(P_{\psi}),{\mathcal{I}}_{\mathcal{X}}(P_{\phi}))=0$,
where $F(\rho,\sigma):=\mathrm{tr}[\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}}]^{2}$
is the fidelity between states $\rho$ and $\sigma$.

###### Proof.

For each outcome $x$, we may coarse-grain $\mathsf{E}$ into a binary
observable
$\\{\mathsf{E}(x),\mathsf{E}(\overline{x}):=\mathds{1}_{\\!{\,{\mathcal{S}}}}-\mathsf{E}(x)\\}$.
By Theorem 6.2, first-kindness implies that
$\mathsf{E}(x)=\sum_{z}p(x|z)\mathsf{G}(z)$ and
$\mathsf{E}(\overline{x})=\sum_{z}p(\overline{x}|z)\mathsf{G}(z)$, with
$p(\overline{x}|z):=1-p(x|z)$, where $\\{\mathsf{G}(z):z\in{\mathcal{Z}}\\}$
is a norm-1 observable with properties given in Proposition 6.2, while
$\\{p(x|z)\\}$ is a family of non-negative numbers satisfying
$\sum_{x}p(x|z)=1$ for each $z$. Now let us define
$p_{\max}(x):=\max\\{p(x|z)\\}$ and $p_{\min}(x):=\min\\{p(x|z)\\}$ as the
maximum and minimum values of the set $\\{p(x|z)\\}$. We may thus define the
sets $Z_{\max}:=\\{z\in{\mathcal{Z}}:p(x|z)=p_{\max}(x)\\}$ and
$Z_{\min}:=\\{z\in{\mathcal{Z}}:p(x|z)=p_{\min}(x)\\}$. Using such sets, we
may define $\mathsf{G}(Z_{\max}):=\sum_{z\in Z_{\max}}\mathsf{G}(z)$ and
$\mathsf{G}(Z_{\min}):=\sum_{z\in Z_{\min}}\mathsf{G}(z)$. Since $\mathsf{G}$
is norm-1, then we may also define $\mathsf{P}(Z_{\max}):=\sum_{z\in
Z_{\max}}\mathsf{P}(z)$ and $\mathsf{P}(Z_{\min}):=\sum_{z\in
Z_{\min}}\mathsf{P}(z)$, where $\mathsf{P}(z)$ is the projection onto the
eigenvalue-1 eigenspace of $\mathsf{G}(z)$. Since $\mathsf{E}(x)$ is assumed
to be non-trivial, then it must hold that $Z_{\max}\cap Z_{\min}=\emptyset$.
If this were not so, it would hold that all $p(x|z)$ are the same, in which
case $\mathsf{E}(x)\propto\mathds{1}_{\\!{\,{\mathcal{S}}}}$. Consequently,
$\mathsf{P}(Z_{\max})\mathsf{P}(Z_{\min})=\mathds{O}$.

Now let us note that

$\displaystyle\|\mathsf{E}(x)\|=\sup_{\|\psi\|=1}\langle\psi|\mathsf{E}(x)\psi\rangle=\sup_{\|\psi\|=1}\sum_{z}p(x|z)\langle\psi|\mathsf{G}(z)\psi\rangle.$

We may now show that a unit vector $\psi$ satisfies
$\mathsf{E}(x)\psi=\|\mathsf{E}(x)\|\psi$ if and only if
$\mathsf{P}(Z_{\max})\psi=\psi$. Let us first prove the only if statement. For
any unit vector $\psi$, it holds that
$\langle\psi|\mathsf{E}(x)\psi\rangle\leqslant p_{\max}(x)$, which follows
from the fact that $p(x|z)$ are positive numbers and that
$\\{\langle\psi|\mathsf{G}(z)\psi\rangle\\}$ is a probability distribution,
with the upper bound being saturated when
$\langle\psi|\mathsf{G}(Z_{\max})\psi\rangle=1$. But this in turn is satisfied
only if $\langle\psi|\mathsf{P}(Z_{\max})\psi\rangle=1$, in which case
$\mathsf{P}(Z_{\max})\psi=\psi$. As such, it follows that
$\|\mathsf{E}(x)\|=p_{\max}(x)$, and the unit vector $\psi$ satisfies
$\mathsf{E}(x)\psi=\|\mathsf{E}(x)\|\psi$ only if
$\mathsf{P}(Z_{\max})\psi=\psi$. The if statement is trivial.

By similar arguments as above, we may show that
$\|\mathds{1}_{\\!{\,{\mathcal{S}}}}-\mathsf{E}(x)\|=\|\mathsf{E}(\overline{x})\|=p_{\max}(\overline{x})=1-p_{\min}(x)$,
and that the unit vector $\phi$ satisfies
$(\mathds{1}_{\\!{\,{\mathcal{S}}}}-\mathsf{E}(x))\phi=\|\mathds{1}_{\\!{\,{\mathcal{S}}}}-\mathsf{E}(x)\|\phi$
if and only if $\mathsf{P}(Z_{\min})\phi=\phi$. Since $\mathsf{E}(x)$ is non-
trivial, then as argued above $\psi$ and $\phi$ are orthogonal, and perfectly
distinguishable by a $\mathsf{G}$ measurement. By Proposition 6.2 it holds
that ${\mathcal{I}}_{\mathcal{X}}(P_{\psi})$ and
${\mathcal{I}}_{\mathcal{X}}(P_{\phi})$ are also perfectly distinguishable by
a $\mathsf{G}$ measurement, that is,
$F({\mathcal{I}}_{\mathcal{X}}(P_{\psi}),{\mathcal{I}}_{\mathcal{X}}(P_{\phi}))=0$.

∎

###  6.4 Measurements of the first kind, distinguishability, and the Wigner-
Araki-Yanase theorem

We shall now use the results in the preceding section to obtain quantitative
bounds that complement those of Theorem 3.1 and Theorem 4.2, providing further
necessary conditions that must be fulfilled for an observable if it is to
admit a first-kind or repeatable measurement in the presence of a conservation
law. To this end, let us first provide a generalisation of Theorem 2 in Ref.
[27], which we shall use in the sequel:

######  Lemma 6.5.

Let
${\mathcal{M}}:=({\mathcal{H}_{\\!{\,{\mathcal{A}}}}},\xi,{\mathcal{E}},\mathsf{Z})$
be a measurement scheme for an $\mathsf{E}$-instrument ${\mathcal{I}}$ acting
in ${\mathcal{H}_{\\!{\,{\mathcal{S}}}}}$, and assume that ${\mathcal{E}}$
conserves an additive quantity
$N={N_{\\!{\,{\mathcal{S}}}}}\otimes\mathds{1}_{\\!{\,{\mathcal{A}}}}+\mathds{1}_{\\!{\,{\mathcal{S}}}}\otimes{N_{\\!{\,{\mathcal{A}}}}}$
on average. For any pair of orthogonal unit vectors
$\psi,\phi\in{\mathcal{H}_{\\!{\,{\mathcal{S}}}}}$, the following will hold:

$\displaystyle|\langle\psi|{N_{\\!{\,{\mathcal{S}}}}}\phi\rangle|\leqslant\|{N_{\\!{\,{\mathcal{A}}}}}\|F({\mathcal{I}}_{\mathcal{X}}(P_{\psi}),{\mathcal{I}}_{\mathcal{X}}(P_{\phi}))+\|{N_{\\!{\,{\mathcal{S}}}}}\|F(\Lambda(P_{\psi}),\Lambda(P_{\phi})),$
(44)

where $\Lambda$ is the conjugate channel to ${\mathcal{I}}_{\mathcal{X}}$
defined in Eq. (8), and $F(\rho,\sigma)$ is the fidelity between states $\rho$
and $\sigma$.

###### Proof.

Let us consider the augmented Hilbert space
${\mathcal{H}_{\\!{\,{\mathcal{A}}}}}\otimes{\mathcal{K}}$ so that
$\xi\in{\mathcal{S}}({\mathcal{H}_{\\!{\,{\mathcal{A}}}}})$ admits the
purification $\xi=\mathrm{tr}_{\\!{\,{\mathcal{K}}}}[P_{\varphi}]$, with the
unit vector
$\varphi\in{\mathcal{H}_{\\!{\,{\mathcal{A}}}}}\otimes{\mathcal{K}}$.
Moreover, if ${\mathcal{K}}$ is sufficiently large, then by Stinespring’s
dilation theorem the channel ${\mathcal{E}}^{*}$ can be expressed as
${\mathcal{E}}^{*}(A)=V^{*}(A\otimes\mathds{1}_{\\!{\,{\mathcal{K}}}})V$ for
all
$A\in{\mathcal{L}}({\mathcal{H}_{\\!{\,{\mathcal{S}}}}}\otimes{\mathcal{H}_{\\!{\,{\mathcal{A}}}}})$,
where
$V:{\mathcal{H}_{\\!{\,{\mathcal{S}}}}}\otimes{\mathcal{H}_{\\!{\,{\mathcal{A}}}}}\to{\mathcal{H}_{\\!{\,{\mathcal{S}}}}}\otimes{\mathcal{H}_{\\!{\,{\mathcal{A}}}}}\otimes{\mathcal{K}}$
is an isometry. By additivity of $N$, and orthogonality of $\psi,\phi$, we
have

$\displaystyle\langle\psi\otimes\varphi|N\phi\otimes\varphi\rangle=\langle\psi|{N_{\\!{\,{\mathcal{S}}}}}\phi\rangle.$

On the other hand, average conservation of $N$ by ${\mathcal{E}}$ implies that

$\displaystyle
N={\mathcal{E}}^{*}(N)=V^{*}({N_{\\!{\,{\mathcal{S}}}}}\otimes\mathds{1}_{\\!{\,{\mathcal{A}}}}\otimes\mathds{1}_{\\!{\,{\mathcal{K}}}})V+V^{*}(\mathds{1}_{\\!{\,{\mathcal{S}}}}\otimes{N_{\\!{\,{\mathcal{A}}}}}\otimes\mathds{1}_{\\!{\,{\mathcal{K}}}})V.$

We therefore have

$\displaystyle|\langle\psi|{N_{\\!{\,{\mathcal{S}}}}}\phi\rangle|\leqslant|\langle\psi\otimes\varphi|V^{*}({N_{\\!{\,{\mathcal{S}}}}}\otimes\mathds{1}_{\\!{\,{\mathcal{A}}}}\otimes\mathds{1}_{\\!{\,{\mathcal{K}}}})V\phi\otimes\varphi\rangle|+|\langle\psi\otimes\varphi|V^{*}(\mathds{1}_{\\!{\,{\mathcal{S}}}}\otimes{N_{\\!{\,{\mathcal{A}}}}}\otimes\mathds{1}_{\\!{\,{\mathcal{K}}}})V\phi\otimes\varphi\rangle|.$
(45)

For any observable $\mathsf{G}:=\\{\mathsf{G}(z)\\}$ acting in
${\mathcal{H}_{\\!{\,{\mathcal{A}}}}}$, we may write

$\displaystyle|\langle\psi\otimes\varphi|V^{*}({N_{\\!{\,{\mathcal{S}}}}}\otimes\mathds{1}_{\\!{\,{\mathcal{A}}}}\otimes\mathds{1}_{\\!{\,{\mathcal{K}}}})V\phi\otimes\varphi\rangle|$
$\displaystyle\qquad=|\sum_{z}\langle\psi\otimes\varphi|V^{*}({N_{\\!{\,{\mathcal{S}}}}}\otimes\mathsf{G}(z)\otimes\mathds{1}_{\\!{\,{\mathcal{K}}}})V\phi\otimes\varphi\rangle|$
$\displaystyle\qquad\leqslant\|{N_{\\!{\,{\mathcal{S}}}}}\|\sum_{z}|\langle\psi\otimes\varphi|V^{*}(\mathds{1}_{\\!{\,{\mathcal{S}}}}\otimes\mathsf{G}(z)\otimes\mathds{1}_{\\!{\,{\mathcal{K}}}})V\phi\otimes\varphi\rangle|$
$\displaystyle\qquad\leqslant\|{N_{\\!{\,{\mathcal{S}}}}}\|\sum_{z}|\langle\psi\otimes\varphi|V^{*}(\mathds{1}_{\\!{\,{\mathcal{S}}}}\otimes\mathsf{G}(z)\otimes\mathds{1}_{\\!{\,{\mathcal{K}}}})V\psi\otimes\varphi\rangle|^{\frac{1}{2}}|\langle\phi\otimes\varphi|V^{*}(\mathds{1}_{\\!{\,{\mathcal{S}}}}\otimes\mathsf{G}(z)\otimes\mathds{1}_{\\!{\,{\mathcal{K}}}})V\phi\otimes\varphi\rangle|^{\frac{1}{2}}$
$\displaystyle\qquad=\|{N_{\\!{\,{\mathcal{S}}}}}\|\sum_{z}\mathrm{tr}[\mathds{1}_{\\!{\,{\mathcal{S}}}}\otimes\mathsf{G}(z){\mathcal{E}}(P_{\psi}\otimes\xi)]^{\frac{1}{2}}\mathrm{tr}[\mathds{1}_{\\!{\,{\mathcal{S}}}}\otimes\mathsf{G}(z){\mathcal{E}}(P_{\phi}\otimes\xi)]^{\frac{1}{2}}$
$\displaystyle\qquad=\|{N_{\\!{\,{\mathcal{S}}}}}\|\sum_{z}\mathrm{tr}[\mathsf{G}(z)\Lambda(P_{\psi})]^{\frac{1}{2}}\mathrm{tr}[\mathsf{G}(z)\Lambda(P_{\phi})]^{\frac{1}{2}}.$

In the third line we have used the Cauchy-Schwarz inequality, in the fourth
line we used Stinespring’s dilation theorem together with the fact that
$\varphi$ is a purification of $\xi$, and in the final line we use the
definitions of the partial trace and the conjugate channel $\Lambda$. Now,
note that the fidelity satisfies
$F(\rho,\sigma)=\min_{\mathsf{G}}\sum_{z}\mathrm{tr}[\mathsf{G}(z)\rho]^{\frac{1}{2}}\mathrm{tr}[\mathsf{G}(z)\sigma]^{\frac{1}{2}}$.
Therefore, choosing $\mathsf{G}$ so as to obtain the fidelity, we have

$\displaystyle|\langle\psi\otimes\varphi|V^{*}({N_{\\!{\,{\mathcal{S}}}}}\otimes\mathds{1}_{\\!{\,{\mathcal{A}}}}\otimes\mathds{1}_{\\!{\,{\mathcal{K}}}})V\phi\otimes\varphi\rangle|\leqslant\|{N_{\\!{\,{\mathcal{S}}}}}\|F(\Lambda(P_{\psi}),\Lambda(P_{\phi})).$

Using similar steps, we may also write

$\displaystyle|\langle\psi\otimes\varphi|V^{*}(\mathds{1}_{\\!{\,{\mathcal{S}}}}\otimes{N_{\\!{\,{\mathcal{A}}}}}\otimes\mathds{1}_{\\!{\,{\mathcal{K}}}})V\phi\otimes\varphi\rangle|\leqslant\|{N_{\\!{\,{\mathcal{A}}}}}\|F({\mathcal{I}}_{\mathcal{X}}(P_{\psi}),{\mathcal{I}}_{\mathcal{X}}(P_{\phi})).$

By Eq. (45), we thus obtain Eq. (44). ∎

We are now ready to prove our main result in this section:

######  Theorem 6.3.

Consider a measurement scheme
${\mathcal{M}}:=({\mathcal{H}_{\\!{\,{\mathcal{A}}}}},\xi,{\mathcal{E}},\mathsf{Z})$
for a nontrivial observable $\mathsf{E}$ with the instrument ${\mathcal{I}}$
acting in ${\mathcal{H}_{\\!{\,{\mathcal{S}}}}}$. Assume that ${\mathcal{I}}$
is a measurement of the first kind, and that ${\mathcal{E}}$ conserves an
additive quantity
$N=N_{\\!{\,{\mathcal{S}}}}\otimes\mathds{1}_{\\!{\,{\mathcal{A}}}}+\mathds{1}_{\\!{\,{\mathcal{S}}}}\otimes
N_{\\!{\,{\mathcal{A}}}}$ on average. For each outcome $x$ associated with a
non-trivial effect $\mathsf{E}(x)$, let $\mathcal{K}_{\max}(x)$ and
$\mathcal{K}_{\min}(x)$ be subspaces of ${\mathcal{H}_{\\!{\,{\mathcal{S}}}}}$
defined by

$\displaystyle\mathcal{K}_{\max}(x):=\\{\psi\in{\mathcal{H}_{\\!{\,{\mathcal{S}}}}}:\
\mathsf{E}(x)\psi=\|\mathsf{E}(x)\|\psi\\},$
$\displaystyle\mathcal{K}_{\min}(x):=\\{\phi\in{\mathcal{H}_{\\!{\,{\mathcal{S}}}}}:\
(\mathds{1}_{\\!{\,{\mathcal{S}}}}-\mathsf{E}(x))\phi=\|\mathds{1}_{\\!{\,{\mathcal{S}}}}-\mathsf{E}(x)\|\phi\\}.$

For all unit vectors $\psi\in\mathcal{K}_{\max}(x)$ and
$\phi\in\mathcal{K}_{\min}(x)$, it holds that

$\displaystyle|\langle\psi|{N_{\\!{\,{\mathcal{S}}}}}\phi\rangle|\leqslant\|{N_{\\!{\,{\mathcal{S}}}}}\|\left(\|\mathsf{E}(x)\|^{\frac{1}{2}}(1-\|\mathds{1}_{\\!{\,{\mathcal{S}}}}-\mathsf{E}(x)\|)^{\frac{1}{2}}+(1-\|\mathsf{E}(x)\|)^{\frac{1}{2}}\|\mathds{1}_{\\!{\,{\mathcal{S}}}}-\mathsf{E}(x)\|^{\frac{1}{2}}\right).$
(46)

###### Proof.

For each outcome $x$ associated with a non-trivial effect $\mathsf{E}(x)$, we
may coarse-grain $\mathsf{E}$ into a binary observable
$\\{\mathsf{E}(x),\mathsf{E}(\overline{x}):=\mathds{1}_{\\!{\,{\mathcal{S}}}}-\mathsf{E}(x)\\}$.
By Corollary 6.2, it holds that for any unit vectors
$\psi\in\mathcal{K}_{\max}(x)$ and $\phi\in\mathcal{K}_{\min}(x)$, $\psi$ and
$\phi$ are orthogonal, and
$F({\mathcal{I}}_{\mathcal{X}}(P_{\psi}),{\mathcal{I}}_{\mathcal{X}}(P_{\phi}))=0$.
As such, given the average conservation of $N$ by the interaction channel
${\mathcal{E}}$, Lemma 6.5 implies that the following inequality must hold:

$\displaystyle|\langle\psi|{N_{\\!{\,{\mathcal{S}}}}}\phi\rangle|$
$\displaystyle\leqslant\|{N_{\\!{\,{\mathcal{S}}}}}\|F\left(\Lambda(P_{\psi}),\Lambda(P_{\phi})\right)$
$\displaystyle\leqslant\|{N_{\\!{\,{\mathcal{S}}}}}\|\sum_{a=x,\overline{x}}\mathrm{tr}[\mathsf{Z}(x)\Lambda(P_{\psi})]^{\frac{1}{2}}\mathrm{tr}[\mathsf{Z}(x)\Lambda(P_{\phi})]^{\frac{1}{2}}$
$\displaystyle=\|{N_{\\!{\,{\mathcal{S}}}}}\|\sum_{a=x,\overline{x}}\langle\psi|\mathsf{E}(a)\psi\rangle^{\frac{1}{2}}\langle\phi|\mathsf{E}(a)\phi\rangle^{\frac{1}{2}}$
$\displaystyle=\|{N_{\\!{\,{\mathcal{S}}}}}\|\left(\|\mathsf{E}(x)\|^{\frac{1}{2}}(1-\|\mathds{1}_{\\!{\,{\mathcal{S}}}}-\mathsf{E}(x)\|)^{\frac{1}{2}}+(1-\|\mathsf{E}(x)\|)^{\frac{1}{2}}\|\mathds{1}_{\\!{\,{\mathcal{S}}}}-\mathsf{E}(x)\|^{\frac{1}{2}}\right).$

The second line uses the fact that for any states $\rho,\sigma$, it holds that
$F(\rho,\sigma)\leqslant\sum_{a}\mathrm{tr}[\mathsf{F}(a)\rho]^{\frac{1}{2}}\mathrm{tr}[\mathsf{F}(a)\sigma]^{\frac{1}{2}}$
for any observable $\mathsf{F}$. The third line uses the fact that $\Lambda$
is the conjugate channel to ${\mathcal{I}}_{\mathcal{X}}$ defined in Eq. (8),
and so it holds that
$\mathrm{tr}[\mathsf{Z}(a)\Lambda(\rho)]=\mathrm{tr}[\mathsf{E}(a)\rho]$ for
all $\rho$ and $a=x,\overline{x}$. To see how the final line is obtained, note
that we have $\langle\psi|\mathsf{E}(x)\psi\rangle=\|\mathsf{E}(x)\|$ and
$\langle\phi|(\mathds{1}_{\\!{\,{\mathcal{S}}}}-\mathsf{E}(x))\phi\rangle=\|\mathds{1}_{\\!{\,{\mathcal{S}}}}-\mathsf{E}(x)\|$
by construction. For the first term, i.e., $a=x$, we obtain
$\langle\psi|\mathsf{E}(a)\psi\rangle=\|\mathsf{E}(x)\|$ and
$\langle\phi|\mathsf{E}(a)\phi\rangle=\langle\phi|(\mathds{1}_{\\!{\,{\mathcal{S}}}}-(\mathds{1}_{\\!{\,{\mathcal{S}}}}-\mathsf{E}(x)))\phi\rangle=1-\langle\phi|(\mathds{1}_{\\!{\,{\mathcal{S}}}}-\mathsf{E}(x))\phi\rangle=1-\|\mathds{1}_{\\!{\,{\mathcal{S}}}}-\mathsf{E}(x)\|$.
The second term for $a=\overline{x}$ is obtained in a similar manner.

∎

Let us note that if $\mathsf{E}$ commutes with ${N_{\\!{\,{\mathcal{S}}}}}$,
then Theorem 6.3 imposes no restrictions on first-kindness. To see this, let
us note that for any (possibly trivial) effect $\mathsf{E}(x)$, and for any
unit vectors $\psi,\phi$ satisfying $\mathsf{E}(x)\psi=\|\mathsf{E}(x)\|\psi$
and
$(\mathds{1}_{\\!{\,{\mathcal{S}}}}-\mathsf{E}(x))\phi=\|\mathds{1}_{\\!{\,{\mathcal{S}}}}-\mathsf{E}(x)\|\phi$,
it holds that
$\langle\psi|\mathsf{E}(x){N_{\\!{\,{\mathcal{S}}}}}\phi\rangle=\|\mathsf{E}(x)\|\langle\psi|{N_{\\!{\,{\mathcal{S}}}}}\phi\rangle$
and
$\langle\psi|{N_{\\!{\,{\mathcal{S}}}}}\mathsf{E}(x)\phi\rangle=(1-\|\mathds{1}_{\\!{\,{\mathcal{S}}}}-\mathsf{E}(x)\|)\langle\psi|{N_{\\!{\,{\mathcal{S}}}}}\phi\rangle$.
It follows that if $[\mathsf{E}(x),{N_{\\!{\,{\mathcal{S}}}}}]=\mathds{O}$
then either (i) $\langle\psi|{N_{\\!{\,{\mathcal{S}}}}}\phi\rangle=0$, or (ii)
$\|\mathsf{E}(x)\|+\|\mathds{1}_{\\!{\,{\mathcal{S}}}}-\mathsf{E}(x)\|=1$.
Condition (i) implies that the lower bound of Eq. (46) vanishes, and so no
constraint is imposed. On the other hand, condition (ii) implies that
$\|\mathsf{E}(x)\|+\|\mathds{1}_{\\!{\,{\mathcal{S}}}}-\mathsf{E}(x)\|=1+(p_{\max}(x)-p_{\min}(x))=1$,
where we recall from Corollary 6.2 that $p_{\max}(x)=\|\mathsf{E}(x)\|$ and
$p_{\min}(x)=1-\|\mathds{1}_{\\!{\,{\mathcal{S}}}}-\mathsf{E}(x)\|$ are the
largest and smallest values from the set $\\{p(x|z)\\}$ given by Theorem 6.2.
Such equality is satisfied if and only if $p_{\max}(x)=p_{\min}(x)=\lambda$,
in which case by Eq. (43) it follows that
$\mathsf{E}(x)=\lambda\mathds{1}_{\\!{\,{\mathcal{S}}}}$ is a trivial effect.
But Eq. (46) only applies to non-trivial effects, and so no constraints are
imposed in such a case.

On the other hand, if for any non-trivial $\mathsf{E}(x)$ not commuting with
${N_{\\!{\,{\mathcal{S}}}}}$ it holds that
$\langle\psi|{N_{\\!{\,{\mathcal{S}}}}}\phi\rangle\neq 0$ for some
$\psi\in\mathcal{K}_{\max}(x)$ and $\phi\in\mathcal{K}_{\min}(x)$, then
$\mathsf{E}$ admits a first-kind measurement only if either (a)
$\|\mathsf{E}(x)\|<1$ or (b)
$\|\mathds{1}_{\\!{\,{\mathcal{S}}}}-\mathsf{E}(x)\|<1$. This is so because if
both (a) and (b) are violated, i.e., if
$\|\mathsf{E}(x)\|=\|\mathds{1}_{\\!{\,{\mathcal{S}}}}-\mathsf{E}(x)\|=1$,
then the upper bound of Eq. (46) vanishes. Such an $\mathsf{E}$ admits a
first-kind measurement only if at most one effect has $1$ as an eigenvalue.
Equivalently, if some effect $\mathsf{E}(x)$ has $1$ as an eigenvalue, then it
cannot also have eigenvalue $0$. This implies that such an observable must be
highly unsharp. Moreover, we obtain the following result as an immediate
consequence:

######  Corollary 6.3.

Consider a measurement scheme
${\mathcal{M}}:=({\mathcal{H}_{\\!{\,{\mathcal{A}}}}},\xi,{\mathcal{E}},\mathsf{Z})$
for an $\mathsf{E}$-instrument ${\mathcal{I}}$ acting in
${\mathcal{H}_{\\!{\,{\mathcal{S}}}}}$, and assume that ${\mathcal{E}}$
conserves an additive quantity
$N=N_{\\!{\,{\mathcal{S}}}}\otimes\mathds{1}_{\\!{\,{\mathcal{A}}}}+\mathds{1}_{\\!{\,{\mathcal{S}}}}\otimes
N_{\\!{\,{\mathcal{A}}}}$ on average. If ${\mathcal{I}}$ is repeatable, then

$\displaystyle[\mathsf{E},\mathsf{P}({\mathcal{X}}){N_{\\!{\,{\mathcal{S}}}}}\mathsf{P}({\mathcal{X}})]=\mathds{O},$

where we define
$\mathsf{P}({\mathcal{X}}):=\sum_{x\in{\mathcal{X}}}\mathsf{P}(x)$, with
$\mathsf{P}(x)$ the projection onto the eigenvalue-1 eigenspace of
$\mathsf{E}(x)$.

###### Proof.

By Proposition 4.1, repeatability implies that $\mathsf{E}$ is a norm-1
observable, with $\\{\mathsf{P}(x)\\}$ mutually orthogonal projection
operators that project onto the eigenvalue-1 eigenspaces of $\mathsf{E}$. For
every $x$, define
$\mathsf{P}(x)^{\perp}:=\mathsf{P}({\mathcal{X}})-\mathsf{P}(x)$. It is
trivial to verify that
$\operatorname{supp}(\mathsf{P}(x))\equiv{\mathcal{K}}_{\max}(x)$ and
$\operatorname{supp}(\mathsf{P}(x)^{\perp})\equiv{\mathcal{K}}_{\min}(x)$ as
defined in Theorem 6.3. Any
$\varphi\in\operatorname{supp}(\mathsf{P}({\mathcal{X}}))$ may be written as
$\varphi=\alpha\psi+\beta\phi$, where $\psi\in{\mathcal{K}}_{\max}(x)$ and
$\phi\in{\mathcal{K}}_{\min}(x)$ are unit vectors and
$\alpha,\beta\in\mathds{C}$. Since repeatability implies first-kindness, and
$\|\mathsf{E}(x)\|=1$ for all $x$, then by Theorem 6.3 it follows that
$\langle\varphi|\mathsf{P}(x){N_{\\!{\,{\mathcal{S}}}}}\mathsf{P}(x)^{\perp}\varphi\rangle=\alpha^{*}\beta\langle\psi|{N_{\\!{\,{\mathcal{S}}}}}\phi\rangle=0$
for all $\varphi\in\operatorname{supp}(\mathsf{P}({\mathcal{X}}))$. It follows
that
$\mathsf{P}(x){N_{\\!{\,{\mathcal{S}}}}}\mathsf{P}(x)^{\perp}=\mathds{O}$,
which implies that
$\mathsf{P}(x){N_{\\!{\,{\mathcal{S}}}}}\mathsf{P}({\mathcal{X}})=\mathsf{P}(x){N_{\\!{\,{\mathcal{S}}}}}\mathsf{P}(x)$.
Since the right hand side is self adjoint, and
$\mathsf{P}(x)=\mathsf{P}(x)\mathsf{P}({\mathcal{X}})=\mathsf{P}({\mathcal{X}})\mathsf{P}(x)$,
it follows that
$[\mathsf{P}(x),\mathsf{P}({\mathcal{X}}){N_{\\!{\,{\mathcal{S}}}}}\mathsf{P}({\mathcal{X}})]=\mathds{O}$.
But since
$\mathsf{P}(x)=\mathsf{E}(x)\mathsf{P}({\mathcal{X}})=\mathsf{P}({\mathcal{X}})\mathsf{E}(x)$,
we have
$[\mathsf{E}(x),\mathsf{P}({\mathcal{X}}){N_{\\!{\,{\mathcal{S}}}}}\mathsf{P}({\mathcal{X}})]=\mathds{O}$.
This completes the proof. ∎

If $\mathsf{E}$ is sharp, then
$\mathsf{P}({\mathcal{X}})=\mathds{1}_{\\!{\,{\mathcal{S}}}}$, in which case
Corollary 6.3 reduces to the repeatability part of the original WAY theorem.
When $\mathsf{E}$ is unsharp, however, so that
$\mathsf{P}({\mathcal{X}})<\mathds{1}_{\\!{\,{\mathcal{S}}}}$, then the above
result can be seen as a strengthening of our generalisation of WAY in Theorem
4.2, in the finite-dimensional setting, providing further necessary conditions
for repeatability beyond unsharpness and coherence. Moreover, note that the
condition for repeatability given by Theorem 6.1 was that $P\mathsf{E}P$ must
commute with $P{N_{\\!{\,{\mathcal{S}}}}}P$, where $P$ is the minimal support
projection on the fixed points of the $\mathsf{E}$-channel
${\mathcal{I}}_{\mathcal{X}}$. Clearly, such $P$ is contingent on the specific
repeatable instrument that we are considering, whereas the condition given in
Corollary 6.3 depends only on the observable, and is hence much more general.
For example, consider the case where for each outcome $x$, the eigenvalue-1
eigenspace of $\mathsf{E}(x)$ has dimension larger than $1$. We may construct
a repeatable instrument with operations satisfying
${\mathcal{I}}_{x}(\rho)=\mathrm{tr}[\mathsf{E}(x)\rho]P_{\psi_{x}}$ for all
states $\rho$ and outcomes $x$, with $\psi_{x}$ some specific eigenvalue-1
eigenstate of $\mathsf{E}(x)$. It is simple to verify that, in such a case,
$P=\sum_{x}P_{\psi_{x}}<\mathsf{P}({\mathcal{X}})$. Since
$P\mathsf{E}(x)P=P_{\psi_{x}}$ and
$P{N_{\\!{\,{\mathcal{S}}}}}P=\sum_{x,y}\langle\psi_{x}|{N_{\\!{\,{\mathcal{S}}}}}\psi_{y}\rangle|\psi_{x}\rangle\langle\psi_{y}|$,
the only necessary condition given by Theorem 6.1 will be that
$\langle\psi_{x}|{N_{\\!{\,{\mathcal{S}}}}}\psi_{y}\rangle=0$ must hold for
all $x\neq y$, which is clearly much weaker than the conditions given by
Theorem 6.3 and hence Corollary 6.3.

##  7 Conclusions

We have provided a number of general and operational bounds which capture
measurement disturbance, with emphasis on the setting in which there is a
conservation law. We obtained a new, quantitative version of the WAY theorem,
which generalises previous work in several respects, going beyond normal
measurement schemes and not assuming that the observable to be measured is
sharp. We saw that the large apparatus coherence played a key role for
measurability and non-disturbance, pointing to the requirement of “large”
apparatus. This points further to possible deep connections between the WAY
theorem and the rapidly developing theory of quantum reference frames,
analysed so far only when the conserved quantity has a conjugate phase [42,
41]. The work presented surrounding the WAY theorem was also studied in the
novel setting of sequential measurements for general pairs of observables. The
quantitative bounds are further refined by the analysis of the fixed point
structure of the measurement channel in settings which have received scant
attention. Such analysis indicates that, though necessary, large coherence in
the conserved quantity is not sufficient for non-disturbance. We provide
further quantitative bounds indicating scenarios in which non-disturbance
cannot be achieved irrespective of the apparatus coherence.

Our work suffers from the drawback that many physically arising conserved
quantities are unbounded; this case should also be systematically studied.
This is a technically challenging endeavour and we save it for future work.

###### Acknowledgements.

M.H.M. acknowledges funding from the European Union’s Horizon 2020 research
and innovation program under the Marie Skłodowska-Curie grant agreement No
801505, as well as from the Slovak Academy of Sciences under MoRePro project
OPEQ (19MRP0027). T.M. acknowledges financial support from JSPS KAKENHI Grant
No.JP20K03732.

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