# Parameter estimation via indefinite causal structures

Lorenzo M. Procopio  Weizmann Institute of Science, Rehovot 7610001, Israel
<EMAIL_ADDRESS>
###### Abstract

Quantum Fisher information is the principal tool used to give the ultimate
precision bound on the estimation of parameters for quantum channels. In this
work, we present analytical expressions for the quantum Fisher information
with three noisy channels for the case where the channels are in superposition
of causal orders. We found that the quantum Fisher information increases as
the number of causal orders increases for certain combinations. We also show
that certain combinations of causal orders attain higher precision on bounds
than others for the same number of causal orders. Based on our results, we
chose the best combinations of causal orders with three channels for probing
schemes using indefinite causal structures.

##  1 Introduction

In the field of quantum metrology, the quantum channel identification problem
attempts to estimate the true value of parameters from quantum channels [1].
The main task is to find the best strategy to efficiently determine the
parameters of a channel with a minimum of resources while maintaining the
desired precision. Initially, quantum entanglement was proposed as a strategy
to estimate the isotropic depolarization parameter within a noisy channel [1].
Other strategies, such as unentangled probes, have been also proposed for
channel identification [2]. Quantum Fisher information is a metric used to
learn about the value of parameters of quantum channels. To assess and compare
different probing strategies, the quantum Fisher information is used as a
figure of merit [3]. In all the above strategies, the order of the application
of channels are in a definite causal order, in which the connections between
channels are classically connected.

A new paradigm has recently been proposed in the field of quantum
communications according to which the connections between quantum channels can
be in a superposition of trajectories in space [4] or time [5]. By placing the
quantum channels in a superposition of trajectories, one obtains an indefinite
causal structure known as the quantum switch, which has been shown to be
useful for new applications in quantum discrimination of channels [6], quantum
computation [7], quantum communication complexity [8], quantum metrology [9],
and quantum thermodynamics [10]. These theoretical investigations have motived
experimental demonstrations with two channels [11, 12, 13, 14]. An additional
experiment recently built the quantum switch with more than two channels [15].

Following the above investigations, the quantum switch was proposed as a new
probing strategy for parameter estimation of quantum channels [16]. This
strategy consists of placing copies of two or $N$ channels in indefinite
causal order limited to only two different causal orders. Refs. [9, 17] also
showed a gain of the quantum Fisher information by using superposition of
causal orders. However, the studies in Refs. [9, 17] were also limited to two
causal orders. An open question that remains is where there is additional gain
in the quantum Fisher information when the number of causal orders is
increased. In this work, we address this question by exploring the quantum
3-switch [18] and performing fine control on selected combinations of causal
orders. Such fine control was not accessible to the two-causal order case
studied previously [19]. We found that the quantum Fisher information
increases as the number of causal orders increases for certain combinations of
causal orders within specific regions of noise. We also found that certain
combinations of causal orders are less efficient for increasing quantum Fisher
information for the same number of causal orders. Based on our results, we
chose the best combinations of causal orders with three channels for probing
strategies using indefinite causal structures.

The structure of the paper is as follows. In section 2, we review the
mathematical structure of the quantum switch with three noisy channels. Then
in section 3 we review the basics of the quantum Fisher information and
explain the procedure to calculate the quantum Fisher information for all
combinations of alternatives orders with three channels. Section 4 presents
the analytical expressions for the quantum Fisher information for the number
of causal orders. Section 5 presents the discussion of our results. Finally,
Section 6 sums up our conclusions.

##  2 The Quantum 3-Switch

In Ref. [18], it was introduced the quantum switch with three quantum channels
${\mathcal{N}}_{1}$, ${\mathcal{N}}_{2}$ and ${\mathcal{N}}_{3}$ can be in
$\sum_{m=2}^{N!}\binom{N!}{m}$ superpositions of $m$ causal orders with $N=3$,
where $\binom{n}{r}=\frac{n!}{r!(n-r)!}$ is the binomial coefficient. A noisy
channel $\mathcal{N}$ on a qudit ($d$-dimensional) input state $\sigma$ can be
modeled as

$\mathcal{N}(\sigma)=\theta\sigma+(1-\theta){\tr}[\sigma]\frac{\mathbb{1}}{d},$
(1)

where $\theta$ is the depolorazing parameter and $\mathbb{1}$ is the identity
operator, which represents a maximally mixed state. We use the Kraus
decomposition $\mathcal{N}(\sigma)=\sum_{i}K_{i}\sigma K_{i}^{\dagger}$ to
mathematically represent the action of a channel ${\mathcal{N}}$ on the
quantum state $\sigma$ such that
$\sum_{i}K_{i}K_{i}^{\dagger}=\mathbb{1}_{t}$. For three noisy channels
$\mathcal{N}_{1}$, $\mathcal{N}_{2}$ and $\mathcal{N}_{3}$, the control system
$\sigma_{c}=\left|\psi_{c}\right>\left<\psi_{c}\right|$ coherently controls
$\sigma$. The quantum state of the control system writes

$\displaystyle\left|\psi_{c}\right>$
$\displaystyle=\sum_{k=1}^{6}\sqrt{p_{k}}\left|k\right>,$ (2)

with $k\in[\\![1;6!]\\!]$ and $p_{k}$ is the probability associated one
combination of orders as in [19] such that $\sum_{k=1}^{6}p_{k}=1$. Setting
the control state as in equation (2), yields a superposition of several causal
orders with their respective weights $p_{k}$. To select specific combinations
of causal orders, one only needs to fix specific values to the $p_{k}$. If the
Kraus operators of the channels $\mathcal{N}_{1}$, $\mathcal{N}_{2}$ and
$\mathcal{N}_{3}$ are $\\{K_{i}^{(1)}\\}$, $\\{K_{j}^{(2)}\\}$ and
$\\{K_{k}^{(3)}\\}$ respectively, then the Kraus operators $\mathcal{K}_{ijk}$
of the full quantum 3-switch channel is

$\displaystyle\mathcal{K}_{ijk}=\sum_{k=1}^{6}\pi_{k}(K_{i}^{(1)}K_{j}^{(2)}K_{k}^{(3)})\left|k\right>\left<k\right|,$
(3)

where $\pi_{n}$ is a permutation of the symmetric group
$S_{6}=\\{\pi_{k}|k\in[\\![1;6!]\\!]\\}$. The action of the quantum $3$-switch
${\mathcal{S}}({\mathcal{N}}_{1},{\mathcal{N}}_{2},{\mathcal{N}}_{3})$ over an
input $\sigma\otimes\sigma_{c}$ can be identified as the output state
$\rho_{{\rm
out}}(\theta_{1},\theta_{2},\theta_{3})=\mathcal{S}({\mathcal{N}}_{1},{\mathcal{N}}_{2},{\mathcal{N}}_{3})\left(\sigma\otimes\sigma_{c}\right)$
of the quantum 3-switch and can be expressed through the Kraus operators
$\mathcal{K}_{ijk}$ as

$\rho_{{\rm
out}}(\theta_{1},\theta_{2},\theta_{3})=\sum_{ijk}\mathcal{K}_{ijk}\left(\sigma\otimes\sigma_{c}\right.)\mathcal{K}_{ijk}^{\dagger}.$
(4)

Following the procedure described in Ref. [18], we found that the output of
the quantum 3-switch is a $6\times 6$ block-symmetry matrix. For sake of
simplicity, we study the one-parameter case, that is, we take three copies of
the same depolarizing channel $\mathcal{N}(\sigma)$, i.e.
$\theta_{1}=\theta_{2}=\theta_{3}=\theta$, thus for this case the quantum
3-switch output can be written as

$\begin{array}[]{lll}{\rho_{\rm
out}^{(m)}}(\theta)=\frac{1}{m}\left(\begin{array}[]{cccccc}\epsilon_{11}{A}&\epsilon_{12}{B}&\epsilon_{13}{B}&\epsilon_{14}{D}&\epsilon_{15}{D}&\epsilon_{16}{F}\\\
\epsilon_{12}{B}&\epsilon_{22}{A}&\epsilon_{23}{D}&\epsilon_{24}{F}&\epsilon_{25}{B}&\epsilon_{26}{D}\\\
\epsilon_{13}{B}&\epsilon_{23}{D}&\epsilon_{33}{A}&\epsilon_{34}{B}&\epsilon_{35}{F}&\epsilon_{36}{D}\\\
\epsilon_{14}{D}&\epsilon_{24}{F}&\epsilon_{34}{B}&\epsilon_{44}{A}&\epsilon_{45}{D}&\epsilon_{46}{B}\\\
\epsilon_{15}{D}&\epsilon_{25}{B}&\epsilon_{35}{F}&\epsilon_{45}{D}&\epsilon_{55}{A}&\epsilon_{56}{B}\\\
\epsilon_{16}{F}&\epsilon_{26}{D}&\epsilon_{36}{D}&\epsilon_{46}{B}&\epsilon_{56}{B}&\epsilon_{66}{A}\end{array}\right),\end{array}$
(5)

where we have introduced the parameter $\epsilon_{ij}=m\sqrt{p_{i}p_{j}}$,
with $p_{i}=1/m$, such that $0\leq\epsilon_{ij}\leq 1$, which reflects the
degree of superposition between two definite causal orders determined by the
probabilities $p_{i}$ and $p_{j}$, and the parameter $m$ stands for the number
of causal orders involved in the superposition, with $m=1,\dots,3!$. The
matrices $A,B,D$ and $F$ from the block matrix (5) are linear combinations of
the identity $\mathbb{1}$ and the input state $\sigma$. These matrices can be
written as

${X}=f_{X}(d,\theta)\mathbb{1}+g_{X}(d,\theta)\sigma,$ (6)

where $X\in\\{{A},{D},{B},{F}\\}$ and $f_{X}(d,\theta)$ and $g_{X}(d,\theta)$
are functions of the dimension $d$ and the depolarising parameter $\theta$,
see Appendix A. Note that the matrices $X$ commute. This fact simplifies
calculations as the matrices $X$ can be treated as matrix elements. Given that
the input state $\sigma$ of the quantum 3-switch can have a spectral
decomposition
$\sigma=\sum_{i=1}^{d}\gamma_{i}^{\sigma}\left|i\right>\left<i\right|$, the
matrices $X$ can have also a spectral decomposition as
$X=\sum_{i=1}^{d}\gamma_{i}^{x}\left|i\right>\left<i\right|$, where the
eigenvalue $\gamma_{i}^{x}$ for ${x}=\\{{a},{d},{b},{f}\\}$ corresponds to
eigenvalues of matrices $A,B,D$ and $F$ respectively. These eigenvalues depend
on $\theta$ and $d$.

##  3 Quantum Fisher Information

To find bounds on the precision of $\theta$ we use the Cramer-Rao inequality
which establishes a lower bound on the variance Var of an estimator
$\hat{\theta}$

${\rm Var}(\hat{\theta})\geqslant\frac{1}{J(\theta)},$ (7)

where $J(\theta)$ is the quantum Fisher information defined as

$J(\theta)=\tr[\rho(\theta)L^{2}],$ (8)

where the score operator $L$ is the Hermitian symmetric logarithmic derivative
(SLD) [3] that satisfy the following equation

$2\frac{\partial\rho(\theta)}{\partial\theta}=L\rho(\theta)+\rho(\theta)L.$
(9)

The Cramer-Rao inequality tells us that the larger $J(\theta)$ is, the lower
Var$(\hat{\theta})$ will be. In other words, the precision to estimate the
parameter $\theta$ is improved if the quantum Fisher information is larger.
Thus, increasing the quantum Fisher information with different strategies is
an important task in quantum metrology. Here we use the quantum 3-switch to
improve the parameter estimation of $\theta$. For the input state we consider
an input $\sigma\otimes\sigma_{c}$ where $\sigma$ is considered to be a pure
state. The output state ${\rho_{\rm
out}^{(m)}}(\theta)=\mathcal{S}({\mathcal{N}}_{,}{\mathcal{N}}_{,}\mathcal{N})\left(\sigma\otimes\sigma_{c}\right)$
is the one-parameter family of density operators which are used to calculate
the quantum Fisher information using equation (8), i.e., we take
$\rho(\theta)={\rho_{\rm out}^{(m)}}(\theta)$. It has been shown that the
maximum quantum Fisher information is attained when the input state $\sigma$
is a pure sate [1], that is, $\sigma=\left|i\right>\left<i\right|$. Without
loss of generality, we take the input state
$\sigma=\left|1\right>\left<1\right|$. The identity $\mathbb{1}$ in equation
(6) has a spectral decomposition as
$\mathbb{1}=\sum_{i=1}^{d}\left|i\right>\left<i\right|$ so that the matrices
$X$ have the following spectral decomposition

${X}=(f_{X}(d,\theta)+g_{X}(d,\theta))\left|1\right>\left<1\right|+\sum_{i=2}^{d}f_{X}(d,\theta)\left|i\right>\left<i\right|,$
(10)

Comparing equation (10) with
$X=\sum_{i=1}^{d}\gamma_{i}^{x}\left|i\right>\left<i\right|$ we found that the
eigenvalues $\gamma_{i}^{x}$ of matrices $X$ are

$\begin{array}[]{ll}\gamma_{1}^{x}=f_{X}(d,\theta)+g_{X}(d,\theta)\vspace{2ex},\forall\hskip
2.15277pti=1,\\\ \gamma_{i}^{x}=f_{X}(d,\theta)\hskip 8.61108pt\forall\hskip
2.15277pti\geqslant 2.\end{array}$ (11)

Our principal task is to determine the quantum Fisher information $J(\theta)$
using equation (8) for the one-parameter output channels ${\rho_{\rm
out}^{(m)}}(\theta)$. To do that, we need first to specify the number of
involved causal orders $m$ to define properly the structure of the matrix (5).
Once we specify ${\rho_{\rm out}^{(m)}}(\theta)$ for a given $m$, we calculate
the corresponding score operator $L$ from equation (9). For a given $m$, the
structure of the score operator $L$ should have the same mathematical
structure as ${\rho_{\rm out}^{(m)}}(\theta)$. We specify this relation by
having a superscript $m$ in $L$, i.e. $L^{(m)}$. We also specify the
corresponding quantum Fisher information for a given $m$ as $J^{(m)}(\theta)$.
The parameters $m$ and $\epsilon_{ij}$ in matrix (5) play an important role to
select the desired superposition of causal orders.

The number of combinations to superimpose $m$ definite causal orders with
three channels is $\binom{3!}{m}$, where $\binom{n}{r}=\frac{n!}{r!(n-r)!}$ is
the binomial coefficient. If we assign to each combination the corresponding
output state ${\rho_{\rm out}^{(m)}}(\theta)$, we will have $\binom{3!}{m}$
different output states ${\rho_{\rm out}^{(m)}}(\theta)$. In total, there will
be $\sum_{m=2}^{3!}\binom{3!}{m}=57$ different output states ${\rho_{\rm
out}^{(m)}}(\theta)$ and therefore 57 different score operators $L^{(m)}$ to
be calculated. However, in Ref. [20], it was found that the transmission of
classical information with three channels in superposition of causal orders
has a threefold behaviour. These behaviours are based on the properties of
three equivalent classes of quantum switch matrices ${\rho_{\rm
out}^{(m)}}(\theta)$. For a given $m$, there are three different equivalent
classes of matrices ${\rho_{\rm out}^{(m)}}(\theta)$ which have the same
matrix invariants and eigenvalues. Based on this fact, we choose only one
representative element ${\rho_{\rm out}^{(m)}}(\theta)$ of each classe to
calculate the corresponding quantum Fisher information $J^{(m)}(\theta)$. This
avoids to calculate the 57 different score operators $L^{(m)}$ since elements
of ${\rho_{\rm out}^{(m)}}(\theta)$ of the same group have the same
mathematical properties. In the next sections we calculate the representative
operators $L^{(m)}$ of each group and then we obtain the corresponding quantum
Fisher information $J^{(m)}(\theta)$ for each $m$. We also analyze the
relative quantum Fisher information gain defined as

$J_{\rm rel}^{(m)}(\theta)=\frac{J^{(m)}(\theta)-J_{{\rm
def}}(\theta)}{J_{{\rm def}}(\theta)},$ (12)

where $J^{(m)}(\theta)$ is the quantum Fisher information associated to
$\theta$ when the channels are in an indefinite causal order with $m$
alternative causal orders and $J_{{\rm def}}(\theta)$ is the quantum Fisher
information for channels in a definite causal order.

##  4 Quantum Fisher information for $m$ causal orders

Following the procedure described above, we derivate analytical expressions
for the quantum Fisher information for each $m$ causal order.

Causal order $m=2$. For this case, for the all combinations with $m=2$, we
have only four nonzero matrix elements in ${\rho_{\rm out}^{(2)}}(\theta)$ and
the rest of elements are zero. Here and for the next cases, we solve the
problem to find $L^{(m)}$ with the corresponding matrices ${\rho_{\rm
out}^{(m)}}(\theta)$, but to avoid to write $6\times 6$ matrices with most of
its elements zeros, we write ${\rho_{\rm out}^{(m)}}(\theta)$ as $m\times m$
matrices. The problem to calculate $L^{(m)}$ should be equivalent using both
type of matrices. The representative matrices ${\rho_{\rm out}^{(2)}}(\theta)$
of each group can be written as

$\begin{array}[]{lll}{\rho}_{X}^{(2)}(\theta)=\left(\begin{array}[]{cc}Ap_{1}&\frac{1}{2}\epsilon_{12}X\\\
\frac{1}{2}\epsilon_{12}X&Ap_{2}\\\ \end{array}\right),\end{array}$ (13)

that is, there are three different quantum switch matrices,
${\rho}_{B}^{(2)}(\theta)$, ${\rho}_{D}^{(2)}(\theta)$ and
${\rho}_{F}^{(2)}(\theta)$ which represent the three equivalent classes of
quantum 3-switch matrices that can be found with two causal orders in
superposition. Given the block form of matrices (13), the score operator
$L^{(2)}$ has also three different score operators
${L}_{B}^{\scaleto{(2)}{5pt}},{L}_{D}^{\scaleto{(2)}{5pt}}$ and
${L}_{F}^{\scaleto{(2)}{5pt}}$ with a corresponding block form

$\begin{array}[]{ll}L^{(2)}=\left(\begin{array}[]{cc}L_{11}&L_{12}\\\
L_{12}&L_{22}\\\ \end{array}\right).\end{array}$ (14)

By introducing the output states (13) and the score operator (14) into (9), we
have the following equation systems

$\begin{array}[]{ll}2AL_{11}p_{1}+BL_{12}\epsilon_{12}=2p_{1}\partial_{\theta}A,\vspace{3ex}\\\
2AL_{22}p_{2}+XL_{12}\epsilon_{12}=2p_{2}\partial_{\theta}A,\vspace{3ex}\\\
AL_{12}+\frac{\epsilon_{12}}{2}XL_{11}+\frac{\epsilon_{12}}{2}XL_{22}=\epsilon_{12}\partial_{\theta}X,\vspace{3ex}\\\
\end{array}$ (15)

whose solutions are

$\begin{array}[]{ll}L_{11}=\text{$A$}^{-1}(\partial_{\theta}\text{$A$}-2p_{1}\text{$X$}\Phi_{\text{$X$}}),\vspace{3ex}\\\
L_{22}=\text{$A$}^{-1}(\partial_{\theta}\text{$A$}-2p_{j}\text{$X$}\Phi_{\text{$X$}})\vspace{3ex},\\\
L_{12}=\epsilon_{1j}\Phi_{\text{$X$}},\vspace{3ex}\\\ \end{array}$ (16)

where
$\Phi_{X}=\frac{{A}\partial_{\theta}\text{${X}$}-{X}\partial_{\theta}\text{${A}$}}{{A}^{2}-{X}^{2}}$.
By introducing solutions (16) into (8) we found that the quantum Fisher
information is

$J^{(2)}_{X}(d,\theta)=\tr[A^{-1}(\partial_{\theta}A)^{2}]+\epsilon_{1j}\tr[A^{-1}(A^{2}-X^{2})\Phi^{2}_{X})],$
(17)

where the term
$\tr[A^{-1}(\partial_{\theta}A)^{2}]=\sum_{i=1}^{d}\frac{(\partial_{\theta}\gamma_{i}^{a})^{2}}{\gamma_{i}^{a}}$
in (17) is the quantum Fisher information associated with $\theta$ when the
three channels are in a definite order, that is

$J^{{\rm N3}}_{{\rm
def}}(d,\theta)\equiv\tr[A^{-1}(\partial_{\theta}A)^{2}]=\frac{9(d-1)\theta^{4}}{\left(1-\theta^{3}\right)\left((d-1)\theta^{3}+1\right)}.$
(18)

The second term in equation (17) is associated with $\theta$ when the three
channels are in an indefinite order

$J^{(2)}_{{{\rm
ind,X}}}(d,\theta)\equiv\tr[A^{-1}(A^{2}-X^{2})\Phi^{2}_{X})]=\sum_{i=1}^{d}\frac{(\gamma_{i}^{a}\partial_{\theta}\gamma_{i}^{x}-\gamma_{i}^{x}\partial_{\theta}\gamma_{i}^{a})^{2}}{\gamma_{i}^{a}((\gamma_{i}^{a})^{2}-(\gamma_{i}^{x})^{2})}.$
(19)

By substituting the eigenvalues (11) of matrices $B,D$ and $F$ in (19), we can
calculate the quantum Fisher information (17) as a function of the parameters
$d$ and $\theta$. Equations (36), (37) and (38) from Appendix B show the
corresponding quantum Fisher information expressions $J^{(2)}_{{{\rm
ind,X}}}(d,\theta)$. Figure 1 shows the quantum Fisher information as a
function of $\theta$ and $d$. For complete depolarizing channels, i.e.
$\theta$=0, in the limit when $d\to\infty$, we have
$J^{(2)}_{B}(d)>J^{(2)}_{D}(d)>J^{(2)}_{F}(d)$.

Causal order $m=3$. For this case we have also three equivalent classes of
quantum 3-switch. We also expect to have three different behaviours for the
quantum Fisher information as we found in the case of $m=2$ causal orders.
However, by calculating the score operators from the system of equations given
by equation (8), only the representative element

$\begin{array}[]{lll}{\rho}_{8}^{\scaleto{(3)}{5pt}}(\theta)=\frac{1}{3}\left(\begin{array}[]{cccccc}A&D&D\\\
D&A&D\\\ D&D&A\\\ \end{array}\right),\end{array}$ (20)

can be used to calculate readily the score operator

${L}_{8}^{\scaleto{(3)}{5pt}}=\left(\begin{array}[]{cccccc}{L}^{\scaleto{(3)}{5pt}}&{L}_{D}^{\scaleto{(3)}{5pt}}&{L}_{D}^{\scaleto{(3)}{5pt}}\\\
{L}_{D}^{\scaleto{(3)}{5pt}}&{L}^{\scaleto{(3)}{5pt}}&{L}_{D}^{\scaleto{(3)}{5pt}}\\\
{L}_{D}^{\scaleto{(3)}{5pt}}&{L}_{D}^{\scaleto{(3)}{5pt}}&{L}^{\scaleto{(3)}{5pt}}\\\
\end{array}\right),$ (21)

where the subscript 8 stands for the label of the quantum 3-switch matrix
associated to a specific combination of causal order, see [20]. By introducing
the output state (20) and the score operator (21) into (9), we found that the
block matrix elements from (21) are

${L}^{\scaleto{(3)}{5pt}}=\frac{A\partial_{\theta}A+D\partial_{\theta}A-2D\partial_{\theta}D}{A^{2}+AD-2D^{2}},$
(22)

${L}_{D}^{\scaleto{(3)}{5pt}}=\frac{A\partial_{\theta}D-D\partial_{\theta}A}{A^{2}+AD-2D^{2}}.$
(23)

Substituting equations (22) and (23) into (8), we found that the quantum
Fisher information for $m=3$ causal orders is

$J^{(3)}_{8}(d,\theta)={\rm
Tr}\left[\frac{(\partial_{\theta}A)^{2}+2(\partial_{\theta}D)^{2}}{(A-D)(A+2D)}A\right]+{\rm
Tr}\left[\frac{\partial_{\theta}A^{2}-4\partial_{\theta}A\partial_{\theta}D}{(A-D)(A+2D)}D\right].$
(24)

By substituting the eigenvalues (11) of matrices $A$ and $D$ in (24), we
obtain the quantum Fisher information in terms of $\theta$ and $d$, see
equation (39) from Appendix B.

Causal order $m=4$. For this case, we only calculate the quantum Fisher
information for the following representative element

$\begin{array}[]{lll}{\rho}_{5}^{\scaleto{(4)}{5pt}}(\theta)=\frac{1}{4}\left(\begin{array}[]{cccccc}A&B&D&F\\\
B&A&F&D\\\ D&F&A&B\\\ F&D&B&A\\\ \end{array}\right).\end{array}$ (25)

Likewise, given the block form of matrix (25), the score operator
${L}^{\scaleto{(4)}{5pt}}$ has the corresponding block form

${L}_{5}^{\scaleto{(4)}{5pt}}=\left(\begin{array}[]{cccccc}{L}^{\scaleto{(4)}{5pt}}&{L}_{B}^{\scaleto{(4)}{5pt}}&{L}_{D}^{\scaleto{(4)}{5pt}}&{L}_{F}^{\scaleto{(4)}{5pt}}\\\
{L}_{B}^{\scaleto{(4)}{5pt}}&{L}^{\scaleto{(4)}{5pt}}&{L}_{F}^{\scaleto{(4)}{5pt}}&{L}_{D}^{\scaleto{(4)}{5pt}}\\\
{L}_{D}^{\scaleto{(4)}{5pt}}&{L}_{F}^{\scaleto{(4)}{5pt}}&{L}^{\scaleto{(4)}{5pt}}&{L}_{B}^{\scaleto{(4)}{5pt}}\\\
{L}_{F}^{\scaleto{(4)}{5pt}}&{L}_{D}^{\scaleto{(4)}{5pt}}&{L}_{B}^{\scaleto{(4)}{5pt}}&{L}^{\scaleto{(4)}{5pt}}\\\
\end{array}\right).$ (26)

By introducing the output states (25) and the score operator (26) into (9), we
found that

$\begin{array}[]{ll}{L}^{\scaleto{(4)}{5pt}}=\frac{1}{4}\left(\frac{\partial_{\theta}\Phi_{\alpha}}{\Phi_{\alpha}}+\frac{\partial_{\theta}\Phi_{\beta}}{\Phi_{\beta}}+\frac{\partial_{\theta}\Phi_{\gamma}}{\Phi_{\gamma}}+\frac{\partial_{\theta}\Phi_{\sigma}}{\Phi_{\sigma}}\right),\hskip
12.91663pt{L}_{B}^{\scaleto{(4)}{5pt}}=\frac{1}{4}\left(\frac{\partial_{\theta}\Phi_{\alpha}}{\Phi_{\alpha}}+\frac{\partial_{\theta}\Phi_{\beta}}{\Phi_{\beta}}-\frac{\partial_{\theta}\Phi_{\gamma}}{\Phi_{\gamma}}-\frac{\partial_{\theta}\Phi_{\sigma}}{\Phi_{\sigma}}\right),\vspace{3ex}\\\
{L}_{D}^{\scaleto{(4)}{5pt}}=\frac{1}{4}\left(-\frac{\partial_{\theta}\Phi_{\alpha}}{\Phi_{\alpha}}+\frac{\partial_{\theta}\Phi_{\beta}}{\Phi_{\beta}}-\frac{\partial_{\theta}\Phi_{\gamma}}{\Phi_{\gamma}}+\frac{\partial_{\theta}\Phi_{\sigma}}{\Phi_{\sigma}}\right),\hskip
4.30554pt{L}_{F}^{\scaleto{(4)}{5pt}}=\frac{1}{4}\left(-\frac{\partial_{\theta}\Phi_{\alpha}}{\Phi_{\alpha}}+\frac{\partial_{\theta}\Phi_{\beta}}{\Phi_{\beta}}+\frac{\partial_{\theta}\Phi_{\gamma}}{\Phi_{\gamma}}-\frac{\partial_{\theta}\Phi_{\sigma}}{\Phi_{\sigma}}\right),\end{array}$
(27)

where $\Phi$’s are linear combinations of matrices $A,B,D$ and $F$

$\begin{array}[]{ll}\Phi_{\alpha}=A+B-D-F,\\\ \Phi_{\beta}=A+B+D+F,\\\
\Phi_{\gamma}=A-B-D+F,\\\ \Phi_{\sigma}=A-B+D-F.\end{array}$ (28)

For this case, we found that the quantum Fisher information (8) can be
calculated as

$\begin{array}[]{ll}J^{(4)}_{5}(d,\theta)=\frac{1}{4}\left(\text{Tr}\left[(\Phi_{\alpha})^{-1}\left(\partial_{\theta}\Phi_{\alpha}\right)^{2}\right]+\text{Tr}\left[(\Phi_{\beta})^{-1}\left(\partial_{\theta}\Phi_{\beta}\right)^{2}\right]+\right.\vspace{3ex}\\\
\hskip
51.6665pt\left.\text{Tr}\left[(\Phi_{\gamma})^{-1}\left(\partial_{\theta}\Phi_{\gamma}\right)^{2}\right]+\text{Tr}\left[(\Phi_{\sigma})^{-1}\left(\partial_{\theta}\Phi_{\sigma}\right)^{2}\right]\right).\end{array}$
(29)

The corresponding analytical expression of (29) in terms of $\theta$ and $d$
can be found in equation (40) from Appendix B.

Causal order $m=5$. For this case, we were unable to calculate directly the
components of the score operators using the system of equations given by (9).
Other methods can be used to determine the score operator, for example see
reference [21].

Causal order $m=6$. For this case, we are able to calculate directly the score
operator from the system of equations given by (9). The quantum 3-switch
output by taking account all the six alternative causal orders is

$\begin{array}[]{lll}{\rho}^{\scaleto{(6)}{5pt}}(\theta)=\frac{1}{6}\left(\begin{array}[]{cccccc}A&B&B&D&D&F\\\
B&A&D&F&B&D\\\ B&D&A&B&F&D\\\ D&F&B&A&D&B\\\ D&B&F&D&A&B\\\
F&D&D&B&B&A\end{array}\right),\end{array}$ (30)

Likewise, given the block form (30), the score operator
${L}^{\scaleto{(6)}{5pt}}$ has the corresponding block form

$L^{\scaleto{(6)}{5pt}}=\left(\begin{array}[]{cccccc}L^{\scaleto{(6)}{5pt}}&L_{B}^{\scaleto{(6)}{5pt}}&L_{B}^{\scaleto{(6)}{5pt}}&L_{D}^{\scaleto{(6)}{5pt}}&L_{D}^{\scaleto{(6)}{5pt}}&L_{F}^{\scaleto{(6)}{5pt}}\\\
L_{B}^{\scaleto{(6)}{5pt}}&L^{\scaleto{(6)}{5pt}}&L_{D}^{\scaleto{(6)}{5pt}}&L_{F}^{\scaleto{(6)}{5pt}}&L_{B}^{\scaleto{(6)}{5pt}}&L_{D}^{\scaleto{(6)}{5pt}}\\\
L_{B}^{\scaleto{(6)}{5pt}}&L_{D}&L^{\scaleto{(6)}{5pt}}&L_{B}^{\scaleto{(6)}{5pt}}&L_{F}^{\scaleto{(6)}{5pt}}&L_{D}^{\scaleto{(6)}{5pt}}\\\
L_{D}&L_{F}^{\scaleto{(6)}{5pt}}&L_{B}^{\scaleto{(6)}{5pt}}&L^{\scaleto{(6)}{5pt}}&L_{D}^{\scaleto{(6)}{5pt}}&L_{B}^{\scaleto{(6)}{5pt}}\\\
L_{D}&L_{B}^{\scaleto{(6)}{5pt}}&L_{F}^{\scaleto{(6)}{5pt}}&L_{D}^{\scaleto{(6)}{5pt}}&L^{\scaleto{(6)}{5pt}}&L_{B}^{\scaleto{(6)}{5pt}}\\\
L_{F}^{\scaleto{(6)}{5pt}}&L_{D}^{\scaleto{(6)}{5pt}}&L_{D}^{\scaleto{(6)}{5pt}}&L_{B}^{\scaleto{(6)}{5pt}}&L_{B}^{\scaleto{(6)}{5pt}}&L^{\scaleto{(6)}{5pt}}\end{array}\right).$
(31)

By introducing the output states (30) and (31) into (9), we found that

$\begin{array}[]{ll}{L}^{\scaleto{(6)}{5pt}}=\frac{1}{6}\left(2\frac{\partial_{\theta}\Phi_{\alpha}}{\Phi_{\alpha}}+2\frac{\partial_{\theta}\Phi_{\gamma}}{\Phi_{\gamma}}+\frac{\partial_{\theta}\Phi_{\mu}}{\Phi_{\mu}}+\frac{\partial_{\theta}\Phi_{\eta}}{\Phi_{\eta}}\right),\hskip
12.91663pt{L}_{B}^{\scaleto{(6)}{5pt}}=\frac{1}{6}\left(\frac{\partial_{\theta}\Phi_{\alpha}}{\Phi_{\alpha}}-\frac{\partial_{\theta}\Phi_{\gamma}}{\Phi_{\gamma}}-\frac{\partial_{\theta}\Phi_{\mu}}{\Phi_{\mu}}+\frac{\partial_{\theta}\Phi_{\eta}}{\Phi_{\eta}}\right),\vspace{3ex}\\\
{L}_{D}^{\scaleto{(6)}{5pt}}=\frac{1}{6}\left(-\frac{\partial_{\theta}\Phi_{\alpha}}{\Phi_{\alpha}}-\frac{\partial_{\theta}\Phi_{\gamma}}{\Phi_{\gamma}}+\frac{\partial_{\theta}\Phi_{\mu}}{\Phi_{\mu}}+\frac{\partial_{\theta}\Phi_{\eta}}{\Phi_{\eta}}\right),\hskip
12.91663pt{L}_{F}^{\scaleto{(6)}{5pt}}=\frac{1}{6}\left(-2\frac{\partial_{\theta}\Phi_{\alpha}}{\Phi_{\alpha}}+2\frac{\partial_{\theta}\Phi_{\gamma}}{\Phi_{\gamma}}-\frac{\partial_{\theta}\Phi_{\mu}}{\Phi_{\mu}}+\frac{\partial_{\theta}\Phi_{\eta}}{\Phi_{\eta}}\right),\end{array}$
(32)

where

$\begin{array}[]{ll}\Phi_{\alpha}=A+B-D-F,\\\ \Phi_{\eta}=A+2B+2D+F,\\\
\Phi_{\gamma}=A-B-D+F,\\\ \Phi_{\mu}=A-2B+2D-F.\end{array}$ (33)

Thus, taking account all causal orders with three channels we found that the
quantum Fisher information for $m=6$ is

$\begin{array}[]{ll}J^{(6)}(d,\theta)=\frac{1}{3}\text{Tr}\left[(\Phi_{\alpha})^{-1}\left(\partial_{\theta}\Phi_{\alpha}\right)^{2}\right]+\frac{1}{6}\text{Tr}\left[(\Phi_{\eta})^{-1}\left(\partial_{\theta}\Phi_{\eta}\right)^{2}\right]+\\\
\hskip
51.6665pt\frac{1}{3}\text{Tr}\left[(\Phi_{\gamma})^{-1}\left(\partial_{\theta}\Phi_{\gamma}\right)^{2}\right]+\frac{1}{6}\text{Tr}\left[(\Phi_{\mu})^{-1}\left(\partial_{\theta}\Phi_{\mu}\right)^{2}\right],\end{array}$
(34)

whose analytical expression of (34) in terms of $\theta$ and $d$ can be found
in equation (41) from Appendix B.

(a) (b) (c) (d)

Figure 1:  Comparison between the quantum Fisher informations of different
number of causal orders $J^{(m)}$. Figures (a) and (b) show the quantum Fisher
information $J^{(m)}$ for different $m$ causal orders as a function of the
depolorazing parameter $\theta$ for two different dimensions of the input
state $\sigma$, $d=2$ and $d\to\infty$ respectively. Figures (c) and (d) show
the relative quantum Fisher gain $J^{(m)}_{\rm rel}$ for dimensions $d=2$ and
$d\to\infty$ respectively. $J^{\rm N3}_{\rm def}$ is the quantum Fisher
information of three channels in definite causal order. The gray zone is for
delimiting the region of the quantum Fisher information $J^{\rm N2}$ of the
quantum 2-switch. $J^{\rm N2}$ was calculated from Ref. [16].

##  5 Discussion

In Figure 1, we compare the quantum Fisher information with $m$ causal orders
as a function of the noise parameter $\theta$. As you can see in Figure 1(a),
for certain combinations of causal orders, the quantum Fisher information is
lower for the quantum 3-switch than the quantum 2-switch in the noise region
$0<\theta<0.5$. We also see that the quantum Fisher information $J^{(4)}_{5}$
is always higher than the quantum Fisher information for the two-case scenario
in any noise region. This allows us to choose the best combination of causal
orders with three channels to minimize the resources needed to implement the
quantum switch, and to exceed the value of the quantum Fisher information of
the two-channel scenario. Figure 1(a) also shows that $J^{(6)}$ is higher than
any quantum Fisher information of different combination of causal orders. This
confirms the fact that if we take all the alternatives of causal orders in
superposition, we will have the maximum gain in the quantum Fisher
information. In addition, the quantum Fisher information with three channels
$J^{(6)}$ is always higher than the quantum Fisher information $J^{N2}$ for
two channels. This may well indicate that the quantum Fisher information
increases as the number of channels increases. However, in Ref. [5], it has
been shown that the classical capacity increases monotonically with the number
of channels, but it asymptotically reaches a limit for larger number of
channels. The question whether the quantum Fisher information increases or has
a limit as the number of channels increases is a fundamental question that
future investigations could be done. In Figure 1(b), as the dimension $d$ of
the input state $\sigma$ tends to infinity, the quantum Fisher information for
$m<6$ causal orders is lower than the quantum Fisher information for the
quantum 2-switch in the noisy regions $0<\theta<0.3$. Remarkably, $J^{(6)}$ is
always higher than the two-channel scenario at large dimensions of $d$ for any
noise region. Figures 1(c)-1(d) show the relative quantum Fisher information
gain defined in equation (12). Finally, we notice that the quantum Fisher
information $J^{(6)}$ was also calculated in Ref. [22] following a different
procedure. Ref. [22] only studies three combinations among the 57 possible
combinations to superimpose different causal orders. Our procedure allowed us
to study the Fisher information for any combination of causal orders through
the quantum switch matrix (5) and to provide analytical and compact
expressions for the quantum Fisher information.

##  6 Conclusions

We studied the quantum Fisher information for any combination of causal orders
with three channels. Our procedure allows us to calculate the quantum Fisher
information in closed analytical expressions for any causal order. We found
that the quantum Fisher information increases as the number of causal orders
increases within specific regions of noise and with specific combinations of
causal orders. This allows us to choose the best indefinite causal structures
to obtain the maximum possible values of the quantum Fisher information with
less experimental resources. Our work can be extended to further study
indefinite causal structures for estimating the value of parameters of other
quantum channels of communications.

The author acknowledges the support of Israel Science Foundation and thanks
Nadia Belabas, Francisco Delgado and Alastair Abbott for discussions and
valuable comments on this manuscript.

##  Appendix A Functions $f_{X}$ and $g_{X}$

The functions $f_{X}$ and $g_{X}$ are the coefficients of equation (6):

$\begin{array}[]{ll}f_{A}(d,\theta)=\frac{1}{d}(1-\theta^{3}),\hskip
12.91663ptg_{A}(d,\theta)=\theta^{3}\vspace{3ex},\\\
f_{D}(d,\theta)=\frac{1}{d}\left(-2\theta^{3}+\theta^{2}+\theta\right),\hskip
12.91663ptg_{D}(d,\theta)=\frac{1}{d^{2}}\left(\left(d^{2}+1\right)\theta^{3}-\theta^{2}-\theta+1\right),\vspace{3ex}\\\
f_{B}(d,\theta)=-\frac{1}{d^{3}}(\theta-1)\left(\left(d^{2}+1\right)\theta^{2}+2\left(d^{2}-1\right)\theta+1\right),\hskip
12.91663ptg_{B}(d,\theta)=\frac{\theta}{d^{2}}\left(\left(d^{2}+1\right)\theta^{2}-2\theta+1\right),\vspace{3ex}\\\
f_{F}(d,\theta)=\frac{1}{d^{3}}\left((1-\theta)^{3}-3d^{2}(\theta-1)\theta^{2}\right),\hskip
12.91663ptg_{F}(d,\theta)=\frac{1}{d^{2}}\left(d^{2}\theta^{3}+3(\theta-1)^{2}\theta\right),\vspace{3ex}\\\
\end{array}$ (35)

where $\theta$ is the depolorazing parameter and $d$ the dimension of the
input state $\sigma$.

##  Appendix B Analytical expressions of the quantum Fisher Information

In this section we present explicitly the expressions for the quantum Fisher
information $J^{(m)}$ for $m$ causal orders. We start presenting the case
where there are $m=2$ causal orders:

$\begin{array}[]{ll}J^{(2)}_{{{\rm
ind,B}}}(d,\theta)=\frac{(d-1)}{d}\left(\frac{\left(d^{2}\left(\theta^{2}-2\theta-2\right)-3\theta^{2}+3\right)^{2}}{\left(\theta^{2}+\theta+1\right)\left(d^{4}\left(2\theta^{2}+3\theta+1\right)+2d^{2}\theta\left(\theta^{2}+\theta-2\right)+(\theta-1)^{3}\right)}\right.+\vspace{3ex}\\\
\hskip
51.6665pt\left.\frac{\left(d^{2}(\theta-3)\theta^{2}+2d\left(\theta^{3}-1\right)-3(\theta+1)(\theta-1)^{2}\right)^{2}}{\left((d-1)\theta^{3}+1\right)\left(2d^{4}\theta^{3}-d^{3}\left(2\theta^{4}+\theta^{2}-2\theta-1\right)+d^{2}\left(2\theta^{4}-5\theta^{2}+2\theta+1\right)-d(\theta-1)^{3}(\theta+1)+(\theta-1)^{4}\right)}\right),\end{array}$
(36)

$\begin{array}[]{ll}J^{(2)}_{{{\rm
ind,D}}}(d,\theta)=\frac{(d-1)}{d}\left(\frac{\left((d-1)\theta^{3}+3(d-1)\theta^{2}+3\theta+1\right)^{2}}{(\theta+1)\left((d-1)\theta^{3}+1\right)\left(2d^{2}\theta^{3}+d\left(-3\theta^{3}+\theta^{2}+\theta+1\right)+(\theta-1)^{2}(\theta+1)\right)}\right.+\vspace{3ex}\\\
\hskip
81.8053pt\left.\frac{\left(\theta^{2}+4\theta+1\right)^{2}}{\left(\theta^{2}+\theta+1\right)\left(3\theta^{3}+5\theta^{2}+3\theta+1\right)}\right),\end{array}$
(37)

$\begin{array}[]{ll}J^{(2)}_{{{\rm
ind,F}}}(d,\theta)=\frac{9(d-1)}{d^{7}}\left(\frac{(d-1)\left(\theta^{2}-1\right)^{2}((d-1)\theta+1)^{4}}{\left((d-1)\theta^{3}+1\right)\left(\frac{\left((d-1)\theta^{3}+1\right)^{2}}{d^{2}}-\frac{((d-1)\theta+1)^{6}}{d^{6}}\right)}\right.+\vspace{3ex}\\\
\hskip
81.8053pt\left.\frac{d^{6}\left(d^{2}\theta(\theta+2)+\theta^{2}-1\right)^{2}}{\left(\theta^{2}+\theta+1\right)\left(d^{4}\left(8\theta^{3}+6\theta^{2}+3\theta+1\right)+6d^{2}(\theta-1)\theta^{2}+(\theta-1)^{3}\right)}\right),\end{array}$
(38)

where the labels $B,D$ and $F$ correspond to the block matrix elements from
(5). Likewise, $\theta$ is the depolorazing parameter and $d$ the dimension of
the input state $\sigma$. Each expression correspond to the evaluation of
quantum Fisher information (17) for each equivalent class of the quantum
switch matrices with $m=2$ causal orders. For $m=3$ causal orders, the
complete expression for the quantum Fisher information (24) is

$\begin{array}[]{ll}J^{(3)}_{8}(d,\theta)=\frac{(d-1)\left(3d^{2}\left(2\theta^{3}-33\theta^{2}-21\theta-2\right)\theta^{3}+2d(3\theta+1)^{2}\left(5\theta^{3}-2\theta^{2}-2\theta-1\right)-6\left(\theta^{3}+3\theta^{2}-3\theta-1\right)^{2}\right)}{d(\theta-1)(\theta+1)\left(5\theta^{2}+3\theta+1\right)\left(3d^{2}\theta^{3}+d\left(-5\theta^{3}+2\theta^{2}+2\theta+1\right)+2(\theta-1)^{2}(\theta+1)\right)}.\end{array}$
(39)

where the subscript 8 stands for the label of the quantum 3-switch matrix
associated to a specific combination of causal order, see [20]. For $m=4$
causal orders, the complete expression for the quantum Fisher information (29)
is

$\begin{array}[]{ll}J^{(4)}_{5}(d,\theta)=\frac{1}{4}(d-1)\left[\frac{(3d\theta+d-6\theta+6)^{2}}{d^{3}(d\theta+d-2\theta+2)}+\frac{(1-9\theta)^{2}}{d^{2}(3\theta+1)}+\frac{9-9\theta}{d^{2}}-\frac{9\left(d^{2}\left(-7\theta^{2}+2\theta+1\right)-2(\theta-1)^{2}\right)^{2}}{d^{3}(\theta-1)\left(d^{2}\left(7\theta^{2}+4\theta+1\right)+2(\theta-1)^{2}\right)}\right.+\vspace{3ex}\\\
\hskip
81.8053pt\left.\frac{9(d-1)\left(4d^{2}\theta^{2}+d\left(-3\theta^{2}+2\theta+1\right)+2(\theta-1)^{2}\right)^{2}}{d^{3}\left(4d^{3}\theta^{3}+d^{2}\left(-7\theta^{3}+3\theta^{2}+3\theta+1\right)+d(5\theta+1)(\theta-1)^{2}-2(\theta-1)^{3}\right)}\right.+\vspace{3ex}\\\
\hskip
81.8053pt\left.\frac{\left(3d^{2}\theta+d^{2}+6\theta-6\right)^{2}}{d^{3}\left(d^{2}(\theta+1)+2(\theta-1)\right)}+\frac{(1-9\theta)^{2}}{3d\theta+d}+\frac{9-9\theta}{d}\right].\end{array}$
(40)

Finally, for $m=6$ causal orders, the expression for the quantum Fisher
information (34) is

$\begin{array}[]{ll}J^{(6)}(d,\theta)=\frac{(d-1)}{6d^{3}}\left[-\frac{\left(9(\theta-1)^{2}-6d^{2}\left(-5\theta^{2}+\theta+1\right)\right)^{2}}{(\theta-1)\left(d^{2}\left(10\theta^{2}+7\theta+1\right)+3(\theta-1)^{2}\right)}+\frac{\left(2d^{2}+9\theta-9\right)^{2}}{d^{2}+3\theta-3}\right.+\vspace{3ex}\\\
\hskip
81.8053pt\left.\frac{9(d-1)\left(6d^{2}\theta^{2}+d\left(-4\theta^{2}+2\theta+2\right)+3(\theta-1)^{2}\right)^{2}}{6d^{3}\theta^{3}+d^{2}\left(-10\theta^{3}+3\theta^{2}+6\theta+1\right)+d(7\theta+2)(\theta-1)^{2}-3(\theta-1)^{3}}\right.+\vspace{3ex}\\\
\hskip
81.8053pt\left.\frac{2d(d+1)(1-9\theta)^{2}}{3\theta+1}+\frac{(2d-9\theta+9)^{2}}{d-3\theta+3}-18d(d+1)(\theta-1)\right].\end{array}$
(41)

## References

## References

  * [1] Fujiwara, A.  Quantum channel identification problem.  In _Asymptotic Theory Of Quantum Statistical Inference: Selected Papers_ , 487–493 (World Scientific, 2005).
  * [2] Frey, M., Collins, D. & Gerlach, K.  Probing the qudit depolarizing channel.  _Journal of Physics A: Mathematical and Theoretical_ 44, 205306 (2011).
  * [3] Paris, M. G.  Quantum estimation for quantum technology.  _International Journal of Quantum Information_ 7, 125–137 (2009).
  * [4] Abbott, A. A., Wechs, J., Horsman, D., Mhalla, M. & Branciard, C.  Communication through coherent control of quantum channels.  _Quantum_ 4, 333 (2020).
  * [5] Chiribella, G. & Kristjánsson, H.  Quantum shannon theory with superpositions of trajectories.  _Proceedings of the Royal Society A_ 475, 20180903 (2019).
  * [6] Chiribella, G.  Perfect discrimination of no-signalling channels via quantum superposition of causal structures.  _Physical Review A_ 86, 040301 (2012).
  * [7] Chiribella, G., D’Ariano, G. M., Perinotti, P. & Valiron, B.  Quantum computations without definite causal structure.  _Physical Review A_ 88, 022318 (2013).
  * [8] Guérin, P. A., Feix, A., Araújo, M. & Brukner, Č.  Exponential communication complexity advantage from quantum superposition of the direction of communication.  _Physical review letters_ 117, 100502 (2016).
  * [9] Zhao, X., Yang, Y. & Chiribella, G.  Quantum metrology with indefinite causal order.  _Physical Review Letters_ 124, 190503 (2020).
  * [10] Felce, D. & Vedral, V.  Quantum refrigeration with indefinite causal order.  _Physical review letters_ 125, 070603 (2020).
  * [11] Procopio, L. M. _et al._ Experimental superposition of orders of quantum gates.  _Nature communications_ 6, 7913 (2015).
  * [12] Rubino, G. _et al._ Experimental verification of an indefinite causal order.  _Science advances_ 3, e1602589 (2017).
  * [13] Goswami, K. _et al._ Indefinite causal order in a quantum switch.  _Physical review letters_ 121, 090503 (2018).
  * [14] Wei, K. _et al._ Experimental quantum switching for exponentially superior quantum communication complexity.  _Physical review letters_ 122, 120504 (2019).
  * [15] Taddei, M. M. _et al._ Computational Advantage from the Quantum Superposition of Multiple Temporal Orders of Photonic Gates.  _PRX Quantum_ 2, 010320 (2021).
  * [16] Frey, M.  Indefinite causal order aids quantum depolarizing channel identification.  _Quantum Information Processing_ 18, 96 (2019).
  * [17] Mukhopadhyay, C., Gupta, M. K. & Pati, A. K.  Superposition of causal order as a metrological resource for quantum thermometry.  _arXiv preprint arXiv:1812.07508_ (2018).
  * [18] Procopio, L. M., Delgado, F., Enríquez, M., Belabas, N. & Levenson, J. A.  Communication enhancement through quantum coherent control of n channels in an indefinite causal-order scenario.  _Entropy_ 21, 1012 (2019).
  * [19] Procopio, L. M., Delgado, F., Enríquez, M., Belabas, N. & Levenson, J. A.f  Sending classical information via three noisy channels in superposition of causal orders.  _Physical Review A_ 101, 012346 (2020).
  * [20] Procopio, L. M., Delgado, F., Enriquez, M. & Belabas, N.  Multifold behavior of the information transmission by the quantum 3-switch.  _Quantum Information Processing_ 20, 1–29 (2021).
  * [21] Šafránek, D.  Simple expression for the quantum fisher information matrix.  _Physical Review A_ 97, 042322 (2018).
  * [22] Frey, M.  Quantum network probing with indefinite routing.  _Quantum Information Processing_ 20, 24 (2021).


# On locally finite orthomodular lattices

Dominika Burešová* and Pavel Pták  *  Czech Technical University in Prague
Faculty of Electrical Engineering
Prague
Czech Republic<EMAIL_ADDRESS>** Czech Technical University in Prague
Faculty of Electrical Engineering
Department of Mathematics
Prague
Czech Republic<EMAIL_ADDRESS>
###### Abstract.

Let us denote by $\mathcal{LF}$ the class of all orthomodular lattices (OMLs)
that are locally finite (i.e., $L\in\mathcal{LF}$ provided each finite subset
of $L$ generates in $L$ a finite subOML). We first show in this note how one
can obtain new locally finite OMLs from the initial ones and enlarge thus the
class $\mathcal{LF}$. We find $\mathcal{LF}$ considerably large though,
obviously, not all OMLs belong to $\mathcal{LF}$. We then study states on the
OMLs of $\mathcal{LF}$. We show that local finiteness may to a certain extent
make up for distributivity. We for instance show that if $L\in\mathcal{LF}$
and if for any finite subOML $K$ there is a state $s:K\to\enskip[0,1]$ on $K$,
then there is a state on the entire $L$. We also consider further algebraic
and state properties of $\mathcal{LF}$ relevant to quantum logic theory.

###### Key words and phrases:

Boolean algebra, orthomodular lattice, local finiteness, state

###### 2010 Mathematics Subject Classification:

06C15; 03G05; 03G12

This work was supported by FWF: Project I 4579-N and GAČR: Project 20-09869L

##  1\. Notions and results

Let $L=(P,\,\leq,\,^{\prime},\,0,\,1)$ be an orthomodular lattice (abbr. OML -
see e.g. [Kal] for the definition and basic properties of OMLs). As known, an
OML does not have to be locally finite. For instance, the free OML over three
generators is not locally finite (this OML is infinite, see [Har]). Also, the
lattice $L(H)$ of the projectors in a Hilbert space $\mathcal{H}$ is not
locally finite (see e.g. [Kal], where one can also learn basic properties of
OMLs). Let us denote by $\mathcal{LF}$ the class of all locally finite OMLs.
The basic information of $\mathcal{LF}$ is listed below.

######  Theorem 1.1.

(i) Each finite OML and each Boolean algebra belongs to $\mathcal{LF}$.
(ii) $\mathcal{LF}$ is closed under the formation of finite cartesian
products. A consequence: If $L\in\mathcal{LF}$ and if we denote by $cf(L)$ the
difference between the number of generators of $L$ and the cardinality of $L$,
then $cf(L)$ can in general be greater than any natural number even for a
finite $L$. ($\mathcal{LF}$ is not closed under the formation of infinite
cartesian products as we shall comment on later.)
(iii) $\mathcal{LF}$ is closed under the formation of arbitrary horizontal
sums. A consequence: There is an infinite non-Boolean set-representable
locally finite OML.
(iv) $\mathcal{LF}$ is closed under the formation of epimorphic images. A
consequence: The class $\mathcal{LF}$ for the objects and the epimorphic
homomorphisms for the morphisms forms a category.
(v) Let $L\in\mathcal{LF}$ and let $B$ be a Boolean algebra. Then $L$ can be
OML-embedded into an OML $K$, $K\in\mathcal{LF}$ such that the centre $C(K)$
of $K$ is equal to $B$.

###### Proof.

The statements (i) and (ii) are obvious, a finitely generated Boolean algebra
possesses finitely many atoms.

As for (iii), recall (see [Kal]) that if $L_{\alpha},\,\alpha\in I$, is a
collection of OMLs, then by the horizontal sum $Hor(L_{\alpha},\,\alpha\in I)$
one means the OML obtained by identifying zeros and ones of all
$L_{\alpha},\,\alpha\in I$. Obviously, if $L_{\alpha}\in\mathcal{LF}$ for any
$\alpha\in I$, then $Hor(L_{\alpha},\,\alpha\in I)\in\mathcal{LF}$, too.

Further, verifying (iv), let $K\in\mathcal{LF}$ and $e:K\to L$ be an
epimorphism in OMLs. Let {$a_{i},\,i\leq n$} $\subseteq L$. Take $b_{i}\in K$
with $e(b_{i})=a_{i}$. Then {$b_{i},\,i\leq n$} generates a finite subOML of
$K$, some $\tilde{K}$, and so $e(\tilde{K})$ is a finite subOML of $L$. Since
$e(\tilde{K})$ contains {$a_{i},\,i\leq n$}, property (iv) follows.

Finally, approaching (v), recall that the centre $C(K)$ means the set of all
“absolutely compatible” elements of $K$ (see [Kal]). Let us first take the
horizontal sum $Hor(L,\,MO_{2})$, where
$MO_{2}=\\{0,\,1,\,a,\,a^{\prime},\,b,\,b^{\prime}\\}$. Then the centre
$C(Hor(L,\,MO_{2}))$ is trivial, $C(Hor(L,\,MO_{2}))=\\{0,\,1\\}$, and,
obviously, $L$ can be embedded in $Hor(L,\,MO_{2})$. To complete the argument,
let us form the so-called Boolean sum of $Hor(L,\,MO_{2})$ and $B$ (see e.g.
[Pt1]). Recall the construction of Boolean sum, the rest is then easily seen.
Let us view $B$ as a collection $\mathcal{B}$ of subsets of a set,
$B=(D,\,\mathcal{B})$ (the Stone Theorem). Let $\mathcal{P}$ be the collection
of all partitions of $B$ (by a partition of $B$ we mean a family
$P=\\{A_{i}|A_{i}\in B,\,i\leq n$} of disjoint non-empty sets $A_{i}$ with
$\bigcup\limits_{i\leq n}A_{i}=D$).
If $L$ is an OML and $B=(D,\,\mathcal{B})$ is a Boolean algebra, Boolean sum
of $L$ and $B$ is the set of all functions $f:D\to L$ with the property that
there is a partition {$A_{i},\,i\leq n$} of $B$ such that $f$ is a constant on
any $A_{i},\,i\leq n$ (the “B step functions” on $D$ with values in $L$).
Obviously, Boolean sum of $Hor(L,\,MO_{2})$ and $B$ satisfies property (v). ∎

Let us pass to the considerations concerning states. Let us recall that
$s:L\to$ $[0,1]$ (resp. $s:L\to$ {$0,1$}) is called a state on $L$ (resp. a
two-valued state on $L$), if
(i) $s(1)=1$, and
(ii) $s(a\cup b)=s(a)+s(b)$ provided $a\leq b^{\prime}$.

Denote by $\mathcal{S}(L)$ (resp. $\mathcal{S}_{2}(L)$) the set of all states
on $L$ (resp. the set of all two-valued states on $L$). The characterization
of the state space of an OML has been obtained in [Shu] (for the interplay of
$\mathcal{S}(L)$ with algebraic properties of $L$ implicitely related to this
paper, see [NaPtRo]). A peculiar circumstance is that $\mathcal{S}(L)$ may be
empty (see [Gre]) that disqualifies certain OMLs for an interpretation as
quantum logics (see [PtPul] and [DvPu]). For a locally finite $L$, if this
phenomenon occurs (i.e, if $\mathcal{S}(L)=\emptyset$), it has to occur for a
finite subOML of $L$. This is shown in our next result. (It may be noted that
in the “hidden variable” interpretation (see [Gud]), if there is no hidden
variable on $L$ (i.e., if $\mathcal{S}_{2}(L)=\emptyset$), then there is no
hidden variable on a certain finite subOML of $L$).

######  Theorem 1.2.

Let $L\in\mathcal{LF}$. (i) If $\mathcal{S}(K)\neq\emptyset$ (resp.
$\mathcal{S}_{2}(K)\neq\emptyset$) for any finite subOML $K$ of $L$, then
$\mathcal{S}(L)\neq\emptyset$ (resp. $\mathcal{S}_{2}(L)\neq\emptyset$).
(ii) (A compactness theorem) Let $P\subseteq L$ and let $q\in L$. Suppose that
for any subOML $K$ of $L$ that is generated by a finite subset $F$ of $P$
there is a state $\tilde{s}\in\mathcal{S}_{2}(L)$ with $\tilde{s}(q)=1$ and
$\tilde{s}(k)=0$ for any $k\in F$. Then there is a state
$s\in\mathcal{S}_{2}(L)$ with $s(q)=1$ and $s(p)=0$ for any $p\in P$.

###### Proof.

(i) By a standard application of Tychonoff’s theorem on the product of compact
topological spaces, the set $[0,1]^{L}$ (resp.{$0,1$}L) of all functions
$f:L\to$ $[0,1]$ (resp. all functions $L\to$ {$0,1$}) is compact in the
topology of pointwise convergence. Let us consider, for each finite subOML $K$
of $L$, the set of all functions $g:L\to$ $[0,1]$ that are states on $K$ when
restricted to $K$. Denote this set of functions by $\mathcal{T}(K)$.
Obviously, each $\mathcal{T}(K)$ is compact, too. Since the family of all
finite subOMLs of $L$ is directed when ordered by inclusion, we see that
{$\mathcal{T}(K),\,K$ finite subOML of $L$} is a centered family of closed
subsets of $[0,1]^{L}$. By our assumption, each $\mathcal{T}(K)$ is non-void.
If $\mathcal{K}$ denotes the set of all finite subOMLs of $L$, then the
intersection $\bigcap\limits_{K\in\mathcal{K}}\mathcal{T}(K)$ is non-void (the
compactness of $[0,1]^{L}$ applies). Let
$s\in\bigcap\limits_{K\in\mathcal{K}}\mathcal{T}(K)$. Then
$s\in\mathcal{S}(L)$ and this completes the proof.
(ii) Suppose that $F$ is a finite subset of $P$ and suppose that $K$ is
generated by $F$. Let us set $\mathcal{V}(K)$ for all states
$\mathcal{S}_{2}(L)$ with $f(q)=1$ and $f(k)=0$ for any $k\in F$. Then
{$\mathcal{V}(K)$, $K$ generated by a finite subset of $P$} is a centered
family. By applying the previous approach of “let the compactness do it”, we
obtain a state $s\in\mathcal{S}_{2}(L)$ with the required properties.

∎

As expected, the local finiteness is essential in the above results.

######  Theorem 1.3.

Let $L(\mathbb{R}^{3})$ be the OML of all projections (linear subspaces) of
$\mathbb{R}^{3}$. Then $\mathcal{S}_{2}(L(\mathbb{R}^{3}))$ = $\emptyset$ but
each finite subOML $K$ of $L(\mathbb{R}^{3})$ is set-representable (and
therefore $K$ possesses an order determining set of two-valued states).

###### Proof.

It is well known (see [KoSpe], for a stronger version see [VorNav]) that
$\mathcal{S}_{2}(L(\mathbb{R}^{3}))=\emptyset$. On the other hand, if $K$ is a
subOML of $L(\mathbb{R}^{3})$ and $K$ contains $3$ atoms (three one-
dimensional subspaces) that are in a general configuration, meaning they are
linearly independent and no two atoms are orthogonal, then $K$ is infinite.
This can be easily calculated (or obtained as a consequence of the classic
result by A.F. Möbius - $K$ has so many directions that their normalizations
form a dense subset of the unit sphere in $\mathbb{R}^{3}$). The only
essentially different situation to be argued is the case when $K$ is generated
by $3$ atoms $a_{1}$, $a_{2}$ and $a_{3}$ such that $a_{1}$ is orthogonal to
the plane $a_{2}\lor a_{3}$.The OML $K$ is then generated by $a_{2}$ and
$a_{3}$. Then $K$ can be viewed as an epimorphic image of the free OML over
two generators. The latter free OML is known to be set-representable and it
consists of 96 elements (see e.g. [Kal] and [Har]). Since any epimorphic image
of a set-representable OML is again set-representable (see e.g. [May]), $K$ is
set-representable and therefore $K$ contains an order determining set of two-
valued states (see e.g. [Pt2]). This concludes the proof. ∎

It may be worth making a short comment, as a fact of a methodological matter,
that the local finiteness approach allows one to have less standard proofs of
the following classic results for Boolean algebras.

######  Theorem 1.4.

Let $B$ a Boolean algebra.
(i) If $A$ is a Boolean subalgebra of $B$ and $s\in\mathcal{S}(A)$, then $s$
can be extended to $B$ as a state.
(ii) If $a\neq 0$, then there is such a prime ideal $I$ on $B$ that $a\in I$.

###### Proof.

(i) (See e.g. [HoTar], among other proofs the Krein-Millman theorem provides
an apparently new proof strategy, see [BurPt]). Let $s\in\mathcal{S}(A)$. For
each finite Boolean subalgebra $K$ of $A$ it is easy to construct a state
$t:B\to$ $[0,1]$ that agrees with $s$ on $K$. So the (compact) set
$\mathcal{U}(K)=\\{t\in\mathcal{S}(B),\,t=s\text{ on }K\\}$ is non-void. If we
denote by $\mathcal{K}$ the set of all finite Boolean subalgebras of $A$, we
see that $\bigcap\limits_{K\in\mathcal{K}}\mathcal{U}(K)$ is non-empty. Take
$v\in\bigcap\limits_{K\in\mathcal{K}}\mathcal{U}(K)$. Then
$v\in\mathcal{S}(B)$ and $v$ extends $s$.

(ii) (The classical Stone Theorem). Consider the collection $\mathcal{G}$ of
all finite Boolean subalgebras of $B$ that contain the element $a$. If
$K\in\mathcal{G}$, then there is a two-valued state
$\tilde{s}\in\mathcal{S}_{2}(K)$ such that $\tilde{s}(a)=0$. By the
compactness argument used above, there is a state $s\in\mathcal{S}_{2}(B)$
with $s(a)=0$. Then $I=\\{a\in B|s(a)=0\\}$ is the required ideal.

∎

In the conclusion, let us ask if $\mathcal{LF}$ contains all set-representable
OMLs. Of course, the free OML over three generators is not set-representable.
In [Har] the author shows that this OML contains the free OML over countably
many generators and if the latter OML were set-representable, then so would be
all “small” OMLs. This is obviously not the case (see [Gre]). However, we can
answer the above formulated question to the negative in other ways
contributing thus to our theme and adding to Theorem 1.1. The first way
utilizes the so-called Kalmbach embedding of lattices. Recall that by [Kal]
each lattice $L$ can be lattice-theoretically embedded into an OML. Let us
denote it by $K(L)$. It can be shown (see [HarNav]) that $K(L)$ is always set-
representable. Hence if $L$ is not locally finite as a lattice, and such a
lattice is easy to find, then $K(L)$ is a set-representable OML that is not
locally finite. Another example related to Theorem 1.1(ii) could be obtained
by showing that the class $\mathcal{LF}$ is not closed under the formation of
the product of countably many (finite set-representable) OMLs. Indeed, it
suffices to find finite set-representable OMLs $L_{n}$, $n\in\mathbb{N}$, such
that $L_{n}$ is generated by three elements $a_{1}^{n},a_{2}^{n}$ and
$a_{3}^{n}$ and, moreover, the cardinality of $L_{n}$ is greater than or equal
to $n$ (one can for instance use the Kalmbach embedding again). Let us take
the Cartesian product $\prod\limits_{n\in\mathbb{N}}L_{n}$. Then the triple
$b_{i}\in\prod\limits_{n\in\mathbb{N}}L_{n},\,i\leq 3$, defined by
$b_{1}=(a_{1}^{n}),\ b_{2}=(a_{2}^{n}),\ b_{3}=(a_{3}^{n}),\,n\in\mathbb{N}$,
generates an infinite subset of $\prod\limits_{n\in\mathbb{N}}L_{n}$ (the
$n$-th coordinate “supplies” at least $n$ elements). Since the product of set-
representable OMLs is again set-representable (see e.g. [May]), we see that
the product $\prod\limits_{n\in\mathbb{N}}L_{n}$ is the example we looked for.
In the final note, let us observe that the analogous questions on the local
finiteness are pertinent for the recently studied orthomodular structures
endowed with an intrinsic symmetric difference. The lattices of this class are
denoted by ODLs (the OMLs with an “XOR” operation; see [BiNav], [BurPt],
[Mat], [MatPt], [Su], etc.). The results similar to our findings of Theorem
1.1 and Theorem 1.2 can be also proved for ODLs. A remarkable result related
to the previous paragraph has already been obtained in [Su]: There is an
infinite set-representable ODL that is generated by finitely many elements. So
the ODLs do not have to be locally finite even for the set-representable ones.

## References

  * [BiNav] BIKCHENTAEV, A.—NAVARA, M.: States on symmetric logics, Mathematica Slovaca 66 (2016), 359–366.
  * [BurPt] BUREŠOVÁ, D.—PTÁK, P.: Quantum logics that are symmetric-difference-closed, International Journal of Theoretical Physics 60 (2021), 3919–3926.
  * [DvPu] DVUREČENSKIJ, A.—PULMANNOVÁ, S.: New Trends in Quantum Structures, Kluwer Academic Publishers, Dordrecht (2000).
  * [Gre] GREECHIE, R.: Orthomodular lattices admitting no states, Journal of Combinatorial Theory A 10 (1971), 119–132.
  * [Gud] GUDDER, S.P.: Stochastic Methods in Quantum Mechanics, North-Holland, Amsterdam (1979).
  * [Har] HARDING, J.: The free orthomodular lattice on countably many generators is a subalgebra of the free orthomodular lattice on three generators, Algebra Universalis 48 (2002), 171–182.
  * [HarNav] HARDING, J.—NAVARA, M.: Embeddings into ortomodular spaces with given centers, state spaces and automorphism groups, Order 17 (2000), 239–254.
  * [HoTar] HORN, A.—TARSKI, A.: Measures in Boolean algebras, Trans. Amer. Math. Soc. 64 (1948), 467–497.
  * [Kal] KALMBACH, G., Orthomodular Lattices, Academic Press (1983).
  * [KoSpe] KOCHEN, S.—SPECKER, E.: The problem of hidden variables in quantum mechanics, Indiana University Mathematics Journal 17 (1968), 59–87.
  * [Mat] MATOUŠEK, M.: Orthocomplemented lattices with a symmetric difference, Algebra Universalis 60 (2009), 185–215.
  * [MatPt] MATOUŠEK, M.—PTÁK, P.: Orthocomplemented posets with a symmetric difference, Order 26 (2009), 1–21.
  * [May] MAYET, R.: Varieties of orthomodular lattices related to states, Algebra Universalis 20 (1985), 368–396.
  * [NaPtRo] NAVARA, M.—PTÁK, P.—ROGALEWICZ, V.: Enlargements of quantum logics, Pacific Journal of Mathematics 135 (1988), 361–369.
  * [Pt1] PTÁK, P.: Summing of Boolean algebras and logics, Demonstratio Math. 19 (1986), 349–358.
  * [Pt2] PTÁK, P.: Concrete quantum logics, International Journal of Theoretical Physics 39 (2000), 827–839.
  * [PtPul] PTÁK, P.—PULMANNOVÁ, S.: Orthomodular Structures as Quantum Logics, Kluwer Academic Publishers, Dordrecht (1991).
  * [Shu] SHULTZ, F.W.: A characterization of state spaces of orthomodular lattices, Journal of Combinatorial Theory A 17 (1974), 317–328.
  * [Su] SU, J.: On set-representable orthocomplemented difference lattices, Order 37 (2020), 621–636.
  * [VorNav] VORÁČEK, V.—NAVARA, M.: Generalized Kochen-Specker theorem in three dimensions, Foundation of Physics 51 (2021).