††thanks: These authors contributed equally to this work.††thanks: These
authors contributed equally to this work.††thanks: These authors contributed
equally to this work.

# Experimental Masking of Real Quantum States

Rui-Qi Zhang  Key Laboratory of Quantum Information,University of Science and
Technology of China, CAS, Hefei 230026, P. R. China  CAS Center For Excellence
in Quantum Information and Quantum Physics  Zhibo Hou  Key Laboratory of
Quantum Information,University of Science and Technology of China, CAS, Hefei
230026, P. R. China  CAS Center For Excellence in Quantum Information and
Quantum Physics  Zihao Li  State Key Laboratory of Surface Physics and
Department of Physics, Fudan University, Shanghai 200433, China  Institute for
Nanoelectronic Devices and Quantum Computing, Fudan University, Shanghai
200433, China  Center for Field Theory and Particle Physics, Fudan University,
Shanghai 200433, China  Huangjun Zhu<EMAIL_ADDRESS>State Key
Laboratory of Surface Physics and Department of Physics, Fudan University,
Shanghai 200433, China  Institute for Nanoelectronic Devices and Quantum
Computing, Fudan University, Shanghai 200433, China  Center for Field Theory
and Particle Physics, Fudan University, Shanghai 200433, China  Guo-Yong Xiang
<EMAIL_ADDRESS>Key Laboratory of Quantum Information,University of
Science and Technology of China, CAS, Hefei 230026, P. R. China  CAS Center
For Excellence in Quantum Information and Quantum Physics  Chuan-Feng Li  Key
Laboratory of Quantum Information,University of Science and Technology of
China, CAS, Hefei 230026, P. R. China  CAS Center For Excellence in Quantum
Information and Quantum Physics  Guang-Can Guo  Key Laboratory of Quantum
Information,University of Science and Technology of China, CAS, Hefei 230026,
P. R. China  CAS Center For Excellence in Quantum Information and Quantum
Physics

###### Abstract

Masking of quantum information is a way of hiding information in correlations
such that no information is accessible to any local observer. Although the set
of all quantum states as a whole cannot be masked into bipartite correlations
according to the no-masking theorem, the set of real states is maskable and is
a maximal maskable set. In this work, we experimentally realize a masking
protocol of the real ququart by virtue of a photonic quantum walk. Our
experiment clearly demonstrates that quantum information of the real ququart
can be completely hidden in bipartite correlations of two-qubit hybrid
entangled states, which are encoded in two different degrees of freedom of a
single photon. The hidden information is not accessible from each qubit alone,
but can be faithfully retrieved with a fidelity of about $99\%$ from
correlation measurements. By contrast, any superset of the set of real density
matrices cannot be masked.

_Introduction._ —Hiding information in correlations is a useful idea that
plays crucial roles in a wide spectrum of subjects. Although classical
information can be completely hidden in quantum correlations of a bipartite
system, quantum information of an arbitrary state cannot be completely hidden
in a similar way according to the no-hiding theorem [1] and no-masking theorem
[2]. These no-go theorems offer valuable insights on the power and limitation
of quantum information processing, which are complementary to the famous no-
cloning theorem [3, 4, 5] and no-broadcasting theorem [6]. Meanwhile, they are
of intrinsic interest to many active research areas, including quantum
communication, quantum secret sharing [7, 8, 9], information scrambling, and
the black-hole information paradox [10, 11, 12, 13].

Notwithstanding the limitations set by the no-hiding and no-masking theorems
[2], it is possible to hide and mask quantum information in certain restricted
sets of quantum states [14, 15, 16, 17, 18, 19] (see also Ref. [20] for
information masking in the multipartite scenario). Notably, very recently Ref.
[19] showed that the no-hiding and no-masking theorems break down in real
quantum mechanics, in sharp contrast with complex quantum mechanics.
Information about real density matrices (with respect to the computational
basis) can be completely hidden in bipartite correlations. Moreover, the set
of real states is a maximal maskable set in the sense that it is not contained
in any other maskable set. These results give a twist to the problem of
information masking, which is of intrinsic interest to the resource theory of
imaginarity [21, 22] and foundational studies on quantum mechanics [23, 24,
25] in addition to the research areas mentioned above. Despite these
theoretical progresses, experimental works on quantum information masking are
quite rare [26, 27] (see Refs. [28, 29] on the test of the no-hiding theorem).
Moreover, all masking protocols implemented so far are restricted to qubit
systems, which is a severe limitation.

In this work, we experimentally realize for the first time the masking
protocol of the real ququart proposed in Ref. [19] using a photonic system. To
achieve this goal, we devise a four-step photonic quantum walk to realize the
desired masking isometry, which turns any pure state of the real ququart into
a two-qubit hybrid entangled state. Our experimental results clearly
demonstrate that quantum information of the real ququart can be completely
hidden in bipartite correlations, and no information can be retrieved from
each subsystem alone. In addition, the encoded quantum information can be
faithfully retrieved from correlation measurements. By contrast, the output
state associated with any input state that is not real cannot be maximally
entangled, so partial information has to leak to each subsystem. Moreover, the
concurrence of the output state is determined by the robustness of imaginarity
of the input state, which is of key interest to the resource theory of
imaginarity [21, 22]. Here, we encode two-qubit hybrid entangled states in two
different degrees of freedom (DoFs) of a single photon [30, 31], but it can be
generalized to two-photon two-qubit states by simply cascading the protocol of
quantum state fission [32] after our photonic quantum walk.

_Masking of quantum information._ —A set $\mathscr{S}$ of quantum states on
the Hilbert space $\mathcal{H}$ of dimension $d$ is maskable if there exists
an isometry $M$ from $\mathcal{H}$ to $\mathcal{H}_{A}\otimes\mathcal{H}_{B}$,
such that both $\tau_{A}={\rm tr}_{B}(M\rho M^{\dagger})$ and $\tau_{B}={\rm
tr}_{A}(M\rho M^{\dagger})$ are independent of $\rho$ for all
$\rho\in\mathscr{S}$ [2]. In this case, no information about the original
state $\rho$ can be retrieved from subsystem $A$ or $B$ alone, and all
information spreads over the bipartite correlations between $A$ and $B$. Let
$\mathscr{D}(\mathcal{H})$ and $\mathscr{P}(\mathcal{H})$ be the sets of all
density matrices and all pure states (rank-1 projectors) on $\mathcal{H}$; let
$\mathscr{D}^{\mathrm{R}}(\mathcal{H})$ and
$\mathscr{P}^{\mathrm{R}}(\mathcal{H})$ be the sets of all real density
matrices and all real pure states with respect to the computational basis
$\\{|j\rangle\\}_{j=0}^{d-1}$ of $\mathcal{H}$. The no-masking theorem states
that $\mathscr{D}(\mathcal{H})$ and $\mathscr{P}(\mathcal{H})$ are not
maskable [2]. However, this theorem does not apply to restricted sets of
quantum states in general. Notably, the set
$\mathscr{D}^{\mathrm{R}}(\mathcal{H})$ is maskable and is actually a maximal
maskable subset of $\mathscr{D}(\mathcal{H})$ [19]. Here a maskable set is
maximal if it is not a proper subset of any other maskable set. Such maximal
maskable sets are particularly appealing because they reflect the potential
and limitation of quantum information masking as well as the distinction
between quantum information processing and classical information processing.

Since any mixed state is a convex mixture of pure states, masking of
$\mathscr{D}^{\mathrm{R}}(\mathcal{H})$ is equivalent to the masking of
$\mathscr{P}^{\mathrm{R}}(\mathcal{H})$. According to Ref. [19], the set
$\mathscr{P}^{\mathrm{R}}(\mathcal{H})$ can be masked by virtue of a set
$\\{U_{j}\\}_{j=1}^{d-1}$ of $d-1$ unitary Hurwitz-Radon (HR) matrices acting
on $\mathcal{H}_{A}$ [33, 34], which is characterized by the equation

$U_{j}U_{k}+U_{k}U_{j}=-2\delta_{jk}\mathbbm{1}_{A}.$ (1)

Let $|\Phi\rangle=\sum_{r=0}^{m-1}|rr\rangle$ be the canonical maximally
entangled state in $\mathcal{H}_{A}\otimes\mathcal{H}_{B}$, assuming that both
$\mathcal{H}_{A}$ and $\mathcal{H}_{B}$ have dimension $m$. Let
$U_{0}:=\openone_{A}$ and define

$|\Phi_{j}\rangle:=(U_{j}\otimes\openone_{B})|\Phi\rangle,\quad
j=0,1,\ldots,d-1.$ (2)

Then the isometry $M$ from $\mathcal{H}$ to
$\mathcal{H}_{A}\otimes\mathcal{H}_{B}$ defined by the map
$j\mapsto|\Phi_{j}\rangle$ is a masker for
$\mathscr{P}^{\mathrm{R}}(\mathcal{H})$ and
$\mathscr{D}^{\mathrm{R}}(\mathcal{H})$. Note that, for any normalized vector
$\bm{c}:=(c_{0},c_{1},...,c_{d-1})$, the ket
$|\psi(\bm{c})\rangle=\sum_{j=0}^{d-1}{c_{j}|j\rangle}$ is mapped to
$[U(\bm{c})\otimes\openone_{B}]|\Phi\rangle$ under the masker $M$, where
$U({\bm{c}}):=\sum_{j}{c_{j}U_{j}}$. Moreover, $U(\bm{c})$ is a unitary
operator for any normalized real vector $\bm{c}$ thanks to the characteristic
of the HR matrices presented in Eq. (1).

_Masking of the real ququart._ —In the case of $d=4$, to mask
$\mathscr{D}^{\mathrm{R}}(\mathcal{H})$, we need a set of three HR matrices. A
simple example can be constructed from Pauli matrices:
$\\{U_{j}\\}_{j=1}^{3}:=\\{\mathrm{i}\sigma_{z},\mathrm{i}\sigma_{x},\mathrm{i}\sigma_{y}\\}$,
which can also be written as $\\{\mathrm{i}Z,\mathrm{i}X,\mathrm{i}Y\\}$. With
this choice, we can define an isometry
$M:\mathcal{H}\mapsto\mathcal{H}_{A}\otimes\mathcal{H}_{B}$ as follows:

$M|j\rangle=-\mathrm{i}|\Phi_{j}\rangle:=-\mathrm{i}(U_{j}\otimes\mathbbm{1}_{B})|\Phi\rangle,\quad
j=0,1,2,3,$ (3)

where $|\Phi\rangle:=(|00\rangle+|11\rangle)/\sqrt{2}$ and
$U_{0}:=\mathbbm{1}_{A}$. Here the choice of the phase factor $-\mathrm{i}$ is
not essential but is convenient for the following discussion. Under this
isometry, the computational basis of $\mathcal{H}$ is mapped to the two-qubit
magic basis [35]; accordingly, all real pure states are mapped to maximally
entangled states. So the isometry $M$ is a masker for
$\mathscr{P}^{\mathrm{R}}(\mathcal{H})$ and
$\mathscr{D}^{\mathrm{R}}(\mathcal{H})$ [19]. By contrast, the image of any
state in
$\mathscr{D}(\mathcal{H})\setminus\mathscr{P}^{\mathrm{R}}(\mathcal{H})$ is
not maximally entangled. Moreover, when $\rho$ is pure, the concurrence of the
output state $M\rho M^{\dagger}$ is determined by the robustness of
imaginarity of $\rho$ [19],

$C(M\rho M^{\dagger})=\sqrt{1-\mathscr{I}_{\mathrm{R}}^{2}(\rho)},$ (4)

where
$\mathscr{I}_{R}(\rho)=\|\rho-\rho^{\mathrm{T}}\|_{1}/2=\sqrt{1-\operatorname{tr}(\rho\rho^{\mathrm{T}})}$
[22].

Figure 1:  Schematic diagram of quantum information masking (upper plot) and
realization of the masker $M$ of the real ququart based on a quantum walk
(lower plot). The coin operators featuring in the figure are given by
$X=\left(\begin{smallmatrix}0&1\\\ 1&0\end{smallmatrix}\right)$,
$Z=\left(\begin{smallmatrix}1&0\\\ 0&-1\end{smallmatrix}\right)$,
${C}_{1}=\frac{1}{\sqrt{2}}\left(\begin{smallmatrix}\mathrm{i}&1\\\
-\mathrm{i}&1\end{smallmatrix}\right)$, and
${C}_{2}=\frac{1}{\sqrt{2}}\left(\begin{smallmatrix}1&\mathrm{i}\\\
-1&\mathrm{i}\end{smallmatrix}\right)$.  Figure 2:  Experimental setup. The
heralded single-photon source (labeled by S) on top is realized by spontaneous
parametric down-conversion in a type-I $\beta$-barium-borate (BBO) crystal.
The rest setup consists of three modules: state-preparation module (labeled by
P), masking module (labeled by Mask), and measurement module (labeled by
Meas). The masking module implements the masker defined in Eq. (3) according
to the quantum-walk scheme illustrated in Fig. 1. The photon’s spatial modes
labeled at the bottom correspond to the walker’s positions defined in Fig. 1.
Two combinations of wave plates of the form HWP-QWP-HWP in the masking module
realize the coin operators $C_{1}$ and $C_{2}$ defined in the caption of Fig.
1. HWP: half-wave plate; QWP: quarter-wave plate; BD: beam-displacer; PBS:
polarizing beam-splitter; SPCM: single-photon counting module; K9: K9 glass
plate.

To realize the masker $M$ of the real ququart, here we design a simple scheme
based on a quantum walk as illustrated in Fig. 1, which can be implemented in
a photonic system. In a quantum walk [36, 37, 38, 39], the state $|x,c\rangle$
of the walker-coin joint system is characterized by two indices $x$ and $c$,
where $x=\dots,-1,0,1,\dots$ denotes the position of the walker on a one-
dimensional chain, and $c=0,1$ labels the state of the coin qubit, which
determines the moving direction of the walker in the next step. Each step of
the quantum walk can be described by a unitary operator of the form
$U(t)=\mathcal{T}C(t)$, where $C(t)=\sum_{x}|x\rangle\langle x|\otimes
C(x,t)$, with $C(x,t)$ being position-dependent coin operators, and
$\mathcal{T}$ is the conditional translation operator,

$\mathcal{T}=\sum_{x}{(|x-1\rangle\langle x|\otimes|0\rangle\langle
0|+|x+1\rangle\langle x|\otimes|1\rangle\langle 1|)}.$ (5)

The ququart can be encoded into the initial state (with $t$ = 0) of the
walker-coin system using the path degree of freedom (DoF). To be concrete, a
general pure state $|\psi\rangle=\sum_{j=0}^{3}{a_{j}|j\rangle}$ of the
ququart with $\sum_{j=0}^{3}{|a_{j}|^{2}}=1$ is encoded as follows:

$|\Psi\rangle=(a_{0}|-3\rangle+a_{1}|-1\rangle+a_{2}|1\rangle+a_{3}|3\rangle)\otimes|1\rangle.$
(6)

After the first step of the quantum walk, the joint state evolves into

$|\Psi_{1}\rangle=a_{0}|-2,1\rangle+a_{1}|-2,0\rangle+a_{2}|2,1\rangle+a_{3}|2,0\rangle.$
(7)

After the second step, the joint state evolves into

$\displaystyle|\Psi_{2}\rangle=$
$\displaystyle\frac{\mathrm{i}a_{0}+a_{1}}{\sqrt{2}}|-3,0\rangle+\frac{\mathrm{i}a_{0}-a_{1}}{\sqrt{2}}|-1,1\rangle$
$\displaystyle+\frac{a_{2}+\mathrm{i}a_{3}}{\sqrt{2}}|1,0\rangle+\frac{a_{2}-\mathrm{i}a_{3}}{\sqrt{2}}|3,1\rangle.$
(8)

Following a similar procedure, the final state after the quantum walk reads

$\displaystyle|\Psi^{\prime}\rangle=$
$\displaystyle\frac{a_{1}-\mathrm{i}a_{0}}{\sqrt{2}}|1,0\rangle-\frac{a_{1}+\mathrm{i}a_{0}}{\sqrt{2}}|-1,1\rangle$
$\displaystyle+\frac{a_{2}-\mathrm{i}a_{3}}{\sqrt{2}}|1,1\rangle+\frac{a_{2}+\mathrm{i}a_{3}}{\sqrt{2}}|-1,0\rangle$
$\displaystyle=$
$\displaystyle-\mathrm{i}\bigl{(}a_{0}|\Phi_{0}^{\prime}\rangle+a_{1}|\Phi_{1}^{\prime}\rangle+a_{2}|\Phi_{2}^{\prime}\rangle+a_{3}|\Phi_{3}^{\prime}\rangle\bigr{)},$
(9)

where

$\displaystyle|\Phi_{0}^{\prime}\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}(|+1,0\rangle+|-1,1\rangle),$ (10)
$\displaystyle|\Phi_{1}^{\prime}\rangle$
$\displaystyle=\frac{\mathrm{i}}{\sqrt{2}}(|+1,0\rangle-|-1,1\rangle),$
$\displaystyle|\Phi_{2}^{\prime}\rangle$
$\displaystyle=\frac{\mathrm{i}}{\sqrt{2}}(|+1,1\rangle+|-1,0\rangle),$
$\displaystyle|\Phi_{3}^{\prime}\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}(|+1,1\rangle-|-1,0\rangle).$

Now $|\Psi^{\prime}\rangle$ can be regarded as a two-qubit state on the
Hilbert space $\mathcal{H}_{A}\otimes\mathcal{H}_{B}$, where $A$ denotes the
effective walker qubit composed of positions $+1$ and $-1$, and $B$ denotes
the coin qubit. Here $\\{|\Phi_{j}^{\prime}\rangle\\}_{j=0}^{3}$ is exactly
the two-qubit magic basis in $\mathcal{H}_{A}\otimes\mathcal{H}_{B}$, so the
quantum walk illustrated in Fig. 1 indeed realizes the masker $M$ defined in
Eq. (3).

_Experimental setup._ —The setup for masking the real ququart is shown in Fig.
2. It contains four modules: a heralded single-photon source, a state-
preparation module, a masking module, and a measurement module. Here we use
the path DoF to encode the real ququart and the polarization DoF to encode the
coin state employed in the quantum walk within the masking module.

In the module of single-photon source, an ultraviolet laser with central
wavelength of 404nm is used to pump a type-I phase-matched beta-barium borate
(BBO) crystal to generate a photon pair in a product (polarization) state via
spontaneous parametric down-conversion [40]. One photon is measured as a
trigger to herald the generation of its twin photon, which is then transmitted
to the state-preparation module.

In the state-preparation module, a polarization beam splitter (PBS) first
prepares the photon in the horizontal-polarization state $|H\rangle$ (in
contrast with the vertical-polarization state $|V\rangle$). Then two beam
displacers (BDs) and three half-wave plates (HWPs) transform the photon state
into

$(a_{0}|-3\rangle+a_{2}|1\rangle)\otimes|H\rangle+(a_{1}|-1\rangle+a_{3}|3\rangle)\otimes|V\rangle,$
(11)

where the four real parameters $a_{i}$ for $i=0,...,3$ are controlled by three
HWPs (see Appendix A). Finally, we prepare the ququart state

$(a_{0}|-3\rangle+a_{1}|-1\rangle+a_{2}|1\rangle+a_{3}|3\rangle)\otimes|V\rangle$
(12)

of the form in Eq. (6) by inserting a HWP at 45∘ (which realizes the $X$ or
NOT gate) on paths $-3$ and $1$ to turn $|H\rangle$ into $|V\rangle$. Note
that the first two $X$ gates on paths $-1$ and $3$ in the masking module in
Fig. 1 can also be implemented by a HWP at 45∘; moreover, the four paths can
share a same HWP placed at the beginning of the masking module in Fig. 2. The
removable quarter-wave plate (QWP) in the preparation module is used only to
prepare pure states with complex coefficients (see Appendix B).

After state preparation, the ququart is sent into the masking module, which
realizes the quantum walk illustrated in Fig. 1. The two QWP-HWP-QWP
combinations realize the coin operators $C_{1}$ and $C_{2}$, respectively,
while the $0^{\circ}$ HWPs realize the $Z$ gates. The output of the masking
module is a hybrid entangled state between the path qubit and the polarization
qubit of the heralded photon.

The measurement module consists of two QWP-HWP combinations designed to
control the measurement settings employed for measuring the path qubit and
polarization qubit (see Appendix C). In addition, a K9 plate is used to
compensate for different path lengths among the interference arms. Finally,
the heralded photon is collected by four single-photon counting modules
(SPCMs).

_Experimental results._ —To demonstrate the performance of the masking
protocol realized using the photonic quantum walk as described above, we
select the following four probe states:

$\begin{gathered}|0\rangle,\quad\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle),\\\
\frac{1}{\sqrt{3}}(|0\rangle+|1\rangle+|2\rangle),\quad\frac{1}{2}(|0\rangle+|1\rangle+|2\rangle+|3\rangle),\end{gathered}$
(13)

which form a complete but non-orthogonal basis in $\mathcal{H}$. To verify
that the masking module indeed transforms these input states according to the
theoretical prediction described in Eqs. (6)-(10), we perform fidelity
estimation based on quantum state verification (QSV) [41, 42, 43] (see
Appendix D) on the output states from the masking module. It turns out the
fidelity between each output state and the ideal masked state is above 98%, as
shown in Fig. 3. To further certify that no information of the input state can
be inferred from each reduced state of the output state, we then perform
quantum state tomography on the path and polarization qubits, respectively. In
each case the average purity obtained in the experiment is very close to 0.5
as shown in Fig. 3, which implies that the reduced states of both subsystems
are nearly completely mixed and little information about the original state
can be retrieved from each subsystem alone; see Appendix E for more details.
These experimental results clearly demonstrate that the masking module in Fig.
2 successfully realizes the masker $M$ in Eq. (3) within small experimental
errors.

Figure 3:  Experimental characterization of the masking module in Fig. 2. Each
blue bar represents the fidelity between the actual output state and the ideal
output state associated with each probe state as marked at the bottom of the
figure. Each red bar represents the average purity of the two reduced states
of the output state. The number in the parentheses above the blue (red) bar
indicates the $95\%$ confidence interval of each fidelity estimator (the
standard deviation of each purity estimator). The methods for quantifying the
confidence interval and the standard deviation are detailed in Appendices D
and E.

Next, we show that the information of the input state can be faithfully
retrieved from the bipartite correlations of the output state of the masker
$M$. To be concrete, the density matrix of any real state can be reconstructed
from the correlation statistics of nine product Pauli measurements
$\sigma_{j}\otimes\sigma_{k}$ for $j,k=x,y,z$ on the output state. In the
experiment, each tensor product $\sigma_{j}\otimes\sigma_{k}$ is measured 4000
times to determine its mean value. The specific decoding method is detailed in
Appendix F. As an example, the decoding result on the input state
$(|0\rangle+|1\rangle+|2\rangle+|3\rangle)/2$ is shown in Fig. 4, and the
decoding fidelity is 98.9%, which is consistent with the result of QSV shown
in Fig. 3.

Figure 4:  Experimental decoding of the masked state from correlation
measurements. Here the input state is given by
$(|0\rangle+|1\rangle+|2\rangle+|3\rangle)/2$. (a) Mean values
$\langle\sigma_{i}\otimes\sigma_{j}\rangle$ for $i,j\in\\{x,y,z\\}$. (b)
Matrix elements of the reconstructed density matrix (only real part is shown
since the imaginary part is zero by reconstruction; see Appendix F). Bars
without color represent the original input state, while bars with color
represent the reconstructed state. Figure 5:  Relation between the concurrence
of the output state and the robustness of imaginarity of the input state,
which has the form $(|0\rangle+e^{i\phi}|1\rangle)/{\sqrt{2}}$ with
$0\leq\phi\leq\pi/2$. The error bars represent the standard deviations of the
concurrence estimations, which are determined via re-sampling (see Appendix
E).

Finally, we verify the relation between the concurrence of the output state
and the robustness of imaginarity of the input state as presented in Eq. (4).
To be concrete, we choose input states of the form

$|\psi(\phi)\rangle=\frac{1}{\sqrt{2}}(|0\rangle+e^{\mathrm{i}\phi}|1\rangle),\quad
0\leq\phi\leq\frac{\pi}{2}.$ (14)

The robustness of imaginarity of the input state and the concurrence of the
output state are respectively given by

$\\!\\!\mathscr{I}_{R}(|\psi(\phi)\rangle)=\sin\phi,\quad
C(M|\psi(\phi)\rangle)=\cos\phi.$ (15)

In the experiment, the concurrence is determined by virtue of the formula
$\mathcal{C}(|\Psi\rangle):=\sqrt{2(1-\operatorname{tr}(\varrho_{\mathrm{P}}^{2}))}$,
where $\varrho_{\mathrm{P}}$ is the reduced state of the path qubit of the
output state as determined by quantum state tomography (see Appendix E). The
experimental results shown in Fig. 5 agree very well with the theoretical
prediction, which implies that partial information of the input state is
accessible to each subsystem once the robustness of imaginarity becomes
nonzero. These results also corroborate the conclusion that
$\mathcal{D}^{\mathrm{R}}(\mathcal{H})$ is a maximal maskable set.

_Summary._ —We experimentally realized a masking protocol of the real ququart
using a photonic quantum walk. This is the first experiment on quantum
information masking beyond a qubit system. Our experiment clearly demonstrates
that quantum information of the real ququart can be completely hidden in
bipartite correlations of hybrid entangled states, which is not accessible to
each subsystem alone, but can be faithfully retrieved from correlation
measurements. By contrast, any superset of the set of real density matrices
cannot be masked. Furthermore, the entanglement of the output state is tied to
the robustness of imaginarity of the input state. These results manifest a
sharp distinction between real quantum mechanics and complex quantum mechanics
in hiding and masking quantum information. Moreover, they offer valuable
insights on the potential and limitation of quantum information masking, which
are of intrinsic interest to many active research areas.

It should be pointed out that the two DoFs sharing the masked information in
our experiment reside in the same location (photon), and the meaning of
“masking” is slightly different from the common literature. Here the
information is masked into the correlations between two DoFs instead of two
space-separated particles. Nevertheless, the underlying mathematical
structures in the two scenarios are identical. So the main conclusions
obtained in our proof-of-principle experiment should apply to both scenarios.
Also, in principle our experiment can be generalized to realize quantum
information masking in two photons by cascading the protocol of quantum state
fission [32] after the photonic quantum walk.

## ACKNOWLEDGMENTS

The work at the University of Science and Technology of China is supported by
the National Natural Science Foundation of China (Grants Nos. 61905234,
11974335, 11574291, and 11774334), the Key Research Program of Frontier
Sciences, CAS (Grant No. QYZDYSSW-SLH003) and the Fundamental Research Funds
for the Central Universities (Grant No. WK2470000026). The work at Fudan
University is supported by the National Natural Science Foundation of China
(Grant No. 11875110) and Shanghai Municipal Science and Technology Major
Project (Grant No. 2019SHZDZX01).

## APPENDIX A: Preparation of pure states with real coefficients

Figure A1:  The preparation module of the experimental setup.

In this section we show that the state-preparation module in Fig. 2 in the
main text (also shown in Fig. A1) can prepare an arbitrary pure state of the
real ququart. Suppose the angles of the optical axes of H1, H2, and H3 in Fig.
A1 are $h_{1}$, $h_{2}$, and $h_{3}$ (with respect to the horizontal
direction), respectively. The input heralded photon is initially prepared in
the state $|1\rangle\otimes|H\rangle$, which is turned into the following
state

$|1\rangle\otimes[\cos(2h_{1})|H\rangle+\sin(2h_{1})|V\rangle]$ (A1)

by H1. The BD after H1 coherently routes the heralded photon to paths $-3$ and
$1$ according to the polarization state of the heralded photon, which yields
the state

$|-3\rangle\otimes\cos(2h_{1})|H\rangle+|1\rangle\otimes\sin(2h_{1})|V\rangle.$
(A2)

Then H2 and H3 transform the photon state into

$\displaystyle|-3\rangle\otimes\bigl{[}\cos(2h_{1})\cos(2h_{2})|H\rangle+\cos(2h_{1})\sin(2h_{2})|V\rangle\bigr{]}$
$\displaystyle+|1\rangle\otimes\bigl{[}\sin(2h_{1})\sin(2h_{3})|H\rangle-\sin(2h_{1})\cos(2h_{3})|V\rangle\bigr{]}.$
(A3)

Next, the second BD transforms the state of the heralded photon into

$\displaystyle\bigl{[}\cos(2h_{1})\cos(2h_{2})|-3\rangle+\sin(2h_{1})\sin(2h_{3})|1\rangle\bigr{]}\otimes|H\rangle$
$\displaystyle+$
$\displaystyle\bigl{[}\cos(2h_{1})\sin(2h_{2})|-1\rangle-\sin(2h_{1})\cos(2h_{3})|3\rangle\bigr{]}\otimes|V\rangle.$
(A4)

Finally, by inserting $45^{\circ}$ HWPs on paths $-3$ and $1$, we can prepare
the following state,

$\displaystyle\bigl{[}$
$\displaystyle\cos(2h_{1})\cos(2h_{2})|-3\rangle+\cos(2h_{1})\sin(2h_{2})|-1\rangle$
(A5)
$\displaystyle+\sin(2h_{1})\sin(2h_{3})|1\rangle-\sin(2h_{1})\cos(2h_{3})|3\rangle\bigr{]}\otimes|V\rangle,$

which has the form of Eq. (6) in the main text. Incidentally, this operation
can also be performed in the masking module, as pointed out in the main text.
To prepare the state in Eq. (6), we need to choose the parameters $h_{1}$,
$h_{2}$, and $h_{3}$ so as to satisfy the following equations:

$\displaystyle\cos(2h_{1})\cos(2h_{2})$ $\displaystyle=a_{0},$ (A6)
$\displaystyle\cos(2h_{1})\sin(2h_{2})$ $\displaystyle=a_{1},$
$\displaystyle\sin(2h_{1})\sin(2h_{3})$ $\displaystyle=a_{2},$
$\displaystyle-\sin(2h_{1})\cos(2h_{3})$ $\displaystyle=a_{3}.$

## APPENDIX B: Preparation of pure states with complex coefficients

In this section we show that the state-preparation module in Fig. 2 in the
main text (also shown in Fig. A1) can also prepare certain pure states with
complex coefficients as presented in Eq. (14) in the main text. By inserting
Q1 into path $-3$ as marked in Fig. A1, we can prepare the states defined in
Eq. (14). Such pure states with complex coefficients are required to
demonstrate the relation between the concurrence of the output state and the
robustness of imaginarity of the input state as presented in Eq. (4) in the
main text.

To be specific, denote the angles of the optical axes of H1 and Q1 from the
horizontal direction by $h_{1}$ and $q_{1}$. Now we set $q_{1}=45^{\circ}$ and
$h_{1}=0^{\circ}$, then the state of the heralded photon after Q1 reads [44]

$\displaystyle|-3\rangle\otimes\frac{1}{\sqrt{2}}\bigl{\\{}[\cos(45^{\circ}-2h_{2})+\mathrm{i}\sin(45^{\circ}-2h_{2})]|H\rangle$
$\displaystyle+[\cos(45^{\circ}-2h_{2})-\mathrm{i}\sin(45^{\circ}-2h_{2})]|V\rangle\bigr{\\}}.$
(A7)

If we set $h_{2}=\frac{\phi}{4}+22.5^{\circ}$ (here $\phi$ is the parameter
appearing in Eq. (14) in the main text), then the photon state after Q1
reduces to

$|-3\rangle\otimes\frac{1}{\sqrt{2}}(|H\rangle+\mathrm{e}^{\mathrm{i}\phi}|V\rangle)$
(A8)

up to a global phase. After the action of the second BD, the photon state
becomes

$\frac{1}{\sqrt{2}}(|-3\rangle|H\rangle+\mathrm{e}^{\mathrm{i}\phi}|-1\rangle|V\rangle).$
(A9)

Finally, by inserting a 45∘ HWP (which realizes the $X$ or NOT gate) on paths
$-3$ and $1$, we can prepare the following state

$\frac{1}{\sqrt{2}}(|-3\rangle+\mathrm{e}^{\mathrm{i}\phi}|-1\rangle)\otimes|V\rangle,$
(A10)

which agrees with Eq. (14) in the main text. Note that this HWP can also be
considered as a part of the masking module.

## APPENDIX C: Implementation of local projective measurements

Figure A2:  The measurement module of the experimental setup.

In this section, we show that the measurement module in Fig. 2 (also shown in
Fig. A2) can perform an arbitrary local projective measurement on the path-
polarization two-qubit system. Suppose we want to perform the projective
measurement onto the product basis

$\\{|\varphi_{0}\rangle|\psi_{0}\rangle,|\varphi_{0}\rangle|\psi_{1}\rangle,|\varphi_{1}\rangle|\psi_{0}\rangle,|\varphi_{1}\rangle|\psi_{1}\rangle\\},$
(A11)

where

$\displaystyle|\varphi_{0}\rangle$
$\displaystyle=\cos\gamma|0\rangle+\mathrm{e}^{\mathrm{i}\zeta}\sin\gamma|1\rangle,$
(A12) $\displaystyle|\varphi_{1}\rangle$
$\displaystyle=\sin\gamma|0\rangle-\mathrm{e}^{\mathrm{i}\zeta}\cos\gamma|1\rangle,$
$\displaystyle|\psi_{0}\rangle$
$\displaystyle=\cos\alpha|0\rangle+\mathrm{e}^{\mathrm{i}\beta}\sin\alpha|1\rangle,$
$\displaystyle|\psi_{1}\rangle$
$\displaystyle=\sin\alpha|0\rangle-\mathrm{e}^{\mathrm{i}\beta}\cos\alpha|1\rangle.$

To simplify the notation, here we temporarily use $|0\rangle$ ($|1\rangle$) to
refer to “path 1” (“path $-1$”) for the path qubit, and to “horizontal
polarization” (“vertical polarization”) for the polarization qubit. Then the
output state from the masking module can be expressed as follows,

$|\Psi\rangle=a_{0}|\varphi_{0}\rangle|\psi_{0}\rangle+a_{1}|\varphi_{0}\rangle|\psi_{1}\rangle+a_{2}|\varphi_{1}\rangle|\psi_{0}\rangle+a_{3}|\varphi_{1}\rangle|\psi_{1}\rangle.$
(A13)

We first choose the angles of the optical axes of Q2 and H4 in Fig. A2 so as
to implement the transformation $|\psi_{0}\rangle\to|0\rangle,\
|\psi_{1}\rangle\to|1\rangle$ on the polarization qubit (see Supplemental
Material of Ref. [44] on how to calculate the angles), which turns
$|\Psi\rangle$ into

$\displaystyle|\Psi^{\prime}\rangle$
$\displaystyle=(a_{0}|\varphi_{0}\rangle+a_{2}|\varphi_{1}\rangle)\otimes|0\rangle+(a_{1}|\varphi_{0}\rangle+a_{3}|\varphi_{1}\rangle)\otimes|1\rangle.$
(A14)

Then the two BDs split and re-combine the two light beams, which turns
$|\Psi^{\prime}\rangle$ into

$|\Psi^{\prime\prime}\rangle=|0\rangle\otimes(a_{1}|\varphi_{0}\rangle+a_{3}|\varphi_{1}\rangle)+|1\rangle\otimes(a_{0}|\varphi_{0}\rangle+a_{2}|\varphi_{1}\rangle).$
(A15)

Next, we set the angles of the optical axes of Q3 and H5 so as to realize the
transformation $|\varphi_{0}\rangle\to|0\rangle,\
|\varphi_{1}\rangle\to|1\rangle$, which turns $|\Psi^{\prime\prime}\rangle$
into

$|\Psi^{\prime\prime\prime}\rangle=a_{1}|0\rangle|0\rangle+a_{3}|0\rangle|1\rangle+a_{0}|1\rangle|0\rangle+a_{2}|1\rangle|1\rangle.$
(A16)

Now, the probabilities that the heralded photon is found by SPCM 0, 1, 2, 3
are $|a_{1}|^{2}$, $|a_{3}|^{2}$, $|a_{0}|^{2}$, $|a_{2}|^{2}$, respectively.
In this way, we can realize the local projective measurement onto the product
basis in Eq. (A11).

## APPENDIX D: Fidelity estimation based on quantum state verification

To estimate the fidelity between a two-qubit state $\varrho$ and the Bell
state $|\Phi\rangle=(|00\rangle+|11\rangle)/\sqrt{2}$, we can apply the idea
of quantum state verification according to Ref. [41]. To be specific, we can
randomly perform one of the three local projective tests associated with the
three test projectors $P_{XX}^{+}$, $P_{YY}^{-}$, $P_{ZZ}^{+}$, respectively,
where

$\displaystyle P_{XX}^{+}=\frac{\mathbbm{1}+X\otimes X}{2},$ (A17)
$\displaystyle P_{YY}^{-}=\frac{\mathbbm{1}-Y\otimes Y}{2},$ $\displaystyle
P_{ZZ}^{+}=\frac{\mathbbm{1}+Z\otimes Z}{2},$

and $\mathbbm{1}$ stands for the identity operator acting on the two-qubit
system. To optimize the performance, each test is performed with probability
$1/3$. The resulting verification operator reads

$\Omega=\frac{1}{3}(P_{XX}^{+}+P_{YY}^{-}+P_{ZZ}^{+})=|\Phi\rangle\langle\Phi|+\frac{1}{3}(\mathbbm{1}-|\Phi\rangle\langle\Phi|).$
(A18)

Now suppose that the fidelity between $\varrho$ and $|\Phi\rangle\langle\Phi|$
is $F(\varrho,|\Phi\rangle)=\langle\Phi|\varrho|\Phi\rangle=1-\epsilon$, where
$\epsilon$ is the infidelity. Then the probability that $\varrho$ can pass
each test on average is given by

$p_{\mathrm{succ}}={\rm tr}(\varrho\Omega)=1-\frac{2}{3}\epsilon,$ (A19)

which implies that

$\epsilon=\frac{3}{2}(1-p_{\mathrm{succ}}).$ (A20)

If $\varrho$ passes $S$ tests after $N$ tests in total in a given experiment,
then $\hat{p}_{\mathrm{succ}}:=S/N$ is an unbiased estimator for
$p_{\mathrm{succ}}$, from which we can construct an unbiased estimator for the
infidelity,

$\hat{\epsilon}=\frac{3}{2}(1-\hat{p}_{\mathrm{succ}}).$ (A21)

To calculate the 95% confidence interval of $\hat{\epsilon}$, here we assume
that all the states $\varrho$ in the $N$ runs are independent and identically
distributed. A common choice for the 95% confidence interval of
$\hat{p}_{\mathrm{succ}}$ is the Agresti-Coull interval [45], which has the
form

$\tilde{p}-\kappa_{\alpha}(\tilde{p}\tilde{q})^{1/2}\tilde{N}^{-1/2}\leq
p_{\mathrm{succ}}\leq\tilde{p}+\kappa_{\alpha}(\tilde{p}\tilde{q})^{1/2}\tilde{N}^{-1/2},$
(A22)

where

$\tilde{p}=\frac{\tilde{S}}{\tilde{N}},\quad\tilde{q}=1-\tilde{p},\quad\tilde{S}=S+\frac{\kappa_{\alpha}^{2}}{2},\quad\tilde{N}=N+\kappa_{\alpha}^{2},$
(A23)

and $\kappa_{\alpha}:=z_{\alpha/2}$ is the ($1-\alpha/2$)th quantile of the
standard normal distribution. Accordingly, the 95% confidence interval of
$\hat{\epsilon}$ reads

$\epsilon_{\rm low}\leq\epsilon\leq\epsilon_{\rm high},$ (A24)

where

$\displaystyle\epsilon_{\rm
low}=\frac{3}{2}\bigl{[}1-\tilde{p}-\kappa_{0.05}(\tilde{p}\tilde{q})^{1/2}\tilde{N}^{-1/2}\bigr{]},$
(A25) $\displaystyle\epsilon_{\rm
high}=\frac{3}{2}\bigl{[}1-\tilde{p}+\kappa_{0.05}(\tilde{p}\tilde{q})^{1/2}\tilde{N}^{-1/2}\bigr{]}.$

When the target state changes from $|\Phi\rangle$ to, say,
$|\Phi^{\prime}\rangle=(U\otimes\mathbbm{1}_{B})|\Phi\rangle$, the above
approach still applies as long as the set of test projectors is replaced by
$\\{P_{X^{\prime}X}^{+},P_{-Y^{\prime}Y}^{+},P_{Z^{\prime}Z}^{+}\\}$
accordingly, where $X^{\prime}=UXU^{\dagger}$, $Y^{\prime}=UYU^{\dagger}$,
$Z^{\prime}=UZU^{\dagger}$.

In our experiment, the total number of tests is chosen to be $N=5000$ to
reduce the statistical fluctuation. The values of $S$ associated with the four
probe states indicated in Fig. 3 in the main text are $4986$, $4969$, $4946$,
and $4963$, respectively. The estimation errors of the fidelities shown in
Fig. 3 are determined by virtue of these numbers and the formula

$\displaystyle\max\\{\hat{\epsilon}-\epsilon_{\rm low},\epsilon_{\rm
high}-\hat{\epsilon}\\}.$ (A26)

Note that the estimation error of each fidelity is the same as the estimation
error of each infidelity.

## APPENDIX E: Quantum state tomography of the path qubit and polarization
qubit

Figure A3:  Experimental results on the reduced density matrices of the path
qubit ($\varrho_{\rm path}$) and polarization qubit ($\varrho_{\rm pola}$),
respectively, of the output states of the masker defined in Eq. (3) in the
main text. Here “re” denotes the real part, and “im” denotes the imaginary
part. The bottom line marks the input probe states associated with the output
states. The experimental results demonstrate that all the reduced density
matrices are nearly maximally mixed.

To further characterize the performance of the masking module, we perform
quantum state tomography on the path qubit and polarization qubit of the
output states associated with the four input states $|0\rangle$,
$(|0\rangle+|1\rangle)/\sqrt{2}$, $(|0\rangle+|1\rangle+|2\rangle)/\sqrt{3}$,
and $(|0\rangle+|1\rangle+|2\rangle+|3\rangle)/2$, respectively [cf. Eq. (13)
in the main text]. To determine each reduced density matrix of each output
state, $X$, $Y$, $Z$ measurements are performed 4000 times, respectively. Then
the density matrix is reconstructed using the maximum-likelihood method
described in Ref. [46]. The reconstruction results on the reduced density
matrices are shown in Fig. A3.

By virtue of the above results we can compute the average purity of the two
reduced density matrices of each output state, as shown in Fig. 3 in the main
text. To determine the estimation error of the average purity, we re-sample
$100$ times the photon-counting results used in the tomography according to
the Poisson distribution, and then calculate the standard deviation of the
average purity using the re-sampled data.

Since the output state is pure, its concurrence is determined by the purity of
each reduced density matrix, which in turn can be determined by quantum state
tomography. The concurrence shown in Fig. 5 in the main text is inferred from
the purity of the density matrix of the path qubit using a similar way as
mentioned above except that here $X$, $Y$, $Z$ measurements are performed
10000 times, respectively. To determine the estimation error of the
concurrence, we re-sample $100$ times the photon-counting results used in the
tomography of the path qubit according to the Poisson distribution, and then
calculate the standard deviation of the concurrence using the re-sampled data.

## APPENDIX F: Decoding input ququart from correlation measurements

In the main text, we introduced a simple masker
$M:\mathcal{H}\mapsto\mathcal{H}_{A}\otimes\mathcal{H}_{B}$ for the real
ququart associated with the Hilbert space $\mathcal{H}$, where both
$\mathcal{H}_{A}$ and $\mathcal{H}_{B}$ are qubit Hilbert spaces. To be
specific, $M$ is the isometry defined by its action on the computational
basis:

$M|j\rangle=-\mathrm{i}(U_{j}\otimes\mathbbm{1}_{B})|\Phi\rangle,\qquad
j=0,1,2,3,$ (A27)

where $U_{0}:=\mathbbm{1}_{A}$,
$\\{U_{j}\\}_{j=1}^{3}:=\\{\mathrm{i}\sigma_{z},\mathrm{i}\sigma_{x},\mathrm{i}\sigma_{y}\\}$
forms a set of Hurwitz-Radon matrices on $\mathcal{H}_{A}$, and
$|\Phi\rangle=(|00\rangle+|11\rangle)/\sqrt{2}$ is the canonical maximally
entangled state in $\mathcal{H}_{A}\otimes\mathcal{H}_{B}$. Here we shall show
that the information of the real input state can be faithfully decoded from
the bipartite correlations of the output state of the masker $M$.

Recall that any two-qubit state $\varrho$ on
$\mathcal{H}_{A}\otimes\mathcal{H}_{B}$ can be expressed as

$\varrho=\frac{1}{4}\bigg{(}\mathbbm{1}+\bm{a}\cdot{\bm{\sigma}}\otimes\mathbbm{1}_{B}+\mathbbm{1}_{A}\otimes\bm{b}\cdot{\bm{\sigma}}+\sum_{j,k}{T_{jk}}\,\sigma_{j}\otimes\sigma_{k}\bigg{)},$
(A28)

where $\bm{\sigma}=(\sigma_{x},\sigma_{y},\sigma_{z})$, $\bm{a}$ and $\bm{b}$
are the Bloch vectors of the two reduced states, respectively, and $T$ is the
correlation matrix. The entries of the correlation matrix can be determined by
suitable Pauli measurements according to the following equation

$T_{jk}=\langle\sigma_{j}\otimes\sigma_{k}\rangle={\rm
tr}[\varrho(\sigma_{j}\otimes\sigma_{k})],\quad j,k\in\\{x,y,z\\}.$ (A29)

To decode the input state, we first consider the case in which the input real
state is pure. A general real pure state in $\mathcal{H}$ can be written as
$|\psi(\bm{c})\rangle=\sum_{j}c_{j}|j\rangle$, where
$\bm{c}=(c_{0},c_{1},c_{2},c_{3})$ is a normalized real vector. The
corresponding output state of the masker $M$ has the form

$|\Psi(\bm{c})\rangle:=M|\psi(\bm{c})\rangle=-\mathrm{i}[U(\bm{c})\otimes\mathbbm{1}_{B}]|\Phi\rangle,$
(A30)

where $U(\bm{c}):=\sum_{j=0}^{3}{c_{j}U_{j}}$. Note that

$|\Phi\rangle\langle\Phi|=\frac{1}{4}(\mathbbm{1}+\sigma_{x}\otimes\sigma_{x}-\sigma_{y}\otimes\sigma_{y}+\sigma_{z}\otimes\sigma_{z}),$
(A31)

and

$\displaystyle U(\bm{c})\sigma_{x}U(\bm{c})^{\dagger}$
$\displaystyle=(c_{0}^{2}-c_{1}^{2}+c_{2}^{2}-c_{3}^{2})\sigma_{x}+2(c_{2}c_{3}-c_{0}c_{1})\sigma_{y}+2(c_{0}c_{3}+c_{1}c_{2})\sigma_{z},$
(A32) $\displaystyle U(\bm{c})\sigma_{y}U(\bm{c})^{\dagger}$
$\displaystyle=2(c_{0}c_{1}+c_{2}c_{3})\sigma_{x}+(c_{0}^{2}-c_{1}^{2}-c_{2}^{2}+c_{3}^{2})\sigma_{y}+2(c_{1}c_{3}-c_{0}c_{2})\sigma_{z},$
$\displaystyle U(\bm{c})\sigma_{z}U(\bm{c})^{\dagger}$
$\displaystyle=2(c_{1}c_{2}-c_{0}c_{3})\sigma_{x}+2(c_{1}c_{3}+c_{0}c_{2})\sigma_{y}+(c_{0}^{2}+c_{1}^{2}-c_{2}^{2}-c_{3}^{2})\sigma_{z}.$

So the density operator of $|\Psi(\bm{c})\rangle$ reads

$|\Psi(\bm{c})\rangle\langle\Psi(\bm{c})|=\frac{1}{4}\bigg{[}\mathbbm{1}+\sum_{j,k}{T(\bm{c})_{jk}}\,\sigma_{j}\otimes\sigma_{k}\bigg{]},$
(A33)

where the correlation matrix $T(\bm{c})$ has the form

$T(\bm{c})=\left(\begin{matrix}c_{0}^{2}-c_{1}^{2}+c_{2}^{2}-c_{3}^{2}&-2(c_{0}c_{1}+c_{2}c_{3})&2(c_{1}c_{2}-c_{0}c_{3})\\\
2(c_{2}c_{3}-c_{0}c_{1})&-(c_{0}^{2}-c_{1}^{2}-c_{2}^{2}+c_{3}^{2})&2(c_{1}c_{3}+c_{0}c_{2})\\\
2(c_{0}c_{3}+c_{1}c_{2})&-2(c_{1}c_{3}-c_{0}c_{2})&c_{0}^{2}+c_{1}^{2}-c_{2}^{2}-c_{3}^{2}\end{matrix}\right).$
(A34)

In conjunction with the normalization condition
$c_{0}^{2}+c_{1}^{2}+c_{2}^{2}+c_{3}^{2}=1$, we can deduce that

$\displaystyle c_{0}^{2}=\frac{1}{4}(1+T_{11}-T_{22}+T_{33}),\qquad$
$\displaystyle c_{1}^{2}=\frac{1}{4}(1-T_{11}+T_{22}+T_{33}),$ (A35)
$\displaystyle c_{2}^{2}=\frac{1}{4}(1+T_{11}+T_{22}-T_{33}),\qquad$
$\displaystyle c_{3}^{2}=\frac{1}{4}(1-T_{11}-T_{22}-T_{33}),$

and

$\displaystyle c_{2}c_{3}=\frac{1}{4}(T_{21}-T_{12}),\qquad$ $\displaystyle
c_{1}c_{3}=\frac{1}{4}(T_{23}-T_{32}),$ $\displaystyle
c_{1}c_{2}=\frac{1}{4}(T_{13}+T_{31}),$ (A36) $\displaystyle
c_{0}c_{1}=-\frac{1}{4}(T_{21}+T_{12}),\qquad$ $\displaystyle
c_{0}c_{2}=\frac{1}{4}(T_{23}+T_{32}),$ $\displaystyle
c_{0}c_{3}=\frac{1}{4}(T_{31}-T_{13}).$

By linearity the above result can be generalized to any mixed input state
$\rho$ that has a real density matrix. In this case, in analogy to Eq. (A34),
the correlation matrix of the output state $M\rho M^{\dagger}$ reads

$T=\left(\begin{matrix}\rho_{00}-\rho_{11}+\rho_{22}-\rho_{33}&-2(\rho_{01}+\rho_{23})&2(\rho_{12}-\rho_{03})\\\
2(\rho_{23}-\rho_{01})&-(\rho_{00}-\rho_{11}-\rho_{22}+\rho_{33})&2(\rho_{13}+\rho_{02})\\\
2(\rho_{03}+\rho_{12})&-2(\rho_{13}-\rho_{02})&\rho_{00}+\rho_{11}-\rho_{22}-\rho_{33}\end{matrix}\right).$
(A37)

Accordingly, we have

$\displaystyle\rho_{00}=\frac{1}{4}(1+T_{11}-T_{22}+T_{33}),\qquad$
$\displaystyle\rho_{11}=\frac{1}{4}(1-T_{11}+T_{22}+T_{33}),$ (A38)
$\displaystyle\rho_{22}=\frac{1}{4}(1+T_{11}+T_{22}-T_{33}),\qquad$
$\displaystyle\rho_{33}=\frac{1}{4}(1-T_{11}-T_{22}-T_{33}),$

and

$\displaystyle\rho_{23}=\frac{1}{4}(T_{21}-T_{12}),\qquad$
$\displaystyle\rho_{13}=\frac{1}{4}(T_{23}-T_{32}),$
$\displaystyle\rho_{12}=\frac{1}{4}(T_{13}+T_{31}),$ (A39)
$\displaystyle\rho_{01}=-\frac{1}{4}(T_{21}+T_{12}),\qquad$
$\displaystyle\rho_{02}=\frac{1}{4}(T_{23}+T_{32}),$
$\displaystyle\rho_{03}=\frac{1}{4}(T_{31}-T_{13}),$

which generalize Eqs. (A35) and Eq. (A36). Therefore, the density matrix of
any real input state of the ququart can be faithfully decoded from the
correlation matrix of the output state, which can easily be determined by
virtue of product Pauli measurements $\sigma_{j}\otimes\sigma_{k}$ for
$j,k\in\\{x,y,z\\}$.

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