#  A unified view of direct measurement of quantum states, processes, and
measurements

Kazuhisa Ogawa<EMAIL_ADDRESS>Center for Quantum Information and
Quantum Biology (QIQB), Osaka University, Osaka 560-0043, Japan  Takumi
Matsuura  Graduate School of Information Science and Technology, Hokkaido
University, Sapporo 060-0814, Japan  Akihisa Tomita  Graduate School of
Information Science and Technology, Hokkaido University, Sapporo 060-0814,
Japan

###### Abstract

The dynamics of a quantum system are characterized by three components:
quantum state, quantum process, and quantum measurement. The proper
measurement of these components is a crucial issue in quantum information
processing. Recently, direct measurement methods have been proposed and
demonstrated wherein each complex matrix element of these three components is
obtained separately, without the need for quantum tomography of the entire
matrix. Since these direct measurement methods have been proposed
independently, no theoretical framework has been presented to unify them
despite the time symmetry of quantum dynamics. In this study, we propose a
theoretical framework to systematically derive direct measurement methods for
these three components. Following this framework and further utilizing the
basis-shift unitary transformation, we have derived the most efficient direct
measurement method using qubit probes. Additionally, we have experimentally
demonstrated the feasibility of the direct measurement method of quantum
states using optical pulse trains.

##  I Introduction

The dynamics of a quantum system are characterized by three key components:
the quantum state, the quantum process, and the quantum measurement. Each of
these components is represented by specific quantum elements: the density
operator $\hat{\rho}$, the completely positive and trace-preserving (CPTP) map
$\mathcal{M}$, and the positive-operator-valued measure (POVM) element
$\hat{E}$, respectively. These components are typically represented by complex
matrices on a fixed basis, and the appropriate measurement of these matrix
elements is a fundamental issue in various quantum applications. The standard
method for measuring quantum states is known as quantum state tomography (QST)
Vogel and Risken (1989); Smithey  _et al._ (1993); Breitenbach  _et al._
(1997); White  _et al._ (1999); James  _et al._ (2001). In QST, projective
measurements are performed on different bases, and each matrix element of the
density operator $\hat{\rho}$ is estimated by post-processing all the measured
data. Similarly, for quantum processes and quantum measurements, methods such
as quantum process tomography (QPT) Poyatos  _et al._ (1997); Chuang and
Nielsen (1997); Childs  _et al._ (2001); Mitchell  _et al._ (2003) and quantum
measurement tomography (QMT) Amri  _et al._ (2011); Keith  _et al._ (2018);
Cattaneo  _et al._ (2023) are utilized. Analogous to QST, the CPTP map
$\mathcal{M}$ and the POVM element $\hat{E}$ can be determined by preparing
various initial states, conducting projection measurements on various bases,
and subsequently post-processing all the measurement data. In quantum
tomography, as the dimension of the Hilbert space of the measured quantum
system increases, the number of measurements and the complexity of the
reconstruction algorithm also increase. This may render the quantum tomography
impractical for certain applications. However, for some quantum applications
Horodecki  _et al._ (2009); Xu  _et al._ (2020); Pallister  _et al._ (2018);
Flammia and Liu (2011); da Silva  _et al._ (2011); Huang  _et al._ (2020), it
is not imperative to reconstruct the complete matrix; rather, only specific
parts of it need to be measured directly. In such scenarios, it becomes
crucial to develop methods for directly measuring only certain matrix elements
of the quantum component under examination.

Recently, a new class of measurement methods, referred to as direct
measurements (DM), has been emerged, exhibiting distinct characteristics from
various quantum tomography methods. In DM, each matrix element of the quantum
component under investigation can be estimated within a single measurement
setup, eliminating the need for post-processing the results of measurements
across different initial states or measurement bases. Initially, DM was
applied to pure-state wavefunctions Lundeen  _et al._ (2011) using the
Aharonov–Albert–Vaidman (AAV) weak measurement Aharonov  _et al._ (1988).
Subsequent studies expanded the scope of DM to encompass objects such as
pseudo-probability distributions Bamber and Lundeen (2014), density operators
Thekkadath  _et al._ (2016), nonlocal entangled states Pan  _et al._ (2019),
quantum processes Kim  _et al._ (2018); Gaikwad  _et al._ (2023), and quantum
measurements Xu  _et al._ (2021a). Notably, certain DM techniques have been
implemented not through AAV weak measurements, but via standard indirect
measurements employing strong interactions and projective measurements on the
probe system, resulting in higher measurement efficiency, as experimentally
demonstrated Vallone and Dequal (2016); Calderaro  _et al._ (2018); Ogawa  _et
al._ (2021); Zhou  _et al._ (2021); Feng  _et al._ (2021); Li  _et al._
(2022); Zhang  _et al._ (2020); Chen  _et al._ (2021); Zhang  _et al._ (2019);
Xu  _et al._ (2021b).

In prior studies, DM methods have been independently proposed for each of the
three quantum components. However, given the time-reversal symmetry of quantum
time evolution, it is reasonable to consider direct measurement methods for
these components from a more unified perspective. Previously, we introduced an
intuitive diagrammatic representation of DM for wavefunctions using qubit
probes, elucidating how DM operates regardless of measurement strength Ogawa
_et al._ (2019). In this research, we expand upon this framework and
demonstrate the systematic design of efficient DM systems for quantum states,
quantum processes, and quantum measurements. This framework can be viewed as a
generalized Hadamard test and offers guidance for designing quantum circuits
to obtain desired complex observables. Leveraging this framework, we introduce
a novel DM method employing the basis-shift unitary transformation, which
proves to be more efficient per shot measurement compared to previous DM
methods. Moreover, we validate the feasibility of this proposed DM method for
quantum states (density operators) by experimentally applying it to optical
pulse trains encoded in discrete time degrees of freedom.

This paper is structured as follows: In Section II, we introduce the
generalized Hadamard test along with its diagrammatic representation,
illustrating its applicability in deriving DM methods for various quantum
components. In Section III, we detail the experimental demonstration of
applying the newly derived direct measurement method for density operators to
optical pulse trains. Finally, in Section IV, we provide a discussion and
summary of this study.

##  II Theory

###  II.1 Generalized Hadamard test and its diagrammatic representation

To systematically derive DM methods for complex matrix elements of quantum
components, we first explore the generalized Hadamard test. This test is
implemented through a quantum circuit comprising a qubit probe system and a
measured system, as depicted in Fig. 1(a). The initial states of the probe
system and the measured system are $|0\rangle\langle 0|$ and $\hat{\rho}$,
respectively. The quantum gates include the first Hadamard gate $\hat{H}$, two
controlled gates ($|0\rangle$-controlled $\hat{A}$ and $|1\rangle$-controlled
$\hat{B}$), the phase gate $\hat{S}^{b}$ (an identity gate for $b=0$ and a
phase gate for $b=1$), and a second Hadamard gate $\hat{H}$. Following the
passage through these quantum gates, we measure the expectation value of the
tensor product $\hat{Z}\otimes\hat{E}$, composed of the Pauli-Z operator
$\hat{Z}$ of the probe system and a POVM element $\hat{E}$ of the measured
system. Depending on whether $b$ is 0 or 1, the resulting expectation value is
obtained by this measurement is given by

$\displaystyle\langle\hat{Z}\otimes\hat{E}\rangle$
$\displaystyle=\left\\{\begin{array}[]{ll}\mathrm{Re}\left[\mathrm{tr}(\hat{A}\hat{\rho}\hat{B}^{\dagger}\hat{E})\right]&(b=0),\vspace{1mm}\\\
\mathrm{Im}\left[\mathrm{tr}(\hat{A}\hat{\rho}\hat{B}^{\dagger}\hat{E})\right]&(b=1).\end{array}\right.$
(3)

In particular, the case $\hat{A}=\hat{E}=\hat{I}$ corresponds to the usual
Hadamard test. By combining these two results, the complex value
$\mathrm{tr}(\hat{A}\hat{\rho}\hat{B}^{\dagger}\hat{E})$ is obtained by the
generalized Hadamard test.

The value $\mathrm{tr}(\hat{A}\hat{\rho}\hat{B}^{\dagger}\hat{E})$ obtained
through the generalized Hadamard test can be intuitively grasped using the
diagrammatic representation introduced in Ref. Ogawa  _et al._ (2019), as
depicted in Fig. 1(b). In this representation, certain elements such as the
initial state $|0\rangle$, quantum gates $\hat{H}$ and $\hat{S}^{b}$, and the
quantum measurement $\hat{Z}$ in the probe system are not explicitly depicted
as they are assumed, while only the initial state of the measured system, the
control gates dependent on the probe state, and the POVM elements of the
measurement are displayed on the diagram. The complex value
$\mathrm{tr}(\hat{A}\hat{\rho}\hat{B}^{\dagger}\hat{E})$ acquired via the
generalized Hadamard test can be interpreted as the trace of the operator
$\hat{A}\hat{\rho}\hat{B}^{\dagger}\hat{E}$, which forms a cyclic product of
the operators $\hat{\rho}$, $\hat{A}$, $\hat{E}$, and $\hat{B}$ in the diagram
(noting that the operator $\hat{B}$ in the bottom row of the diagram must take
the Hermitian conjugate). Leveraging this relationship, we can derive a DM
method for a desired complex value by appropriately arranging the operators on
the diagram.

Figure 1:  (a) Quantum circuit illustrating the generalized Hadamard test. The
expectation values of $\hat{Z}\otimes\hat{E}$ for $b=0,1$ correspond to the
real and imaginary parts of
$\mathrm{tr}(\hat{A}\hat{\rho}\hat{B}^{\dagger}\hat{E})$ respectively. (b)
Diagrammatic representation for intuitive comprehension of the generalized
Hadamard test. The top and bottom rows of the diagram depict the time
evolution of the measured system when the probe system is $|0\rangle$ and
$|1\rangle$, respectively. The initial states of the probe system are set to
$|+\rangle$, and finally $\hat{X}$\- and $\hat{Y}$-basis measurements are
conducted; however, these are not explicitly depicted in the diagrammatic
representation. The complex value
$\mathrm{tr}(\hat{A}\hat{\rho}\hat{B}^{\dagger}\hat{E})$ obtained from the
measurement can be intuitively understood as the trace of the operator
$\hat{A}\hat{\rho}\hat{B}^{\dagger}\hat{E}$, forming a cyclic product of the
operators $\hat{\rho}$, $\hat{A}$, $\hat{E}$, and $\hat{B}$ in the diagram.

###  II.2 Derivation of direct measurement methods for quantum components

Using the diagrammatic representation, we derive a generalized-Hadamard-test-
based DM method for each quantum component—quantum states, measurement POVM
elements, and quantum processes. First, to directly measure the matrix element
$\rho_{ij}:=\langle
i|\hat{\rho}|j\rangle=\mathrm{tr}(\hat{\rho}|j\rangle\langle i|)$ of the
quantum state $\hat{\rho}$, the diagrammatic representation suggests that we
can select $\hat{A}$, $\hat{E}$, and $\hat{B}$ such that
$\hat{B}^{\dagger}\hat{E}\hat{A}\propto|j\rangle\langle i|$ holds. For
example, as in Refs. Thekkadath  _et al._ (2016); Calderaro  _et al._ (2018),
if we choose $\hat{A}=|i\rangle\langle i|$, $\hat{B}=|j\rangle\langle j|$, and
$\hat{E}=|\mathrm{MUB}\rangle\langle\mathrm{MUB}|$, the above condition is
satisfied, yielding
$\mathrm{tr}(\hat{A}\hat{\rho}\hat{B}^{\dagger}\hat{E})=\rho_{ij}/d$, where
$|\mathrm{MUB}\rangle$ represents the mutually unbiased basis of $|i\rangle$
and $|j\rangle$, and $\langle i|\mathrm{MUB}\rangle=\langle
j|\mathrm{MUB}\rangle=1/\sqrt{d}$ (where $d$ is the dimension of the measured
system). As another example, to maximize the coefficient of $|j\rangle\langle
i|$ and enhance measurement efficiency, it is preferable to choose $\hat{A}$
and $\hat{B}$ as unitary operators. For instance, selecting
$\hat{A}=\hat{U}_{\mathrm{shift}}(j-i)=\sum_{k}|k+j-i\ \mathrm{mod}\
d\rangle\langle k|$, $\hat{B}=\hat{1}$, and $\hat{E}=|j\rangle\langle j|$, as
illustrated in Fig. 2(a), eliminates the coefficient of $1/d$, resulting in
$\mathrm{tr}(\hat{A}\hat{\rho}\hat{B}^{\dagger}\hat{E})=\rho_{ij}$. Here,
$\hat{U}_{\mathrm{shift}}(n)$ represents the basis-shift unitary
transformation defined by

$\displaystyle\hat{U}_{\mathrm{shift}}(n):=\sum_{k}|k+n\ \mathrm{mod}\
d\rangle\langle k|,$ (4)

which shifts the basis number by $+n$. Hence, the generalized Hadamard test
can be utilized to determine the expectation value of a non-physical
observable that does not meet the definition of a measurement POVM element,
such as $|j\rangle\langle i|$ for the state $\hat{\rho}$. In the latter part
of this paper, we will conduct experiments to demonstrate the DM method of the
quantum state using this basis-shift unitary transformation
$\hat{U}_{\mathrm{shift}}(j-i)$.

Next, if we aim to directly measure the matrix element $E_{ij}:=\langle
i|\hat{E}|j\rangle=\mathrm{tr}(\hat{E}|j\rangle\langle i|)$ of the measurement
POVM element $\hat{E}$, the diagrammatic representation suggests that we can
select $\hat{A}$, $\hat{\rho}$, and $\hat{B}$ in a manner that
$\hat{A}\hat{\rho}\hat{B}^{\dagger}\propto|j\rangle\langle i|$ holds. For
instance, as demonstrated in Refs. Xu  _et al._ (2021a, b), if we set
$\hat{\rho}=|\mathrm{MUB}\rangle\langle\mathrm{MUB}|$,
$\hat{A}=|j\rangle\langle j|$, and $\hat{B}=|i\rangle\langle i|$, the
aforementioned condition is satisfied, resulting in
$\mathrm{tr}(\hat{A}\hat{\rho}\hat{B}^{\dagger}\hat{E})=E_{ij}/d$. As another
example, if $\hat{A}$ and $\hat{B}$ are chosen as unitary operators,
$\hat{\rho}=|i\rangle\langle i|$,
$\hat{A}=\hat{U}_{\mathrm{shift}}(j-i)=\sum_{k}|k+j-i\ \mathrm{mod}\
d\rangle\langle k|$, and $\hat{B}=\hat{1}$, as shown in Fig. 2(b), then the
coefficient of $1/d$ vanishes and
$\mathrm{tr}(\hat{A}\hat{\rho}\hat{B}^{\dagger}\hat{E})=E_{ij}$ is obtained.
Therefore, the generalized Hadamard test can also yield the expectation value
of a measurement POVM element $\hat{E}$ for an unphysical quantum state, which
does not adhere to the definition of a density operator such as
$|j\rangle\langle i|$.

Moreover, DM of matrix elements of quantum processes can also be achieved
using the generalized Hadamard test. The transformation of a quantum process
$\mathcal{M}$ for a state $\hat{\rho}$ can be expressed in the following
operator-sum representation:

$\displaystyle\mathcal{M}(\rho)=\sum_{ijkl}\chi_{ijkl}|l\rangle\langle
i|\hat{\rho}|j\rangle\langle k|.$ (5)

Here we consider DM of the matrix element $\chi_{ijkl}\in\mathbb{C}$.
$\chi_{ijkl}$ is expressed as follows:

$\displaystyle\chi_{ijkl}=\mathrm{tr}\big{[}\mathcal{M}(|i\rangle\langle
j|)|k\rangle\langle l|\big{]}.$ (6)

In other words, $\chi_{ijkl}$ can be interpreted as the expectation value of
the unphysical measurement POVM element $|k\rangle\langle l|$ for the
unphysical quantum state $|i\rangle\langle j|$ with time evolution
$\mathcal{M}$. Therefore, $\chi_{ijkl}$ can be measured using the generalized
Hadamard test with gates positioned before and after the quantum process
$\mathcal{M}$, as depicted in Fig. 2(c). Each operator is selected such that
$\hat{A}\hat{\rho}\hat{B}^{\dagger}\propto|i\rangle\langle j|$ and
$\hat{D}^{\dagger}\hat{E}\hat{C}\propto|k\rangle\langle l|$. For instance,
when using projection operators, we can opt for $\hat{A}=|i\rangle\langle i|$,
$\hat{B}=|j\rangle\langle j|$, $\hat{C}=|l\rangle\langle l|$,
$\hat{D}=|k\rangle\langle k|$, and
$\hat{\rho}=\hat{E}=|\mathrm{MUB}\rangle\langle\mathrm{MUB}|$. The result of
this measurement is given as

$\displaystyle\mathrm{tr}\big{[}\mathcal{M}(\hat{A}\hat{\rho}\hat{B}^{\dagger})\hat{D}^{\dagger}\hat{E}\hat{C}\big{]}=\frac{\chi_{ijkl}}{d^{2}}.$
(7)

On the other hand, in the scenario of employing the basis-shift unitary
transformation $\hat{U}_{\mathrm{shift}}$, we can choose
$\hat{A}=\hat{U}_{\mathrm{shift}}(i-j)=\sum_{m}|m+i-j\ \mathrm{mod}\
d\rangle\langle m|$, $\hat{B}=\hat{1}$, $\hat{\rho}=|j\rangle\langle j|$,
$\hat{C}=\hat{U}_{\mathrm{shift}}(k-l)=\sum_{m}|m+k-l\rangle\langle m|$,
$\hat{D}=\hat{1}$, and $\hat{E}=|k\rangle\langle k|$. Then the coefficient of
$1/d^{2}$ vanishes and

$\displaystyle\mathrm{tr}\big{[}\mathcal{M}(\hat{A}\hat{\rho}\hat{B}^{\dagger})\hat{D}^{\dagger}\hat{E}\hat{C}\big{]}=\chi_{ijkl}$
(8)

is obtained.

Figure 2:  (a) Diagrammatic representation illustrating the quantum dynamics
enabling the DM of the matrix elements of the quantum state $\hat{\rho}$,
along with the corresponding quantum circuit. (b) Diagrammatic representation
demonstrating the DM of the matrix elements of the measurement POVM element
$\hat{E}$, accompanied by the corresponding quantum circuit. (c) Diagrammatic
representation showcasing the DM of the matrix elements of a quantum process
$\mathcal{M}$, along with the corresponding quantum circuit.

##  III Experiment

In this section, we experimentally demonstrate the DM method for matrix
elements of quantum states depicted in Fig. 2(a). This method is one of the DM
techniques utilizing the basis-shifted unitary transformation derived through
the diagrammatic representation. We employ quantum states of photons encoded
in discrete time degrees of freedom as the quantum states to be measured. In
this experiment, we examine three-dimensional quantum states encoded in a
series of three laser pulse trains. Since the detection probabilities for a
single photon, without quantum entanglement, are proportional to the intensity
measurement result for classical coherent light, this experiment was conducted
using coherent light with output power in the classical regime. This
experimental approach can be adapted for single photons by employing a
suitable single-photon light source and photon detector.

Figure 3:  Experimental setup used in the proposed experiment for DM of
quantum states. Continuous light with a wavelength of 1550 nm emitted from a
fiber laser is injected into a DPM through an optical fiber. In the DPM,
arbitrary complex amplitudes can be realized by three applied voltages
$V_{1}$, $V_{2}$, and $V_{\mathrm{ph}}$. Here, we fix
$V_{\mathrm{ph}}=V_{\pi,\mathrm{ph}}$ and apply the voltages corresponding to
the real and imaginary parts of the desired complex amplitude to $V_{1}$ and
$V_{2}$, respectively, to obtain the desired complex amplitude. After passing
through the DPM, the test pulse enters the DM circuit. The waveforms without
and with AMZI, which has delays of 200 ps and 400 ps, are measured using an
oscilloscope. The measurements are made for four cases of AMZI phase
difference $\phi$: $0^{\circ}$, $90^{\circ}$, $180^{\circ}$, and
$270^{\circ}$.

Figure 3 illustrates the experimental setup. A fiber laser source (NP
Photonics SMPF-2030) emits continuous light at a wavelength of 1550 nm. The
output laser beam is amplified to 53 mW by an erbium-doped fiber amplifier
(EDFA; GIP Technology), with polarization adjustment performed by a
polarization controller (CPC900, Thorlabs; not shown in Fig. 3) to optimize
injection efficiency into the next component, a dual parallel modulator (DPM;
Mach-10 086, COVEGA). Subsequently, the DPM modulates the time distribution of
the laser light into a triple-pulse test waveform, crucial for verifying the
DM method. In the DPM, the output complex amplitude is modulated as

$\displaystyle
E_{\mathrm{out}}\propto\sin\left(\pi\frac{V_{1}}{V_{\pi,1}}\right)+\exp\left({\mathrm{i}}\pi\frac{V_{\mathrm{ph}}}{V_{\mathrm{\pi,ph}}}\right)\sin\left(\pi\frac{V_{2}}{V_{\pi,2}}\right)$
(9)

by applying the three voltages $V_{1}$, $V_{2}$, and $V_{\mathrm{ph}}$ as
illustrated in Fig. 3, where $V_{\pi,i}\ (i=1,2,\mathrm{ph})$ represents the
voltage required for a phase shift of $\pi$ at each voltage application port.
With $V_{\mathrm{ph}}$ fixed at $V_{\mathrm{\pi,ph}}/2$ while assuming
$V_{1}\ll V_{\pi,1}$ and $V_{2}\ll V_{\pi,2}$, we can approximately express:

$\displaystyle
E_{\mathrm{out}}\propto\pi\frac{V_{1}}{V_{\pi,1}}+{\mathrm{i}}\pi\frac{V_{2}}{V_{\pi,2}}.$
(10)

Thus, arbitrary complex amplitude waveforms can be generated by applying RF
voltages to $V_{1}$ and $V_{2}$ that are proportional to the real and
imaginary parts, respectively, of the complex amplitude waveform to be
generated. We have generated five amplitude states from
$|\psi^{\mathrm{test}}_{1}\rangle$ to $|\psi^{\mathrm{test}}_{5}\rangle$,
presented in Fig. 4 as test waveforms.

Figure 4:  (a)–(e) Waveforms of the voltages $V_{1}$ and $V_{2}$ applied to
the DPM to realize the test states
$|\psi_{1}^{\mathrm{test}}\rangle$–$|\psi_{5}^{\mathrm{test}}\rangle$. The
pulse repetition period is 200 ps and the pulse width is 44 ps.

The generalized Hadamard test is implemented using an asymmetric Mach–Zehnder
interferometer (AMZI) with delays of 200 ps and 400 ps. AMZI can introduce not
only delays between two paths but also the phase difference $\phi$. We
conducted measurements for four phase difference cases: $\phi=0^{\circ}$,
$90^{\circ}$, $180^{\circ}$, and $270^{\circ}$. From the difference between
the measurement results for $\phi=0^{\circ}$ and $180^{\circ}$, and those for
$\phi=90^{\circ}$ and $270^{\circ}$, the expectation values of $\hat{X}$ and
$\hat{Y}$ are obtained, respectively. The pulse trains output by AMZI were
detected by a photodetector (PD), and their time waveforms were observed using
an oscilloscope.

Taking the case of $|\psi_{1}^{\mathrm{test}}\rangle$ as an example, we
explain the procedure for obtaining each matrix element from the observed
waveform in this DM method. Initially, we examine the waveform without AMZI,
as depicted in Fig. 5(a), and derive three pulse amplitudes through fitting,
denoted as $A_{0}$, $A_{1}$, and $A_{2}$, respectively (refer to Appendix A
for details regarding the fitting function). Normalizing these values allows
us to obtain the diagonal components of the matrix as follows:

$\displaystyle A:=A_{0}+A_{1}+A_{2}$ (11)
$\displaystyle\rho_{00}=\frac{A_{0}}{A},\quad\rho_{11}=\frac{A_{1}}{A},\quad\rho_{22}=\frac{A_{2}}{A}.$
(12)

Next, as depicted in Fig. 5(b), the waveform shifted by one pulse using the
basis-shift unitary transformation is interfered with the original waveform by
employing AMZI with a delay of 200 ps. We observe the waveform after passing
through AMZI and obtain the amplitudes of the second and third pulses by
fitting. Let $A_{01}(\phi)$ and $A_{12}(\phi)$ represent the amplitudes of the
second and third pulses at the phase difference $\phi$. Using the
normalization constant $A$ obtained above, we obtain the values of $\rho_{01}$
and $\rho_{12}$ as follows:

$\displaystyle\rho_{01}$
$\displaystyle=\frac{[A_{01}(0^{\circ})-A_{01}(180^{\circ})]+{\mathrm{i}}\left[A_{01}(90^{\circ})-A_{01}(270^{\circ})\right]}{A},$
(13) $\displaystyle\rho_{12}$
$\displaystyle=\frac{[A_{12}(0^{\circ})-A_{12}(180^{\circ})]+{\mathrm{i}}\left[A_{12}(90^{\circ})-A_{12}(270^{\circ})\right]}{A}.$
(14)

The values of $\rho_{10}$ and $\rho_{21}$ are obtained as complex conjugates
of $\rho_{01}$ and $\rho_{12}$, respectively.

Subsequently, as depicted in Fig. 5(c), we observe the waveforms after passing
through the AMZI with a delay of 400 ps, equivalent to a basis shift of two
pulses, and extract the amplitude of the third pulse through fitting. Denoting
$A_{02}(\phi)$ as the amplitude of the third pulse at the phase difference
$\phi$, $\rho_{02}$ is obtained as follows:

$\displaystyle\rho_{02}$
$\displaystyle=\frac{[A_{02}(0^{\circ})-A_{02}(180^{\circ})]+{\mathrm{i}}\left[A_{02}(90^{\circ})-A_{02}(270^{\circ})\right]}{A}.$
(15)

Similarly, $\rho_{20}$ is obtained as the complex conjugate of $\rho_{02}$.
This procedure allows us to derive every matrix element of the density
operator of the quantum state. Figs. 6(a)–(e) present the results of DM of the
density matrix for each state. Notably, each matrix element obtained through
the direct measurement method closely approximates the ideal value,
demonstrating the efficacy of the proposed DM method.

Figure 5:  FIG. 5. Experimental procedures for DM of each matrix element
within the density matrix of the state $|\psi_{1}^{\mathrm{test}}\rangle$. In
each graph, red dots represent experimental data points, while blue lines
depict the fitted curves. (a) Measurement outcomes aimed at determining the
diagonal components $\rho_{00}$, $\rho_{11}$, and $\rho_{22}$ of the density
matrix. Without the AMZI, the normalized peak amplitudes directly correspond
to $\rho_{00}$, $\rho_{11}$, and $\rho_{22}$. (b) Measurement results for
obtaining $\rho_{01}$, $\rho_{10}$, $\rho_{12}$, and $\rho_{21}$ within the
density matrix. In the measurement results for the case of inserting AMZI with
200 ps delay, the two peak amplitudes enclosed by the green dashed lines are
normalized, and the difference between these normalized amplitudes when
$\phi=0^{\circ}$ and $\phi=180^{\circ}$, $\phi=90^{\circ}$ and
$\phi=270^{\circ}$ correspond to the real and imaginary parts of the matrix
elements to be obtained, respectively. The $\rho_{10}$ and $\rho_{21}$ are
determined as the complex conjugates of $\rho_{10}$ and $\rho_{12}$,
respectively. (c) Measurement outcomes to determine $\rho_{02}$ and
$\rho_{20}$ within the density matrix. In the measurement results for the case
of inserting AMZI with 400 ps delay, the two peak amplitudes enclosed by the
green dashed lines are normalized, and the difference between these normalized
amplitudes when $\phi=0^{\circ}$ and $\phi=180^{\circ}$, $\phi=90^{\circ}$ and
$\phi=270^{\circ}$ correspond to the real and imaginary parts of $\rho_{02}$,
respectively. $\rho_{20}$ is deduced as the complex conjugate of $\rho_{02}$.

Furthermore, the density matrix $\hat{\rho}^{\mathrm{exp}}$ reconstructed by
this DM method is evaluated by the fidelity:

$\displaystyle
F(\hat{\rho}^{\mathrm{exp}},|\psi^{\mathrm{test}}\rangle)=\sqrt{\langle\psi^{\mathrm{test}}|\hat{\rho}^{\mathrm{exp}}|\psi^{\mathrm{test}}\rangle},$
(16)

where $|\psi^{\mathrm{test}}\rangle$ represents the ideal state. The matrices
presented in Figs. 6(a)–(e), obtained through the current method, do not
adhere to the requirement of non-negativity and unity trace that a density
operator should satisfy. Consequently, we applied a transformation to ensure
that the matrices obtained via the DM method conform to the properties of non-
negativity and unity trace (refer to Appendix B for details regarding the
transformation procedure and the resulting matrices). The fidelities of these
modified matrices with respect to the ideal states are depicted in Fig. 6(f).
Notably, all fidelities exceed 0.98, indicating the proper functionality of
the proposed direct measurement method.

Figure 6:  (a)–(e) Results of DM of the density matrix for the quantum state
$|\psi_{1}^{\mathrm{test}}\rangle$–$|\psi_{5}^{\mathrm{test}}\rangle$ (top
panel) and ideal values (bottom panel). (f) Fidelities of the non-negative and
normalized density matrices obtained by DM to the ideal states.

##  IV Discussion and summary

The DM method utilizing the basis-shift unitary transformation, proposed in
this paper, proves to be the most efficient in terms of DM of matrix elements
using qubit probes. It minimizes the statistical fluctuation of the estimated
value to be measured within a fixed number of shots. This efficiency can be
explained as follows: In measurements with qubit probes, the value to be
estimated, denoted as $A\in[-1,+1]$, is given in proportion to the expectation
value of the Pauli-$Z$ measurement:

$\displaystyle A=k\langle\hat{Z}\rangle=k(p_{+}-p_{-})=k(2p_{+}-1),$ (17)

where $p_{\pm}$ is the probability that the Pauli-$Z$ measurement yields the
outcome $\pm 1$. The proposed DM method with basis-shift unitary
transformation corresponds to the case $k=1$. Let $\bar{Z}_{n}$ be the sample
mean of $Z$ measurements taken $n$ times. The estimator $\tilde{A}_{n}$ of $A$
is given by $\tilde{A}_{n}=k\bar{Z}_{n}$. Using Eq. (17), the sample variance
of $\bar{Z}_{n}$ is given as

$\displaystyle\mathrm{Var}(\bar{Z}_{n})=\frac{\mathrm{Var}(\hat{Z})}{n}=\frac{4p_{+}(1-p_{+})}{n}=\frac{1-k^{-2}A^{2}}{n},$
(18)

and the sample variance of $\tilde{A}_{n}$ is expressed as

$\displaystyle\mathrm{Var}(\tilde{A}_{n})=k^{2}\mathrm{Var}(\bar{Z}_{n})=\frac{k^{2}-A^{2}}{n}.$
(19)

Given that the variance is minimized when $k=1$, it is concluded that the
proposed DM method employing the basis-shift unitary transformation stands as
the most efficient approach for DM of matrix elements using qubit probes. In
contrast, in most previous studies Lundeen  _et al._ (2011); Bamber and
Lundeen (2014); Thekkadath  _et al._ (2016); Pan  _et al._ (2019); Kim  _et
al._ (2018); Gaikwad  _et al._ (2023); Xu  _et al._ (2021a); Vallone and
Dequal (2016); Calderaro  _et al._ (2018); Ogawa  _et al._ (2021); Zhou  _et
al._ (2021); Zhang  _et al._ (2020); Chen  _et al._ (2021); Xu  _et al._
(2021b), $k<1$ since they conduct the projection measurement in MUB for the
basis of the matrix to be obtained. Hence, the efficiency of the proposed DM
method represents a notable advantage. The use of basis-shift unitary
transformation is also expected to improve the measurement efficiency in DM of
processes and measurements.

In conclusion, we have introduced a theoretical framework aimed at unifying
the DM methods for the three components of quantum dynamics: quantum states,
quantum processes, and quantum measurements. Leveraging this framework, DM
methods for each quantum component can be systematically derived.
Particularly, we have developed a DM method utilizing the basis-shift unitary
transformation, which stands out as the most efficient approach when utilizing
qubit probes. Furthermore, we have conducted experimental demonstrations of
the DM method for quantum states using optical pulse trains. Our experiments
have showcased that the matrix elements of the density operator of the initial
state can be obtained separately, and the overall state can be estimated with
high fidelity. Although this experiment utilized optical pulse trains for
quantum communication applications, we anticipate that this DM method will
find applications in various physical systems. We envisage that in the future,
DM techniques will not only facilitate the measurement of states but also
enable the measurement of processes and measurements in diverse quantum
systems.

###### Acknowledgements.

This research was supported by JSPS KAKENHI Grant Number 21K04915.

##  Appendix A Details of experimental results and fitting process

In the experiments detailed in Section III, we produced optical pulse trains
corresponding to the five initial states $|\psi_{1}^{\rm
test}\rangle$–$|\psi_{5}^{\rm test}\rangle$ and detected these pulses using an
oscilloscope following the proposed DM procedure. By fitting a function with a
peak shape to the detected waveforms, we obtained the values of the peaks of
each pulse. The measured waveforms and the fitting results are depicted in
Fig. 7. For each individual peak, the following function is employed as a peak
shape fitting function:

$\displaystyle f_{\rm
fitting}(x)=A\left\\{\exp\left[\frac{-(x-x_{0})^{2}}{2\sigma^{2}}\right]+R\frac{\exp[-(x-x_{0})/\tau_{1}]}{1+\exp[-(x-x_{0})/\tau_{2}]}\right\\},$
(20)

where the first term represents a Gaussian function with a center at $x_{0}$
and a standard deviation of $\sigma$, while the second term represents a peak
shape function with a center at $x_{0}$ and exhibits asymmetric behavior with
its left and right sides decreasing exponentially with decay constants
$\tau_{1}$ and $\tau_{2}^{-1}-\tau_{1}^{-1}$, respectively. The magnitude
ratio of the second term to the first term is denoted by $R$. The second term
is added to express the asymmetric shape of the measured results for each
peak, with a slower decay on the right side. The fitting functions were
obtained by adding three, four, and five instances of the aforementioned peak
functions for cases without the AMZI, with AMZI inserted with a delay of 200
ps, and with AMZI inserted with a delay of 400 ps, respectively. Each peak was
fitted to the measured data to obtain parameters including
$x_{0},\sigma,\tau_{1},\tau_{2},A$, and $R$. These peak values $A$ were then
utilized to estimate each matrix element of the initial state according to the
procedure outlined in the main text.

Figure 7:  (a)–(e) Measured waveforms and fitting results for the five initial
states $|\psi_{1}^{\rm test}\rangle$–$|\psi_{5}^{\rm test}\rangle$. In each
plot, red dots represent experimental measurements, and blue lines depict
fitting curves. Left panels: measurement results without the AMZI. Center
panels: measurement results with AMZI inserted with a 200 ps delay and phase
differences of of $\phi=0^{\circ}$, $90^{\circ}$, $180^{\circ}$, and
$270^{\circ}$. Right panels: measurement results with AMZI inserted with a 400
ps delay and phase differences of $\phi=0^{\circ}$, $90^{\circ}$,
$180^{\circ}$, and $270^{\circ}$.

##  Appendix B Post-processing of density matrices measured by direct
measurement

When each matrix element is obtained individually by DM, the resultant
matrices generally fail to meet the physical requirements of density
operators, namely being non-negative operators with a unity trace. To rectify
this in the experiment, we post-processed the density matrix $A$ obtained by
DM using the following steps: First, to ensure the non-negativity requirement,
$A$ underwent spectral decomposition, and the imaginary part of its
eigenvalues was replaced by zero. If the real part was negative, the real part
was also replaced by zero. Subsequently, to ensure the unit trace requirement,
the entire matrix was normalized to $A\rightarrow A/a$, where $a$ is the sum
of the diagonal components $a:=A_{00}+A_{11}+A_{22}$. The matrices obtained
after this post-processing are displayed in Fig. 8. The fidelities of these
matrices with respect to the ideal states were evaluated as discussed in the
main text.

Figure 8:  (a)–(e) Density matrices of the quantum state
$|\psi_{1}^{\mathrm{test}}\rangle$–$|\psi_{5}^{\mathrm{test}}\rangle$ measured
by DM, post-processed to meet the requirements of a non-negative operator and
unit trace (upper panel), alongside the ideal values (bottom panel).

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