# Quantum state tomography of undetected photons

Jorge Fuenzalida<EMAIL_ADDRESS>Institute for Quantum
Optics and Quantum Information, Austrian Academy of Sciences, Boltzmanngasse
3, Vienna A-1090, Austria  Vienna Center for Quantum Science and Technology
(VCQ), Faculty of Physics, Boltzmanngasse 5, University of Vienna, Vienna
A-1090, Austria  Current address: Institute of Applied Physics, Technical
University of Darmstadt, Schloßgartenstraße 7, 64289 Darmstadt, Germany
Jaroslav Kysela<EMAIL_ADDRESS>Institute for Quantum Optics and
Quantum Information, Austrian Academy of Sciences, Boltzmanngasse 3, Vienna
A-1090, Austria  Vienna Center for Quantum Science and Technology (VCQ),
Faculty of Physics, Boltzmanngasse 5, University of Vienna, Vienna A-1090,
Austria  Krishna Dovzhik  Institute for Quantum Optics and Quantum
Information, Austrian Academy of Sciences, Boltzmanngasse 3, Vienna A-1090,
Austria  Vienna Center for Quantum Science and Technology (VCQ), Faculty of
Physics, Boltzmanngasse 5, University of Vienna, Vienna A-1090, Austria
Gabriela Barreto Lemos  Instituto de Física, Universidade Federal do Rio de
Janeiro, Av. Athos da Silveira Ramos 149, Rio de Janeiro, CP: 68528, Brazil
Armin Hochrainer  Institute for Quantum Optics and Quantum Information,
Austrian Academy of Sciences, Boltzmanngasse 3, Vienna A-1090, Austria  Vienna
Center for Quantum Science and Technology (VCQ), Faculty of Physics,
Boltzmanngasse 5, University of Vienna, Vienna A-1090, Austria  Mayukh Lahiri
<EMAIL_ADDRESS>Department of Physics, Oklahoma State University,
Stillwater 74078, Oklahoma, USA  Anton Zeilinger<EMAIL_ADDRESS>Institute for Quantum Optics and Quantum Information, Austrian Academy of
Sciences, Boltzmanngasse 3, Vienna A-1090, Austria  Vienna Center for Quantum
Science and Technology (VCQ), Faculty of Physics, Boltzmanngasse 5, University
of Vienna, Vienna A-1090, Austria

###### Abstract

The measurement of quantum states is one of the most important problems in
quantum mechanics. We introduce a quantum state tomography technique in which
the state of a qubit is reconstructed, while the qubit remains undetected. The
key ingredients are: (i) employing an additional qubit, (ii) aligning the
undetected qubit with a known reference state by using path identity, and
(iii) measuring the additional qubit to reconstruct the undetected qubit
state. We theoretically establish and experimentally demonstrate the method
with photonic polarization states. The principle underlying our method could
also be applied to quantum entities other than photons.

Introduction.—The process and interpretation of quantum measurements have been
vastly studied since the early days of quantum mechanics [1]. The results of
measurements made on a given quantum system can be succinctly summarized in
the form of a quantum state, which represents the complete description of the
quantum system. The quantum state estimation spans over multiple techniques,
such as projective measurements [2, 3], weak measurements [4, 5, 6, 7], and
device-independent measurements [8, 9, 10]. The quantum state tomography
consists in the identification of a quantum state by measurements on many
identical copies of a quantum system [11]. In the case of photons, the
measurement is destructive via direct measurements [12, 13] or interferometric
measurement [14]. Furthermore, technical limitations might hinder an efficient
detection process. For example, single-photon detectors are not equally
efficient over the whole spectrum. Therefore, standard quantum tomography
approaches may be challenging or even impossible in some cases.

Here, we introduce and implement a quantum tomography method in which the
state of a single qubit is reconstructed without the qubit being detected or
any direct measurement being performed on it 111We note here that for some
quantum systems such as trapped ions, the quantum state can be determined by
detecting fluorescence light, i.e., without performing direct measurement on
the ions [40, 41]. However, such approaches are not applicable to photons and
many other quantum entities like neutrons, electrons, etc.. Instead, we
introduce an additional particle (photon in our case). We then apply the
concept of _path identity_ [16] to build an interferometer, in which only the
additional photon is detected via intensity measurement. We reconstruct the
quantum state of the qubit from the interferometric data. Our method works for
both pure and mixed states, and we experimentally demonstrate it for pure
single-photon polarization states.

In contrast with existing tomographic methods, our method is applicable to
cases in which the qubit cannot be detected for any technical or fundamental
reason. This is because we do not detect the qubit and the additional photon
can be chosen at a frequency for which adequate detectors are available.

Figure 1:  Quantum state tomography with detected and undetected particles.
(a) Quantum state tomography retrieves the state of a quantum system by
measuring an ensemble of identical particles. (b) In our scheme, the target
subsystem for which we aim to obtain $\widehat{\rho}$ remains undetected. What
is measured instead is an auxiliary subsystem that is quantum correlated to
the target one.

Our method of quantum state tomography is based on the quantum phenomenon of
induced coherence without induced emission [17, 18], which can be viewed as
interference by path identity [16]. Mandel and collaborators first
demonstrated this phenomenon using an interferometric arrangement consisting
of two photon-pair sources. They induced coherence between two modes of one
photon by making paths of its partner photon _identical_. They obtained a
single-photon interference pattern detecting the former photon only. The
detection of the partner photon was not required. We henceforth refer to this
phenomenon as “induced coherence” for simplicity. The potential of induced
coherence has already been recognised as it has found use in quantum imaging
[19], spectroscopy [20], holography [21], sensing [22], optical coherence
tomography [23, 24], measurement of two-photon correlations and entanglement
[25, 26, 27, 28]. This phenomenon has also been exploited to investigate the
interplay between the polarization and coherence [29, 30]. Nevertheless, to
date no technique based on induced coherence has been developed to fully
reconstruct the polarization state of single photons.

Principle.—In the standard polarization quantum state tomography [Fig. 1(a)] a
photon is prepared in a general state $\widehat{\rho}$ and then subjected to
intensity measurements performed in various settings given by the Pauli
operators. The recorded results are then used to obtain the Stokes parameters
and the associated density matrix $\widehat{\rho}$ [12]. In our technique
[Fig. 1(b)], the target photon whose state $\widehat{\rho}$ we reconstruct
remains undetected. What is measured instead is an auxiliary photon with which
the target photon interacts. Information about the state $\widehat{\rho}$ is
transferred to the auxiliary photon by employing a path identity approach
[Fig. 2]. Through this step, the state of the target photon changes, and its
representation can be found in the Supplemental Material. To obtain
$\widehat{\rho}$, the auxiliary photon is prepared, projected, and detected in
different configurations. The state $\widehat{\rho}$ is retrieved from the
interference patterns obtained by measuring the auxiliary photon.

Figure 2: Path identity concept of implementation. The crucial point in our
scheme is transferring the state information from the target to the auxiliary
particle which is done by a path identity approach. Measurements on the
auxiliary particle allow us to fully retrieve $\widehat{\rho}$ of the target
particle. $\phi$: phase shifter, U: unitary operation, P: projector, BS: beam
splitter, D: detector.

Method.—

Figure 3: The experimental setup for the proposed quantum tomography method of
polarization states. The pump, the idler and the signal photons are
represented by the green, red and blue lines, respectively. One photon pair is
created in a coherent superposition of three nonlinear crystals (NL1–NL3).
Idler beams are overlapped in order to erase the which-source information, the
lack of which induces coherence of the signal photon. Signal photons emitted
in each crystal are superposed on a beam splitter (BS), split into $H$ and $V$
polarization by a polarizing beam splitter (PBS) and detected by detectors D1
and D2. The recorded intensities of the signal photon exhibit interference as
the independent phase $\phi$ is varied. Our method allows one to identify the
polarization state of the idler photon emitted in crystal NL1 (source Q1) even
though it is never detected. $a$, $b$, $b^{\prime}$, and $c$ stand for optical
paths.

Our aim is to reconstruct the quantum state of a photon, henceforth referred
to as the idler photon, whose density matrix has the general form

$\widehat{\rho}_{I}=\begin{pmatrix}P_{H}&\mathscr{I}\sqrt{P_{H}P_{V}}e^{-i\xi}\\\
\mathscr{I}\sqrt{P_{H}P_{V}}e^{i\xi}&P_{V}\end{pmatrix}.$ (1)

In the formula above, $P_{H}$ ($P_{V}$) is the probability of detecting the
horizontal $H$ (vertical $V$) polarization of the idler photon,
$P_{H}+P_{V}=1$; $0\leq\xi<2\pi$ is a relative phase between $H$ and $V$; and
$0\leq\mathscr{I}\leq 1$ quantifies the purity of the state. It can be readily
checked that $\mathscr{I}=0$ and $\mathscr{I}=1$ correspond to fully mixed and
pure states, respectively.

Our reconstruction technique employs the experimental setup depicted in Fig.
3. We employ an additional photon, referred to as the signal photon, such that
the two photons are in the state

$\displaystyle\widehat{\rho}_{1}=|H_{Sa}\rangle\langle
H_{Sa}|\otimes\widehat{\rho}_{I},$ (2)

where $|H_{S_{a}}\rangle$ represents a horizontally polarized signal photon
propagating along path $a$, and the idler photon $I$ propagates along path
$b^{\prime}$ in the state $\widehat{\rho}_{I}$. Our technique uses two sources
of photon pairs, denoted by Q1 and Q2. The first source comprises a single
nonlinear crystal and produces photons in a separable state. After the state
preparation of the idler photon, this state is given by Eq. (2). The second
source comprises two nonlinear crystals in a cross-crystal configuration and
produces photon pairs in a maximally entangled state
$\widehat{\rho}_{2}=|\psi_{2}\rangle\langle\psi_{2}|$ in path $b$, where

$\displaystyle|\psi_{2}\rangle=\frac{1}{\sqrt{2}}(|H_{Sb}\rangle|H_{Ib}\rangle+|V_{Sb}\rangle|V_{Ib}\rangle).$
(3)

This state plays the role of a reference state of polarization, with respect
to which state $\widehat{\rho}_{I}$ is probed, as explained below.

$Q_{1}$ and $Q_{2}$ are pumped coherently. The pump power for both sources is
low, such that the two sources do not emit photons simultaneously and the
effect of stimulated emission can be neglected. In this case, one photon pair
is in a coherent superposition of being emitted by the sources Q1 and Q2. The
density matrix representing a photon pair in such a scenario is given by Eq.
(4) of the Supplemental material.

The alignment of photons’ paths plays a crucial role in our technique. We
superimpose the two signal-photon paths $a$ and $b$ at a beam splitter and
later the two polarization modes in path $b$ are separated by a polarizing
beam splitter. The final $H$ ($V$) component is collected by detector D1 (D2);
see Fig. 3. When the idler beam emitted from the source Q1 is aligned
precisely such that it overlaps with the idler beam emitted from the source
Q2, the source information is erased, and an interference pattern arises in
intensities detected by D1 and D2. Therefore, we have an effective two-fold
induced coherence in two orthogonal polarization modes $H$ and $V$, where the
level of indistinguishability depends on $\widehat{\rho}_{I}$.

When the pump powers are adjusted appropriately (see Supplemental material),
the intensity (photon counting rate) for $H$ polarization at detector D1 is
given by

$\langle\widehat{R}_{H}\rangle=\frac{1}{4}\left(1+|T_{H}|\sqrt{P_{H}}\cos(\phi)\right),$
(4)

provided that the half-wave plate HWPS is set to $0^{\circ}$, where $T_{H}$ is
the total amplitude transmission coefficient of the $H$ polarization, which
takes into account all optical components placed on path $b^{\prime}$, and
$\phi\in\mathbb{R}$ is the relative phase between the two sources. The rate
$\langle\widehat{R}_{V}\rangle$ for vertical polarization stays constant for
such a setting. To record the interference pattern for $V$ polarization, we
set the HWPS to $45^{\circ}$, such that the signal photon’s state in path $a$
turns to $|V_{Sa}\rangle$. In this case, rate $\langle\widehat{R}_{H}\rangle$
monitored by D1 stays constant, while detector D2 records photon counts whose
rate varies according to

$\langle\widehat{R}_{V}\rangle=\frac{1}{4}\left(1+\mathscr{I}|T_{V}|\sqrt{P_{V}}\cos(\phi+\xi)\right),$
(5)

where $T_{V}$ is the transmission coefficient for the $V$ polarization,
defined analogously to $T_{H}$, and $\xi$ was introduced in Eq. (1). Upon
varying $\phi$ the two expressions above display interference with
visibilities 222The visibility is $\mathcal{V}=(\langle
R_{\mathrm{max}}\rangle-\langle R_{\mathrm{min}}\rangle)/(\langle
R_{\mathrm{max}}\rangle+\langle R_{\mathrm{min}}\rangle)$, where $\langle
R_{\mathrm{max}}\rangle$ and $\langle R_{\mathrm{min}}\rangle$ are the maximum
and minimum intensities, respectively.

$\displaystyle\mathcal{V}_{H}$ $\displaystyle=$
$\displaystyle|T_{H}|\sqrt{P_{H}},$ (6) $\displaystyle\mathcal{V}_{V}$
$\displaystyle=$ $\displaystyle\mathscr{I}|T_{V}|\sqrt{P_{V}}.$ (7)

These formulas quantify the level of indistinguishability in the two
orthogonal polarization modes. We can perform a separate measurement of the
amplitude transmission coefficients $|T_{H}|$ and $|T_{V}|$ (i.e., losses and
alignment). These coefficients determine the maximum achievable visibilities.
The visibility $\mathcal{V}_{H}$ allows us to infer the value of $P_{H}$ and
$P_{V}=1-P_{H}$, the visibility $\mathcal{V}_{V}$ contains information about
the degree of coherence $\mathscr{I}$ and the local phase $\xi$ is equal to
the relative phase shift between interference fringes for
$\langle\widehat{R}_{H}\rangle$ (4) and $\langle\widehat{R}_{V}\rangle$ (5).
This way we can reconstruct state $\widehat{\rho}_{I}$ in Eq. (1).

Figure 4: Experimental results for pure state tomography of photons that are
never detected. Main results of visibility measurements are shown in the left
panels. The red and blue colors represent the horizontal and vertical
polarization, respectively. (a) The idler photon is prepared in different
linear polarization states by HWPI, see the Bloch sphere in the inset. The
signal photon visibility for each linear polarization is plotted against the
angle of the plate HWPI. (b) The idler photon is prepared in different
elliptical polarization states by QWPI, starting in $|H_{Ib^{\prime}}\rangle$.
The signal photon visibility is plotted against the angle of the plate QWPI.
The calibrated experimental data points in (a) and (b) are in agreement with
the theoretical predictions, which are given by the solid lines. The idler
states $|D_{Ib^{\prime}}\rangle$ and $|A_{Ib^{\prime}}\rangle$, designated by
(1) and (2) in (a), and $|R_{Ib^{\prime}}\rangle$ and
$|L_{Ib^{\prime}}\rangle$, designated by (3) and (4) in (b), are all
represented by the same visibilities of the signal photon. Nevertheless, these
four states have different relatives phases. We have included the experimental
density matrices obtained for these four states in the insets (1)–(4).

Implementation.—We demonstrated the described technique of reconstruction of
the idler state experimentally. We used ppKTP crystals emitting photon pairs
by spontaneous parametric down-conversion in a non-degenerate, type-0 process.
Signal (idler) photons had a central wavelength of 842 (780) nm. In the
detection part, we used a 800 nm long-pass filter and a $842\pm 1$ nm
interference filter to select only signal photons.

The tomography procedure needs to be calibrated to compensate for the
misalignment in the experimental implementation. To that end, we measured the
maximum visibilities $T_{H}$ and $T_{V}$ that can be achieved experimentally
for $H$ and $V$ polarization. We experimentally obtained $T_{H}$ = 0.85 $\pm
0.03$ and $T_{V}$ = 0.73 $\pm 0.02$ 333The difference between these maximum
visibilities is caused by the impossibility of imaging the crystal plane of
the source Q1 simultaneously to the crystal planes of the two crystals
comprising the source Q2. However, this difference could be fixed by building
the source Q2 in the Sagnac geometry.. The calibration itself consisted in the
division of each experimentally obtained visibility $\mathcal{V}_{H}$
($\mathcal{V}_{V}$) by $T_{H}$ ($T_{V}$).

A cascade of a half-wave plate (HWPI) and a quarter-wave plate (QWPI) is used
to prepare the polarization state $\widehat{\rho}_{I}$ of the idler photon
generated by the source Q1 in the path $b^{\prime}$. The tomography was
performed by taking measurements in two configurations of the signal photon,
$|H_{Sa}\rangle$ and $|V_{Sa}\rangle$, which were prepared by the plate HWPS.
For the configuration $|H_{Sa}\rangle$ ($|V_{Sa}\rangle$) we recorded the
intensity $\langle\widehat{R}_{H}\rangle$ ($\langle\widehat{R}_{V}\rangle$) at
the detector D1 (D2) while the phase $\phi$ was varied. Fitting a sinusoidal
function to the recorded intensity profiles we extracted the visibilities for
the signal photon as well as the phase shift for each state. The values of
$P_{H}$, $P_{V}$, and $\xi$ were then extracted using Eqs. (4) and (5).
Additionally, we implemented a maximum likelihood estimation algorithm to
reconstruct density matrices from our data 444Additional information about the
explicit form and evolution of the photons’ state, polarization measurement,
calibration, state reconstruction, post-measurement state, and measured
fidelities can be found in the supplemental material..

Experimental results are presented in Fig. 4, where dots represent
experimental data, whereas solid lines represent theoretical prediction. We
performed two measurement scans. In the first scan the entire range of pure
linearly polarized states of the idler photon was traversed by rotating HWPI.
Calibrated experimental results, which are presented in Fig. 4 (a), agree well
555The standard deviation is obtained from sinusoidal function fitted to the
experimental data. with the theoretical predictions given by Eqs. (6) and (7).
Indeed, initially $\mathcal{V}_{H}$ is maximal and $\mathcal{V}_{V}$ is
minimal as the idler state is $H$-polarized. As HWPI is rotated,
$\mathcal{V}_{H}$ ($\mathcal{V}_{V}$) gradually decreases (increases) until it
reaches its minimum (maximum) when the idler photon is $V$-polarized. The mean
fidelity of states reconstructed in this scan is found to be
$\bar{F}_{\mathrm{HWP}}=0.97$.

In the second scan, QWPI was rotated to generate a series of pure elliptically
polarized states ranging from the right-circular polarization to the
horizontal polarization and then further to the left-circular polarization. We
started with $|H_{Ib^{\prime}}\rangle$. As QWPI is rotated and the state of
the idler photon acquires a complex $V$-polarized component, the visibility
for $H$ decreases whereas that of $V$ increases until they both reach the
extreme value of $1/\sqrt{2}$ when the state is right- or left-circular. This
behavior is closely followed by the experimental data, which are presented in
Fig. 4 (b). The states measured in this scan have a mean fidelity of
$\bar{F}_{\mathrm{QWP}}=0.92$.

As has been pointed out in the previous section, the visibilities
$\mathcal{V}_{H}$ (6) and $\mathcal{V}_{V}$ (7) do not identify the idler
photon’s state unambiguously. Any two states $\widehat{\rho}_{I}$ and
$\widehat{\rho}^{\prime}_{I}$ for which
$\mathcal{V}_{H}=\mathcal{V}_{H}^{\prime}$ and
$\mathcal{V}_{V}=\mathcal{V}_{V}^{\prime}$ will be misidentified. For precise
identification also the phase shift between interference patterns in Eqs. (4)
and (5) is needed. This fact is illustrated for states
$|D_{Ib^{\prime}}\rangle$ and $|A_{Ib^{\prime}}\rangle$ (diagonal and anti-
diagonal) in the insets of Fig. 4 (a), and states $|R_{Ib^{\prime}}\rangle$
and $|L_{Ib^{\prime}}\rangle$ (right- and left-circular) in the insets of Fig.
4 (b). Visibilities $\mathcal{V}_{H}$ as well as $\mathcal{V}_{V}$ are
identical for all these states, but the relative phases differ.

The experimental results clearly demonstrate the feasibility of our
reconstruction technique, as long as pure polarization states of photons are
considered. We successfully characterized all the linearly polarized states
and also a specific class of elliptically polarized states. Fidelities of all
the states, shown in a plot in the Supplemental Material, lie above
$F_{\mathrm{min}}=0.72$.

Conclusion.— In this paper we introduced a novel tomography method that allows
one to determine a polarization state of a photon that is never detected. This
photon is a part of one photon pair that is emitted in a coherent
superposition of two spatially separated sources. The first source emits idler
photons in an unknown state that is to be reconstructed. The second source
produces a reference state that contains two orthogonal polarization modes
666We note that the crystal NL3 can in principle be replaced by additional
retarder plates placed in the idler path before NL2.. As a result of the
indistinguishability in the two polarization modes the resulting interference
patterns contain all the information about the idler photon that allows us to
fully reconstruct its state. We demonstrated experimentally this
reconstruction for two classes of idler states—linearly polarized states and
specifically elliptically polarized states. We also presented the theoretical
treatment of the general case when the idler photon’s polarization state is
mixed. The fidelities of states can be increased by improving the phase
stability in our experiment.

The main advantage over traditional techniques stems from the fact that in our
approach only signal photons are detected and these can have a different
wavelength from that of the idler photons. In certain frequency (wavelength)
ranges, efficient detectors are not affordable or available. In our approach,
the high flexibility of the SPDC process allows us to choose the favorable
wavelength that can be efficiently detected [36, 37, 22, 38].

To conclude, this work provides a new tool for the precise identification of
single-photon states with high potential in a wide range of applications. Our
approach can be adapted to the tomography of high-dimensional degrees of
freedom of single photons, such as the orbital angular momentum [39].
Including spatial modes in our technique, i.e. position and momentum, one can
implement polarization quantum imaging with undetected photons [19]. It can
also be used in the characterization of birefringent materials, sensing, and
polarization-sensitive optical coherence tomography [23, 24]. Note that the
post-measurement state of the idler photon still carries partial information
about the state preparation. If the idler detection is not a problem, our
scheme can be used in quantum communication protocols between two parties,
where one party obtains information through interference patterns (signal
photon), and the other party obtains probabilistic information via projective
measurements (idler photon). It has also not escaped our notice that our
method can be adapted to any two-level quantum system, such as atoms, ions, or
molecules.

JF thanks G. Carvacho for helpful discussions. The authors thank R. Blach for
technical support. This work was supported by the Austrian Academy of Sciences
(ÖAW). JF also acknowledges ANID for the financial support (Becas de doctorado
en el extranjero “Becas Chile”/ 2015 – No. 72160487). JK acknowledges the
support of the University of Vienna via the project QUESS (Quantum Experiments
on Space Scale). KD acknowledges the support of the Austrian Science Fund
(FWF): F40 (SFB FoQuS).

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  * Note [3] The difference between these maximum visibilities is caused by the impossibility of imaging the crystal plane of the source Q1 simultaneously to the crystal planes of the two crystals comprising the source Q2. However, this difference could be fixed by building the source Q2 in the Sagnac geometry.
  * Note [4] Additional information about the explicit form and evolution of the photons’ state, polarization measurement, calibration, state reconstruction, post-measurement state, and measured fidelities can be found in the supplemental material.
  * Note [5] The standard deviation is obtained from sinusoidal function fitted to the experimental data.
  * Note [6] We note that the crystal NL3 can in principle be replaced by additional retarder plates placed in the idler path before NL2.
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