# Einstein locality: The key to reveal the dynamics for nonlocality

Sheng Feng<EMAIL_ADDRESS>School of Electrical and Electronic
Information Engineering,
Hubei Polytechnic University, Huangshi, Hubei 435003, China  Center for
Fundamental Physics,
Hubei Polytechnic University, Huangshi, Hubei 435003, China

††preprint: APS/123-QED

The persistent efforts [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] to understand the
nonlocal behavior of entangled systems have brought the blossom of the quantum
information science [11, 12, 13, 14, 15, 16, 17, 18, 19], which exemplifies
the notion that the ongoing in-depth studies of physical laws governing the
behaviors of microscopic objects provide a continuous impetus for the advance
of modern science. Albeit the celebrated loop-free experiments [7, 8, 9, 10]
have firmly verified nonlocality that indicates “physical influence faster
than the speed of light” [4, 20, 21], to grasp the exact meaning of
“influence” and to uncover the dynamic mechanism for the “spooky action at a
distance” nonetheless remain a world-class challenge. Here I report on the
discovery of Einstein locality, i.e., there exists no instantaneous state
collapse of a quantum object due to that of other ones, that proves the key to
unlock the mystery of nonlocality. A theoretical proof for Einstein locality
is provided with the formalism of quantum mechanics and an Einstein locality
model built for nonlocality. Moreover, the existential results of double-slit
experiments involving entangled photons are presented as the evidence for
Einstein locality in the quantum world. The discovery will enrich the
worldview about the behaviors of quantum objects and is anticipated to have
far-reaching impacts on quantum science and technology.

The idea of nonlocality arose from a renowned paper [1] published in 1935 by
Einstein, Podolsky, and Rosen (EPR) as part of the debate between Bohr and
Einstein over the completeness of quantum mechanics [1, 2]. Regarding two
particles in an entangled state, EPR tried to show in the paper that the
measurement-induced state reduction of one particle should cause an
instantaneous state collapse of the other (Einstein nonlocality) in accordance
to quantum mechanics. By assuming no physical influence faster than the speed
of light, they further argued that quantum mechanics is therefore incomplete
unless the uncertainty principle is violated. For a long period of time, the
two sides in the debate nonetheless faced difficulty in finding a way to put
the EPR argument to experimental tests. The turning point occurred in 1964
when Bell announced his famous inequality that provides an excellent
quantitative measure for Bell nonlocality, i.e., the dependence of the outcome
of one event on others at a distance [4, 22].

After decades of hard work by the pioneers who have successfully conducted
Bell experiments [23, 6, 20, 24, 25, 7, 8, 9, 10], the experiments on
nonlocality based on Bell inequality [4] or its variants [5], substantial
advances have been achieved in the fundamental tests of quantum mechanics [26,
22, 27] and the developments of quantum technology [15, 16, 17, 18, 19].
Nevertheless, the mysterious essence of nonlocality still stimulates unfading
research interests among the scientific community [28, 29, 30, 31, 32, 33]
because a full understanding has not been reached. The specific dynamic
mechanism that dictates nonlocality and the accompanying “physical influence
faster than the speed of light” [4, 20, 21] is shrouded in secrecy and to be
unraveled, given that the lack of knowledge about this mechanism sets an upper
limit to what one can learn from Bell experiments. To solve the problem once
for all, one needs to step out of the shadows of EPR and Bell and must be
aware that the fundamental source of the problem might just lie in the
rationales about nonlocality given by EPR and Bell. Put it another way, one
would better shake off semantic arguments about the precise definitions of
locality and reality by EPR and Bell; In stead, one should stick to the
accurate formalism of quantum mechanics and, thereby, inspect the logic chains
of the works by EPR and Bell [1, 4]. Moreover, any results achieved in theory
must be conclusively based upon experimental observations and can provide new
predictions verifiable in experiment. In this spirit, I present the discovery
of Einstein locality that proves the key to uncover the dynamics for the
nonlocally correlated behaviors of quantum objects violating Bell inequality
as observed in Bell experiments [23, 6, 20, 24, 25, 7, 8, 9, 10].

In terms of the formalism of quantum mechanics, the start point of the whole
story about nonlocality is the application of the superposition principle by
EPR to write the wavefunction of a two-particle system as a Schmidt
decomposition of the eigenfunctions of two operators describing a joint
measurement on the system [1]. It is interesting that EPR’s idea of
nonlocality was based on an extremely subtle mistake in their use of the
superposition principle, as shown below, that turns out to be the fundamental
source of the problem. To elucidate the point, let kick off with the
exploration of the superposition principle by considering two electrons, $a$
and $b$, on which a joint spin measurement is performed. When the electrons
are in an entangled state where their spins are perfectly anti-correlated and
the total spin angular momentum of the whole system is zero, this state is
referred to as a singlet state. Suppose that electron $a$ is measured for its
$\hat{\sigma}_{z}$ spin value (namely a $\hat{\sigma}_{z}^{a}$ measurement),
where $\hat{\sigma}_{z}$ is the $z$-component of the Pauli operator, and
electron $b$ is measured for its $\hat{\sigma}_{n}$ spin (a
$\hat{\sigma}_{n}^{b}$ measurement), in which $\hat{\sigma}_{n}$ denotes the
component of the Pauli operator along an arbitrary $\mathbf{n}$-axis with a
polar angle $\alpha$ and an azimuthal angle $\beta$. Then the wavefunction
$|\phi_{c}\rangle$ of the two electrons in the singlet state reads

$\displaystyle|\phi_{c}\rangle$ $\displaystyle=$
$\displaystyle\sum_{p,q}\gamma_{pq}\ |p_{z}\rangle_{a}|q_{n}\rangle_{b}\ ,$
(1)

wherein $p=\pm$, $q=\pm$,
$\gamma_{++}=\gamma_{--}^{*}=2^{-1/2}\sin(\alpha/2)e^{-i\beta}$,
$\gamma_{+-}=-\gamma_{-+}=2^{-1/2}\cos(\alpha/2)$, and $|\pm_{z,n}\rangle$ are
respectively the eigenstates of the $\hat{\sigma}_{z,n}$ operators, i.e.,
$\hat{\sigma}_{z}|p_{z}\rangle=p|p_{z}\rangle$ and
$\hat{\sigma}_{n}|q_{n}\rangle=q|q_{n}\rangle$. $|\cdot\rangle_{a,b}$ stand
for the states of the labeled particles. It must be stressed that Eq. (1)
cannot be expressed as a Schmidt decomposition of $|\phi_{c}\rangle$ for the
joint $\hat{\sigma}_{z}^{a}\hat{\sigma}_{n}^{b}$ measurement unless
$\alpha=0$, which is a crucial but subtle point missed by EPR in the paper
(see Methods Mark1 for detailed explanations).

In Eq. (1), each term on the righthand side represents one possible result of
the joint $\hat{\sigma}_{z}^{a}\hat{\sigma}_{n}^{b}$ measurement on the system
with a $|\gamma_{pq}|^{2}$ probability and the corresponding physical event
will not happen before the $\hat{\sigma}_{z}^{a}\hat{\sigma}_{n}^{b}$
measurement is fully accomplished. For instance, in the case that electron $a$
is projected onto the $|+_{z}\rangle_{a}$ state by a $\hat{\sigma}_{z}^{a}$
measurement, there are two possible events for the whole system: It may be
projected onto either the $|+_{z}\rangle_{a}|+_{n}\rangle_{b}$ or the
$|+_{z}\rangle_{a}|-_{n}\rangle_{b}$ state. But none of these potential events
will occur before the joint measurement is finished, which entails that
electron $b$ will not collapse into the $|+_{n}\rangle_{b}$ or the
$|-_{n}\rangle_{b}$ state unless the $\hat{\sigma}_{n}^{b}$ measurement is
actually performed on it. In other words, electron $b$ will stay in its
initial state until it is projected onto a $|q_{n}\rangle_{b}$ state by the
$\hat{\sigma}_{n}^{b}$ measurement. Hence this particle will not undergo an
instantaneous state collapse due to the measurement-induced state collapse of
electron $a$, which reasonably recognizes the Einstein locality of quantum
mechanics (see Methods Mark2 for more detailed discussions).

In fact, electron $a$ and $b$ are independently projected onto the
$|p_{z}\rangle_{a}$ and $|q_{n}\rangle_{b}$ states respectively by the
$\hat{\sigma}_{z}^{a}$ and $\hat{\sigma}_{n}^{b}$ measurements with the joint
probability of $2^{-1}\cos^{2}(\alpha/2)$ or $2^{-1}\sin^{2}(\alpha/2)$.
Alternatively, one may understand the result of Eq. (1) as follows: The
equation allows one to predict the outcomes of a $\hat{\sigma}_{z}^{a}$
measurement on electron $a$ that is reduced into a $|p_{z}\rangle_{a}$ state,
with the probability of $2^{-1}\cos^{2}(\alpha/2)$ or
$2^{-1}\sin^{2}(\alpha/2)$ dependent on the specific $\hat{\sigma}_{n}^{b}$
measurement that projects electron $b$ onto a $|q_{n}\rangle_{b}$ state. It is
this conditional predictability of the measurement results that gives rise to
Bell nonlocality (Methods Mark3). However, to assimilate the result of
recognising Einstein locality through Eq. (1), one needs a pictorial insight
into the essence of Bell nonlocality from the perspective of Einstein locality
so that no concern will be raised about the consistency between Bell
nonlocality and Einstein locality. For this purpose, a theoretical model is
built based on Einstein locality; The goal is to satisfactorily explain Bell
nonlocality observed in Bell experiments with this model. For a Bell
experiment with an electron singlet system, let use $N_{\pm_{z,n}}^{a,b}$
respectively denote the numbers of counted $|\pm_{n,z}\rangle_{a,b}$ electrons
in the corresponding $\hat{\sigma}_{z,n}^{a,b}$ measurements. Then the
Einstein locality model described below consists of two core elements:
(1) Coherence kernel. Initially each electron is in a reduced state governed
by a density operator
$\hat{\rho}=2^{-1}(|+_{n}\rangle\langle+_{n}|+|-_{n}\rangle\langle-_{n}|)$ for
an arbitrary $\mathbf{n}$-axis;
(2) Correlation kernel. The results of local $\hat{\sigma}_{n}$ measurements
on the electrons are anti-correlated, i.e., $N_{+_{n}}^{a}=N_{-_{n}}^{b}$ and
$N_{-_{n}}^{a}=N_{+_{n}}^{b}$ for an arbitrary $\mathbf{n}$-axis.

Now one should note that both the coherence and correlation requirements by
the two kernels are naturally satisfied by the electron singlet system and
that each single kernel of the model nevertheless does not endow the compound
system with a nonlocal property in any sense. For instance, the correlation
kernel alone guarantees neither Bell nonlocality nor Einstein nonlocality,
because it is trivial to implement $N_{\pm_{n}}^{a}=N_{\mp_{n}}^{b}$ for
arbitrary $\mathbf{n}$-axes in the context of classical optics [23, 34].
Interestingly, as shown in the following, the combination of the two kernels
is nonetheless sufficient for explaining the Bell experiment; The key point is
to identify some coherence property owned by electron $a$, which may be
achieved by considering two measurements on the particle (a
$\hat{\sigma}_{z}^{a}$ measurement and a $\hat{\sigma}_{n}^{a}$ measurement).
From the coherence kernel, it is not difficult to calculate the expectation
value of the $\hat{\sigma}_{z}^{a}\hat{\sigma}_{n}^{a}$ operator,

$\displaystyle<\hat{\sigma}_{z}^{a}\hat{\sigma}_{n}^{a}>$ $\displaystyle=$
$\displaystyle\cos\alpha\ ,$ (2)

which shows some coherence between the $|p_{z}\rangle_{a}$ and
$|q_{n}\rangle_{a}$ states of electron $a$ (the same is true for electron $b$
and see Methods Mark4 for more details).

The $<\hat{\sigma}_{z}^{a}\hat{\sigma}_{n}^{a}>$ value given by Eq. (2) may be
measured in experiment if one could simultaneously carry out the
$\hat{\sigma}_{z}^{a}$ and $\hat{\sigma}_{n}^{a}$ measurements on electron $a$
when it is in its initial state. By the measurements, experimental data might
be produced respectively for $N_{p_{z}}^{a}$ and $N_{q_{n}}^{a}$, which can
then be used to calculate a normalized coincidence counting rate,
$C(N_{p_{z}}^{a},N_{q_{n}}^{a})$, as a function of $N_{p_{z}}^{a}$ and
$N_{q_{n}}^{a}$. Finally one would have the experimentally measured value of
$<\hat{\sigma}_{z}^{a}\hat{\sigma}_{n}^{a}>$ [5],

$\displaystyle<\hat{\sigma}_{z}^{a}\hat{\sigma}_{n}^{a}>$ $\displaystyle=$
$\displaystyle\sum_{p,q}\mbox{sign}(pq)\ C(N_{p_{z}}^{a},N_{q_{n}}^{a})\ ,$
(3)

wherein $\mbox{sign}(\pm\pm)=1$ and $\mbox{sign}(\pm\mp)=-1$.

Figure 1: A Bell experiment with an electron singlet system from the
perspective of the Einstein locality model. To acquire the $N_{p_{z}}^{a}$
data (triangles) electron $a$ will be destroyed after being detected in a
$|p_{z}\rangle_{a}$ state by the $\hat{\sigma}_{z}^{a}$ measurement. Then, the
$N_{-q_{n}}^{a}$ data (circles) cannot be acquired via a
$\hat{\sigma}_{n}^{a}$ measurement on electron $a$ since nobody can count it
twice when it is in the initial state. The key to obtain the $N_{-q_{n}}^{a}$
data is to use the $N_{q_{n}}^{b}$ data (red dots) as the replicas of the
needed $N_{-q_{n}}^{a}$ data because $N_{-q_{n}}^{a}=N_{q_{n}}^{b}$.

However, it is impossible in practice to simultaneously perform the
$\hat{\sigma}_{z}^{a}$ and $\hat{\sigma}_{n}^{a}$ measurements on electron $a$
in the same initial state and, therefore, the experimental result of relevance
to Eq. (3) is unavailable under usual experimental conditions. Fortunately,
for the electron singlet system, the photon correlation makes it possible for
one to achieve the result given by Eq. (3). For example, one may acquire the
$N_{p_{z}}^{a}$ data directly through the $\hat{\sigma}_{z}^{a}$ measurement
on electron $a$ in the initial state. As for the $N_{q_{n}}^{a}$ data, one may
first obtain $N_{q_{n}}^{b}$ data by the independent $\hat{\sigma}_{n}^{b}$
measurement on electron $b$ in its initial state (Einstein locality). Because
$N_{-q_{n}}^{a}=N_{q_{n}}^{b}$ (correlation kernel), then the $N_{q_{n}}^{b}$
data can be used as replicas of the $N_{-q_{n}}^{a}$ data that are otherwise
unavailable but needed for Eq. (3) (Fig. 1). Due to the spin anti-correlation
between the electrons, the result achieved via Eq. (3) using the
$N_{p_{z}}^{a}$ and $N_{-q_{n}}^{a}$ data needs to be multiplied by a factor
of $-1$ if compared with the expectation of Eq. (2); Otherwise, one will have
an experimental value of $<\hat{\sigma}_{z}^{a}\hat{\sigma}_{n}^{b}>$ that is
exactly the negation of $<\hat{\sigma}_{z}^{a}\hat{\sigma}_{n}^{a}>$, i.e.,
$<\hat{\sigma}_{z}^{a}\hat{\sigma}_{n}^{b}>=-<\hat{\sigma}_{z}^{a}\hat{\sigma}_{n}^{a}>=-\cos\alpha$
according to Eq. (2). From this, it then trivial to show the violation of Bell
inequality [4] and, thereby, Bell nonlocality is well explained by the
Einstein locality model (see Methods Mark5 for an in-depth theoretical proof).

Strictly speaking, the theoretical recognition of Einstein locality and the
explanation of Bell nonlocality by the Einstein locality model just prove the
possibility of Einstein locality in reality; Einstein locality may involve
only an alternative interpretation of quantum mechanics unless it can be
physically observed in experiment. Next let present an existential
experimental evidence for Einstein locality in the real quantum world.

Figure 2: A double-slit experiment originally on wave-particle dualism and
complementarity with entangled photons [35]. The far-field detection of photon
$s$ was performed by the D2 detector that was spatially scanned. Curves with
different colors stand for different intensity maxima of the TEM01 beam modes.
BBO: beta-barium borate crystal. PBS: Polarizing beamsplitter. C.C.:
Coincidence counting.

One decade ago, two double-slit experiments [35, 36] were conducted in a row
originally on wave-particle dualism and complementarity by use of entangled
photons, $s$ and $i$. The photons were produced through Type-II spontaneous
parametric down-conversion (SPDC) in a beta-barium borate (BBO) crystal (Fig.
2). The unique feature shared by the two experiments was the use of laser
pumps in TEM01 modes, each displaying two distinct intensity maxima in an
upper-lower geometry. As a consequence, the photon pairs were created in TEM01
modes as well and the corresponding intensity maxima are respectively
indicated by curves with different colors in Fig. 2 (red for the upper maximum
and purple for the lower one). In each experiment, since the average pump
intensity was controlled low enough, only one pair of photons was created
during each measurement time interval. The paired photons with orthogonal
polarizations were separated by a polarizing beamsplitter (PBS) and a lens
(lens1) was used to couple the generated photons into the openings of a near-
field double slit (photon $s$) or directly into a near-field photon detector
D1 (photon $i$).

Under the reported near-field measurement conditions [35, 36], the created
photons as a composite system was in an entangled state described by

$|\phi_{c}\rangle=2^{-1/2}(|u\rangle_{s}|u\rangle_{i}+e^{i\theta}|d\rangle_{s}|d\rangle_{i})\
,$ (4)

in which $|u\rangle$ ($|d\rangle$) denotes the state where a photon is in the
upper (lower) intensity maxima of the TEM01 mode and the $\theta$ phase is
determined by the SPDC process. The quantum correlation delineated by Eq. (4)
originates from the simultaneous generation of the paired photons at the same
spatial point; This correlation was experimentally confirmed with 99$\%$
fidelity when both photons were detected in near fields [35].

Figure 3: Illustration of the dynamic mechanism for nonlocality observed in
the Bell experiment with the electron singlet system. (a) Entangled electrons
are produced with coherence owned by each electron,
$<\hat{\sigma}^{a}_{z}\hat{\sigma}^{a}_{n}>=<\hat{\sigma}^{b}_{z}\hat{\sigma}^{b}_{n}>=\cos\alpha$,
and correlation between them, $\sigma^{a}_{z}=-\sigma^{b}_{z}$ and
$\sigma^{b}_{n}=-\sigma^{a}_{n}$. (b) After the $\hat{\sigma}^{a}_{z}$ and
$\hat{\sigma}^{a}_{n}$ measurements, both electron $a$ and $b$ undergo
decoherence and experimental data retaining correlation are produced for
$\sigma^{a}_{z}=-\sigma^{b}_{z}$ and $\sigma^{b}_{n}=-\sigma^{a}_{n}$,
respectively. (c) Coincidence counting is carried out, automatically
exploiting the correlation $\sigma^{b}_{n}=-\sigma^{a}_{n}$ and producing the
result of $<\hat{\sigma}^{a}_{z}\hat{\sigma}^{b}_{n}>=-\cos\alpha$ that is the
negation of $<\hat{\sigma}^{a}_{z}\hat{\sigma}^{a}_{n}>=\cos\alpha$ due to the
spin anti-correlation.

Then the authors moved detector D2 from the double slit to the far field to
record photon $s$ as a function of its vertical position (Fig. 2), in
coincidence with photon $i$ registered by detector D1 at the lower intensity
maximum of the near-field TEM01 mode. With the idea of Einstein nonlocality
and the observed photon correlation in experiment, it was expected that the
detection of photon $i$ in the lower intensity maximum should cause a state
collapse to photon $s$ [35, 36], i.e., photon $s$ should be present only in
the lower intensity maximum according to Eq. (4). Consequently, photon $s$
must have lost its coherence between the two intensity maxima and should
prevent one from observing interference fringes when scanning detector D2 in
the far field. However, the achieved experimental results were the opposite:
The authors observed far-field interference fringes with a visibility of about
50%, much higher than the expected 1% visibility given the 99% which-slit
information provided by detector D1 [35].

One may understand the presented experimental results [35, 36] based on
Einstein locality as follows. The local detection of photon $i$ in a
$|d\rangle_{i}$ state did not cause state collapse to photon $s$ at the near-
field double slit where photon $s$ still stayed in its initial state.
Therefore, when passing through the double slit, photon $s$ kept its initial
coherence (and necessary power balance) between the intensity maxima in
accordance to Einstein locality. It was this coherence owned by photon $s$
that enabled the observation of its interference fringes in the far field,
which signifies Einstein locality exhibited by the entangled photons (see
Methods Mark6 for more detailed analysis).

So far, a comprehensive account has been provided for the discovery of
Einstein locality with a theoretical proof and an experimental evidence. The
discovery marks an essential distinction between Einstein nonlocality and Bell
nonlocality. The two types of nonlocality are commonly thought by many people
as different manifestations of the same entity [21, 29, 30] and Einstein
nonlocality has been often utilized to provide a physical picture to explain
the underlying principle of Bell nonlocality [34, 37]. This practice has
released a widespread misleading message that Bell experiments have verified
not only Bell nonlocality but also Einstein nonlocality. After all, Bell did
not explicitly distinguish the two types of nonlocality since he originally
intended to derive Bell inequality for the experimental test of the EPR
argument [4]. Now with Einstein locality, the conceptual confusion about
nonlocality may be successfully clarified; Otherwise, it will be detrimental
to the foundation of quantum science and our understanding of the essence of
nonlocality. Imperceptible obstacles could have been created by such confusion
to answer the questions of, for example, whether quantum many-body
entanglement is more fundamental than quantum fields and what is the optimum
hardware for quantum computers [38].

Most importantly, the discovery of Einstein locality hopefully will solve the
nonlocality puzzle by casting light on the dynamic mechanism for the
nonlocally correlated behaviors of entangled objects that violate Bell
inequality (Fig. 3). For the electron singlet system, the two crucial elements
in the mechanism are the coherence owned by the electrons and their spin anti-
correlation; One of these elements alone does not ensure nonlocality violating
Bell inequality, but the combination of the two does. Moreover, a new
worldview may be erected upon Einstein locality about the behaviors of quantum
objects and hence provide an interesting way to interpret what is meant by
relativity: From the viewpoint of classical science [4], entangled objects
exhibit nonlocal correlations that violate Bell inequality as if there were
some real “physical influence faster than the speed of light”; On the other
hand, from the perspective of quantum science, the performance of these
objects can be fully explained by the Einstein locality model where there is
little room for the “spooky action at a distance”. This worldview justifies
that Bell inequality, which was derived in the context of classical science
though, qualifies as a quantitative tool for distinguishing quantum science
from classical science and analysing the quantum behaviors of entangled
objects. Last but not least, the discovery reported here will lay on a more
solid foundation the basic concepts of entanglement [39, 40, 41] and quantum
steering [42, 43, 29, 30], may allow one to dig out something new behind
quantum measurement [44, 45, 46] and decoherence [47, 48], and suggests an
encouraging novel approach towards hardware optimization for quantum computers
[38] (Methods Mark7).

## Methods

### Wavefunction decomposition and Einstein locality

Mark1: According to quantum mechanics, if a single particle in a state
delineated by a wavefunction $|\psi\rangle$ is measured for an observable $A$
that is described by an operator $\hat{A}$, by invoking the superposition
principle, one may write $|\psi\rangle$ in a superposed form,

$|\psi\rangle=\sum_{j=1}a_{j}|a_{j}\rangle\ ,$ (5)

where the complex numbers $a_{j}$ satisfy $\sum_{j}|a_{j}|^{2}=1$ and
$|a_{j}\rangle$ are the eigenstates of operator $\hat{A}$. From Eq. (5), a
vector $\mathbf{A}$
($\mathbf{A}^{T}\equiv\left[a_{1},a_{2},...,a_{j},...\right]$) may be
constructed as a representation of the $|\psi\rangle$ state in a Hilbert space
spanned by the ${|a_{j}\rangle}$ bases and, thereby, vector $\mathbf{A}$ fully
dictates the projection of the particle onto the final $|a_{j}\rangle$ state
by the $\hat{A}$ measurement. If otherwise the particle is measured for an
observable $B$ described by an operator $\hat{B}\neq\hat{A}$, then one should
have

$|\psi\rangle=\sum_{j=1}b_{j}|b_{j}\rangle\ ,$ (6)

in which $\sum_{j}|b_{j}|^{2}=1$, and $|b_{j}\rangle$ are the eigenstates of
operator $\hat{B}$. In this case, a constructed vector $\mathbf{B}$
($\mathbf{B}^{T}\equiv\left[b_{1},b_{2},...,b_{j},...\right]$) governs the
measurement-induced projection of the particle onto the final $|b_{j}\rangle$
state.

Mathematically Eqs. (5) and (6) are equivalent for the same $|\psi\rangle$
state. However, $\mathbf{A}\neq\mathbf{B}$ in general, and they are different
representations of the $|\psi\rangle$ state in different Hilbert spaces.
Consequently, for an $\hat{A}$ measurement on the particle one must choose the
${|a_{j}\rangle}$ expansion bases when writing $|\psi\rangle$ in a superposed
form, so that the outcomes may be appropriately predicted with vector
$\mathbf{A}$; If otherwise the ${|b_{j}\rangle}$ bases are chosen when
applying the superposition principle to the $|\psi\rangle$ state, the
theoretical prediction by vector $\mathbf{B}$ will not agree with the actual
$\hat{A}$ measurement results. Therefore, the choice of expansion bases (CEB)
for the $|\psi\rangle$ state must be measurement oriented whenever invoking
the superposition principle, which here is referred to as the CEB rule that
should be strictly followed in quantum theory.

The generalization of the CEB rule to the case of composite systems is
straightforward. To that end, let consider a compound system consisting of two
particles, $a$ and $b$. Suppose that an $\hat{A}$ measurement is performed on
particle $a$ and a $\hat{B}$ measurement on particle $b$, then the
wavefunction $|\psi_{c}\rangle$ of the system can be written as a
superposition of the $|a_{i}\rangle_{a}|b_{j}\rangle_{b}$ states,

$|\psi_{c}\rangle=\sum_{ij=1}\gamma_{ij}|a_{i}\rangle_{a}|b_{j}\rangle_{b}\ .$
(7)

As a representation of the $|\psi_{c}\rangle$ state in a Hilbert space spanned
by the ${|a_{i}\rangle}_{a}|b_{j}\rangle_{b}$ bases, a matrix
$\mathbf{\Gamma}$ may be constructed with $\gamma_{ij}$ being the matrix
elements to fully describes the measurement-induced projection of the compound
system onto the final $|a_{i}\rangle_{a}|b_{j}\rangle_{b}$ states,
respectively with $|\gamma_{ij}|^{2}$ probabilities
($\sum_{ij}|\gamma_{ij}|^{2}=1$). Now if a different joint measurement is
carried out on the system, the wavefunction $|\psi_{c}\rangle$ must be
expressed as a superposition of expansion bases other than the
$|a_{i}\rangle_{a}|b_{j}\rangle_{b}$ ones, resulting in a different matrix
$\mathbf{\Gamma}^{\prime}$ to correctly predict the measurement outcomes. So,
the measurement-oriented CEB rule for the superposition principle should be
enforced also for the case of composite systems as well.

Unfortunately, when composite systems are concerned, the above CEB rule has
caught very little attention from quantum scientists. This widespread
ignorance of the CEB rule may be illustrated with the electron singlet state,

$|\phi_{c}\rangle=2^{-1/2}(|+_{z}\rangle_{a}|-_{z}\rangle_{b}-|-_{z}\rangle_{a}|+_{z}\rangle_{b})\
,$ (8)

which is a Schmidt decomposition of $|\phi_{c}\rangle$. The
$|\pm_{z}\rangle_{a}|\pm_{z}\rangle_{b}$ bases are the right choice to write
$|\phi_{c}\rangle$ in the superposed form of Eq. (8) to have a correct
prediction for the outcomes of a joint
$\hat{\sigma}_{z}^{a}\hat{\sigma}_{z}^{b}$ measurement on the singlet system,
because $|\pm_{z}\rangle$ are the eigenstates of the $\hat{\sigma}_{z}$
operator. However, if a $\hat{\sigma}_{z}^{a}$ measurement is performed on
electron $a$ and a $\hat{\sigma}_{n}^{b}$ measurement on electron $b$, the CEB
rule requires that $|\phi_{c}\rangle$ cannot be expressed in the form of Eq.
(8) any more. Instead, it must be expanded as a superposition of
$|\pm_{z}\rangle_{a}|\pm_{n}\rangle_{b}$ states, i.e.,

$\displaystyle|\phi_{c}\rangle$ $\displaystyle=$ $\displaystyle\gamma_{++}\
|+_{z}\rangle_{a}|+_{n}\rangle_{b}+\gamma_{+-}\
|+_{z}\rangle_{a}|-_{n}\rangle_{b}$ (9) $\displaystyle+\gamma_{-+}\
|-_{z}\rangle_{a}|+_{n}\rangle_{b}+\gamma_{--}\
|-_{z}\rangle_{a}|-_{n}\rangle_{b}\ ,$

which is nothing but Eq. (1). Here $|\pm_{n}\rangle$ are the eigenstates of
the $\hat{\sigma}_{n}$ operator,
$|+_{n}\rangle=\cos(\alpha/2)|+_{z}\rangle+\sin(\alpha/2)e^{i\beta}|-_{z}\rangle$,
and
$|-_{n}\rangle=-\sin(\alpha/2)e^{-i\beta}|+_{z}\rangle+\cos(\alpha/2)|-_{z}\rangle$.
The connection between Eqs. (8) and (9) is very tricky in quantum theory
because of their equivalence in mathematics. In addition, Eq. (9) reduces into
Eq. (8) when $\alpha=0$ and, therefore, the former can also be considered as a
generalization of the latter to the generic case of $\alpha\neq 0$. However,
as long as $\alpha\neq 0$, from Eqs. (8) and (9) two matrices
$\mathbf{\Gamma}$’s are respectively constructed as the representations of the
$|\phi_{c}\rangle$ state in two Hilbert spaces and the corresponding joint
measurement results are respectively given by these two obviously-different
matrices, from which it follows that Eqs. (8) and (9) have distinctive
physical meanings. This explains why Eq. (1) or (9) cannot be expressed as a
Schmidt decomposition of $|\phi_{c}\rangle$ like Eq. (8) for the joint
$\hat{\sigma}_{z}^{a}\hat{\sigma}_{n}^{b}$ measurement on the singlet system
when $\alpha\neq 0$.

Mark2: In practice, Eq. (8) as a Schmidt decomposition of $|\phi_{c}\rangle$
was almost exclusively used to gain a physical picture for the underlying
principle of Bell experiments involving joint
$\hat{\sigma}_{z}\hat{\sigma}_{n}$ measurements on entangled electrons
($\alpha\neq 0$), which obviously breaks the CEB rule. Surely Eq. (8) alone
cannot provide a valid prediction for the
$\hat{\sigma}_{z}^{a}\hat{\sigma}_{n}^{b}$ measurement results, which hence
requires extra elements to be introduced into the theory as a price paid for
breaking the CEB rule. One of the wanted extra elements is the concept of
Einstein nonlocality that is traced back to the famous EPR paper [1]. It turns
out that the use of the superposition principle without following the CEB rule
was one of the subtlest weaknesses of the EPR paper, as highlighted by Eqs.
(7) and (8) of the paper that did not specify any joint measurements on the
studied composite system [1]. According to the idea of Einstein nonlocality,
electron $a$ and $b$ described by Eq. (8) must be simultaneously left in
$|p_{z}\rangle_{a}$ and $|-p_{z}\rangle_{b}$ (or, $|-q_{n}\rangle_{a}$ and
$|q_{n}\rangle_{b}$) states respectively after the $\hat{\sigma}_{z}^{a}$
($\hat{\sigma}_{n}^{b}$) measurement on electron $a$ ($b$). Nonetheless,
Einstein nonlocality still remains in need of experimental verification, since
Bell experiments can be satisfactorily explained by assuming Einstein locality
as will be shown in this work.

A potential issue with Einstein locality could be its compatibility with the
uncertainty principle when one simultaneously performs a
$\hat{\sigma}_{z}^{a}$ measurement on electron $a$ and a
$\hat{\sigma}_{y}^{b}$ measurement on electron $b$. It might be argued that
arbitrarily precise results for both the $\hat{\sigma}_{z}^{a}$ and
$\hat{\sigma}_{y}^{b}$ measurements are expected from Einstein locality [1]
and, at the same time, the $\sigma_{y}$ value of electron $a$ may be precisely
inferred from the $\hat{\sigma}_{y}^{b}$ measurement on electron $b$ due to
quantum correlation of the singlet system. Then it follows that one will have
precise $\sigma_{y}$ and $\sigma_{z}$ values for electron $a$ and violate the
uncertainty principle. Nonetheless, the above rationale has overlooked the
fact that, whenever electron $a$ is reduced into a $|+_{z}\rangle_{a}$ or
$|-_{z}\rangle_{a}$ state in the $\hat{\sigma}_{z}^{a}$ measurement, electron
$b$ will be randomly projected onto a $|\pm_{y}\rangle_{b}$ state with a 50%
probability by the $\hat{\sigma}_{y}^{b}$ measurement according to Eq. (9). In
other words, the $\hat{\sigma}_{y}^{b}$ measurement on electron $b$ will not
gain useful $\sigma_{y}$ information as soon as electron $a$ is detected in a
$|+_{z}\rangle_{a}$ or $|-_{z}\rangle_{a}$ state. Therefore, the uncertainty
principle will not be violated and it is indeed compatible with Einstein
locality.

In the literature, one may find other interesting trials [31, 32] to solve the
nonlocality issue with the concept of locality. For instance, in the
“consistent histories” approach it is claimed that nonlocality is inconsistent
with Hilbert-space quantum mechanics [31]. But in this treatment, the
wavefunction collapse is considered only as a calculation device that
corresponds to no real physical process, which does not follow the formalism
of quantum mechanics. Since the physical state of a quantum object is fully
described by its wavefunction, the collapse of the wavefunction must be
accompanied by a change of the physical state to the object according to
quantum mechanics. Another trial to tackle the problem of nonlocality [32], by
analysing quantum information processing in the Heisenberg picture by use of
superdense coding protocol as a calculation tool for quantum information
science, asserts quantum locality in the sense of locally accessible
information which is nonetheless just an alternative way to re-express the
results of the no-signalling theorem [49] that unfortunately does not prove
locality.

### Theoretical calculations pertinent to the Einstein locality model

Mark3: In a Bell experiment with an electron singlet system on which a joint
$\hat{\sigma}_{z}^{a}\hat{\sigma}_{n}^{b}$ measurement is carried out, the
expectation value of the $\hat{\sigma}_{z}^{a}\hat{\sigma}_{n}^{b}$ operator
may be calculated without much difficulty through Eq. (1),

$\displaystyle<\hat{\sigma}_{z}^{a}\hat{\sigma}_{n}^{b}>$ $\displaystyle=$
$\displaystyle\langle\psi_{c}|\hat{\sigma}_{z}^{a}\hat{\sigma}_{n}^{b}|\psi_{c}\rangle$
(10) $\displaystyle=$ $\displaystyle
2^{-1}\sin^{2}(\alpha/2){}_{a}\langle+_{z}|{}_{b}\langle+_{n}|\hat{\sigma}_{z}^{a}\hat{\sigma}_{n}^{b}|+_{z}\rangle_{a}|+_{n}\rangle_{b}$
$\displaystyle+$ $\displaystyle
2^{-1}\cos^{2}(\alpha/2){}_{a}\langle+_{z}|{}_{b}\langle-_{n}|\hat{\sigma}_{z}^{a}\hat{\sigma}_{n}^{b}|+_{z}\rangle_{a}|-_{n}\rangle_{b}$
$\displaystyle+$ $\displaystyle
2^{-1}\cos^{2}(\alpha/2){}_{a}\langle-_{z}|{}_{b}\langle+_{n}|\hat{\sigma}_{z}^{a}\hat{\sigma}_{n}^{b}|-_{z}\rangle_{a}|+_{n}\rangle_{b}$
$\displaystyle+$ $\displaystyle
2^{-1}\sin^{2}(\alpha/2){}_{a}\langle-_{z}|{}_{b}\langle-_{n}|\hat{\sigma}_{z}^{a}\hat{\sigma}_{n}^{b}|-_{z}\rangle_{a}|-_{n}\rangle_{b}$
$\displaystyle=$ $\displaystyle\sin^{2}(\alpha/2)-\cos^{2}(\alpha/2)$
$\displaystyle=$ $\displaystyle-\cos\alpha\ .$

The result of Eq. (10) is a direct consequence of the conditional
predictability of the measurement results with a probability of
$2^{-1}\cos^{2}(\alpha/2)$ or $2^{-1}\sin^{2}(\alpha/2)$ dependent on the
specific joint $\hat{\sigma}_{z}^{a}\hat{\sigma}_{n}^{b}$ measurement. Using
this result, it is straightforward to show the violation of Bell inequality
that signifies Bell nonlocality exhibited by the electron singlet system [4].

Mark4: The above mathematical calculation does not provide a pictorial insight
into the nature of Bell nonlocality. In this work an Einstein locality model
is built as presented in the main text body to draw a physical picture for the
underlying principle of the Bell experiment. In the following, let show the
coherence property owned by electron $a$. For two measurements on electron $a$
(a $\hat{\sigma}_{z}^{a}$ measurement and a $\hat{\sigma}_{n}^{a}$
measurement), from the coherence kernel it follows that the expectation value
of the $\hat{\sigma}_{z}^{a}\hat{\sigma}_{n}^{a}$ operator reads

$\displaystyle<\hat{\sigma}_{z}^{a}\hat{\sigma}_{n}^{a}>$ $\displaystyle=$
$\displaystyle\mbox{Tr}(\hat{\rho}\hat{\sigma}_{z}^{a}\hat{\sigma}_{n}^{a})$
(11) $\displaystyle=$ $\displaystyle
2^{-1}({}_{a}\langle+_{n}|\hat{\sigma}_{z}^{a}\hat{\sigma}_{n}^{a}|+_{n}\rangle_{a}+{}_{a}\langle-_{n}|\hat{\sigma}_{z}^{a}\hat{\sigma}_{n}^{a}|-_{n}\rangle_{a})$
$\displaystyle=$ $\displaystyle\cos\alpha\ ,$

in which $\hat{\sigma}_{n}^{a}|q_{n}\rangle_{a}=q|q_{n}\rangle_{a}$ are
exploited in the second step and then $|\pm_{n}\rangle_{a}$ is expanded as the
superposition of $|\pm_{z}\rangle_{a}$ followed by using
$\hat{\sigma}_{z}^{a}|p_{z}\rangle_{a}=p|p_{z}\rangle_{a}$ in the
calculations. Eq. (11) shows how one may arrive at Eq. (2). Meanwhile, if
inserting an identity operator $\sum_{s}|s_{z}\rangle\langle s_{z}|$ ($s=\pm$)
into Eq. (11) between $\hat{\sigma}_{z}^{a}$ and $\hat{\sigma}_{n}^{a}$, one
may also have

$\displaystyle<\hat{\sigma}_{z}^{a}\hat{\sigma}_{n}^{a}>$ $\displaystyle=$
$\displaystyle 2^{-1}\sum_{p,q}\mbox{sign}(pq)\ |{}_{a}\langle
p_{z}|q_{n}\rangle_{a}|^{2}\ .$ (12)

In the calculations,
$\hat{\sigma}_{n}^{a}|q_{n}\rangle_{a}=q|q_{n}\rangle_{a}$ and
$\hat{\sigma}_{z}^{a}|p_{z}\rangle_{a}=p|p_{z}\rangle_{a}$ are exploited
again. The results of Eqs. (11) and (12) may be understood as follows: With
respect to the eigenstates of $\hat{\sigma}_{n}^{a}$ and
$\hat{\sigma}_{z}^{a}$ onto which electron $a$ can be projected from its
initial $\hat{\rho}$ state by the corresponding measurements, i.e.,
$\hat{\sigma}_{n}^{a}:\hat{\rho}\rightarrow|q_{n}\rangle\langle q_{n}|$ and
$\hat{\sigma}_{z}^{a}:\hat{\rho}\rightarrow|p_{z}\rangle\langle p_{z}|$,
electron $a$ exhibits a property of coherence between the $|p_{z}\rangle_{a}$
and $|q_{n}\rangle_{a}$ states in the sense that the averaged value of
$\mbox{sign}(pq)|{}_{a}\langle p_{z}|q_{n}\rangle_{a}|^{2}$, i.e., the average
of the overlap between the $|p_{z}\rangle_{a}$ and $|q_{n}\rangle_{a}$ states
weighted with $\mbox{sign}(pq)$, is proportional to $\cos\alpha$. The same
truth holds for both electrons, which explains the name of the coherence
kernel in the model.

Mark5: In what follows, let show in theory that the combination of the two
kernels constitutes a sufficient condition for violating Bell inequality (Bell
nonlocality). To this aim, one must recall that electron $a$ in the
$\hat{\rho}$ state may be projected either onto a $|p_{z}\rangle$ state with a
$p$ outcome by the $\hat{\sigma}_{z}^{a}$ measurement or onto a
$|q_{n}\rangle$ state with a $q$ outcome by the $\hat{\sigma}_{n}^{a}$
measurement. Now note that, for the $\hat{\sigma}_{z}^{a}\hat{\sigma}_{n}^{a}$
operator, its expectation value may be determined by Eq. (12) where the
$|{}_{a}\langle p_{z}|q_{n}\rangle_{a}|^{2}$ term, as the intersection of the
$|p_{z}\rangle_{a}$ and $|q_{n}\rangle_{a}$ states, is proportional to the
probability that “electron $a$ has a theoretically expected $p$ result for the
$|p_{z}\rangle_{a}$ state in the $\hat{\sigma}_{z}^{a}$ measurement and a
theoretically expected $q$ result for the $|q_{n}\rangle_{a}$ state in the
$\hat{\sigma}_{n}^{a}$ measurement”. This probability in terms of experiment
[5] should be characterized by the normalized coincidence counting rate
$C(N_{p_{z}}^{a},N_{q_{n}}^{a})$. Consequently,
$C(N_{p_{z}}^{a},N_{q_{n}}^{a})=2^{-1}|{{}_{a}}\langle
p_{z}|q_{n}\rangle_{a}|^{2}$ with $2^{-1}$ being a normalization coefficient.
Plugging this into Eq. (12) leads to

$\displaystyle<\hat{\sigma}_{z}^{a}\hat{\sigma}_{n}^{a}>$ $\displaystyle=$
$\displaystyle\sum_{p,q}\mbox{sign}(pq)\ C(N_{p_{z}}^{a},N_{q_{n}}^{a})\ ,$
(13)

which is nothing but Eq. (3). By invoking the correlation kernel of the model,
one may substitute $N_{q_{n}}^{a}=N_{-q_{n}}^{b}$ into Eq. (13) and obtain

$\displaystyle<\hat{\sigma}_{z}^{a}\hat{\sigma}_{n}^{a}>$ $\displaystyle=$
$\displaystyle-\sum_{p,q}\mbox{sign}(pq)\ C(N_{p_{z}}^{a},N_{q_{n}}^{b})$ (14)
$\displaystyle=$ $\displaystyle-<\hat{\sigma}_{z}^{a}\hat{\sigma}_{n}^{b}>\ ,$

in which $q\rightarrow-q$ is used. The last step comes from the standard way
to obtain the value of $<\hat{\sigma}_{z}^{a}\hat{\sigma}_{n}^{b}>$ in
experiment [5] and the normalized coincidence counting rate
$C(N_{p_{z}}^{a},N_{q_{n}}^{b})$ equals the probability that the measurement
results are $p$ and $q$ respectively for the electron $a$ and $b$ in the joint
$\hat{\sigma}_{z}^{a}\hat{\sigma}_{n}^{b}$ measurement. With Eqs. (11) and
(14), one finally arrives at Eq. (10) and proves the violation of Bell
inequality (Bell nonlocality) for the Bell experiment where the joint
$\hat{\sigma}_{z}^{a}\hat{\sigma}_{n}^{b}$ measurement is performed on the
electron singlet system. In the proof of Bell nonlocality, Eq. (11) is derived
from the coherence kernel of the model and Eq. (14) from the correlation
kernel. Albeit each kernel is local in the sense of Einstein locality, the
combination of the two kernels ensures Bell nonlocality as signified by the
violation of Bell inequality.

### Analysis on the evidence of Einstein locality

Mark6: To understand the amazing experimental results reported by Menzel et
al. [35, 36], one must note that they are repeatable and not contradictory to
other experiments on similar topic [50, 51]. Usually, the detection of photons
should be implemented in a manner such that their correlation remains as
produced. If so, when being detected, the entangled photons should be
projected onto a $|u\rangle_{s}|u\rangle_{i}$ or $|d\rangle_{s}|d\rangle_{i}$
state with equal probability according to Eq. (4). Indeed, this quantum
correlation between the photons was observed when they were collected in near
fields. What distinguishes the two experiments from others [50, 51] is that
photon $s$ was collected in a far field while photon $i$ was detected in a
near field. Under this featured photon-detection condition, one should not
expect to retain in measurement the photon correlation given by Eq. (4) that
is invalidated by the CEB rule due to the far-field detection of photon $s$.
The reason is that both the $|u\rangle_{s}$ and $|d\rangle_{s}$ bases are not
the right choice to write $\phi_{c}$ in the superposed form of Eq. (4) since
none of the $|u\rangle_{s}$ and $|d\rangle_{s}$ states is the eigenstate of
the far-field measurement operator, the exact expression of which is of little
relevance here though, as evidenced by that fact that a far-field measurement
cannot not project photon $s$ onto a $|u\rangle_{s}$ or $|d\rangle_{s}$ state
in near field.

Bearing the above analysis in mind, one may appreciate Einstein locality in an
attempt [52, 53] to explain the two experiments using the concept of fair
sampling [21]. It was shown [52, 53] that the results reported by Menzel et
al. were a consequence of different samplings in the measurements of which-
slit information and interference fringes: The detection of photon $i$
provided which-slit information in the near field whereas the heralded photon
$s$ built an interference pattern in the far field. Through numerical
simulation it was confirmed that [52], conditioned on the near-field detection
of photon $i$ in a $|d\rangle_{i}$ state, photon $s$ passed through both slits
and hence did not carry much which-slit information despite some power
imbalance between the two slits. From the simulation results it immediately
follows that photon $s$ indeed stayed in its initial $\hat{\rho}$ state while
photon $i$ collapsed into a $|d\rangle_{i}$ state, which justifies Einstein
locality for the compound system of entangled photons.

It was claimed [35] that the observed interference fringes was due to the
TEM01 mode function, including the “unoccupied” intensity maximum, imprinted
on photon $s$ in form of a superposition of two wave vectors resulting from a
special choice of the pump modes. However, the said mode function was only one
of the necessary, but not sufficient, conditions for the interference fringes
to appear in the far field. Other necessary conditions include coherence and
enough power balance between the two humps of the TEM01 mode. The observed
interference fringes signify that both intensity maxima of the TEM01 mode must
have been “occupied” by photon $s$ with necessary coherence and power balance.
Therefore, the detection of photon $i$ could not provide much which-slit
information about photon $s$ that remained in its initial $\hat{\rho}$ state
at the double slit (Einstein locality); Neither wave-particle dualism nor
complementarity was really violated in the experiments.

### Impacts on quantum science and technology

Mark7: The concepts of entanglement and quantum steering were introduced into
quantum theory by Schr$\ddot{\mbox{o}}$dinger [42] as a provoked response of
his interest to the EPR paper [1]. From Einstein locality it follows that the
essence of entanglement and quantum steering lies in the conditional
predictability of the experiment outcomes depending on the specific joint
measurements on the compound systems. Nevertheless,
Schr$\ddot{\mbox{o}}$dinger’s original understanding [42] of entanglement and
quantum steering was obviously against Einstein locality. Therefore, it is
desired to distinguish Schr$\ddot{\mbox{o}}$dinger’s version of entanglement
and steering from their modern version [37], so that the two important
concepts will be more consolidated. Since Bell nonlocality ensures
entanglement and quantum steering [37], the Einstein locality model suffices
to explain not only Bell nonlocality but also entanglement and quantum
steering in their modern forms [37, 29, 30].

In addition, the discovery of Einstein locality may provide valuable hints for
possible new science behind quantum measurement [44, 45, 46, 54] and
decoherence [47, 48] that are two closely-related subjects of relevance to
nonlocality. New treatments to quantum measurement problems [44, 45, 46] are
wanted to integrate the idea of Einstein locality into the relevant theory. As
for decoherence, the destruction of coherence between the states of a system
associated with observables sensed by environment [47, 39], it is of great
significance for the foundations of quantum science and problems in quantum
engineering. It is believed that a system in a “Schr$\ddot{\mbox{o}}$dinger
cat” state is extremely fragile because of decoherence. Nevertheless, the
Einstein locality model may provide a new perspective on the fragility of
“Schr$\ddot{\mbox{o}}$dinger cat” states. To illustrate the idea, let consider
a multipartite system of three photons, $a$, $b$, and $c$, in an entangled
state [55, 56],

$|\phi_{abc}\rangle=2^{-1/2}(|h\rangle_{a}|h\rangle_{b}|h\rangle_{c}+|v\rangle_{a}|v\rangle_{b}|v\rangle_{c})\
,$ (15)

wherein $|h\rangle$ and $|v\rangle$ stand for the horizontal and vertical
polarization states respectively. Eq. (15) entails that the system is subject
to a joint linear-polarization measurement, as required by the CEB rule. It is
trivial to show that the reduced density operator for each photon is
$\hat{\rho}=2^{-1}(|h\rangle\langle h|+|v\rangle\langle v|)$. So, the Einstein
locality model for the electron singlet state can be generalized to the case
of three-photon entanglement with just a modification for the correlation
kernel: $N_{h}^{a}=N_{h}^{b}=N_{h}^{c}$ and $N_{v}^{a}=N_{v}^{b}=N_{v}^{c}$,
i.e., the results of local polarization measurements on the photons along the
horizontal and vertical directions are correlated ($N_{h,v}^{a,b,c}$ stand for
the numbers of counted $|h\rangle$ or $|v\rangle$ photons in the corresponding
measurements).

Figure 4: A typical decoherence process in the Einstein locality model. One
particle of a three-photon system ($a$, $b$, and $c$) undergoes decoherence
and hence a state collapse. The system is initially in a
$\hat{\rho}_{abc}=|\phi_{abc}\rangle\langle\phi_{abc}|$ state and, after the
decoherence, the subsystem of photon $a$ and $b$ is in a $\hat{\rho}_{ab}$
state where the states of the two photons and their correlation remain. Each
circle represents an initial $\hat{\rho}$ state for a photon and the black dot
stands for a collapsed state. The red lines indicate photon correlations and
the red dash lines denote ruined correlations.

Now if one of the three photons, say photon $c$, suffers from decoherence and
hence a state collapse due to the environment. According to the Einstein
locality model, the states of photon $a$ and $b$ and the correlation between
them remain unchanged despite the state collapse of photon $c$ (Fig. 4). Then
it immediately follows that the subsystem of photon $a$ and $b$ must be in a
state of $\hat{\rho}_{ab}\neq\hat{\rho}^{\ r}_{ab}$, where $\hat{\rho}^{\
r}_{ab}=2^{-1}(|h\rangle_{a}|h\rangle_{b}\langle h|_{b}\langle
h|_{a}+|v\rangle_{a}|v\rangle_{b}\langle v|_{b}\langle v|_{a})$ denotes a
mixed state in which the photons are completely uncorrelated. In accordance to
Einstein locality and the pristine correlation between photon $a$ and $b$, one
may speculate that the two photons are probably in an entangled state of
$\hat{\rho}_{ab}=|\phi_{ab}\rangle\langle\phi_{ab}|$ where $|\phi_{ab}\rangle$
reads

$|\phi_{ab}\rangle=2^{-1/2}(|h\rangle_{a}|h\rangle_{b}+e^{i\phi}|v\rangle_{a}|v\rangle_{b})\
.$ (16)

Here $\phi$ is an arbitrary phase to be determined in experiment.

If Eq. (16) is verified in experiment, a novel approach may be paved towards
hardware optimization for quantum computing. To see the point, suppose that a
computing task is assigned to a two-qubit quantum computer built with photon
$a$ and $c$ and with photon $b$ as a backup qubit. In the case that photon $c$
suffers from decoherence due to the environment, the computer may stay on its
task after the backup qubit of photon $b$ is used as a substitute for photon
$c$. If this strategy is successfully adopted in practice, the stringent
requirements for protecting all the qubits of a quantum computer from
decoherence could be substantially relaxed, which will benefit the ongoing
research efforts for building commercial quantum computers. Albeit the
validity of Eq. (16) is still subject to experimental justification, possible
new science behind decoherence certainly cannot be denied with preconceptions.
For an experiment on the validity of Eq. (16), the value of the unknown phase
$\phi$ needs to be determined and then compensated at the beginning of the
experiment. In addition, one must note that Einstein locality forbids any
feedback information from the ruined photon $c$ for the measurement of the
quantum state of photon $a$ and $b$. Specifically, the detection of photon $c$
cannot be used as a triggering signal for the acquisition of the
$N_{h,v}^{a,b}$ data.

## Acknowledgments

This work was financially supported by the National Natural Science Foundation
of China (grant number 12074110). The author is grateful to K. Wu and L. Chen
for preparing some of the figures.

## Competing interests

The author declares that he has no competing interests.

## Data and materials availability

All data are available in the main text or the supplementary materials.

## References

  * Einstein  _et al._ [1935] A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777–780 (1935).
  * Bohr [1935] N. Bohr, Phys. Rev. 48, 696–702 (1935).
  * Bohm [1952] D. Bohm, Phys. Rev. 85, 166–179 (1952).
  * Bell [1964] J. S. Bell, Physics 1, 195–200 (1964).
  * Clauser  _et al._ [1969] J. F. Clauser, M. H. Horne, A. Shimony, and R. A. Holt, Phys. Rev. Lett. 23, 880–884 (1969).
  * Aspect  _et al._ [1982] A. Aspect, J. Dalibard, and G. Roger, Phys. Rev. Lett. 49, 1804–1807 (1982).
  * Giustina  _et al._ [2015] M. Giustina, M. A. M. Versteegh, S. Wengerowsky, J. Handsteiner, A. Hochrainer, K. Phelan, F. Steinlechner, J. Kofler, J.-A. Larsson, C. Abellan, W. Amaya, V. Pruneri, M. W. Mitchell, J. Beyer, T. Gerrits, A. E. Lita, L. K. Shalm, S. W. Nam, T. Scheidl, R. Ursin, B. Wittmann, and A. Zeilinger, Phys. Rev. Lett. 115, 250401 (2015).
  * Shalm  _et al._ [2015] L. K. Shalm, E. Meyer-Scott, B. G. Christensen, P. Bierhorst, M. A. Wayne, M. J. Stevens, T. Gerrits, S. Glancy, D. R. Hamel, M. S. Allman, K. J. Coakley, S. D. Dyer, C. Hodge, A. E. Lita, V. B. Verma, C. Lambrocco, E. Tortorici, A. L. Migdall, Y. Zhang, D. R. Kumor, W. H. Farr, F. Marsili, M. D. Shaw, J. A. Stern, C. Abellan, W. Amaya, V. Pruneri, T. Jennewein, M. W. Mitchell, P. G. Kwiat, J. C. Bienfang, R. P. Mirin, E. Knill, and S. W. Nam, Phys. Rev. Lett. 115, 250402 (2015).
  * Hensen  _et al._ [2015] B. Hensen, H. Bernien1, A. E. Dreau, A. Reiserer, N. Kalb, M. S. Blok, J. Ruitenberg, R. F. L. Vermeulen, R. N. Schouten, C. Abellan, W. Amaya, V. Pruneri, M. W. Mitchell, M. Markham, D. J. Twitchen, D. Elkouss, S. Wehner, T. H. Taminiau, and R. Hanson, Nature 526, 682–686 (2015).
  * The BIG Bell Test Collaboration [2018] The BIG Bell Test Collaboration, Nature 557, 212–216 (2018).
  * Nielsen and Chuang [2000] M. A. Nielsen and I. L. Chuang,  _Quantum Computation and Quantum Information_ (Cambridge Univ. Press, 2000).
  * Scarani  _et al._ [2009] V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Du$\check{\mbox{s}}$ek, N. L$\ddot{\mbox{u}}$tkenhaus, and M. Peev, Rev. Mod. Phys. 81, 1301–1350 (2009).
  * Ladd  _et al._ [2010] T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O’Brien, Nature 464, 45–53 (2010).
  * G. Benenti and Strini [2019] D. R. G. Benenti, G. Casati and G. Strini,  _Principles of quantum computation and information_ (World Scientific Publishing Co. Pte. Ltd, 2019).
  * Arute  _et al._ [2019] F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, R. Biswas, S. Boixo, F. G. S. L. Brandao, D. A. Buell, B. Burkett, Y. Chen, Z. J. Chen, B. Chiaro, R. Collins, W. Courtney, A. Dunsworth, E. Farhi, B. Foxen, A. Fowler, C. Gidney, M. Giustina, R. Graff, K. Guerin, S. Habegger, M. P. Harrigan, M. J. Hartmann, A. Ho, M. Hoffmann, T. Huang, T. S. Humble, S. V. Isakov, E. Jeffrey, Z. Jiang, D. Kafri, K. Kechedzhi, J. Kelly, P. V. Klimov, S. Knysh, A. Korotkov, F. Kostritsa, D. Landhuis, M. Lindmark, E. Lucero, D. Lyakh, S. Mandra, J. R. McClean, M. McEwen, A. Megrant, X. Mi, K. Michielsen, M. Mohseni, J. Mutus, O. Naaman, M. Neeley, C. Neill, M. Y. Niu, E. Ostby, A. Petukhov, J. C. Platt, C. Quintana, E. G. Rieffel, P. Roushan, N. C. Rubin, D. Sank, K. J. Satzinger, V. Smelyanskiy, K. J. Sung, M. D. Trevithick, A. Vainsencher, B. Villalonga, T. White, Z. J. Yao, P. Yeh, A. Zalcman, H. Neven, and J. M. Martinis, Nature 574, 505–510 (2019).
  * Wu et al. [2021] Y. L. Wu et al., Phys. Rev. Lett. 127, 180501 (2021).
  * Ma  _et al._ [2023] S. Ma, G. Y. Liu, P. Peng, B. C. Zhang, S. Jandura, J. Claes, A. P. Burgers, G. Pupillo, S. Puri, and J. D. Thompson, Nature 622, 279–284 (2023).
  * Wang  _et al._ [2023] W. Y. Wang, R. Wang, C. Q. Hu, V. Zapatero, L. Qian, B. Qi, M. Curty, and H.-K. Lo, Phys. Rev. Lett. 130, 220801 (2023).
  * Zapatero  _et al._ [2023] V. Zapatero, W. Y. Wang, and M. Curty, Quantum Sci. Technol. 8, 025014 (2023).
  * Salart  _et al._ [2008] D. Salart, A. Baas, C. Branciard, N. Gisin, and H. Zbinden, Nature 454, 861–864 (2008).
  * Giustina  _et al._ [2013] M. Giustina, A. Mech, S. Ramelow, B. Wittmann, J. Kofler, J. Beyer, A. Lita, B. Calkins, T. Gerrits, S. Nam, R. Ursin, and A. Zeilinger, Nature 497, 227–230 (2013).
  * Brunner  _et al._ [2014] N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, Rev. Mod. Phys. 86, 419–478 (2014).
  * Freedman and Clauser [1972] S. J. Freedman and J. F. Clauser, Phys. Rev. Lett. 28, 938–941 (1972).
  * Christensen  _et al._ [2013] B. G. Christensen, K. T. McCusker, J. B. Altepeter, B. Calkins, T. Gerrits, A. E. Lita, A. Miller, L. K. Shalm, Y. Zhang, S. Nam, N. Brunner, C. C. Lim, N. Gisin, and P. G. Kwiat, Phys. Rev. Lett. 111, 130406 (2013).
  * Erven  _et al._ [2014] C. Erven, K. F. E. Meyer-Scott, J. Lavoie, B. L. Higgins, Z. Yan, C. J. Pugh, J.-P. Bourgoin, R. Prevedel, L. K. Shalm, L. Richards, N. Gigov, R. Laflamme, G. Weihs, T. Jennewein, , and K. J. Resch, Nat. Photonics 8, 292–296 (2014).
  * Hirsch  _et al._ [2013] F. Hirsch, M. T. Quintino, J. Bowles, and N. Brunner, Phys. Rev. Lett. 111, 160402 (2013).
  * Popescu [2014] S. Popescu, Nat. Phys. 10, 264–270 (2014).
  * Stapp [2009] H. P. Stapp,  _Einstein Locality_ (In: D. Greenberger, K. Hentschel, F. Weinert, (eds) Compendium of Quantum Physics, Springer, Berlin, Heidelberg, 2009).
  * Reid and He [2019] M. D. Reid and Q. Y. He, Phys. Rev. Lett. 123, 120402 (2019).
  * Uola  _et al._ [2020] R. Uola, A. C. S. Costa, H. C. Nguyen, and O. G$\ddot{\mbox{u}}$hne, Rev. Mod. Phys. 92, 015001 (2020).
  * Griffiths [2020] R. B. Griffiths, Phys. Rev. A 101, 022117 (2020).
  * B$\acute{\mbox{e}}$dard [2021] C. A. B$\acute{\mbox{e}}$dard, Quantum Rep. 3, 272–285 (2021).
  * Cabello [2021] A. Cabello, Phys. Rev. Lett. 127, 070401 (2021).
  * Cramer [1986] J. G. Cramer, Rev. Mod. Phys. 58, 647–687 (1986).
  * Menzel  _et al._ [2012] R. Menzel, D. Puhlmann, A. Heuer, and W. P. Schleich, Proc. Natl. Acad. Sci. U.S.A. 109, 9314–9319 (2012).
  * Menzel  _et al._ [2013] R. Menzel, A. Heuer, D. Puhlmann, K. Dechoum, M. Hillery, M. J. A. Sp$\ddot{\mbox{a}}$hn, and W. P. Schleich, J. Mod. Opt. 60, 86–94 (2013).
  * Wiseman  _et al._ [2007] H. M. Wiseman, S. J. Jones, and A. C. Doherty, Phys. Rev. Lett. 98, 140402 (2007).
  * [38] 125 questions: Exploration and discovery, a booklet produced by the Science/AAAS Custom Publishing Office, 2021\.
  * Almeida  _et al._ [2007] M. P. Almeida, F. de Melo, M. Hor-Meyll, A. Salles, S. P. Walborn, P. H. S. Ribeiro, and L. Davidovich, Science 316, 579–582 (2007).
  * Horodecki  _et al._ [2009] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Rev. Mod. Phys. 81, 865–942 (2009).
  * Tendick  _et al._ [2020] L. Tendick, H. Kampermann, and D. Bruß, Phys. Rev. Lett. 124, 050401 (2020).
  * Schr$\ddot{\mbox{o}}$dinger [1935] E. Schr$\ddot{\mbox{o}}$dinger, Proc. Cambridge Philos. Soc. 31, 555–563 (1935).
  * Walborn  _et al._ [2011] S. P. Walborn, A. Salles, R. M. Gomes, F. Toscano, and P. H. Souto Ribeiro, Phys. Rev. Lett. 106, 130402 (2011).
  * Braginsky  _et al._ [1992] V. B. Braginsky, F. Y. Khalili, and K. S. Thorne,  _Quantum Measurement_ (Cambridge University Press, 1992).
  * Wiseman and Milburn [2009] H. M. Wiseman and G. J. Milburn,  _Quantum Measurement and Control_ (Cambridge University Press, 2009).
  * Busch  _et al._ [2016] P. Busch, P. Lahti, J.-P. Pellonp$\ddot{\mbox{a}}$$\ddot{\mbox{a}}$, and K. Ylinen,  _Quantum Measurement_ (Springer Cham, 2016).
  * Zurek [2003] W. H. Zurek, Rev. Mod. Phys. 75, 715–775 (2003).
  * Wilkinson and Blundell [2020] J. M. Wilkinson and S. J. Blundell, Phys. Rev. Lett. 125, 087201 (2020).
  * Ghirardi  _et al._ [1980] C. G. Ghirardi, A. Rimini, and T. Weber, Lettere al Nuovo Cimento 27, 293–298 (1980).
  * Kaiser  _et al._ [2012] F. Kaiser, T. Coudreau, P. Milman, D. B. Ostrowsky, and S. Tanzilli, Science 338, 637–640 (2012).
  * Bienfait  _et al._ [2020] A. Bienfait, Y. P. Zhong, H.-S. Chang, M.-H. Chou, C. R. Conner, E. Dumur, J. Grebel, G. A. Peairs, R. G. Povey, K. J. Satzinger, , and A. N. Cleland, Phys. Rev. X 10, 021055 (2020).
  * Bolduc  _et al._ [2014] E. Bolduc, J. Leach, F. M. Miatto, G. Leuchs, and R. W. Boyd, Proc. Natl. Acad. Sci. U.S.A. 111, 12337–12341 (2014).
  * Leach  _et al._ [2016] J. Leach, E. Bolduc, F. M. Miatto, K. Pich$\acute{\mbox{e}}$, G. Leuchs, and R. W. Boyd, Sci. Rep. 6, 19944 (2016).
  * Pan  _et al._ [2019] W.-W. Pan, X.-Y. Xu, Y. Kedem, Q.-Q. Wang, Z. Chen, M. Jan, K. Sun, J.-S. Xu, Y.-J. Han, C.-F. Li, and G.-C. Guo, Phys. Rev. Lett. 123, 150402 (2019).
  * Greenberger  _et al._ [1990] D. M. Greenberger, M. A. Horne, A. Shimony, and A. Zeilinger, Am. J. Phys. 58, 1131–1143 (1990).
  * Bouwmeester  _et al._ [1999] D. Bouwmeester, J.-W. Pan, M. Daniell, H. Weinfurter, and A. Zeilinger, Phys. Rev. Lett. 82, 1345–1348 (1999).


# How to administer an antidote to Schrödinger’s cat

Juan-Rafael Álvarez  Mark IJspeert  Oliver Barter  Ben Yuen  now at
Metamaterials Research Centre, University of Birmingham, Edgbaston,
Birmingham, B15 2TT, UK  Thomas D. Barrett  now at InstaDeep, London, UK.
Dustin Stuart  Jerome Dilley  now at Google.  Annemarie Holleczek  now at
Robert Bosch GmbH, Postfach 13 55, 74003 Heilbronn, Germany  Axel Kuhn
<EMAIL_ADDRESS>University of Oxford, Clarendon Laboratory, Parks
Road, Oxford OX1 3PU, UK

###### Abstract

In his 1935 Gedankenexperiment, Erwin Schrödinger imagined a poisonous
substance which has a 50% probability of being released, based on the decay of
a radioactive atom. As such, the life of the cat and the state of the poison
become entangled, and the fate of the cat is determined upon opening the box.
We present an experimental technique that keeps the cat alive on any account.
This method relies on the time-resolved Hong-Ou-Mandel effect: two long,
identical photons impinging on a beam splitter always bunch in either of the
outputs. Interpreting the first photon detection as the state of the poison,
the second photon is identified as the state of the cat. Even after the
collapse of the first photon’s state, we show their fates are intertwined
through quantum interference. We demonstrate this by a sudden phase change
between the inputs, administered conditionally on the outcome of the first
detection, which steers the second photon to a pre-defined output and ensures
that the cat is always observed alive.

##  I Introduction

One of the most intriguing principles of quantum mechanics is that of
superposition, which states that a quantum system, before being measured, can
be interpreted to be in two simultaneous states at once. In his 1935
Gedankenexperiment (Schrödinger, 1935), Erwin Schrödinger illustrated the
paradoxical nature of superposition by depicting a cat in a box whose state
(dead or alive) is entangled with a vicious device releasing a poisonous
substance upon a 50% probable radioactive decay. Only upon opening the box for
the first time, is it possible to determine the state of the combined system
of the cat and the radioactive device; with no decay together with an intact
vile and a cat observed alive only half of the time.

The randomness of quantum measurements introduces a fundamental distinction
with respect to classical measurements. The process of measurement is central
to many open theoretical questions (Adlam, 2021), where the measurement
induces apparent contradictions between the predictions of quantum mechanics
and the appearance of sharp measurement outcomes (Landsman, 2017). In
addition, many systems and applications involve a form of quantum control that
relies on the quantum nature of measurements (Lloyd, 1997).

This poses the question: can a partial measurement of a quantum system be made
such that it triggers a sequence of events that coerces the remainder of the
system into a desired state? In the example of Schrödinger’s cat, such an
ability would ensure that the cat is always observed alive, whether or not the
poison has been released. Such an approach would normally be implemented in
the form of a feedback loop, by which a device obtains information about the
trajectory of a physical system in order to modify it in real time (Zhang et
al., 2017; Habib et al., 2002).

In quantum systems, feedback can be of two types: the first, measurement-based
quantum feedback, occurs when a measurement outcome defines a subsequent
action on the original system. The second, coherent quantum feedback, involves
no measurements but provides control using coherent interaction with an
auxiliary quantum system.

Both feedback types exist in a range of applications: the generation of
amplitude squeezed states in a semiconductor laser (Yamamoto et al., 1986),
the improvement of single-shot phase measurements in quantum metrology (Armen
et al., 2002), the stabilisation of a combined atom-cavity quantum state
(Smith et al., 2002), and the preparation and stabilisation of Fock states in
a high-Q microwave cavity with weak measurements (Sayrin et al., 2011; Zhou et
al., 2012).

In this work, we take advantage of a measurement-based feedback protocol to
deliver a photonic state with a desired property: that of always exiting
through the output of a beam splitter of the experimentalist’s choosing.
Specifically, the feedback process is applied in a time-resolved two-photon
quantum interference experiment (Legero et al., 2003) in which the time span
between the first and second photon detection is long enough ($\sim
500\,\mathrm{ns}$) for a phase change to be applied on the second photon. This
allows us to alternate between the behaviours of bosonic and fermionic
interference to control the routing of a single photon as desired.

Figure 1: Experimental arrangement: Two long, double-hump photons interfere in
a beam splitter. Photons are emitted at a repetition rate of $1\,\mathrm{MHz}$
from an atomic source driven by a laser pulse controlled by an acousto-optic
modulator (AOM), whose phase is changed using an electro-optic modulator
(EOM). The AOM pulses are generated by an arbitrary waveform generator (AWG).
A delay line with an optical path length of $300\,\mathrm{m}$ ensures the
simultaneous arrival of two sequentially emitted photons. The emitted photons
have such long wave packets, that the first time bin of both photons
interferes before the relative phase $\phi$ of the second photon in the second
time bin is set. Furthermore, the emitted photons have random polarisations.
Using a polarising beam splitter (PBS), photons are routed into random paths,
and two photons simultaneously impinge on the input ports $A$ and $B$ of a
beam splitter (BS) in 25% of all possible cases. The relative polarisation
between both photons is changed using a half wave plate (HWP). Photons then
interfere in the BS and exit through the ports $C$ and $D$, upon which
measurements are performed using single photon counting modules (SPCMs). The
outcome of a measurement in the first time bin is used to perform feedback on
the value of the phase, $\phi$ during the second half of the driving pulse.

##  II Theory

Let us consider two photons arriving simultaneously at the input ports $A$ and
$B$ of a beam splitter (BS), as illustrated in the Interference box of Fig. 1.
The joint probability of detecting two photons at times $t_{0}$ and
$t_{0}+\tau$ at detectors placed at outputs $C$ and $D$ of the BS,
respectively, can be written as (Legero et al., 2003, 2006)

$\begin{aligned} P_{\text{joint
}}\left(t_{0},\tau\right)=\left|\hat{E}_{D}^{+}\left(t_{0}+\tau\right)\hat{E}_{C}^{+}\left(t_{0}\right)\hat{a}_{A}^{\dagger}\hat{a}_{B}^{\dagger}\left|\text{0}_{A}0_{B}\right\rangle\right|^{2}\end{aligned},$
(1)

where $\hat{a}_{A}^{\dagger}$ and $\hat{a}_{B}^{\dagger}$ correspond to the
creation operators of a photon in the ports $A$ and $B$ ($\hat{a}_{A}$ and
$\hat{a}_{B}$ would correspond to the annihilation operators),
$\hat{E}_{C}^{+}+\hat{E}_{C}^{-}$ and $\hat{E}_{D}^{+}+\hat{E}_{D}^{-}$ are
the electric field operators at the output ports of the beam splitter, and
$\left|0_{A}0_{B}\right\rangle$ corresponds to the vacuum state on the input
ports $A$ and $B$. In this context, $\tau$ can be positive or negative, and
$\tau<0$ corresponds to the detector in port $D$ clicking before that in port
$C$. We emphasise that $\tau$ is a detection time difference giving rise to a
time-resolved Hong-Ou-Mandel (HOM) signal (Legero et al., 2003, 2006), and
must not be confused with the photon arrival time difference $\left(\Delta
t\right)$ used in many other HOM experiments (Hong et al., 1987). Here, we are
only interested in the case for which the two photon wavepackets arrive
simultaneously ($\Delta t=0$).

The electric field operators can be written as the sum of spatio-temporal
functions in distinct modes $k$,
$\zeta_{k}\left(t\right)=\epsilon_{k}\left(t\right)\exp\left(-i\phi_{k}\left(t\right)\right)$:

$\hat{E}^{+}(t)=\sum_{k}\zeta_{k}(t)\hat{a}_{k}\text{,
\quad}\hat{E}^{-}(t)=\sum_{k}\zeta_{k}^{*}(t)\hat{a}_{k}^{\dagger},$ (2)

where $\epsilon_{k}\left(t\right)$ corresponds to the photon amplitude in mode
$k$ and $\phi_{k}\left(t\right)$ to its phase.

Since the absolute photon detection times are irrelevant, we calculate the
joint detection probability as a function of $\tau$ only:

$P_{\text{joint}}\left(\tau\right)=\int_{-\infty}^{\infty}dt_{0}P_{\text{joint}}\left(t_{0},\tau\right).$
(3)

For two linearly polarised photons with a relative polarisation angle
$\theta$, Eq. 1 can be written as (Legero et al., 2006)

$\begin{aligned}
P_{\text{joint}}\left(t_{0},\tau\right)=P_{\text{joint}}^{(HV)}\left(t_{0},\tau\right)-\cos^{2}\theta\text{
}F\left(t_{0},\tau\right)\end{aligned},$ (4)

where

$P_{\text{joint}}^{(HV)}\left(t_{0},\tau\right)=\frac{1}{4}\left(\left|\epsilon_{A}\left(t_{0}\right)\epsilon_{B}\left(t_{0}+\tau\right)\right|^{2}+\right.\\\
\left.\left|\epsilon_{A}\left(t_{0}+\tau\right)\epsilon_{B}\left(t_{0}\right)\right|^{2}\right)$
(5)

and

$F\left(t_{0},\tau\right)=\frac{\epsilon_{A}\left(t_{0}\right)\epsilon_{B}\left(t_{0}+\tau\right)\epsilon_{A}\left(t_{0}+\tau\right)\epsilon_{B}\left(t_{0}\right)}{2}\times\\\
\cos\left(\phi_{A}\left(t_{0}\right)-\phi_{A}\left(t_{0}+\tau\right)+\phi_{B}\left(t_{0}+\tau\right)-\phi_{B}\left(t_{0}\right)\right).$
(6)

When both input photons have orthogonal polarisations
$\left(\theta=\pi/2\right)$, $P_{\text{joint}}\left(\tau\right)$ can be
written as the convolution of their spatio-temporal squared amplitudes,
$P_{\text{joint}}\left(\tau\right)=\frac{1}{2}\left(\left|\varepsilon_{A}\right|^{2}*\left|\varepsilon_{B}\right|^{2}\right)(\tau),$
rendering the choice of $\phi_{A}$ and $\phi_{B}$ irrelevant. This is in sharp
contrast to the case for which both input photons have parallel polarisations
$\left(\theta=0\right)$, where $\phi_{A}$ and $\phi_{B}$ become relevant. The
expected behaviour of $P_{\text{joint}}\left(\tau\right)$ for the particular
case of two photons of perpendicular polarisation with the temporal shape
$\epsilon\left(t\right)=\sin^{2}\left(2\pi t/\delta t\right)$, is shown as a
dashed curve in Fig. 3(a) for $\delta t=401\,\mathrm{ns}$. The three peaks
that are shown in Fig. 3(a) correspond to the detection of coincidences in
detectors $C$ and $D$ delayed by varying time differences
$\tau\in\left[-401,401\right]\text{ ns}$.

We subdivide the overall duration of the photons into two distinct intervals
(bins) of equal length labeled $I_{1}$ (early) and $I_{2}$ (late).
Additionally, let us assume that, for the photon arriving through $A$,
$\phi_{A}\left(t\right)=0$ for all times, and that, for the photon arriving
through $B$, $\phi_{B}\left(t\right)=0$ for $t\in I_{1}$, and
$\phi_{B}\left(t\right)=\phi$ for $t\in I_{2}$. In the Schrödinger picture
(Legero et al., 2006), the state entering the BS is given by

$\left|\Psi_{\text{in}}\right\rangle=\frac{1}{2}\left(\hat{a}_{A1}^{\dagger}+\hat{a}_{A2}^{\dagger}\right)\left(\hat{a}_{B1}^{\dagger}+\text{e}^{\text{i}\phi}\hat{a}_{B2}^{\dagger}\right)\left|0\right\rangle.$
(7)

Here, $\hat{a}_{Aj}^{\dagger}$ and $\hat{a}_{Bj}^{\dagger}$ correspond to the
photon creation operators in ports $A$ and $B$ in the time intervals $I_{j}$,
with $j\in\\{1,2\\}$, and $\left|0\right\rangle$ is the vacuum state in the
basis of all the temporal and beam splitter input paths:
$\left|0\right\rangle=\left|0_{A1}0_{A2}0_{B1}0_{B2}\right\rangle$. For the
cases $\phi=0$ and $\phi=\pi$, the theoretical predictions of
$P_{\text{joint}}\left(\tau\right)$ are shown as dashed curves in Figs. 3(b)
and (c). For the case of Fig. 3(b), both input photons are identical and
feature photon bunching in the output detectors. For this reason, the expected
behaviour would not include any coincidences between detectors $C$ and $D$. In
contrast, in Fig. 3(c) the photon entering through port $B$ acquired a
$\pi$-phase change from $I_{1}$ to $I_{2}$. This is the only difference
between the two photons. Therefore, if both photons are detected during either
$I_{1}$ or $I_{2}$, the photons are indistinguishable and no correlations are
found with $\tau\simeq 0$. However, if the photons are detected in different
intervals, the change in phase drives them to different outputs, resulting in
a coincidence probability that is twice as large as the orthogonal
polarisation case for $\tau=\pm 201\,\mathrm{ns}$.

Figs. 3(b) and (c) exhibit a form that corresponds to a different type of
interference between the two photons depending on the value of $\phi$. Bosonic
($\phi=0$) and fermionic interference ($\phi=\pi$) is found with photons
either bunching in the same outputs or avoiding each other and giving rise to
coincidences, respectively. To see this, consider the following argument,
using the Schrödinger picture: the operators after the beam splitter in the
output channels $C$ and $D$ in $I_{j}$ are linked to those before it by the
standard unitary relation
$\hat{a}_{Aj,Bj}^{\dagger}=\left(\hat{a}_{Cj}^{\dagger}\pm\hat{a}_{Dj}^{\dagger}\right)/\sqrt{2}.$
Therefore, after the beam splitter, Eq. 7 results in:

$\left|\Psi_{\text{out}}\right\rangle=\frac{1}{2\sqrt{2}}\left(\left(\hat{a}_{C1}^{\dagger}\hat{a}_{C1}^{\dagger}+\text{e}^{\text{i}\phi}\hat{a}_{C2}^{\dagger}\hat{a}_{C2}^{\dagger}\right.\right.\\\
-\left.\hat{a}_{D1}^{\dagger}\hat{a}_{D1}^{\dagger}-\text{e}^{\text{i}\phi}\hat{a}_{D2}^{\dagger}\hat{a}_{D2}^{\dagger}\right)\\\
\left.+\left(\text{e}^{\text{i}\phi}+1\right)\left(\hat{a}_{C1}^{\dagger}\hat{a}_{C2}^{\dagger}-\hat{a}_{D1}^{\dagger}\hat{a}_{D2}^{\dagger}\right)\right.\\\
\left.+\left(\text{e}^{\text{i}\phi}-1\right)\left(\hat{a}_{C2}^{\dagger}\hat{a}_{D1}^{\dagger}-\hat{a}_{C1}^{\dagger}\hat{a}_{D2}^{\dagger}\right)\right)\left|0\right\rangle.$
(8)

Here, $\left|0\right\rangle$ is represented in the basis of all the output
ports and temporal modes:
$\left|0\right\rangle=\left|0_{C1}0_{C2}0_{D1}0_{D2}\right\rangle$. The first
four terms account for both photons arriving at the same detector in the same
time bin. These terms lead to the standard boson bunching reported in the
canonical HOM effect (Hong et al., 1987). The last four terms correspond to
cross-correlations between detectors and/or time bins. Unless the single
photon detectors are number resolving, it is not possible to measure outcomes
where both detections occur at the same detector in the same time bin. By
eliminating the terms of the form
$\hat{a}_{Xi}^{\dagger}\hat{a}_{Xi}^{\dagger}$ and rearranging the terms, the
observable sub-state from Eq. 8 becomes

$\left|\tilde{\Psi}_{\text{out}}\right\rangle=\frac{1}{2\sqrt{2}}\left(\hat{a}_{C1}^{\dagger}\left(\left(e^{i\phi}+1\right)\hat{a}_{C2}^{\dagger}-\left(e^{i\phi}-1\right)\hat{a}_{D2}^{\dagger}\right)\right.\\\
\left.+\hat{a}_{D1}^{\dagger}\left(\left(e^{i\phi}-1\right)\hat{a}_{C2}^{\dagger}-\left(e^{i\phi}+1\right)\hat{a}_{D2}^{\dagger}\right)\right)\left|0\right\rangle.$
(9)

In this sub-state, the phase appears explicitly. By setting $\phi=0$, Eq. 9
reads

$\left|\tilde{\Psi}_{\text{out}}^{\left(\phi=0\right)}\right\rangle=\frac{1}{\sqrt{2}}\left(\hat{a}_{C1}^{\dagger}\hat{a}_{C2}^{\dagger}-\hat{a}_{D1}^{\dagger}\hat{a}_{D2}^{\dagger}\right)\left|0\right\rangle.$
(10)

This state again leads to the canonical HOM effect, yet across the time bins
this time. Therefore, both photons arrive at in the same detector, but in
separate time intervals. In contrast, setting $\phi=\pi$, Eq. 9 now gives

$\left|\tilde{\Psi}_{\text{out}}^{\left(\phi=\pi\right)}\right\rangle=\frac{1}{2\sqrt{2}}\left(\hat{a}_{C1}^{\dagger}\hat{a}_{D2}^{\dagger}-\hat{a}_{C2}^{\dagger}\hat{a}_{D1}^{\dagger}\right)\left|0\right\rangle,$
(11)

which is strikingly different to Eq. 10, as both photons exhibit fermionic
behaviour by arriving at different detectors.

Switching between the bosonic and fermionic behaviour by choice of $\phi$
allows one to put the system into a pre-defined quantum state conditioned on
the random outcome of the first measurement. This is implemented in the form
of a feedback mechanism, where a photon detection at either detector in
$I_{1}$ reveals the required change of $\phi$ to steer the remaining photon in
$I_{2}$ to a specific detector. For instance, if detector $D$ clicks in
$I_{1}$, the second detection would normally occur in detector $D$, due to the
canonical HOM effect. However, as the second halves of the photons have not
reached the input ports at the moment of detection, we can instantly change
the phase between $I_{1}$ and $I_{2}$ for one of the incoming photons to
change the port in which the second detection occurs. By choosing $\phi=\pi$
and projecting the output state in Eq. 11 onto the measured state
$\left|0_{C1}1_{D1}\right\rangle=\hat{a}_{D1}^{\dagger}\left|0_{C1}0_{D1}\right\rangle$
(involving only the first time bin, since the second one has not yet
occurred), the state reduces to

$\bigg{\langle}0_{C1}0_{D1}\bigg{|}\hat{a}_{D1}\bigg{|}\tilde{\Psi}_{\text{out}}^{\left(\phi=\pi\right)}\bigg{\rangle}=\frac{1}{\sqrt{2}}\hat{a}_{C2}^{\dagger}\left|0_{C2}0_{D2}\right\rangle.$
(12)

This state clearly shows that, when a measurement occurs in $I_{2}$, any click
will be recorded in detector $C$. Analogously, if detector $C$ clicks in
$I_{1}$ and $\phi=0$ is chosen, both photons are identical in their two halves
and any detection in $I_{2}$ will always occur at detector $C$. This reduces
the quantum state to

$\bigg{\langle}0_{C1}0_{D1}\bigg{|}\hat{a}_{C1}\bigg{|}\tilde{\Psi}_{\text{out}}^{\left(\phi=0\right)}\bigg{\rangle}=\frac{1}{\sqrt{2}}\hat{a}_{C2}^{\dagger}\left|0_{C2}0_{D2}\right\rangle.$
(13)

Again, when a measurement occurs in $I_{2}$, any click will be recorded by
detector $C$. Eqs. 12 and 13 imply that $\phi$ can be used as a parameter for
feedback control, to steer the remaining photon to a desired output.

Returning to Eqs. (3) and (4) with a value of $\phi$ conditional on the
measurement of a photon in $I_{1}$, we observe the expected behaviour of
$P_{\text{joint}}\left(\tau\right)$, as illustrated in Fig. 3(d). Fig. 3(d)
shows increased coincidences with $\tau<0$, for which a detection in $D$
occurs before a detection in $C$. There are significantly less coincidences
with $\tau>0$, as this would correspond to detector $C$ firing first and
detector $D$ second, which cannot occur with an active feedback that always
prompts the second detection in $C$. Fig. 3 shows the subset of probabilities
for cross-detector correlations only. For this reason, the value $\int
P_{\text{joint}}\left(\tau\right)d\tau$ is bounded by $1/2$. The theoretical
curves for same-detector correlations are shown in the Supplementary Material.

##  III Experimental methods

To demonstrate this phenomenon experimentally, we generate photons in an atom-
cavity system using a standard V-STIRAP scheme (Kuhn and Ljunggren†, 2010)
coupling the hyperfine levels of the $D_{2}$ line of ${}^{87}\text{Rb}$.
Specifically, the
$\left|e\right\rangle=\left|5^{2}\text{S}_{1/2},F=1\right\rangle$ and
$\left|g\right\rangle=\left|5^{2}\text{S}_{1/2},F=2\right\rangle$ ground
states are coupled in a $\Lambda$-scheme to the excited state
$\left|x\right\rangle=\left|5^{2}\text{P}_{3/2},F^{\prime}=3\right\rangle$
using a driving laser $\Omega(t)$, as illustrated in Fig. 2. The cavity has a
decay rate of $\kappa=2\pi\times 12\,\mathrm{MHz}$ and a maximum coupling
strength of $g_{0}=2\pi\times 15\,\mathrm{MHz}$. With the system initially
prepared in $\left|e,0\right\rangle$, the laser adiabatically drives the
system to $\left|g,1\right\rangle$, whereupon the photon is emitted from the
cavity mode, leaving the system in $\left|g,0\right\rangle$, decoupled from
further evolution. Atoms are loaded into the cavity with an atomic fountain,
from a magneto-optic trap (MOT) located directly under the cavity.

The spatio-temporal profile $\zeta\left(t\right)$ of the emitted photons can
be directly controlled by shaping the driving pulse (Nisbet-Jones et al.,
2011) using an acousto-optic modulator (AOM). For our purposes, it is most
crucial that the phase of the driving laser, which we control via a separate
electro-optic modulator (EOM), is directly mapped to the phase of the emitted
photon. Fig. 2(b) shows the driving pulse shape and Fig. 2(c) the resulting
doubly-humped photon profile, evenly distributed across two $225\,\mathrm{ns}$
time-bins, for a total coherence time of $\delta t=450\,\mathrm{ns}$.__

The experimental sequence for the production of photons is as follows: photons
are emitted at a repetition rate of $1\,\mathrm{MHz}$, with $450\,\mathrm{ns}$
of the cycle used for the production of a single photon. The remaining
$550\,\mathrm{ns}$ of the cycle are used to optically repump the atom to
$\left|e\right\rangle$ to repeat the process. The photons impinge on a
polarising beam splitter (PBS), and are randomly routed into two paths, one of
which incorporates a fibre loop of $300\,\mathrm{m}$ of optical path length to
induce a $1\,\mathrm{\mu s}$ delay. This ensures that a pair of subsequently
emitted photons arrives simultaneously at the beam splitter, which is the only
case of interest. A half wave plate (HWP) sets the relative polarisation of
both photons.

Figure 2: __ Photon production scheme for the generation of double-hump
photons. Fig (a) shows the driving scheme used to generate single photons for
a single ${}^{87}\mathrm{Rb}$ atom inside of an optical cavity, following a
coherent STIRAP process between the
$\left|F=1\right\rangle\equiv\left|e\right\rangle$ and
$\left|F=2\right\rangle\equiv\left|g\right\rangle$ ground states of
${}^{87}\mathrm{Rb}$ via the virtually excited state
$\left|F^{\prime}=3\right\rangle\equiv\left|x\right\rangle$ (Nisbet-Jones et
al., 2011). (b) shows the pre-calculated (Vasilev et al., 2010) pulse
necessary for the photon shape to follow the form $\sin^{2}(2\pi\tau)$ with
$0<\tau<1$, where $\tau=t/450\,\mathrm{ns}$, as can be seen theoretically
(solid line) and experimentally (dots) in (c). Our cavity has a decay rate of
$\kappa=2\pi\times 12\,\mathrm{MHz}$ and coupling strength of $g=2\pi\times
15\,\mathrm{MHz}$.

Note from Fig. 2(c), that the first halves of the emitted photons span
$67.5\,\mathrm{m}$ ($\sim 225\,\mathrm{ns}$), despite the optical path length
between the cavity and the detectors being only $1.5\,\mathrm{m}$ or
$301.5\,\mathrm{m}$, depending on the path taken. This means that the first
time interval of the two photon state is already being measured before the
second half of the second photon leaving the cavity (travelling along the
shorter path) has been fully generated. The length of the interfering photons
is sufficient for a feedback loop to alter the phase $\phi$ of the second half
of the photon under production, conditioned on the measurement outcome within
the first time interval (Fig. 1.)

The feedback is implemented using a home-built circuit controller and single
photon counting modules (SPCMs) with a quantum efficiency of 60-65% and a
resolution of $<300\,\mathrm{ps}$ (Excelitas SPCM-AQRH-780-14-FC). The total
feedback loop latency, from the EOM via the cavity to the SPCM, then back to
the EOM via the controller circuitry is $97.0\pm 0.2\,\mathrm{ns}$. Therefore,
a conditioned phase change cannot be realised in time if a photon detection in
$I_{2}$ occurs less than 97 ns after a detection in $I_{1}$. The resulting
error rate is limited to 0.2% for a $\sin^{4}\left(t\right)$ photon intensity
envelope. All photon counts are recorded with 81ps accuracy using a qutools
quTAU time-to-digital converter (TDC). Detections within the
$450\,\mathrm{ns}$ photon window are further processed for a dark count
correction for each SPCM. A detailed description on the error rates, time
budgets, the implementation of the logical feedback circuit and background
correction is provided in the Supplemental Material.

##  IV Results

Figure 3: Coincidence probability densities and sliding histograms (whose bin
width is larger than its bin separation) showing the theoretical (dashed) and
experimental (solid) values for $P_{\text{joint}}\left(\tau\right)$. This
illustrates the time-resolved HOM interference of two photons under three
different conditions. (a) shows the random routing of photons with
perpendicular polarisations, which serves as reference. (b) shows the
interference between indistinguishable photons ($\phi=0$) with parallel
polarisations _,_ resulting in almost no coincidences between the two
detectors. (c) shows the trace obtained for photons with phase shifts of
$\phi=\pi$ without feedback. (d) shows the asymmetric pattern observed under
feedback control, where the feedback has prevented (most) correlations in the
right satellite peak, steering the second photon to a pre-defined output. This
ensures that the cat is always observed alive. The data shown here has been
corrected for correlations involving background noise, as explained in the
Supplemental Material.

There are two different ways to look at the measured data. One has been
discussed previously and corresponds to Fig. 3, which shows the coincidence
probability as a function of the time-difference between two detections.
However, our main interest is whether both detections are registered within
the same or opposite time intervals ($I_{1}$ and $I_{2}$). Therefore, the same
data can be represented in a cross-correlation diagram (Fig. 4) showing the
four possible values of the coincidence probability between detectors $C$ and
$D$ firing in either $I_{1}$ or $I_{2}$. In contrast to Fig. 3, this allows to
further differentiate between coinciding detections in $I_{1}$ and $I_{2}$.

The cross detection probability of interfering photons with perpendicular
polarisations is shown in Fig. 3(a). This gives rise to the random routing of
the simultaneously arriving photons, such that the resulting time-resolved
coincidence rate yields the autocorrelation function of the photons’ intensity
profile. The cross-correlations are depicted in Fig. 4(a). For the case of
photons with orthogonal polarisations, these are identical and theoretically
equal to $1/8$.

For photons with parallel polarisations, the cross correlation probability is
shown in Fig. 3(b) for $\phi=0$. This measurement follows the predictions of
photon bunching in the canonical HOM effect, for which we expect no
coincidences. Photons bunch in the same output regardless of the actual
detection time. Thus, the probability of cross channel detections is expected
to be zero for all time differences (Fig. 4(b)).

Fig. 3(c) shows the cross detection probability for photons with parallel
polarisations when $\phi=\pi$. In this case, the photons antibunch (i.e., they
are found in different output channels if detected in different time
intervals). The maximum probability to find photon-photon correlations occurs
at $\tau=\pm\delta t$, where $\delta t$ is the length of the photons. This
maximum probability is twice the reference value of Fig. 3(a), which is in
accordance with the cross-channel detections shown in Fig 4(c). Nonetheless,
the likelihood of correlated photon detections in the same time interval is
close to its theoretical value of zero.

Finally, the cross detection probability shown in Fig. 3(d) represents the
case of active feedback on $\phi$, for which we expect any photon recorded in
$I_{2}$ be detected at $C$. Therefore, the number of "$D\text{ before }C$"
correlations reaches a maximum while we barely see any "$C\text{ before }D$"
correlations, as shown in Fig. 4(d). Our experimental results demonstrate that
classical feedback control of a quantum excitation spanning multiple systems
(RF pulse driving AOM $\rightarrow$ driving laser $\rightarrow$ atom
$\rightarrow$ cavity $\rightarrow$ quantum field modes) can be achieved,
resulting in a photonic state with the property of always exiting through the
same output of a beam splitter.

Some differences between theoretical expectations and experimental results are
visible in Figs. 3 and 4. These can be attributed to a partial loss of
coherence or depolarisation of the interfering photons, most evident from the
presence of correlations in the side lobes of Fig. 3 (b). The mutual coherence
between photons is often characterised by the HOM visibility, defined as

$V_{\text{HOM}}=1-\frac{N_{\parallel}}{N_{\perp}}$ (14)

where $N_{\parallel}$ and $N_{\bot}$ are the total number of correlations
observed for interfering photons with parallel and perpendicular
polarisations, respectively. Using the results obtained with $\phi=0$ for
$N_{\parallel}$ we find $V_{\text{HOM}}=0.78\pm 0.04$. However, we measure a
reduced visibility of $V_{\text{ref}}=0.61\pm 0.04$ if we restrict our
analysis to those correlations with detections across time intervals $I_{1}$
and $I_{2}$. This serves as a reference for all effects discussed here, as
these affect only the correlations across both time intervals.

The visibility under phase control upon switching between $\phi=0$ and
$\phi=\pi$ reads

$V_{\phi}=\frac{N_{\pi}-N_{0}}{N_{\pi}+N_{0}},$ (15)

where $N_{0}$ and $N_{\pi}$ represent the coincidence counts between pairs of
photons with relative phase shifts of $\phi=0$ and $\pi$, respectively. Again,
we only count coincidences across $I_{1}$ and $I_{2}$, and find
$V_{\phi}=0.56\pm 0.06$, in agreement with $V_{\text{ref}}$ within error bars.
This validates the robustness of the phase switch as no further loss of
coherence is induced111$V_{\text{ref}}$ and $V_{\phi}$ are expected to yield
the same value, assuming $N_{\parallel}=N_{0}$ and
$N_{\perp}=\left(N_{0}+N_{\pi}\right)/2$ due to the random splitting of
photons..

Since the feedback relies on a conditional phase switch, we quantify its
visibility by comparing coincidence counts between intervals where a
measurement in $D$ occurs before one in $C$ ($N_{D1C2}$), and vice-versa
($N_{C1D2}$):

$V_{\text{feed}}=\frac{N_{D1C2}-N_{C1D2}}{N_{D1C2}+N_{C1D2}}.$ (16)

For our experiment, we find $V_{\text{feed}}=0.56\pm 0.06$, which is identical
to $V_{\phi}$ and equally within error bars of $V_{\text{ref}}$. We therefore
conclude that the feedback works as expected, without introducing any loss of
coherence.

Figure 4: The correlations from Fig. 3 are now shown according to the exact
detector-time-bin detections. In (a), an increased number in the correlations
between $C_{1}$ and $D_{1}$ with respect to $C_{2}$ and $D_{2}$ is explained
due to a larger number of photons in the first time bin, as shown in the
Supplementary Material. In (d), we observe an increase in the correlations
between $C_{2}$ and $D_{1}$ with respect to (a). This implies that a detection
in $D$ in $I_{1}$ has been used to steer the measurement of a photon in
$I_{2}$ to $C$, and serves as a demonstration that the second photon is sent
to a pre-defined output state, i.e., the cat is always observed alive. A
limitation in visibility is evident when comparing the data to their
theoretical values, shown dashed. The data shown here has been corrected for
correlations involving background noise, as explained in the Supplementary
Material.

##  V Conclusions

We have demonstrated a technique for steering the measurement of a quantum
superposition towards a definitive result, a result which can be interpreted
as ensuring that Schrödinger’s cat is always observed alive. This was achieved
by using a feedback mechanism that enforces either bosonic or fermionic
behaviour on interfering photons with long coherence lengths.

We emphasize that a classical interpretation of the described experiment
fails. Classically, one might expect the feedback control to be successful
only when the first detection corresponds to the photon in the delay arm.
Otherwise, if the first detection was of the photon in the short arm, the
phase change of the driving laser would have no effect (as the photon under
generation has already been detected). In a classical description, one would
expect a random routing of the second photon regardless of the phase switch,
and a reduction of the feedback visibility to zero, which is clearly not the
case.

This result constitutes an elementary step towards introducing active control
into processes such as quantum random walks and optical networks (Peruzzo et
al., 2010). Generalisations of the technique demonstrated here are suitable
candidates in photonic switchyards requiring multiple photon streams for
studying multi-mode interferometry (Barrett et al., 2019), where the
deterministic routing of photons would be performed using feedback operations.

## Acknowledgements

J.R.A. acknowledges Alejandra Valencia, David Guzmán and Sebastián Murgueitio
Ramírez for useful discussions.

We dedicate this paper to the memory of Bruce W. Shore, who sadly passed away
on 9 January 2021. Bruce inspired us to keep questioning the fundamental
principles of the light-matter interaction, and without his deep insight, laid
out convincingly in his memoir on our changing views of photons (Shore, 2020),
we would never have accomplished the present work.

## Funding

European Union Horizon 2020 (Marie Sklodowska-Curie 765075-LIMQUET).
EPSRC through the quantum technologies programme (NQIT hub, EP/M013243/1).

## References

  * Schrödinger (1935) E. Schrödinger, Naturwissenschaften 23, 807 (1935), ISSN 1432-1904.
  * Adlam (2021) E. Adlam, _Foundations of Quantum Mechanics_ , Elements in the Philosophy of Physics (Cambridge University Press, 2021).
  * Landsman (2017) K. Landsman, in _Foundations of Quantum Theory: From Classical Concepts to Operator Algebras_ , edited by K. Landsman (Springer International Publishing, Cham, 2017), Fundamental Theories of Physics, pp. 435–457, ISBN 978-3-319-51777-3.
  * Lloyd (1997) S. Lloyd, arXiv:quant-ph/9703042 (1997), eprint quant-ph/9703042.
  * Zhang et al. (2017) J. Zhang, Y.-x. Liu, R.-B. Wu, K. Jacobs, and F. Nori, Physics Reports 679, 1 (2017), ISSN 0370-1573.
  * Habib et al. (2002) S. Habib, K. Jacobs, and H. Mabuchi, p. 10 (2002).
  * Yamamoto et al. (1986) Y. Yamamoto, N. Imoto, and S. Machida, Physical Review A 33, 3243 (1986), ISSN 0556-2791.
  * Armen et al. (2002) M. A. Armen, J. K. Au, J. K. Stockton, A. C. Doherty, and H. Mabuchi, Physical Review Letters 89 (2002).
  * Smith et al. (2002) W. P. Smith, J. E. Reiner, L. A. Orozco, S. Kuhr, and H. M. Wiseman, Physical Review Letters 89 (2002).
  * Sayrin et al. (2011) C. Sayrin, I. Dotsenko, X. Zhou, B. Peaudecerf, T. Rybarczyk, S. Gleyzes, P. Rouchon, M. Mirrahimi, H. Amini, M. Brune, et al., Nature 477, 73 (2011).
  * Zhou et al. (2012) X. Zhou, I. Dotsenko, B. Peaudecerf, T. Rybarczyk, C. Sayrin, S. Gleyzes, J. M. Raimond, M. Brune, and S. Haroche, Physical Review Letters 108 (2012).
  * Legero et al. (2003) T. Legero, T. Wilk, A. Kuhn, and G. Rempe, Applied Physics B: Lasers and Optics 77, 797 (2003), eprint quant-ph/0308024.
  * Legero et al. (2006) T. Legero, T. Wilk, A. Kuhn, and G. Rempe, Advances In Atomic, Molecular, and Optical Physics 53, 253 (2006).
  * Hong et al. (1987) C. K. Hong, Z. Y. Ou, and L. Mandel, Phys. Rev. Lett. 59, 2044 (1987).
  * Kuhn and Ljunggren† (2010) A. Kuhn and D. Ljunggren†, Contemporary Physics 51, 289 (2010), ISSN 0010-7514.
  * Nisbet-Jones et al. (2011) P. B. R. Nisbet-Jones, J. Dilley, D. Ljunggren, and A. Kuhn, New Journal of Physics 13, 103036 (2011), ISSN 1367-2630.
  * Vasilev et al. (2010) G. S. Vasilev, D. Ljunggren, and A. Kuhn, New J. Phys. 12, 063024 (2010), ISSN 1367-2630.
  * Peruzzo et al. (2010) A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Worhoff, et al., Science 329, 1500 (2010).
  * Barrett et al. (2019) T. D. Barrett, A. Rubenok, D. Stuart, O. Barter, A. Holleczek, J. Dilley, P. B. R. Nisbet-Jones, K. Poulios, G. D. Marshall, J. L. O’Brien, et al., Quantum Science and Technology 4, 025008 (2019), ISSN 2058-9565.
  * Shore (2020) B. W. Shore, _Our Changing Views of Photons: A Tutorial Memoir_ (Oxford University Press, New York, 2020), 1st ed., ISBN 978-0-19-886285-7.