Steering of Quantum Walks through Coherent Control of High-dimensional Bi-photon Quantum Frequency Combs with Tunable State Entropies Raktim Haldar1 2 3Robert Johanning1 2 3Philip R ubeling1 2 3Anahita Khodadad Kashi1 2 3

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Steering of Quantum Walks through Coherent Control of High-dimensional Bi-photon
Quantum Frequency Combs with Tunable State Entropies
Raktim Haldar,1, 2, 3, Robert Johanning,1, 2, 3 Philip R¨ubeling,1, 2, 3 Anahita Khodadad Kashi,1, 2, 3
Thomas Bækkegaard,1, 2, 4 Surajit Bose,1, 2, 3 Nikolaj Thomas Zinner,4, 5 and Michael Kues1, 2, 3,
1Institute of Photonics, Leibniz University Hannover,
Nienburger Straße 17, D-30167 Hannover, Germany
2Hannover Centre for Optical Technologies, Leibniz University Hannover,
Nienburger Straße 17, D-30167 Hannover, Germany
3Cluster of Excellence PhoenixD (Photonic, Optics,
and Engineering – Innovation Across Disciplines),
Leibniz University Hannover, Hannover, Germany
4Kvantify APS, DK-2300, Copenhagen S, Denmark
5Department of Physics and Astronomoy, Aarhus University, Aarhus C DK-8000, Denmark
Quantum walks are central to a wide range of applications such as quantum search, quantum
information processing, and entanglement transport. Gaining control over the duration and the
direction of quantum walks (QWs) is crucial to implementing dedicated processing. However, in
current systems, it is cumbersome to achieve in a scalable format. High-dimensional quantum
states, encoded in the photons’ frequency degree of freedom in on-chip devices are great assets
for the scalable generation and reliable manipulation of large-scale complex quantum systems.
These states, viz. quantum frequency combs (QFCs) accommodating huge information in a single
spatial mode, are intrinsically noise tolerant, and suitable for transmission through optical fibers,
thereby promising to revolutionize quantum technologies. Existing literature aimed to generate
maximally entangled QFCs excited from continuous-wave lasers either from nonlinear microcavities
or from waveguides with the help of filter arrays. QWs with flexible depth/duration have been
lately demonstrated from such QFCs. In this work, instead of maximally-entangled QFCs, we
generate high-dimensional quantum photonic states with tunable entropies from periodically
poled lithium niobate waveguides by exploiting a novel pulsed excitation and filtering scheme.
We confirm the generation of QFCs with normalized entropies from 0.35 to 1 by performing
quantum state tomography with high fidelities. These states can be an excellent testbed for several
quantum computation and communication protocols in nonideal scenarios and enable artificial
neural networks to classify unknown quantum states. Further, we experimentally demonstrate the
steering and coherent control of the directionality of QWs initiated from such QFCs with tunable
entropies. Our findings offer a new control mechanism for QWs as well as novel modification means
for joint probability distributions.
Keywords—Quantum walk, entanglement, entropy, quantum frequency combs, Bell-state, High-
dimensional quantum states.
I. INTRODUCTION
Quantum information processing (QIP) aided by pho-
tonics is a promising approach due to the availability
of proficient manipulation tools at room temperature
and the robustness to decoherence1,2. High-dimensional
quantum systems, i.e., multi-level systems employing qu-
dits as basic computational units are inherently more
resilient to noise, fault-tolerant, and by leveraging the
elegance of high-dimensional quantum algorithms en-
hance the performance of quantum computers3–5. High-
dimensional quantum states exploiting the photons’
frequency degrees of freedom, so-called quantum fre-
quency combs (QFCs), can be generated on-chip, ma-
nipulated through off-the-shelf telecommunication com-
ponents, carried over long distances via commercial op-
tical fibers, and can further help in realizing large-scale
complex quantum states. Thus, QFCs constitute highly
coveted scalable resources for QIP and quantum com-
munication (QC)6–10. QFCs that have been previously
demonstrated in on-chip optical microresonators (MRs)
through spontaneous four-wave mixing (SFWM)7,8 or
by judiciously carving the continuous-wave spontaneous
parametric down-conversion (SPDC) spectra from peri-
odically poled lithium niobate (PPLN) waveguides, form
maximally entangled d-dimensional Bell-states11. Ide-
ally, maximally entangled quantum states (MES) are
regarded as the most crucial resources for QIP and
QCs. However, in practice, separable, weakly en-
tangled, and non-maximally entangled quantum states
(NMES) or the partially entangled states (PES) are ex-
tremely useful tools for quantum computing in dissipa-
tive environments12. In the presence of decoherence,
it is challenging to retain the quality of a MES intact,
whereas, state preparation, manipulation, and main-
tenance of the entanglement through a noisy channel
are easier for non-maximally entangled pure and mixed
states12. For instance, without filtering and entangle-
ment concentration, conventional teleportation protocols
with MES shared as a resource between the sender and
arXiv:2210.06305v1 [quant-ph] 12 Oct 2022
2
FIG. 1. Generation scheme of non-maximally entangled quantum frequency comb (QFC) from a femtosecond
pulse excitation. A QFC can be generated through spontaneous parametric down-conversion (SPDC) by pumping a period-
ically poled lithium niobite (PPLN) waveguide and by using an equidistant filter array. When the waveguide is pumped with
a CW-excitation of angular frequency (ωp), the generated signal (s) and idler (i) around the degenerate angular frequency
(ωd) have a one-to-one correspondence, which means that the detected frequency of the signal photon explicitly dictates the
frequency of the corresponding idler photon. Therefore, the QFC is maximally entangled. On the other hand, if the waveguide
is excited through a broadband pulse (center frequency ωp), the one-to-one correspondence is lost and the QFC becomes non-
maximally entangled. This pulse excitation scheme is similar to the multi-chromatic excitation of an optical resonator, which
also yields non-maximally entangled QFCs.
the receiver cannot guarantee unit fidelity for an un-
known state13,14. In the presence of amplitude damp-
ing throughout an imperfect channel, utilizing partially
entangled resources are advantageous for quantum cor-
relation distribution and storage over MES15,16. Sim-
ilarly, the entanglement lower bound ensuring the op-
timal fidelity for local cloning of a pure bi-partite sys-
tem, and the roles of quantum correlation on the fidelity
of cloning and deletion are theoretically established17,18.
Non-maximally entangled states are known to reduce
the required detector efficiencies for loophole-free tests
of Bell inequalities19,20. Besides increasing the acces-
sibility of Hilbert space, such non-maximally entangled
states can be implemented in nonlocality (e.g., Hardy’s
proof) tests without inequalities, thereby enabling us to
answer fundamental questions of quantum mechanics19.
Recently, PESs are employed to examine the effective-
ness of entanglement concentration, distillation proto-
cols, and random number generation schemes in prac-
tical scenarios21–23. Therefore, we expect that high-
dimensional quantum photonic states with tunable en-
tropies suitable for long-haul communications would also
offer excellent testbeds for other QIP protocols such
as teleportation, cloning, deletion, purification, error-
correction, and would be suitable for noisy-intermediate
scale quantum (NISQ) technologies24. Apart from that,
for quantum algorithms, such as the high-dimensional
Deutsch algorithm, amplitude encoding for quantum
algorithms25,26, or for producing pure heralded photons,
often high-dimensional product (separable) states are re-
quired. These are not trivial to realize, as it requires
broad energy transitions with a bandwidth that is acces-
sible with optical filters27,28.
Moreover, quantifying the entropy of an entangled re-
source and determining its efficacy for a certain QIP or
a cryptographic protocol could be of paramount interest.
Nevertheless, unlike energy, entanglement does not corre-
spond to any observable and there is no straightforward
experimental procedure to obtain the entanglement of
an unknown quantum system. Especially measuring en-
tanglement of a strongly correlated many-body quantum
system remains an intractable problem to date29. There-
fore, it is necessary to generate quantum states with a
range of entropies, for instance as resources to train neu-
ral networks, able to directly classify unknown quantum
states and their suitability in QC channels30.
Recently, we proposed pumping schemes in an MR31
based QFC that facilitate the excitation of multiple anti-
3
diagonal coincidence lines that reduces the frequency cor-
relation, thereby allowing us to tune the entropy of the
QFC. In this work, we demonstrate a novel pumping
and filtering scheme32 in PPLN waveguide-based QFC
to maneuver its entropy, by tailoring its joint spectral
intensity (JSI). Further, gaining control over the direc-
tion of quantum walk (QW), although very critical to
QIP and quantum information transport33, has been in-
deed difficult34,35, and often impossible due to its in-
herent stochastic nature. Unidirectional one and two-
dimensional QW from separable two-particle states have
been shown theoretically36. Nevertheless, the experimen-
tal demonstration of obtaining a definitive control over
the directions of QWs in photonic systems is still scarce.
To address this issue, we study quantum walks initiated
from such high-dimensional quantum states with variable
entropies37 and demonstrate control.
Quantum superpositions enable QWs38,39 to po-
tentially speedup certain computational tasks such
as database searches, tests of graph isomorphism,
ranking nodes in a network40, quantum many-body
simulations39, boson sampling41, universal quantum
computing42,43, and quantum state preparation44. En-
tanglement generation, localization, and quantum infor-
mation transport through QW even find applications
in exotic fields of studies, notably, in explaining the
energy transfer mechanism within photosynthesis45, in
neural network46, for topology identification47, and in
neuroscience48. Being robust and immune to decoher-
ence at room temperature, QW realized in photonic
platforms are advantageous over other platforms such
as cold atom, Bose-Einstein condensates (BEC), opti-
cal lattices, trapped ions, etc.49. However, QW imple-
menting spatial49, polarization40, angular momentum50
degrees of freedom of photon either require large over-
head to alter the depth of the QW40 or necessitate
modifying the physical layout to attain the tunability
of the duration of QW49,51. Recently, QWs exhibiting
enhanced ballistic transport (bosonic) or strong energy
confinement (fermionic) have been demonstrated52 using
high-dimensional bi-photon quantum frequency combs
(QFCs)52, which do not require any change of the device
arrangement. However, no control over the directions
of the demonstrated QWs52 could be achieved, which
were initiated from the maximally entangled states. Re-
cently, Floquet engineered discrete- and continuous-time
QWs and their control have been reported numerically
by using time-dependent coins34, and by tweaking the
node-coupling coefficients35. The role of space-dependent
coins53, and initial conditions54 on QWs are also stud-
ied extensively. Lately, the directionality of QW in BEC
has been observed55,56. Quantum photonic states having
tunable entropies extend our accessibility to the Hilbert
space. Consequently, richer dynamics of QWs instigated
from such states are expected yet have not been observed
to date. For the first time to the best of our knowledge,
here we experimentally demonstrate the coherent con-
trol of the direction and steering of quantum walk ini-
tiated from a high-dimensional bi-photon quantum fre-
quency comb with tunable state entropies leading to a
completely new paradigm of QWs. Procuring precise
control over a nonclassical stochastic process involving
a high-dimensional Hilbert space may have immense im-
plications in, e.g., quantum search, transport of quantum
information, and atomic interference.
II. DESCRIPTION AND CHARACTERISTICS
OF NON-MAXIMALLY ENTANGLED QFCS
Biphoton quantum frequency combs providing a route
to generate complex high-dimensional states10 can play
a major role in discrete-variable photonic quantum com-
puting. They are either generated by monochromatically
exciting one of the resonating modes of a microresonator
(MR) or by pumping a nonlinear waveguide typically be-
low the parametric threshold6with the help of an ar-
ray of equidistant optical filters. We adopted the lat-
ter approach at the moderate expense of reduced bright-
ness and second-order correlation (g(2)) because of the
increased design flexibility of choosing the QFC free-
spectral range (FSR), also necessary to synchronize with
the RF-driving frequency to perform the quantum state
tomography (QST).
A scheme for the generation of non-maximally en-
tangled QFCs from pulse excitation is shown in Fig.
1. A programmable filter (PF1) configuration is used
to discretize the SPDC spectra from a femtosecond
(fs) laser-driven PPLN-waveguide to create a bi-photon
QFC8. The distance between the adjacent bandpass
filters (BPFs) defines the free spectral range (FSR) of
the QFC. The femtosecond (fs)-laser allows producing a
broad single-mode frequency bandwidth for each photon
with respect to CW-pumping52,57,58. Together with a
special filter, this may result in multiple anti-diagonal
lines in the JSI. Due to the presence of multiple lines,
one particular idler mode is spectrally connected to sev-
eral signal modes (instead of one, as it would be in a
maximally entangled state). This effectively reduces the
entanglement of the system as explained in Fig. 1. One
can create a variety of JSIs with several frequency anti-
correlation line configurations by changing the spectral
profile of the excitation or by altering the FSRs. To
simulate such versatile JSIs, we develop a mathemati-
cal model (see Supplementary Information), where we
assume that the BPF corresponding to the 0-th (central)
mode is placed matching with the center of the quasi-
phase-matching (QPM) bandwidth (i.e., the degenerate
angular frequency (fd) = pump-frequency (fp)/2) of the
PPLN.
Initially we model the corresponding JSI of the QFC
assuming there exists an equal number of frequency anti-
correlation lines (i.e., symmetrical) surrounding the cen-
tral antidiagonal. The quantum-state representing the
pth adjacent anti-diagonals with respect to the central
anti-diagonal (p= 0) of such QFC having Nnumber of
4
total frequency bins can be approximated by,
|ψqi=1
q(N1)
2(|q| − 1)
(N1)/2
X
k=|q|
|k+qis|−kii;q < 0
|kis|−kii;q= 0
|kis|− (kq)ii;q > 0
(1)
Where, q < 0, q= 0, q > 0 represent the lower, central,
and upper diagonals, respectively. The complete state-
vector of the QFC |ψQFCican be written as the summa-
tion of each normalized diagonal element weighted by the
complex diagonal contribution aq. Fig. 2 (b)-(d) show
an examples of the JSIs calculated from Eq. (1), where
the amplitudes of the three diagonals are (b) equiprob-
able (i.e., a1=a2=a3), and following a Gaussian dis-
tribution with standard deviations (c) σ= 1, and (d)
σ= 2, respectively. Thereafter, we simulate QFCs with
five frequency anticorrelation lines in the JSI with Gaus-
sian distribution having standard deviations (e) σ= 1,
(f) σ= 2, and (g) asymmetric distribution with respect
to the central antidiagonal, where two out of five lines are
dominant. By diagonalizing the reduced density matrix
of rank robtained from Eq. (1), we can compute the von
Neumann entropy SA=Pλmlog2(λm). Since entropy
increases with dimension, we introduced the normalized
entropy SN=Pλmlogr(λm), where λmare the eigen-
values of ˆρ. In Fig. 2 (i), we plot SNwith respect to the
total number of modes Nwhile varying the number of
diagonals Din the JSI from 3 to N. Note that, if the
frequency modes are equiprobable, for D= 1, the QFC is
maximally entangled with SN= 1. From Fig. 2 (i), it can
also be seen that with an increasing number of diagonals,
the entropy of the QFC reduces. The entropy of the QFC
increases with increasing N. We further study the effect
of the JSI spectral bandwidth, i.e., the combined effect of
the pump spectral distribution and the QPM bandwidth
on entropy. We assume that the amplitudes aqof the JSI-
diagonals follow a Gaussian distribution with standard
deviation σ. As depicted in Fig. 2 (ii), for small values
of σ, there is effectively a single JSI-line corresponding
to the maximally entangled QFC, whereas, σ→ ∞ cre-
ates the QFC with equal amplitudes, i.e., for aq= 1.
Note that, we can also omit the degenerate mode (central
mode) of the QFC by the programmable filter. In that
case, the rank of the reduced density matrix rcoincides
with the dimension (d) of the QFC (mathematical details
can be found in the Supplementary Materials).
III. EXPERIMENTAL DEMONSTRATION OF
QFC WITH TUNABLE ENTROPIES
The experimental set-up (Fig: 3 (a)) used a fs-laser
(pulse width 80 fs, 50 MHz repetition rate, central
wavelength (λp778.6±0.2 nm) to excite a PPLN
waveguide, generating broadband time-frequency entan-
gled photon pairs with 40 nm bandwidth through
SPDC (see the Supplementary). The waveguide output
was passed through a programmable filter (PF1) to define
the frequency modes of the QFC with N= 17 frequency
bins of 10 GHz bandwidth each. The output of the PF1
was collected and split by another programmable filter
(PF2) to measure photon coincidences. We have two dif-
ferent PPLN chips having overall generation bandwidths
of 80 GHz and 40 GHz. Note that the generation
bandwidth depends upon the spectral envelope of the
pump and the phase-matching bandwidth. Therefore, we
can alter the number of antidiagonal lines (and entropy
of the system) either by changing the pump bandwidth,
phase-matching bandwidth, or by changing the FSR of
the PF1, which is demonstrated in Fig. 3. As observed
in Fig. 3 (b) and 3 (c), we obtain about five and three
dominant frequency anti-correlation lines from two indi-
vidual PPLN chips with 80 GHz and 40 GHz gen-
eration bandwidths, respectively. BPFs of the PF1 have
been placed by 25 GHz apart in both cases. We also
attained two anti-diagonal lines in the JSI-diagram for
the PPLN-chip with 40 GHz bandwidth when the PF1
filters are set to be 50 GHz apart (Fig. 3 (d)). There-
fore, the proposed excitation and filtering scheme enable
the engineering of quantum frequency states from max-
imally entangled (D/N 1) to nearly separable states
(D/N 1), covering a wide range of entropies.
IV. QUANTUM STATE TOMOGRAPHY
To corroborate our theoretical findings that how the
number of frequency anti-correlation lines is related to
the entropy, we experimentally reconstructed the density
matrix (ˆρ) of the QFC through full QST. We performed
the QST on the QFCs having dimensions d= 2 and 3,
generated from the 40 GHz bandwidth chip. The de-
generacy mode, i.e., the central frequency-bin (mode 0)
was omitted by the first programmable filter (PF1) for
simplicity. To generate QFCs with different entropies,
the spectral spacing between two adjacent BPFs of the
filter array at PF1 was set to 50 GHz, 100 GHz, and
150 GHz, respectively. Note that the filter separations
were chosen to be the integer multiple of the phase-
modulator’s driving frequency, i.e., 25 GHz. To imple-
ment the projectors (Ψproj) necessary for QST, an op-
tical phase modulator was used to create sidebands via
mode-mixing7. We calculated the fidelity, purity, and
entropy of the QFC from the reconstructed density ma-
trix for each filter setting and compared the values to
the ones for the theoretically predicted quantum states.
The QFC with multiple antidiagonal lines within the JSI
generated through SPDC from a perfectly coherent laser
pulse should ideally be a pure quantum state. The pre-
pared quantum state deviated slightly from being a per-
fectly pure state that corresponds to the predicted QFC
as given by Eqs. (S.21-S.22). We achieved a fidelity close
to unity while the overall noise, including phase fluctua-
tions during the measurements, and the spectral incoher-
ence of the pulse-excitation are taken into account. The
meticulous details regarding the QST and the formation
of the density matrices are provided in the Supplemen-
5
FIG. 2. Joint spectral intensities of non-maximally entangled QFCs and the dependency of their entropies on
different parameters (a) Filter configuration to generate a bi-photon QFC with N= 17 modes. Simulated joint spectral
intensity (JSI) with three frequency anti-correlation lines (b) having equal amplitudes, and having Gaussian distribution of
standard deviations (c) σ= 1, and (d) σ= 2. Five frequency anti-correlation lines having Gaussian distribution of standard
deviations (e) σ= 1, (f ) σ= 2, and (g) asymmetric distribution with two dominant lines. (i) Normalized Von Neumann
entanglement entropy (SN) as a function of the total number of QFC-modes Nwith varying JSI-diagonals Ds in the JSI when
all the diagonals are equiprobable (i.e., aq= 1). (ii) SNwith the standard deviation (σ) when the JSI follows the normal
distribution. BW: bandwidth.
摘要:

SteeringofQuantumWalksthroughCoherentControlofHigh-dimensionalBi-photonQuantumFrequencyCombswithTunableStateEntropiesRaktimHaldar,1,2,3,RobertJohanning,1,2,3PhilipRubeling,1,2,3AnahitaKhodadadKashi,1,2,3ThomasBkkegaard,1,2,4SurajitBose,1,2,3NikolajThomasZinner,4,5andMichaelKues1,2,3,y1Instituteof...

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