81 research outputs found

### Complex Hadamard matrices contained in a Bose-Mesner algebra

A complex Hadamard matrix is a square matrix H with complex entries of
absolute value 1 satisfying $HH^*= nI$, where $*$ stands for the Hermitian
transpose and I is the identity matrix of order $n$. In this paper, we first
determine the image of a certain rational map from the $d$-dimensional complex
projective space to $\mathbb{C}^{d(d+1)/2}$. Applying this result with $d=3$,
we give constructions of complex Hadamard matrices, and more generally, type-II
matrices, in the Bose-Mesner algebra of a certain 3-class symmetric association
scheme. In particular, we recover the complex Hadamard matrices of order 15
found by Ada Chan. We compute the Haagerup sets to show inequivalence of
resulting type-II matrices, and determine the Nomura algebras to show that the
resulting matrices are not decomposable into generalized tensor products.Comment: 28 pages + Appendix A + Appendix

### Bordered complex Hadamard matrices and strongly regular graphs

We consider bordered complex Hadamard matrices whose core is contained in the
Bose-Mesner algebra of a strongly regular graph. Examples include a complex
Hadamard matrix whose core is contained in the Bose-Mesner algebra of a
conference graph due to J. Wallis, F. Sz\"{o}ll\H{o}si, and a family of
Hadamard matrices given by Singh and Dubey. In this paper, we prove that there
are no other bordered complex Hadamard matrices whose core is contained in the
Bose-Mesner algebra of a strongly regular graph.Comment: 21 pages, corrected typ

### Scalable implementation of $(d+1)$ mutually unbiased bases for $d$-dimensional quantum key distribution

A high-dimensional quantum key distribution (QKD) can improve error rate
tolerance and the secret key rate. Many $d$-dimensional QKDs have used two
mutually unbiased bases (MUBs), while $(d+1)$ MUBs enable a more robust QKD.
However, a scalable implementation has not been achieved because the setups
have required $d$ devices even for two MUBs or a flexible convertor for a
specific optical mode. Here, we propose a scalable and general implementation
of $(d+1)$ MUBs using $\log_p d$ interferometers in prime power dimensions
$d=p^N$. We implemented the setup for time-bin states and observed an average
error rate of 3.8% for phase bases, which is lower than the 23.17% required for
a secure QKD against collective attack in $d=4$.Comment: 6 pages, 3 figures, followed by Supplemental Material of 8 pages, 1
figure, 1 tabl

### 10-GHz-clock time-multiplexed non-degenerate optical parametric oscillator network with a variable planar lightwave circuit interferometer

A coherent XY machine (CXYM) is a physical spin simulator that can simulate
the XY model by mapping XY spins onto the continuous phases of non-degenerate
optical parametric oscillators (NOPOs). Here, we demonstrated a large-scale
CXYM with >47,000 spins by generating 10-GHz-clock time-multiplexed NOPO pulses
via four-wave mixing in a highly nonlinear fiber inside a fiber ring cavity. By
implementing a unidirectional coupling from the i-th pulse to the (i+1)-th
pulse with a variable 1-pulse delay planar lightwave circuit interferometer, we
successfully controlled the effective temperature of a one-dimensional XY spin
network within two orders of magnitude.Comment: 5 pages, 4 figure

### Generation of a time-bin Greenberger--Horne--Zeilinger state with an optical switch

Multipartite entanglement is a critical resource in quantum information
processing that exhibits much richer phenomenon and stronger correlations than
in bipartite systems. This advantage is also reflected in its multi-user
applications. Although many demonstrations have used photonic polarization
qubits, polarization-mode dispersion confines the transmission of photonic
polarization qubits through an optical fiber. Consequently, time-bin qubits
have a particularly important role to play in quantum communication systems.
Here, we generate a three-photon time-bin Greenberger-Horne-Zeilinger (GHZ)
state using a 2 x 2 optical switch as a time-dependent beam splitter to
entangle time-bin Bell states from a spontaneous parametric down-conversion
source and a weak coherent pulse. To characterize the three-photon time-bin GHZ
state, we performed measurement estimation, showed a violation of the Mermin
inequality, and used quantum state tomography to fully reconstruct a density
matrix, which shows a state fidelity exceeding 70%. We expect that our
three-photon time-bin GHZ state can be used for long-distance multi-user
quantum communication.Comment: 8 pages, 4 figures, 1 tabl

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