33,540 research outputs found
Congruences and Canonical Forms for a Positive Matrix: Application to the Schweinler-Wigner Extremum Principle
It is shown that a real symmetric [complex hermitian] positive
definite matrix is congruent to a diagonal matrix modulo a
pseudo-orthogonal [pseudo-unitary] matrix in [ ], for any
choice of partition . It is further shown that the method of proof in
this context can easily be adapted to obtain a rather simple proof of
Williamson's theorem which states that if is even then is congruent
also to a diagonal matrix modulo a symplectic matrix in
[]. Applications of these results considered include a
generalization of the Schweinler-Wigner method of `orthogonalization based on
an extremum principle' to construct pseudo-orthogonal and symplectic bases from
a given set of linearly independent vectors.Comment: 7 pages, latex, no figure
Experimental demonstration of a directionally-unbiased linear-optical multiport
All existing optical quantum walk approaches are based on the use of
beamsplitters and multiple paths to explore the multitude of unitary
transformations of quantum amplitudes in a Hilbert space. The beamsplitter is
naturally a directionally biased device: the photon cannot travel in reverse
direction. This causes rapid increases in optical hardware resources required
for complex quantum walk applications, since the number of options for the
walking particle grows with each step. Here we present the experimental
demonstration of a directionally-unbiased linear-optical multiport, which
allows reversibility of photon direction. An amplitude-controllable probability
distribution matrix for a unitary three-edge vertex is reconstructed with only
linear-optical devices. Such directionally-unbiased multiports allow direct
execution of quantum walks over a multitude of complex graphs and in tensor
networks. This approach would enable simulation of complex Hamiltonians of
physical systems and quantum walk applications in a more efficient and compact
setup, substantially reducing the required hardware resources
Joint Entanglement of Topology and Polarization Enables Error-Protected Quantum Registers
Linear-optical systems can implement photonic quantum walks that simulate
systems with nontrivial topological properties. Here, such photonic walks are
used to jointly entangle polarization and winding number. This joint
entanglement allows information processing tasks to be performed with
interactive access to a wide variety of topological features. Topological
considerations are used to suppress errors, with polarization allowing easy
measurement and manipulation of qubits. We provide three examples of this
approach: production of two-photon systems with entangled winding number
(including topological analogs of Bell states), a topologically error-protected
optical memory register, and production of entangled topologicallyprotected
boundary states. In particular it is shown that a pair of quantum memory
registers, entangled in polarization and winding number, with
topologically-assisted error suppression can be made with qubits stored in
superpositions of winding numbers; as a result, information processing with
winding number-based qubits is a viable possibility
Directionally-unbiased unitary optical devices in discrete-time quantum walks
The optical beam splitter is a widely-used device in photonics-based quantum information processing. Specifically, linear optical networks demand large numbers of beam splitters for unitary matrix realization. This requirement comes from the beam splitter property that a photon cannot go back out of the input ports, which we call “directionally-biased”. Because of this property, higher dimensional information processing tasks suffer from rapid device resource growth when beam splitters are used in a feed-forward manner. Directionally-unbiased linear-optical devices have been introduced recently to eliminate the directional bias, greatly reducing the numbers of required beam splitters when implementing complicated tasks. Analysis of some originally directional optical devices and basic principles of their conversion into directionally-unbiased systems form the base of this paper. Photonic quantum walk implementations are investigated as a main application of the use of directionally-unbiased systems. Several quantum walk procedures executed on graph networks constructed using directionally-unbiased nodes are discussed. A significant savings in hardware and other required resources when compared with traditional directionally-biased beam-splitter-based optical networks is demonstrated.Accepted manuscriptPublished versio
Coherent State Quantum Key Distribution with Entanglement Witnessing
An entanglement witness approach to quantum coherent state key distribution
and a system for its practical implementation are described. In this approach,
eavesdropping can be detected by a change in sign of either of two witness
functions, an entanglement witness S or an eavesdropping witness W. The effects
of loss and eavesdropping on system operation are evaluated as a function of
distance. Although the eavesdropping witness W does not directly witness
entanglement for the system, its behavior remains related to that of the true
entanglement witness S. Furthermore, W is easier to implement experimentally
than S. W crosses the axis at a finite distance, in a manner reminiscent of
entanglement sudden death. The distance at which this occurs changes measurably
when an eavesdropper is present. The distance dependance of the two witnesses
due to amplitude reduction and due to increased variance resulting from both
ordinary propagation losses and possible eavesdropping activity is provided.
Finally, the information content and secure key rate of a continuous variable
protocol using this witness approach are given
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