Congruences and Canonical Forms for a Positive Matrix: Application to the Schweinler-Wigner Extremum Principle

It is shown that a $N\times N$ real symmetric [complex hermitian] positive definite matrix $V$ is congruent to a diagonal matrix modulo a pseudo-orthogonal [pseudo-unitary] matrix in $SO(m,n)$ [ $SU(m,n)$], for any choice of partition $N=m+n$. 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 $N$ is even then $V$ is congruent also to a diagonal matrix modulo a symplectic matrix in $Sp(N,{\cal R})$ [$Sp(N,{\cal C})$]. 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