Hadamard transform

About: Hadamard transform is a research topic. Over the lifetime, 7262 publications have been published within this topic receiving 94328 citations.

Multilevel superconducting circuits as two-qubit systems: Operations, state preparation, and entropic inequalities

Abstract: We theoretically study operations with a four-level superconducting circuit as a two-qubit system. Using a mapping on a two-qubit system, we show how to implement iswap gates and Hadamard gates through pulses on transitions between particular pairs of energy levels. Our approach allows one to prepare pure two-qubit entangled states with desired form of reduced density matrices of the same purity and, in particular, arbitrary identical reduced states of qubits. We propose using schemes for the Hadamard gate and two-qubit entangled states with identical reduced density matrices in order to verify $logN$ inequalities for Shannon and R\'enyi entropies for the considered noncomposite quantum system.

The necessity of the Hadamard condition

Abstract: Hadamard states are generally considered as the physical states for linear quantized fields on curved spacetimes, for several good reasons. Here, we provide a new motivation for the Hadamard condition: for “ultrastatic slab spacetimes” with compact Cauchy surface, we show that the Wick squares of all time derivatives of the quantized Klein-Gordon field have finite fluctuations only if the Wick-ordering is defined with respect to a Hadamard state. This provides a converse to an important result of Brunetti and Fredenhagen. The recently proposed “S-J (Sorkin-Johnston) states” are shown, generically, to give infinite fluctuations for the Wick square of the time derivative of the field, further limiting their utility as reasonable states. Motivated by the S-J construction, we also study the general question of extending states that are pure (or given by density matrices relative to a pure state) on a double-cone region of Minkowski space. We prove a result for general quantum field theories showing that such states cannot be extended to any larger double-cone without encountering singular behaviour at the spacelike boundary of the inner region. In the context of the Klein-Gordon field this shows that even if an S-J state is Hadamard within the double cone, this must fail at the boundary.

Construction of Hadamard States by Pseudo-Differential Calculus

Abstract: We give a new construction based on pseudo-differential calculus of quasi-free Hadamard states for Klein–Gordon equations on a class of space-times whose metric is well-behaved at spatial infinity. In particular on this class of space-times, we construct all pure Hadamard states whose two-point function (expressed in terms of Cauchy data on a Cauchy surface) is a matrix of pseudo-differential operators. We also study their covariance under symplectic transformations. As an aside, we give a new construction of Hadamard states on arbitrary globally hyperbolic space-times which is an alternative to the classical construction by Fulling, Narcowich and Wald.

Detection of a multi-sequence spread spectrum signal

Abstract: A detector of a multiple-sequence spread spectrum signal uses a Hadamard transform (106) to simultaneously correlate a received signal comprising two sequences (64) with a plurality of candidate sequences The received signal is stripped of the first sequence (65, 66), and the signal is permuted (via a table lookup) (104) A Hadamard transform is performed on the permuted data and the candidate sequences (106) After transformation, the data is permuted again (112 ) to determine the symbol (sequence) transmitted Alternatively, Fast Fourier Transforms (FFTs) (FIG 3), Winograd Fourier Transform Algorithms (WFTA), or other cyclic correlation algorithms (FIG 5) may be used to compute the transformation In a preferred embodiment, a "pilot" signal is transmitted in quadrature (90 degrees phase offset) with an information-bearing signal And, a block error correcting code (150) (eg, a modified Reed-Solomon code) is transmitted with the information-bearing signal a(t) The block length of the block error correcting code (150) is equal to an integral multiple of the period of the pilot signal The period of the pilot signal is an integral multiple of the information bearing signal Thus, carrier recovery, sequence synchronization, and block code synchronization are all achieved simultaneously by correlating (synchronizing) the received signal with a baseband version of the pilot signal