Bernhard G. Bodmann

Stable phase retrieval with low-redundancy frames

We investigate the recovery of vectors from magnitudes of frame coefficients when the frames have a low redundancy, meaning a small number of frame vectors compared to the dimension of the Hilbert space. We first show that for complex vectors in d dimensions, 4d−4 suitably chosen frame vectors are sufficient to uniquely determine each signal, up to an overall unimodular constant, from the magnitudes of its frame coefficients. Then we discuss the effect of noise and show that 8d−4 frame vectors provide a stable recovery if part of the frame coefficients is bounded away from zero. In this regime, perturbing the magnitudes of the frame coefficients by noise that is sufficiently small results in a recovery error that is at most proportional to the noise level.

Decoherence-Insensitive Quantum Communication by Optimal

The central issue in this paper is to transmit a quantum state in such a way that after some decoherence occurs, most of the information can be restored by a suitable decoding operation. For this purpose, we incorporate redundancy by mapping a given initial quantum state to a messenger state on a larger dimensional Hilbert space via a C* -algebra embedding. Our noise model for the transmission is a phase damping channel which admits a noiseless subsystem or decoherence-free subspace. More precisely, the transmission channel is obtained from convex combinations of a set of lowest rank yes/no measurements that leave a component of the messenger state unchanged. The objective of our encoding is to distribute quantum information optimally across the noise-susceptible component of the transmission when the noiseless component is not large enough to contain all the quantum information to be transmitted. We derive simple geometric conditions for optimal encoding and construct examples of such encodings.

C^{\ast }

We derive necessary conditions for the existence of complex Seidel matrices containing pth roots of unity and having exactly two eigenvalues, under the assumption that p is prime. The existence of such matrices is equivalent to the existence of equiangular Parseval frames with Gram matrices whose off-diagonal entries are a common multiple of the pth roots of unity. Explicitly examining the necessary conditions for p = 5 and p = 7 rules out the existence of many such frames with a number of vectors less than 50, similar to previous results in the cube roots case. On the other hand, we confirm the existence of p 2 × p 2 Seidel matrices containing pth roots of unity, and thus the existence of the associated complex equiangular Parseval frames, for any p > 2. The construction of these Seidel matrices also yields a family of previously unknown Butson-type complex Hadamard matrices.

-Encoding

The objective of this paper is the linear reconstruction of a vector, up to a unimodular constant, when all phase information is lost, meaning only the magnitudes of frame coefficients are known. Reconstruction algorithms of this type are relevant for several areas of signal communications, including wireless and fiber-optical transmissions. The algorithms discussed here rely on suitable rank-one operator valued frames defined on finite-dimensional real or complex Hilbert spaces. Examples of such operator-valued frames are the rank-one Hermitian operators associated with vectors from maximal sets of equiangular lines or maximal sets of mutually unbiased bases. A more general type of examples is obtained by a tensor product construction. We also study erasures and show that in addition to loss of phase, a maximal set of mutually unbiased bases can correct for erased frame coefficients as long as no more than one erasure occurs among the coefficients belonging to each basis, and at least one basis remains without erasures.

Complex equiangular Parseval frames and Seidel matrices containing

We derive fast algorithms for doing signal reconstruction without phase. This type of problem is important in signal processing, especially speech recognition technology, and has relevance for state tomography in quantum theory. We show that a generic frame gives reconstruction from the absolute value of the frame coefficients in polynomial time. An improved efficiency of reconstruction is obtained with a family of sparse frames or frames associated with complex projective 2-designs.

p

This paper investigates the performance of frames for the linear, redundant encoding of vectors when consecutive frame coefficients are lost due to the occurrence of random burst errors. We assume that the distribution of bursts is invariant under cyclic shifts and that the burst-length statistics are known. In analogy with rate-distortion theory, we wish to find frames of a given size, which minimize the mean-square reconstruction error for the encoding of vectors in a complex finite-dimensional Hilbert space. We obtain an upper bound for the mean-square reconstruction error for a given Parseval frame and in the case of cyclic Parseval frames, we find a family of frames which minimizes this upper bound. Under certain conditions, these minimizers are identical to complex Bose-Chaudhuri-Hocquenghem codes discussed in the literature. The accuracy of our upper bounds for the mean-square error is substantiated by complementary lower bounds. All estimates are based on convexity arguments and a discrete rearrangement inequality.

th roots of unity

The reversion of the Born-Neumann series of the Lippmann-Schwinger equation is one of the standard ways to solve the inverse acoustic scattering problem. One limitation of the current inversion methods based on the reversion of the Born-Neumann series is that the velocity potential should have compact support. However, this assumption cannot be satisfied in certain cases, especially in seismic inversion. Based on the idea of distorted wave scattering, we explore an inverse scattering method for velocity potentials without compact support. The strategy is to decompose the actual medium as a known single interface reference medium, which has the same asymptotic form as the actual medium and a perturbative scattering potential with compact support. After introducing the method to calculate the Green’s function for the known reference potential, the inverse scattering series and Volterra inverse scattering series are derived for the perturbative potential. Analytical and numerical examples demonstrate the fea...

Frames for linear reconstruction without phase

Direct inversion of acoustic scattering problems is nonlinear. One way to treat the inverse scattering problem is based on the reversion of the Born–Neumann series solution of the Lippmann–Schwinger equation. An important issue for this approach is the radius of convergence of the Born–Neumann series for the forward problem. However, this issue can be tackled by employing a renormalization technique to transform the Lippmann–Schwinger equation from a Fredholm to a Volterra integral form. The Born series of a Volterra integral equation converges absolutely and uniformly in the entire complex plane. We present a further study of this new mathematical framework. A Volterra inverse scattering series (VISS) using both reflection and transmission data is derived and tested for several acoustic velocity models. For large velocity contrast, series summation techniques (e.g., Cesaro summation, Euler transform, etc) are employed to improve the rate of convergence of VISS. It is shown that the VISS method with summation techniques can provide a relatively good estimation of the velocity profile. The method is fully data-driven in the respect that no prior information of the model is required. Besides, no internal multiple removal is needed. This one dimensional VISS approach is useful for inverse scattering and serves as an important step for studying more complicated and realistic inversions.

Fast algorithms for signal reconstruction without phase

We answer a number of open problems in frame theory concerning the decomposition of frames into linearly independent and/or spanning sets. We prove that in finite dimensional Hilbert spaces, Parseval frames with norms bounded away from 1 can be decomposed into a number of sets whose complements are spanning, where the number of these sets only depends on the norm bound. We also prove, assuming the Kadison-Singer conjecture is true, that this holds for infinite dimensional Hilbert spaces. Further, we prove a stronger result for Parseval frames whose norms are uniformly small, which shows that in addition to the spanning property, the sets can be chosen to be independent, and the complement of each set to contain a number of disjoint, spanning sets.

Burst Erasures and the Mean-Square Error for Cyclic Parseval Frames

The modern use of spectral decomposition has shown that reflection events in practice are always frequency dependent, a phenomenon called reflectivity dispersion. Often, this can be attributed to strong interference effects from neighboring reflection coefficients of the classical type (i.e., parameter discontinuities or jumps). However, an intrinsic frequency dependence from a single layer is possible if the contact is not a jump discontinuity but a gradual transition. We have expanded the normal-incidence theory of a linear velocity transition zone (termed a Wolf ramp) and have shown how it leads to frequency-dependent reflectivity. The development of waveform forward modeling in turn has led to a ramp detection method that we have tested on migrated field data.